Communications in Mathematical Physics - Volume 191

Commun. Math. Phys. 191, 1 – 13 (1998) Communications in Mathematical Physics c Springer-Verlag 1998 Classification ...

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Commun. Math. Phys. 191, 1 – 13 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1998

Classification and Construction of Quantum Communication Systems ¨ Bernd Muller Institut für theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany. E-mail: [email protected] Received: 5 September 1995 / Accepted: 5 November 1996

Abstract: We consider Quantum Communication Systems (QCS’s), introduced by Davies [10] on trace class operators, on general state spaces. These are different from the usual quantum information theory as they deal with input and output being continuous in time. We introduce the concept of a refinement of a QCS, offering the possibility of distinguishing particles, for instance according to their phase or location, and classify refinements of general QCS. We then introduce the notions of a bounded interaction rate and a bounded modulation and give, in special cases, a classification of QCS’s satisfying both conditions. As this classification shows the existence of a class of QCS’s being completely different from the examples studied by Davies, we are able to discuss the cases of an independent detector and a linear modulation. 1. Introduction In classical communication theory the transmission of information is described by means of a Markov kernel P connecting the input and the output system [2], i.e. the input signal w ∈ X (the set of all input signals) leads to a probability distribution E → P (w, E) on the set of possible outputs. Usually one considers the input as generated randomly, so instead of using one signal w ∈ X one takes a probability distribution µ on X. Then µ is transformed into a probability distribution ν on via the formula Z P (w, E)µ(dw). ν(E) = X

In many cases the set of measures on X or is the dual of the C ∗ -algebra C(X) or C() of the continuous complex valued functions on X or , respectively. Then the above map 3∗ : µ → 3∗ µ := ν is the dual of a (completely) positive map 3 : C() → C(X). This definition in terms of C ∗ -algebras and completely positive maps shows an obvious way to generalize the notion of a communication channel to the non-commutative case.

2

B. Müller

The use of general W ∗ - or C ∗ -algebras instead of abelian ones is supposed to take into account the quantum mechanical nature of the physical system which is used for signal transmission. This includes a quantum mechanical formulation of information, (relative) entropy and in particular of the transition kernel P [1, 16, 17, 19–22]. The non-commutative analogue to a message, consisting of the letters a1 , . . . , an then is the transmission of a sequence of states ρ1 , . . . , ρn , which is transformed by the channel into a sequence ω1 , . . . , ωn of output states [13]. The transformation ρi → ωi = 3∗ ρi here is the non-commutative generalization of a Markov kernel. In applications the problem is the identification of the states ωi . This can only be done by means of a measurement, the result of which by definition is classical. So by fixing one type of measurement we still have a classical Markov kernel from input to output, which is also the case in the approach presented below, where the measurement always is a counting experiment. The physical situation we look at in this paper is partly different and not covered by the above considerations. In applications one often wants to transmit a signal function w that is continuous or piecewise continuous in time. This is to be achieved by the modulation of the light beam, which is caused by the input system. The light beam itself is described by a state of the photon system. This implies that a possible input is not a discrete chain of letters or (by use of a quantum code ai → ρi ) states, but a real valued function. However, we emphasize that it is possible to restrict the attention to functions that are constant on the time intervals [nc, (n+1)c) and only take values out of a finite set {a1 , . . . , an }. c represents the time necessary for the transmission of one signal ai , and a1 , . . . , an are the possible letters. This restriction shows that a digital (binary) signal transmission is included in the class of models described below. The way the modulating system influences the state is not specified a priori, yet it is proved in 4.2 that, under some additional assumptions, it can always be formulated as a time dependent perturbation of the interaction free dynamics (cf. Eq. (16)). To give a physical example, we think of a light beam entering a modulating system that influences the light. The modulating system for instance is a semiconductor crystal whose transmission coefficient and refraction index at time t are a function of the value w(t) of the signal w that is to be transmitted. If H is the free Hamiltonian of the photon system, the dynamics of the modulated light might be described by ∂ ρt = −i[H, ρt ] + B(w(t))ρt , (1) ∂t where B(w(t)) contains the influence of the modulating system. The above equation assumes that the state ρ is a trace class operator on some Hilbert space H. We remark that, in the sequel, we use a general state space (for instance a predual space of a W ∗ algebra). That way classical features of the light beam can be described as well [15]. At the other end of the channel the receiver performs a measurement on the state “arriving there”. Usually, one considers a number measurement to count the arriving quanta, though, in general this is not necessary. However, most authors do not take into account that, because of the continuity in time, such a measurement cannot be described by a selfadjoint operator on some Hilbert space, in particular not by only taking the spectral decomposition of the Fock number operator in order to get the relevant observable. In this paper we use the theory of counting experiments developed in [6, 9, 10, 18], where the detecting operation itself is assumed to be a quantum stochastic process. So both input and output are continuous in time, and thus it is not at all clear if the resulting channel is memoryless, which is, though not always physically justified, an assumption made implicitly in many articles by only considering the transmission of one letter. However, it is nevertheless possible in models of the type described below to get a memoryless channel, which will be shown in future work. The main advantage of this

Classification and Construction of Quantum Communication Systems

3

approach is, that it can easily be applied in practice. The use of the powerful theory of open systems [7, 9] reduces very efficiently the set of parameters that have to be taken into account. Though it lacks the generality of the investigations on completely positive channels made for instance in [22], the restriction to the special class of counting experiments allows to get a very concrete meaning of any part of the channel and thus easily shows how to construct physical models. And, we remark that because of the possibility of distinguishing particles according to phase, location, energy, . . . (see 3.1), this restriction is not as strong as one might guess. The paper is organized as follows: In Sect. 2 we give the basic definitions – including the one of a quantum communication system (QCS) – which all go back to Davies [10]. For technical reasons the definition given in [10] differs slightly from Definition 2.1. In particular it is shown how to calculate the transition kernel (Eq. (2)). The connections between systems that can distinguish particles (for instance according to energy, phase or location) and systems that can not are studied in Sect. 3. As for counting processes (CP’s, cf. [18]) we call the corresponding constructions coarse–grainings or refinements. In Theorem 3.1 we give a classification of refinements which generally allows to construct a QCS in two steps: First, one has to construct a one point QCS (i.e. a QCS that cannot distinguish particles) and then it can be refined. In Sect. 4 we introduce the ideas of a bounded interaction rate (for CP’s cf. [6, 8, 9, 18]) and a bounded modulation, and call a QCS bounded if both conditions are fulfilled. For a bounded one point QCS having an interaction free semigroup St (2.2) of type R [18] it is shown in Theorem 4.2 that the QCS always can be formulated via the perturbation theoretic equations (10), (11). This implies that Eq. (1), which so far only seems to be a physically reasonable example, can be derived from rather principal considerations, condensed in Definition 2.1. Conversely Theorem 4.3 implies that (10) and (11) always define a QCS. In particular there is an interaction rate operator J(c) (cf. the results for CP’s [10, 18]). Yet J(c) can depend on a real parameter c. In Proposition 4.4 we show that any decomposition of J(c), i.e. a positive operator valued measure (POVM) J(c, .) on a standard Borel space (X, Σ) with J(c, X) = J(c), defines a refinement of the corresponding QCS. For physical applications this implies that one only has to determine the operators J(c, E) (characterizing the instaneous change of state when a particle of type E is detected) in order to get a refinement of a QCS. Finally in Sect. 5 we introduce two important classes: QCS’s with linear modulation and QCS’s with an independent detector. For the first class it is shown that the modulation operators have to be generators of groups of isometries. Both classes are interesting from the physical point of view and we remark that Davies [10] only studied QCS’s satisfying both conditions. 2. Definition of a QCS The first point we are interested in is the set characterizing the input. As we want to have a model respecting the continuity of time, the messages are supposed to be functions of the time. We thus define Yt to be the set of functions w : [0, t) → R, where w is continuous from the right and has finite limits from the left at any point of the interval. For t = ∞ we additionally require that w is bounded. We consider two topologies on Yt : The first one is the uniform topology, the second one is defined by the Skorohod metric d [23]. Important is that the set of step functions is dense in Yt for both topologies. As one can send one message immediately after another one, we introduce the map v(r), r ∈ [0, t) β s,t : Ys × Yt → Ys+t , β s,t (w, v)(r) = w(r − t), r ∈ [t, t + s).

4

B. Müller

Usually the indices s, t are neglected and we use β(w, v) or even w × v instead of β s,t (w, v). In the sequel we often will vary t but have a fixed message w. This means that we use one message w ∈ Y∞ and only consider the restrictions of w to the interval [0, t). The output of the communication system shall be produced by a detection system counting the arriving quanta. Thus we use the following notations, introduced in the theory of counting processes [6, 18]: Atn Cnt

:= :=

{((x1 , t1 ), . . . , (xn , tn ))|0 ≤ t1 < . . . < tn < t, xi ∈ X}, {(t1 , . . . , tn )|0 ≤ t1 < . . . < tn < t},

for n ∈ N. Any point of the measurable space (X, Σ) is characterizing a certain type (energy, location, . . .) of particle. The output ((x1 , t1 ), . . . , (xn , tn )) ∈ Atn therefore means that a particle of type xi has been detected at time ti . The event that no particle has been detected during the time interval [0, t) we denote by zt , which is independent from (X, Σ). Therefore C0t = At0 := {zt } and Xt := Un≥0 Atn is the set of possible outputs. Again the possibility of recording two sequences after each other corresponds to a map λ : Xt × Xs → Xt+s defined by (((x1 , t1 ), . . . , (xn , tn )), ((y1 , s1 ), . . . , (ym , sm ))) → ((y1 , s1 ), . . . , (ym , sm ), (s1 , t1 + s), . . . , (sn , tn + s)). The physical channel (for a mathematical realization see Eq. (2)) connecting Yt and Xt for instance may be the quantized electromagnetic field, described by its state ω. More general the set of states of a physical transmission system forms a state space (V, K, τ ), i.e. V is a base normed space with positive cone K, and τ ∈ V ∗ equals the norm on K [7, 9, 12]. Measurements generally are described by means of an instrument. Given a state space, an instrument by definition is a positive operator valued measure (POVM) E: 0 → B+ (V), where (, 0) is the set of possible measurement outputs, satisfying E()∗ τ = τ (for definitions and the necessity of these notations, see [7, 18]). It is important that the probability of a measurement result in A ∈ is given by hE(A)ω, τ i = ||E(A)ω||. if the system is in the normalized state ω ∈ K = V+ (||ω|| = hω, τ i = 1) at the beginning of the measurement, and afterwards the system is in the state E(A)ω. In the above situation any measurement (and henceforth the corresponding instrument) depends on the message w and the time t up to which it is performed. This leads to the following definition: Definition 2.1. A quantum communication system (QCS)is a family of instruments Et (ω, ·) on Xt such that (i) limt↓0 Et (w, Xt )ω = ω in norm for all ω ∈ V and w ∈ Y∞ . (ii) w → Et (w, E) is weakly measurable and even weakly continuous for E = Xt . (iii) Et (w, E)Es (v, F ) = Et+s (w × v, λ(E × F )) for w ∈ Yt , v ∈ Ys and measurable E ⊆ Xt , F ⊆ Xs . We use the uniform topology in (ii) to define the σ-field on Yt . We could also take the Skorohod metric, but as the uniform continuity in most cases is easier to verify we use the weaker condition. Recall that for hω, τ i = ||ω|| = 1 the probability for having an output in the set E ⊆ Xt , provided that the message w is sent, is given by hEt (w, E)ω, τ i. In


5

this case the state of the channel directly after the measurement has been transformed into Et (w, E)ω. Therefore (iii) expresses that the time evolution has no memory and is homogeneous. Now for any normalized ω ∈ V+ the map (w, E) → hEt (w, E)ω, τ i =: Pt (w, E)

(2)

is a probability transition kernel from Yt to Xt . Therefore for any choice of ω this map defines a channel in the sense of classical communication theory. Note that the above Markov property (iii) of the time evolution does not imply that this channel is memoryless in the sense of communication theory. Remarks 2.2. (i) If g: R → R is continuous and Et is a QCS, then (E, w) → Et (g ◦ w, E) defines another QCS. (ii) For any constant function, i.e. w(t) ≡ c the family Et (c, .) defines a counting process [6]. w w := Et (w, {zt }) and St+s,s := Et (αs (w), {zt }), (iii) We use the following notations: St,0 w where αs (w)(r) := w(s + r). For 0 ≤ s < t St,s is the interaction free dynamics according to w. For any constant c the family Stc := Et (c, {zt }) is a semigroup. 3. Coarse-Grainings and Refinements If the detector counts the particles without distinguishing them, the space (X, Σ) consists of one point only and Atn is isomorphic to Cnt . In this case we call the corresponding QCS a one point QCS. Let us consider a general QCS Et with measurable (X, Σ). It is obvious that via the projection [ Cnt , ((x1 , t1 ), . . . , (xn , tn )) → (t1 , . . . , tn ) π: Xt → n≥0

the QCS Et is transformed into a one point QCS E˜t , where E˜t is defined by E˜t (w, E) = Et (w, π −1 (E)). Physically this corresponds to the neglection of the possibility of distinguishing particles and we therefore call the resulting one point QCS E˜t the coarsegraining of Et . In most cases the inverse way is more important, as one point QCS can often be constructed by use of semigroup theory (cf. next section). Given a one point QCS E˜t the possibility of distinguishing particles corresponds to a refinement of E˜t . In other words: a refinement of E˜t is another QCS Et with measurable space (X, Σ) whose coarse-graining coincides with E˜t . Let Et be such a refinement of E˜t , then obviously the w of both QCS is the same. Moreover it is interaction free dynamics St,s Et (w, X × [0, t)) = E˜t (w, [0, t))

and

w w Et (w, E × [a, b)) = St,b Eb−a (αa (w), E × [0, b − a))Sa,0

(3) (4)

for all E ∈ Σ and 0 < a < b < t. The subsequent theorem shows that these equations essentially are enough to construct a refinement. In the sequel we will often have to deal with sets of the form A = (E1 × [a1 , b1 )) × · · · × (En × [an , bn )),

(5)

6

B. Müller

P where Ei ∈ and 0 ≤ a1 < b1 < · · · < an < bn < t. Obviously A is a measurable t set in An . We call a finite union of sets of this form a standard set. The standard sets are closed under the formation of finite intersections and generate the σ-field on Atn . Note that it is sufficient to give a QCS on At1 (and At0 ) as the Markov property then determines the QCS on standard sets and henceforth on general sets. Theorem 3.1. Let (X, Σ) be a standard Borel space and E˜t a one point QCS. Then there is a 1:1-correspondence between the following classes: (i) Refinements Et of E˜t with measurable space (x, Σ). (ii) POVM’s Gt (w, .) on X × [0, t) = At1 (Gt (w, E) ∈ B+ (V)) such that (a) w → Gt (w, E) is weakly measurable, (b) Gt (w, X × [0, t)) = E˜t (w, [0, t)) . w w Gb−a (αa (w), E × [0, b − a))Sa,0 for E ∈ Σ and (c) Gt (w, E × [a, b)) = St,b 0 < a < b < t. The correspondence is given by Et (w, A) = Gt (w, A) for measurable A ⊆ X × [0, t) . Proof. The main idea of the proof is the same as in the corresponding proof for CP’s [18], but as the situation here is much more complicated the details need a closer look. The formal bijective map between Gt and Et is given by Gt (w, A) := Et (w, A) for measurable A ⊆ X × [0, t), i.e., Gt is the restriction of Et on the measurable subsets of At1 ⊆ Xt . To show the bijectivity of the map it is obviously enough to show that it is possible to construct Et whenever Gt is given. For fixed t we subdivide the interval [0, t) k n in 2n disjoint intervals Ikn := [ k−1 2n t, 2n t) for k = 1, . . . , 2 and restrict our attention to the set 3ln := {x = ((x1 , t1 ), . . . , (xl , tl )) ∈ Atl | xi ∈ X, for each k there is at most one ti ∈ Ikn }. It is easy to verify that 3ln ⊆ 3ln+1 and Un≥0 3ln = Atl . We say a map φ : {1, . . . , 2n } → {0, 1} is in Qln , iff cardφ−1 ({1}) = l. For each φ ∈ Qln we define Pφ := {ψ ∈ Qln+1 | for each k ∈ φ−1 (1) there is one and only one l ∈ {2k − 1, 2k} with ψ(l) = 1}. Obviously the Pφ are pairwise disjoint. Now let M0n := {zt/2n }, M1n := X × [0, t/2n ), and for n n × · · · × λ(Mφ(2n ) × Mφ(1) ) . . .). If X = {1} we use each φ ∈ Qln let Aφ := λ(Mπ(2n) l l ˜ n instead of 3n . It is easy to verify that Aφ = ∪ψ∈Pφ Aψ and Bφ instead of Aφ and 3

3ln = ∪φ∈Qln Aφ , where the Aφ are pairwise disjoint. The σ-algebra on Aφ is generated n by sets of the form E = λ(E2n × · · · × E1 ) with measurable sets Ei ⊆ Mφ(i) . For these sets we define µ˜ φ (w, E) = Gt,(2n −1)t/2n (w, E2n ) · · · Gt/2n ,0 (w, E1 ), w and Gb,a (w, {zr }) = Sb,a . Then where Gb,a (w, E) = Gb−a (αa (w), E) for E ⊆ Ab−a 1 Q2 n using the fact that i=1 Mφ(i) is Borel isomorphic to Aφ it follows by [7, Theorem 2] that E → µ˜ φ (w, E) can be extended to a POVM on Aφ . Now it is a technical but straightforward calculation, that for measurable E ⊆ Aφ holds: X µ˜ φ (w, E) = µ˜ ψ (w, E ∩ Aψ ).

Thus the definition µln (w, E) := erates a POVM on Atl satisfying

P

ψ∈Pφ φ∈Qln

µ˜ φ (w, E ∩ Aφ ) for measurable E ⊆ Atl gen-


7

(i) µln+1 (w, E) = µln (w, E) for measurable E ⊆ 3ln , (ii) µln (w, Clt ) = µln (w, 3ln ). The last equation holds as 3ln is the disjoint union of the Aφ and µln by definition is concentrated on this union. In particular we have µln+1 (w, E) ≥ µln (w, E) for all measurable E ⊆ Atl . Moreover for a standard set E = (E1 ×[a1 , b1 ))×· · ·×(El ×[al , bl )) a direct calculation shows that E ⊆ 3ln for sufficient large n and then w Gbl −al (αal (w), El ) · · · Gb1 −a1 (αa1 (w), E1 )Saw1 ,0 . µln (E) = St,b l

(6)

Now by definition one calculates X µln (w, Aφ ) = E˜t (w, ∪φ∈Qln Bφ ) ≤ E˜t (w, Clt ). µln (w, Atl ) = l φ∈qn

Monotony and boundedness of this sequence imply that for all measurable E ⊆ Clt and ω ∈ V+ (and therefore all ω ∈ V) limn→∞ µln (w, E)ω =: Etl (w, E)ω exists in norm. By the Vitali–Hahn–Saks Theorem [11] we conclude that E → Etl (w, E) is weakly, hence (because of the positivity and the state space properties) strongly σ-additive. Thus Etl (w, .) is a POVM on Atl with (all limits are in the strong topology) Etl (w, Atl )

= =

lim µln (w, Atl ) n→∞ E˜tl (w, Clt ).

˜ ln ) = lim E˜t (w, ∪φ∈Qln Bφ ) = lim E˜t (w, 3 n→∞

n→∞

P∞ Now we can define Et (w, E)ω := l=0 Etl (w, E ∩Atl )ω. It is straightforward that Etl (w, .) is a POVM with Etl (w, Xt ) = E˜t (w, ∪n=0 Cnt ). By construction w → Et (w, E) is weakly measurable and it remains to show 2.1(iii) for Et being an QCS. But as it suffices to verify this equation on standard sets this follows from Eq. (6). Thus there is a unique QCS Et the restriction to At1 of which is Gt , and the theorem is proven. 4. Classification and Construction of QCS It the sequel V, K, τ is a fixed state space. Let Et be a one point QCS. We say Et has a bounded interaction rate if for all a > 0 there is a K(a) > 0 such that ||Et (w, Xt \{zt })|| 5 K(a)t

(7)

for all w ∈ Y∞ satisfying ||w|| 5 a. The QCS has a bounded modulation if for all a > 0 there is a K(a) > 0 such that for measurable sets E ⊆ C0t ∪ C1t holds ||Et (w, E) − Et (v, E)|| ≤ K(a)||w − v||∞ t

(8)

if ||w||∞ , ||v||∞ , t ≤ a. Finally the QCS is bounded if it has a bounded interaction rate and a bounded modulation. We explain the ideas one has in mind when given these definitions. First, for a bounded interaction rate the probability of detecting one or more particles during a time interval of length t gets (linearly) small with t, which is definitely physically reasonable. The problem is that Eq. (7), being formulated for the instruments, contains a uniform estimation for all states. Comparing with CP’s the “best” physical condition should have the form

8

B. Müller

||Et (w, Xt \{zt })ω|| ≤ K(a, ω)t,

(9)

for ω in some dense subset of V. (If (9) is valid for all ω ∈ V the uniform boundedness principle implies that (7) holds.) Besides the fact that it is certainly the appropriate way to investigate first the simpler structure before turning to a more complicated one, there is another argument that justifies to consider bounded interaction rates. For CP’s (always take into account 2.2(ii)) Eq. (7) leads to the theory of unbounded interaction rates for which a main result is [18] that formally the same equations hold as in the bounded case, one only has to replace bounded operators by unbounded ones. The interpretation of the modulation condition (8) is that similar messages should imply similar outputs, at least for small t. Concerning the physical relevance the same considerations can be made as for the IR-condition, and it is reasonable to expect that in general the operators B(c) in Theorem 4.2 just have to be allowed to be unbounded. However, this is not yet proved and this paper deals with the bounded case. As for CP’s [6, 18] we require that the semigroup St := Et (0, {zt }) has additional properties. We say a semigroup St is of type R if its Farvard class, i.e. the set of all ω ∈ V for which the orbit {St ω|t ≥ 0} is Lipschitz continuous [4], equals the domain D(W ) of its generator W . Recall that this is always true for reflexive spaces. Any pure semigroup on the state space of trace class operators of a Hilbert space is of type R [6, 18]. We denote the Farvard class of St by F (W ). Lemma 4.1. Let St be a contraction semigroup of type R on V with generator (W, D(W )). (i) If A ∈ B(V) and Tt := exp[(W + A)t] is contractive, then Tt is of type R. (ii) If Tt is another contraction semigroup satisfying ||Tt − St || ≤ Kt, then Tt is of type R and there is a unique bounded operator A ∈ B(V) such that Tt = exp[(W + A)t]. Rt Proof. (i) If ω ∈ F (W +A) then the perturbation equation Tt ω = St ω + 0 Tt−s ASs ωds implies ||St ω − ω|| ≤ ||(Tt − St )ω|| + ||Tt ω − ω|| ≤ ||A|| ||ω||t + ||Tt ω − ω||, thus ω ∈ F (W ) = D(W ) = D(W + A), hence F (W + A) ⊆ D(W + A). The inverse inclusion is always true so Tt is of type R. (ii) Let (Z, D(Z)) denote the generator of Tt , then ω ∈ D(Z) implies ||St ω − ω|| ≤ ||Tt ω − ω|| + ||(St − Tt )ω|| ≤ ||Zω||t + Kt, thus ω ∈ D(W ). Hence A := Z − W is well defined on D(Z) and ||A|| ≤ K. Therefore Z = W + A, D(Z) = D(W ) and Tt is of type R by (i). We say a QCS is of type R if its interaction free semigroup St := Et (0, {zt }) is of type R. We always use the notation St = exp(W t). Theorem 4.2. Let Et be a bounded one point QCS of type R. Then there is a unique Family (B(c))c∈R in B(V) such that ∞ Z X w St,s = St−tn B(w(tn )) . . . B(w(t1 ))St1 ωdt1 . . . dtn (10) n=0

t−s s+Cn

0 = Et (0, {zt }) = exp(W t)). All the semigroups St1 c = Et (c, {zt }) for all ω ∈ V (St := St,0 are of type R and we have Stc = exp[(W + B(c))t]. Furthermore there is a unique family (J(c))c∈R in B+ (V) such that


Et (w, E)ω =

∞ Z X n=0

t ∪E Cn

9

w St,t J(w(tn )) . . . J(w(t1 ))Stw1 ,0 ωdt1 . . . dtn . n

(11)

For any a > 0 there is a K(a) > 0 such that for all c, d ∈ R with |c|, |d| ≤ a, ||J(c) − J(d)|| ≤ K(a)|a − d|

and

||B(c) − B(d)|| ≤ K(a)|c − d|.

(12) (13)

Finally we have τ ∈ D(W ∗ ) and for all c ∈ R it holds (W ∗ + B(c)∗ + J(c)∗ )τ = 0.

(14)

Proof. First the modulation equation (8) implies that limt↓0 ||St ω −ω|| = 0 for all ω ∈ V and c ∈ R. Thus Stc is a contractive C0 -semigroup satisfying ||Stc − St || ≤ K(c)t. By Lemma 4.1 Stc is of type R and Stc = exp[(W + B(c))t] with a unique bounded operator B(c). In particular (10) is valid for any constant function w. Because of B(d)ω−B(c)ω = limt→0 1t (Std − Stc )ω the modulation condition implies (13). Thus defining w ω := S˜ t,s

∞ Z X t−s s+Cn

n=0

St−tn B(w(tn )) . . . B(w(t1 ))St1 ωdt1 . . . dtn ,

w one easily verifies that S˜ t,s is a well defined bounded operator satisfying w ˜w w S˜ t,s Ss,r = S˜ t,r

(15)

w w for r < s < t. By perturbation expansion we have S˜ t,s = St,s for constant functions and (15) implies that this is even true for piecewise constant functions. Now by the w is strongly continuous and a direct calculation modulation condition the map w → St,s w . By the density of the piecewise using (13) shows that the same holds for w → S˜ t,s constant functions we have proved (10). The theory of counting processes [18] shows that (11) is true for constant w, if we define J(c) to be the perturbation connecting the two semigroups Stc and Ttc := Et (c, Xt ) (both of type R by Lemma 4.1 and the interaction condition(7)). A similar argument as the above one shows the validity of (12), and (14) follows from [18]. We thus have for any constant c:

Z Et (c, [0, t))ω = 0

t

c c St,s J(c)Ss,0 ωds.

If w is constant on the interval [a, b) ⊆ [0, t), i.e. w(s) = c for s ∈ [a, b), then by (4) Z Et (w, [a, b))ω

=

w w St,b Eb−a (c, [0, b − a))Sa,0 ω=

Z

b

= a

b−a 0

w c c w St,b Sb−a,s J(c)Ss,0 Sa,0 ωds

w w St,b J(w(s))Ss,0 ωds.

Hence, if w is piecewise constant, i.e. w(s) = ci for s ∈ [ai , ai+1 ), where 0 < a = a0 < a1 < . . . < aN +1 = b < t, it follows

10

B. Müller

Et (w, [a, b))ω

=

N X i=0 Z b

= a

Et (w, [ai , ai+1 ))ω =

N Z X i=0

ai+1 ai

w w St,b J(w(s))Ss,0 ωds

w w St,b J(w(s))Ss,0 ωds.

As both sides of this equation are continuous in w it holds for general w. Finally one verifies by some easy substitutions that for standard sets E = [a1 , b1 ) × . . . × [an , bn ), 0 ≤ a1 < b1 < . . . an < bn < t: Z w Et (w, E)ω = St,t J(w(tn )) . . . J(w(t1 ))Stw1 ,0 ωdt1 . . . dtn . n E

Both sides of this equation are σ-additive and coincide on the generating system of standard sets, hence we get (11). We now give the converse of the above theorem. Theorem 4.3. Let St = exp(W t) be a C0 -semigroup on V and (B(c))c∈R a strongly continuous family of bounded operators on V with B(0) = 0 such that all the semigroups Stc := exp[(W +B(c))t] are positive and contractive. For each c ∈ R, let J(c) ∈ B+ (V) be w and Et (w, E) a positive operator such that (14) holds. Then defining the operations St,s by (10) and (11) we get a one point QCS Et . Et is bounded iff (12) and (13) are valid. Proof. The uniform boundedness principle and the strong continuity of c → B(c) and c → J(c) imply that for any r > 0 there is a constant Ar > 0 such that supu≤r (||B(u)||+ w is a well ||J(u)||) = Ar < ∞. Then for fixed w ∈ Y∞ , it is easy to verify that St,s P∞ n w n defined operator with ||St,s || ≤ n=0 Ar (t − s) /k!. Using the Lebesgue Theorem and w . By hypothesis and this uniform estimation we get the strong continuity of w → St,s w perturbation theory the operators St,s are positive and contractive for constant functions w w w w Ss,r = St,r for r < s < t, therefore St,s is w. A direct calculation shows that St,s positive and contractive for piecewise constant w and by continuity even for arbitrary w. It now is obvious that each Et (w, E) is a well defined positive bounded operator. The strong σ-additivity of E → Et (w, E) the Markov equation 2.1(iii) and 2.1(i) are straightforward calculations. 2.1(ii) follows similar to the corresponding proof for the w . Equation (14) and perturbation theory show that Et (w, Xt )∗ τ = τ for map w → St,s constant functions. As above we extend this equality to piecewise constant functions and then to arbitrary w ∈ Y∞ . Now if the QCS Et is bounded (12) and (13) follow as in the proof of Theorem 4.2. The converse result can be obtained by using the explicit expressions for Et (w, E). We yet have shown how one point QCS’s should be constructed. Now we will concentrate on refinements of such a QCS. As in the case of CP’s any decomposition [18] of the operators J(c) will determine a refinement of the QCS. Proposition 4.4. Let (X, Σ) be a standard Borel space and let Et be constructed as in Theorem 4.3. For each c ∈ R let J(c, .) : Σ → B+ (V), E → J(c, E) be a strongly σadditive map with J(c, X) = J(c). Furthermore let c → J(c, E) be strongly continuous for all E ∈ Σ. Then there is a unique refinement E˜t of Et with measurable space (X, Σ) such that for all E ∈ Σ holds: Z t w ˜ sSt, sw J(w(t), E)Ss,0 ds. Et (w, E × [0, t))ω = 0


11

Proof. Uniqueness is obvious. To prove existence we use Theorem 3.1. For measurable A ⊆ X × [0, t) and s ∈ [0, t) define As := {x ∈ X|(x, s) ∈ A} and Z

t

Gt (w, A)ω := 0

w w St,s J(w(s), As )Ss,0 ωds.

As in usual measure theory one verifies that the integral is weakly well defined and σadditive in A. Thus Gt (w, .) is a POVM on X × [0, t) which obviously is measurable in w. By Hypothesis 3.1(ii) (b) holds and 3.1(ii)(c) follows with some easy substitutions. We remark that not all refinements of the QCS E˜t necessarily have this form. The classification theorem for arbitrary CP’s with bounded interaction rate [18] shows that for a general refinement the operators J(c, E) have to be formulated as operators on a -dual space [5]. This theorem can even be used to derive structural results similar to those in Theorem 3.2 for arbitrary bounded QCS’s. We made the additional assumptions in Theorem 4.2 because the existence of the operators B(c) ∈ B(V) could not be proved without assuming that St is of type R. And even then it is not yet shown that an arbitrary non-one point QCS can be formulated on V rather than on some -dual space. However, using the classification of counting processes on Hilbert spaces [6, 18] one can show as in the proof of 4.2, that any refinement of a bounded QCS of type R on the state space of trace class operators must have the form which is given in the above proposition.

5. Special Cases Let us consider the QCS Et constructed in 4.3. There are two different differential w ρ) one equations related to Et . First for the interaction free dynamics (i.e. ρt = St,s formally gets ∂ ρt = (W + B(w(t)))ρt . (16) ∂t For the dynamics characterizing the whole system (ρt = Et (w, Xt )ρ) we have ∂ ρt = (W + B(w(t)) + J(w(t)))ρt . ∂t

(17)

Both equations at least are valid on D(W ) for the dense set of piecewise constant functions at any point where w is continuous. The first one shows that the interaction free dynamics contains some information about the transmitted message via the “modulation” B(w(t)). As W essentially contains the energy of a particle, B(w(t)) may be interpreted as an energy modulation caused by the influence of the input system. It then seems to be physically reasonable – at least for small w(t) – that B(c) has the form B(c) = cB0 ,

(18)

i.e. the modulation is linear. We say the QCS Et has a linear modulation if (18) holds. In order to meet the conditions of Theorem 4.3 the operator B0 must have additional properties. Recall that any norm continuous group of isometries on a state space V is positive.

12

B. Müller

Proposition 5.1. If Et is a bounded QCS as constructed in Theorem 4.3 with B(c) = cB0 , then B0 is the generator of a group of isometries, in particular B0∗ τ = 0. Conversely if B0 generates a group of isometries, W is the generator of a positive semigroup St of contractions and c → J(c) ∈ B+ (V) is a strongly continuous map with J(c)∗ τ = −W ∗ τ for all c, then Eq. (10) and (11) define a QCS with linear modulation. Proof. Let Et be a QCS with linear modulation, then all the operators W (c) = W + cB0 are generators of contraction semigroups, hence dissipative. Thus for any ω ∈ D(W ) = D(W (c)) with tangent functional x it is 0 ≥ hW (c)ω, xi = hW ω, xi + chB0 ω, xi. As this inequality is valid for all c ∈ R, we conclude hB0 ω, xi = 0. Therefore the operators (±B0 , D(W )) are dissipative and by [3, Prop. 3.1.15] even ±B0 is dissipative, hence B0 is the generator of a group of isometries (which is positive). By use of the Trotter–Kato formula one easily sees that the operators W (c) = W + cB0 generate positive contraction semigroups Stc with W (c)∗ τ = W ∗ τ . Thus we can apply Theorem 4.3 to get the desired result. We see that for a QCS with linear modulation the functional J(c)∗ τ does not depend on c. In particular ||J(c)|| = ||J(c)∗ τ || = ||W ∗ τ || is independent of c. We call a QCS Et as constructed in Theorem 4.3 a QCS with independent detector if J(c) = J for some fixed J ∈ B+ (V). In this case Eq. (11) suggests that J describes the instantaneous change of the state when a particle is detected. For physical applications the most interesting cases will be QCS’s with linear modulation and an independent detector, though one cannot expect that the operators J and B0 are bounded. To give some ideas on applications, we add a few comments on existing models that are not yet published. A direct generalization of Davies’ model [10] to macroscopic coherent light [14, 15] has to use W ∗ -algebraic methods, as the creation and annihilation operators in the relevant representation of the CCR cannot be formulated on Fock space. However, as the center of this representation is non-trivial, classical structures appear and can be detected. If the transmission uses squeezed light, the modulation operators on Hilbert space may have the form a∗2 − a2 or another quadratic expression in the creation and annihilation operators. On Fock space (resp. the trace class operators on Fock space) the existence of such a model has been proved and additional properties could be derived. Using Theorem 4.3 for a dependent model on the set Tsa (H) of trace class operators on a Hilbert space H with J(c)ρ = a(c)uρu∗ , W ρ = −i[H, ρ] − a(0)ρ, B(c)ρ = −i[8(c), ρ] − (a(c) − a(0))ρ, where ρ ∈ Tsa (H), u ∈ B(H) is unitary and a : R → R+ and φ : R → Bsa (H) are continuous functions, we get a very simple expression for the intensity registered by the detector: I(t) = hEt (w, Xt )ω, J ∗ (w(t))τ i = a(w(t)), which easily allows a numeric calculation of the channel capacity when using for instance a binary signal transmission. It is not always clear how to interpret a dependent detector although some considerations imply that this might be connected with a nontrivial movement of the detecting system. However we want to emphasize that there exist QCS without linear modulation or independent detector.


13

References 1. Accardi, L.: A new class of quantum states: Examples and applications. In: C. Bendjaballah, O. Hirota, and S., Reynaud, ed., Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin–Heidelberg–New York: Springer-Verlag, 1991, pp. 138–150 2. Ash, R: Information Theory. New York: Interscience, 1965 3. Bratteli, O. and Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics. Volume 1. New York–Heidelberg–Berlin: Springer-Verlag, 1987 4. Butzer, P.L. and Behrens, H.: Semigroups of Operators and Approximation. New York: Springer-Verlag, 1967 5. Clement, P., Diekmann, O., Gyllenberg, M., Heilmann, H.J.A.M., and Thieme, H.R.: Perturbation theory for dual semigroups, I. The sun-reflexive case. Math. Ann. 277, 709–725 (1987) 6. Davies, E.B.: Quantum stochastic processes. Commun. Math. Phys. 15, 277–304 (1969) 7. Davies, E.B.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970) 8. Davies, E.B.: Quantum stochastic processes III. Commun. Math. Phys. 22, 51–70 (1971) 9. Davies, E.B.: Quantum Theory of Open Systems. London–New York: Academic Press, 1976 10. Davies, E.B.: Quantum communication systems. IEEE Trans. on Inf. Th. 23, 530–534 (1977) 11. Dunford, N., and Schwartz, J.T.: Linear Operators, Volume 1. New York–London: Interscience Publishers, 1958 12. Edwards, C.M.: The operational approach to algebraic quantum theory I. Commun. Math Phys. 16, 207–230 (1970) 13. Hall, M.J.W., and O’Rourke, M.J.: Realistic performance of the maximum information channel. Quantum Opt. 5, 161–180 (1993) 14. Honegger, R., and Rapp, A.: General Glauber coherent states on the Weylalgebra and their phase integrals. Physica A 167 945–961 (1990) 15. Honegger, R., and Rieckers, A.: The general form of non-Fock coherent Boson states. Publ. RIMS Kyoto University 26, 397–417 (1990) 16. Ingarden, R.S.: Quantum information theory. Rep. Math. Phys. 10, 43–73 (1976) 17. Lindblad, G.: Quantum entropy and quantum measurements. In: C. Bendjaballah, O. Hirota, and S. Reynaud, ed. Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin– Heidelberg–New York: Springer-Verlag, 1991, pp. 71–80 18. Müller, B.: On generalized quantum stochastic counting processes. To appear in Commun. Math. Phys. 19. Ohya, M.: On compound state and mutual information in quantum information theory. IEEE Trans. on Inf. Th. 29 (5), 770–774 (1983) 20. Ohya, M.: Some aspects of quantum information theory and their applications to irreversible processes. Rep. Math. Phys. 27 (1), 19–47 (1989) 21. Ohya, M.: Information dynamics and its applications to optical communication processes. In: C. Bendjaballah, O. Hirota, and S. Reynaud, ed. Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin–Heidelberg–New York: Springer-Verlag, 1991, pp. 81–92 22. Ohya, M.: Rigorous derivation of error probability in coherent optical communication. In: In: C. Bendjaballah, O. Hirota, and S. Reynaud, ed. Quantum Aspects of Optical Communications. No. 378, Lecture Notes in Physics. Berlin–Heidelberg–New York: Springer-Verlag, 1991, pp. 203–212 23. Parthasarathy, K.R.: Probability measures on metric spaces. New York–London: Academic Press, 1967 Communicated by H. Araki

Commun. Math. Phys. 191, 15 – 29 (1998)

Communications in


Quantum Dynamical R-Matrices and Quantum Frobenius Group G.E. Arutyunov, S.A. Frolov Steklov Mathematical Institute, Gubkin str. 8, GSP-1, 117966, Moscow, Russia. E-mail: [email protected]; [email protected] Received: 24 January 1997 / Accepted: 17 March 1997

Abstract: We propose an algebraic scheme for quantizing the rational RuijsenaarsSchneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over GL(N, C). In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical r-matrix. ¯ Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system. 1. Introduction As soon as the classical dynamical r-matrices first appeared [1] on the scene of integrable many body systems, the problem of their quantization became of real interest. The main hope related to this problem is to find a new algebraic structure that ensures the integrability of the corresponding quantum models. We recall [2] that having a finite-dimensional completely integrable system with the Lax representation dL dt = [M, L] one can always write the Poisson algebra of L-operators in the r-matrix form. However, in general, an r-matrix appears to be a nontrivial function of dynamical variables. At present the classical dynamical r-matrices are known for the rational, trigonometric [1, 3] and elliptic [4, 5] Calogero-Moser (CM) systems, as well as for their relativistic generalizations – rational, trigonometric [6, 7] and elliptic [8, 9] Ruijsenaars-Schneider (RS) systems [10]. The problem of quantizing the dynamical r-matrices is rather nontrivial since, in general, such r-matrices do not satisfy a single closed equation of the Yang-Baxter type, from which they can be uniquely determined. Up to now there exists only one example of a quantum dynamical R-matrix related to the quantum spin CM system [11]. This Rmatrix solves the Gervais-Neveu-Felder equation [12, 13] and has a nice interpretation in terms of quasi-Hopf algebras [14].

16

G.E. Arutyunov, S.A. Frolov

A natural way to understand the origin of dynamical r-matrices is to consider the Hamiltonian reduction procedure [3, 15]. Factorizing a free motion on an initial phase space by the action of some symmetry group, we get nontrivial dynamics on the reduced space. An r-matrix appears in the Dirac bracket, which describes the phase structure of the reduced space. In our recent papers [16, 17] we obtained the elliptic RS model, being the most general one among the integrable systems of the CM and RS types, by using two different reduction schemes. In the first scheme, the affine Heisenberg double was used as the initial phase space and in the second one we considered the cotangent bundle over the centrally extended group of double loops. The aim of this paper is to quantize the reduction scheme leading to the dynamical systems of the RS type. Although the most interesting is the spectral-dependent elliptic case [16, 17], to clarify the general approach in this paper we restrict ourselves to considering the simplest rational model. Our construction is based on a special parametrization of the cotangent bundle over the group GL(N, C). This parametrization is similar to the one considered in [18]. However, instead of Euler angles, we use another system of generators to parametrize the “momentum”. These generators obey a quadratic Poisson algebra described by two r-matrices r and r. ¯ The matrix r solves the N -parametric classical Yang-Baxter equation and is related to the special Frobenius subgroup in GL(N, C). We define a special matrix function L on T ∗ G invariant with respect to the action of this Frobenius subgroup. We call this function the “L-operator” since the Poisson algebra of L literally coincides with the one for the rational RS model [7]. Moreover, performing the Hamiltonian reduction leading to the RS model [19], one can recognize in L the standard L-operator of the rational RS model. Then we pass to the quantization. The quadratic Poisson algebra can be quantized by using the R-matrix approach [20, 21]. The compatibility of the corresponding quantum algebra implies the quantum Yang-Baxter equation for R and some new equations ¯ We solve these equations and get an explicit form for R and R. ¯ involving R and R. Coming back to the original generators of T ∗ G we recover the standard commutation relations of the quantum cotangent bundle. We derive a new quadratic algebra which is satisfied by the “quantum” L-operator: −1 −1 −1 L2 R¯ 12 R12 R¯ 21 = R12 L2 R¯ 12 L1 . L1 R¯ 21

We see that the matrices R and R¯ come in this algebra in a nontrivial way. Thus, the dynamical r-matrices (classical and quantum) appear as the composite objects constructed ¯ from the more elementary blocks R and R. It follows from our construction that the quantum L-operator is factorized in the form L = W P . Here W satisfies the defining relations of the quantum Frobenius group, W1 W2 R12 = R12 W2 W1 , where R, being the quantization of r, is an N -parametric solution to the quantum Yang-Baxter equation. The diagonal matrix P plays the role of a generalized momentum. We find the simplest representation of the L-operator algebra and relate it with the rational RS model. The hyperbolic (trigonometric) CM system is known to be dual to the rational RS model [10]. This duality is explained by the existence of the dual parametrization of T ∗ G. In this parametrization the CM model can be easily quantized and we get the commutation relations satisfied by the corresponding quantum L-operator.

Quantum Dynamical R-Matrices and Quantum Frobenius Group

17

2. Frobenius Algebra and Dynamical r-Matrices In this section we introduce a special parametrization of the cotangent bundle T ∗ G over the matrix group G = GL(N, C). As a manifold the space T ∗ G is naturally isomorphic to G ∗ × G, where G ∗ is dual to the Lie algebra G = Mat(N, C). The standard Poisson structure on T ∗ G can be written in terms of variables (A, g), where A ∈ G ∗ ≈ G and g ∈ G, as follows 1 [C, A1 − A2 ], (2.1) 2 {A1 , g2 } = g2 C, (2.2) (2.3) {g1 , g2 } = 0. P Here we use the standard tensor notation and C = i,j Eij ⊗ Eji is the permutation operator. Any matrix A belonging to an orbit of maximal dimension in G ∗ admits a factorization: (2.4) A = T QT −1 , {A1 , A2 } =

where Q is a diagonal matrix with entries qi , qi 6= qj . We also fix the order of qi by using the action of the Weyl group. It is obvious that the matrix T in Eq. (2.4) is not uniquely defined. Indeed, one can multiply T by an arbitrary diagonal matrix from the right. We remove this ambiguity by imposing the following condition: T e = e,

(2.5)

where e is a column with all ei = 1. The choice of (2.5) is motivated by the study of the reduction procedure leading to the Calogero-type integrable systems [22]. Let us note that the condition (2.5) defines a Lie subgroup F ⊂ G. The corresponding Lie algebra F has a natural basis Fij = Eii − Eij , where Eij are the standard matrix unities. The commutation relations of Fij are [Fij , Fkl ] = δik (Fil − Fij ) + δil (Fkj − Fkl ) + δjk (Fij − Fil ). It is worthwhile to mention that F is not only the Lie algebra but also an associative algebra with respect to the usual matrix multiplication: Fij Fkl = δik Fil + δjk (Fik − Fil ). Let us rewrite the Poisson structure (2.1) in terms of the variables T and Q. It is well known that the center of the Poisson algebra (2.1) is generated by the Casimir elements trAn = trQn . Therefore, the coordinates qi Poisson commute with A, T and Q. The only nontrivial bracket one has to calculate is {T, T }. We have {Tij , Tkl } = To find

δTij δAmn

X

δTij δTkl {Amn , Aps }. δAmn δAps mn,ps

we perform the variation of (2.4): T −1 δT Q − QT −1 δT + δQ = T −1 δAT.

This equation can be easily solved, and we obtain the derivatives

18


X 1 δTij −1 −1 = (Tia Tnj Tam + Tij Tna Tjm ) δAmn qja

(2.6)

δqi −1 = Tni Tim , δAmn

(2.7)

{T1 , T2 } = T1 T2 r12 (q),

(2.8)

a6=j

and

where qij ≡ qi − qj . By using (2.6) we get where the r-matrix

r12 (q) =

X 1 Fij ⊗ Fji qij

(2.9)

i6=j

appears. It is clear that r12 (q) should be a skew-symmetric solution of the classical Yang-Baxter equation (CYBE). The origin of this r-matrix can be easily understood if we notice that F is a Frobenius Lie algebra, i.e. there is a nondegenerate 2-cocycle (coboundary) on F: ω(X, Y ) = tr(Q[X, Y ]), X, Y ∈ F .

(2.10)

According to [23], to any Frobenius Lie algebra one can associate a skew-symmetric solution of the CYBE by inverting the corresponding 2-cocycle. One can check that the cocycle (2.10) corresponds to r12 (q) ∈ F ∧ F. Coming back to (2.4), we see that any orbit of maximal dimension in G ∗ can be supplied with the structure of the Frobenius group. It is worthwhile to note that ω is the Kirillov symplectic form on the coadjoint orbit of the maximal dimension parametrized by Q. Now, following [18], we introduce a special parametrization for the group element g. To this end we consider an element A0 = gAg −1 , which Poisson commutes with A and possesses the Poisson bracket 1 {A01 , A02 } = − [C, A01 − A02 ]. 2 Diagonalizing A0 = U QU −1 with the help of the matrix U , U e = e, we find that g = U P T −1 ,

(2.11)

where P is some diagonal matrix. It is obvious that the Poisson bracket for U is given by (2.12) {U1 , U2 } = −U1 U2 r12 (q) and that {T, U } = 0. To proceed with the calculation of the brackets {U, P }, {T, P }, δU {P, P } and {P, Q}, we should use the derivative δA0ij that is given by (2.6) with the mn replacement T → U , A → A0 . Performing simple computations, we get {P1 , P2 } = 0, {Q1 , P2 } = P2

X

Eii ⊗ Eii .

(2.13)

i

Introducing pi = log Pi , we conclude that {qi , pj } = δij . We also find the Poisson brackets:


{U1 , P2 } = U1 P2 r¯12 (q), {T1 , P2 } = T1 P2 r¯12 (q), where we have introduced a new matrix X 1 Fij ⊗ Ejj . r¯12 (q) = qij

19

(2.14) (2.15)

(2.16)

i6=j

The Jacobi identity leads to a set of equations on the matrices r and r. ¯ However, we postpone the discussion of these equations till the next section, where the quantization of T ∗ G will be given in terms of variables Q, T, P, U . Let us define the L-operator as the following function of phase variables, being invariant under the action of the Frobenius group: L = T −1 gT = T −1 U P.

(2.17)

By using (2.8), (2.12) and (2.13-2.15) one can easily find the Poisson brackets containing L: X Eii ⊗ Eii , (2.18) {Q1 , L2 } = L2 i

{T1 , L2 } = T1 L2 r¯12 (q) − T1 r12 (q)L2 , {L1 , L2 } = r12 (q)L1 L2 + L1 L2 (r¯12 (q) − r¯21 (q) − r12 (q)) + L1 r¯21 (q)L2 − L2 r¯12 (q)L1 .

(2.19) (2.20)

We see that the brackets for L and Q are the ones for the L-operator and coordinates of the rational RS model found in [7]. It is obvious that In = trLn = trg n form a set of mutually commuting functions. Let us note that the L-operator (2.17) has the form L = W P , where W = T −1 U . Since both T and U are elements of the Frobenius group F , the element W also belongs to F . Calculating the Poisson bracket for W , we see that it coincides with the Sklyanin bracket defining on F the structure of a Poisson-Lie group: {W1 , W2 } = [r12 (q), W1 W2 ].

(2.21)

The Poisson relations of W and P are {W1 , P2 } = −P2 [r¯12 (q), W1 ].

(2.22)

A well-known property of the Poisson algebra (2.21) is the existence of a family of mutually commuting functions Jn = trW n . Moreover, it turns out that Jn commute not only with themselves but also with P and Q. In Sect. 3 we show that the same property holds in the quantum case. Now we construct the simplest representation of the Poisson algebra (2.21), (2.22) and (2.13) and relate it with the rational RS model. In fact, this representation corresponds to a zero-dimensional symplectic leaf of the bracket (2.21). To this end we employ the Hamiltonian reduction procedure. The 1-form corresponding to the Poisson structure (2.1-2.3) is α = tr(Ag −1 dg). Recall that G acts on T ∗ G by transformations A → hAh−1 , g → hgh−1 in a Hamiltonian way. By using the 1-form α one can easily get the corresponding moment map µ: µ = gAg −1 − A. Performing the Hamiltonian reduction, we fix its value to be

20


gAg −1 − A = −γ(ee+ − 1), where γ is an arbitrary constant and e+ is a row with all e+i = 1. In terms of (T, L, Q) variables this equation acquires the form T (LQL−1 − Q)T −1 = −γ(ee+ − 1).

(2.23)

Since T e = e and L = W P the last equation can be written as W Q − QW − γW = −γee+ U. Equation (2.24) can be elementary solved and one gets X γ ei bj Eij , W = γ + qij i,j

(2.24)

(2.25)

where b = e+ U . If we recall that W should be an element of F , i.e. W e = e, then we find the coefficients bj : Q 1 a (qaj + γ) Q , (2.26) bj = γ a6=j qaj and thereby

Q Wij =

a6=i (qaj

Q

+ γ)

a6=j qaj

.

(2.27)

One can check that W given by (2.27) has the desired Poisson brackets (2.21) and (2.22). We do not give an explicit proof of this statement since in Sect. 3 we show that the same function W realizes a representation for the corresponding quantum algebra. The relation of our L-operator with the standard Ruijsenaars L-operator [10] is given by the following canonical transformation: qi → qi , Pi →

Y (qai − γ)1/2 a6=i

(qai + γ)1/2

Pi .

3. Quantum L-Operator Algebra In this section we quantize T ∗ G in terms of variables Q, T, P, U and obtain the permutation relations for the quantum L-operator. The algebra of functions on T ∗ G can be easily quantized and one gets an associative algebra generated by A and g subject to the standard relations: 1 ~[C, A1 − A2 ], 2 [A1 , g2 ] = ~g2 C, [g1 , g2 ] = 0,

[A1 , A2 ] =

(3.1) (3.2) (3.3)

where ~ is a quantization parameter. The commutation relations for the generators T, Q, U, P can be straightforwardly written by using the ideology of the Quantum Inverse Scattering Method [20, 21] (we present only nontrivial relations):


21

−1 T1 T2 = T2 T1 R12 (q), U1 U2 = U2 U1 R12 (q), T1 P2 = P2 T1 R¯ 12 (q), U1 P2 = P2 U1 R¯ 12 (q), X Eii ⊗ Eii . [Q1 , P2 ] = ~P2

(3.4) (3.5) (3.6)

i

¯ Here R(q) and R(q) are quantum dynamical R-matrices having the following behavior near ~ = 0: ¯ R(q) = 1 + ~r(q) + o(~), R(q) = 1 + ~r(q) ¯ + o(~). It is worthwhile to mention that the matrix elements of the dynamical R-matrices do not commute only with the momenta pi due to the relations [T1 , Q2 ] = [U1 , Q2 ] = 0. It follows from the compatibility conditions that the R-matrices should satisfy the following set of equations R12 (q)R21 (q) R12 (q)R13 (q)R23 (q) R12 (q)R¯ 13 (q)R¯ 23 (q) R¯ 12 (q)P −1 R¯ 13 (q)P2 2

= = =

1, R23 (q)R13 (q)R12 (q), R¯ 23 (q)R¯ 13 (q)P3−1 R12 (q)P3 , = R¯ 13 (q)P −1 R¯ 12 (q)P3 . 3

(3.7) (3.8) (3.9) (3.10)

Let us demonstrate how to get, for example, Eq. (3.9). This equation follows from (3.5) and the following chain of relations: −1 T1 R12 R¯ 13 R¯ T1 T2 P3 = T1 P3 T2 R¯ 23 = P3 T1 R¯ 13 T2 R¯ 23 = P3 T2 T1 R12 R¯ 13 R¯ 23 = T2 P3 R¯ 23 ¯ = T2 T1 P3 R¯ −1 R¯ −1 R12 R¯ 13 R¯ = T1 T2 R−1 P3 R¯ −1 R¯ −1 R12 R¯ 13 R. 13

23

12

13

23

The solution of the Yang-Baxter Eq. (3.8) can be easily found if one notes that 2 = 0. It is known (see e.g. [24] Prop.6.4.13) that if r ∈ Mat(N, C) ⊗ Mat(N, C) r12 satisfies r3 = 0 and solves the classical Yang-Baxter equation then R = er is a solution of QYBE. Therefore, R12 = e~r12 = 1 + ~r12 is a desired solution of the QYBE. The solution of Eq. (3.9) can be found if one supposes that R¯ has the same matrix structure as r¯ does: X r¯ij (~, q)Fij ⊗ Ejj . (3.11) R¯ 12 (q) = 1 + ~ i6=j

¯ Then the following R-matrix is a solution of Eqs. (3.9) and (3.10): R¯ 12 (q) = 1 +

X i6=j

~ Fij ⊗ Ejj . qij − ~

(3.12)

P −1 (q) = 1 − i6=j q~ij Fij ⊗ Ejj . It is not difficult to verify that R¯ 12 Now we should show that the generators A = T QT −1 and g = U P T −1 satisfy the commutation relations (3.1-3.3). From the relations (3.4-3.6) we get [A1 , A2 ] = T2 T1 (R12 Q1 R21 Q2 − Q2 R12 Q1 R21 )T1−1 T2−1 , X −1 Eii ⊗ Eii )R¯ 12 R12 − R12 Q1 )T2−1 T1−1 , [A1 , g2 ] = g2 T2 T1 (R¯ 12 (Q1 + ~ [g1 , g2 ] =

i −1 −1 −1 ¯ ¯ R12 U2 U1 (R12 P1 R21 P2 R12

−1 −1 − P2 R¯ 12 P1 R¯ 21 )T2−1 T1−1 .

22


By using the following identities, which can be checked by direct computation: R12 Q1 R21 Q2 − Q2 R12 Q1 R21 = ~(CQ1 R21 − R12 Q1 C), X −1 Eii ⊗ Eii )R¯ 12 R12 − R12 Q1 = ~C, R¯ 12 (Q1 + ~ i −1 −1 −1 ¯ P2 R¯ 12 R12 R12 P1 R21

−1 −1 − P2 R¯ 12 P1 R¯ 21 = 0,

(3.13)

one derives the desired commutation relations for A and g. Let us finally present the permutation relations for Q, T and L = T −1 U P , which can be easily obtained by using (3.4-3.6) and (3.13): X Eii ⊗ Eii , (3.14) [Q1 , L2 ] = ~L2 i

T1 R12 L2 = L2 T1 R¯ 12 , −1 −1 −1 L2 R¯ 12 R12 R¯ 21 = R12 L2 R¯ 12 L1 . L1 R¯ 21

(3.15) (3.16)

It is clear that just as in the classical case the quantities In = trg n form a set of mutually commuting operators. Let us show that In can be expressed in terms of L and Q solely and thereby they can be interpreted as quantum integrals of motion for the rational RS model. By using the definition of L we rewrite In as t

t2 In = trg n = trT Ln T −1 = tr12 C12 T1 Ln1 T2−1 = tr12 C12 T1 Ln1 T2−1 ,

where t2 denotes the matrix transposition in the second factor of the tensor product. It follows from (3.15) that t

t

t2 t2 L1 R¯ 21 . L1 T2−1 =T2−1 R12

Applying this relation, we derive t

t2 t2 t2 t2 t2 T1 T2−1 R12 L1 R¯ 21 · · · R12 L1 R¯ 21 . In = tr12 C12 t

Exchanging T1 and T2−1 with the help of Eq. (3.4), one gets t

t2 t2 t2 t2 In = tr12 C12 T2−1 T1 L1 R¯ 21 · · · R12 L1 R¯ 21 . t

t2 t2 t2 t2 t2 Since C12 T2−1 T1 = C12 and R¯ 21 C12 = C12 we finally arrive at t2 t2 t 2 t2 t 2 t2 t 2 In = tr12 C12 L1 R¯ 21 R12 L1 R¯ 21 R12 L1 · · · L1 R¯ 21 R12 L1 .

It is natural to regard this expression for In as a “quantum trace” of the operator Ln . Just as in the classical case the quantum L-operator has the form L = W P , where W = T −1 U satisfies the defining relations of the quantum Frobenius group: R12 W2 W1 = W1 W2 R12 .

(3.17)

The algebra (3.14), (3.16) rewritten in terms of Q, P and W is given by (3.6), (3.17) and by the relation


23

−1 −1 W1 P2 R¯ 12 = P2 R¯ 12 W1 .

(3.18)

This shows that the representation theory for L essentially reduces to the one for the quantum Frobenius group. It is known that the algebra (3.17) admits a family of mutually commuting operators given by [25]: Jn = tr1...n Rˆ 12 Rˆ 23 . . . Rˆ n−1,n W1 . . . Wn , where Rˆ ij = Rij Cij . Now we demonstrate that Jn commutes with P . For the sake of clarity we do it for n = 3. It follows from (3.18) that −1 −1 −1 W1 R¯ 14 R¯ 24 W2 R¯ 24 R¯ 34 W3 R¯ 34 . J3 P4 = Rˆ 12 Rˆ 23 P4 R¯ 14

Equation (3.9) written in terms of Rˆ acquires the form −1 ¯ −1 ˆ P3−1 Rˆ 12 (q)P3 = R¯ 23 R13 R12 R¯ 13 R¯ 23 .

By using this equation we can push P4 on the left. Taking into account that R¯ 12 is diagonal in the second space, we get −1 ¯ −1 ¯ −1 ˆ R24 R34 R12 Rˆ 23 W1 W2 W3 R¯ 34 R¯ 24 R¯ 14 . J3 P4 = P4 R¯ 14

Taking the trace in the first, second and third spaces, one gets the desired property. Now we give an example of the representation of the algebra (3.17), (3.18). Namely, −~

∂

we prove that the W -operator given by Eq. (2.27) realizes this algebra with Pj = e ∂qj . It is obvious that [W1 , W2 ] should be equal to zero since W depends only on the coordinates qi . Thus, the following relation has to be valid: [r12 (q), W1 W2 ] = 0. Substituting the explicit form of r12 (q) we have

1 Wmn qkm qkm qln 1 1 1 − Wml Wmn − Wkn − Wkn qkm qkm qln X 1 (Wkl Wmj − Wkj Wml ). + δln qlj

[r12 (q), W1 W2 ]kl mn = Wkl

1

Wmn −

1

Wkn −

(3.19)

j6=l

First we show that when l 6= n the first line in (3.19) cancels the second one. Since γ bj we get Wij = γ+q ij 1 1 γ γ γ γ 1 − − − γ + qkl qkm γ + qmn qkm γ + qkn qln γ + qmn 1 1 γ γ γ γ 1 = 0. − − γ + qml qkm γ + qmn qkm γ + qkn qln γ + qkn In the case l = n the r.h.s. of (3.19) reduces to

24


− −

1 qkm

(Wkl − Wml )2 +

X 1 (Wkl Wmj − Wkj Wml ) = qlj j6=l

X 1 γ qkm γ γ γ γ 2 b + ( − )bj bl l (γ + qkm )2 (γ + qml )2 qlj γ + qkl γ + qmj γ + qkj γ + qml 2

X Wmj qmk = Wkl . γ + qml γ + qkj j

j6=l

P W Thus, one has to show that for m 6= k the series S = j γ+qmj vanishes. To this end we kj 1 consider the following integral Q I 1 dz a6=m (qa − z + γ) Q , I= 2πi qk − z + γ a (qa − z) where the integration contour is taken around infinity. Since the integrand is nonsingular at z → ∞, we get I = 0. On the other hand, summing up the residues one finds Q X 1 a6=m (qaj + γ) Q = S. I= γ + qkj a6=j qaj j Now we turn to Eq. (3.18). Explicitly it reads as ~ ~ ~2 ~ ~2 −1 − W + Wjl [W , P ] = − + Pj kl j kl qlj − ~ qkj qkj (qlj − ~) qkj (qlj − ~) qkj X ~ ~2 ~2 − Wki − Wli (.3.20) + δjl qkj (qij − ~) qij − ~ qkj (qij − ~) i6=j

For the sake of shortness in (3.20) we adopt a convention that if in some denominator qij becomes zero, the corresponding fraction is also regarded as zero. Thus Eq. (3.20) is equivalent to the following system of equations: ~

Pj−1 [Wkl , Pj ] =

(qkl Wkl − qjl Wjl ), for k 6= l 6= j; qkj (qlj − ~) ~ Wjl , for j 6= l; Pj−1 [Wjl , Pj ] = (qlj − ~) X ~ Wki ; Pk−1 [Wkk , Pk ] = − qik − ~

(3.21) (3.22) (3.23)

i6=k

Pj−1 [Wkj , Pj ] =

~ ~(~ − qkj ) X 1 Wki (Wjj − Wkj ) + qkj qkj qij − ~

(3.24)

i6=j

−

~2 X 1 Wji , for k 6= j. qkj qij − ~ i6=j

In the sequel we shall give an explicit proof only for the latter case since the other three cases are treated quite analogously. The l.h.s. of (3.24) is 1

We are grateful to N.A. Slavnov for explaining to us the technique of treating such series.


Q Pj−1 [Wkj , Pj ]

=

Pj−1

a6=k (qaj

Q

a6=j

+ γ)

qaj

25

Q

a6=j,k (qaj

Q

Pj − Wkj = γ

+ γ − ~)

a6=j (qaj

− ~)

− Wkj .

P As to the r.h.s., one needs to calculate the sum i6=j qij1−~ Wki . For this purpose we evaluate the following integral with the integration contour around infinity: 1 I= 2πi

I

Q

dz z − qj − ~

a6=k (qa

Q

− z + γ)

a (qa

− z)

.

The regularity of the integrand at z → ∞ gives I = 0. On the other hand, summing up the residues one finds 1 I=− ~

Q a6=k Q

(qaj + γ − ~)

−

a6=j (qaj − ~)

X i6=j

1 1 Wki + Wkj . qij − ~ ~

From here one deduces the desired series: X i6=j

1 1 Wki = − qij − ~ ~

and X i6=j

1 1 Wji = − qij − ~ ~

Q a6=k Q

(qaj + γ − ~)

a6=j (qaj − ~)

Q a6=j Q

(qaj + γ − ~)

a6=j (qaj

− ~)

+

1 Wkj ~

(3.25)

+

1 Wjj . ~

(3.26)

Now substituting these sums in the r.h.s. of (3.24) one proves (3.24). It follows from our proof that the L-operator L=

X ij

Q

a6=i (qaj

Q

a6=j

+ γ)

qaj

e

∂ −~ ∂q

j

Eij

realizes the representation of the algebra (3.16). Let us briefly discuss the degeneration of the RS system to the rational CM model. To get the CM model, one should rescale ~ → γ~ and consider the limit γ → 0, L → 1 + γL. Then L is the L-operator of the CM model. From Eqs. 3.14), (3.16) one derives the quantum algebra satisfied by the L-operator of the CM model: [Q1 , L2 ] = ~

X

Eii ⊗ Eii ,

(3.27)

i

[L1 , L2 ] = ~[r12 − r¯12 , L1 ] − ~[r21 − r¯21 , L2 ] + ~2 [r12 − r¯12 , r21 − r¯21 ]. (3.28) The last formula can be written in the following elegant form [L1 + ~(r21 − r¯21 ), L2 + ~(r12 − r¯12 )] = 0.

(3.29)

26


4. Quantum R-Matrix for the Hyperbolic CM System In this section we describe a dual parametrization of T ∗ G, which is related to the hyperbolic CM system. We start with diagonalizing the group element g = V DV −1 and δVij δDi impose the constraint V e = e. The derivatives δgkl and δg are obtained in the same kl manner as in Sect. 2. Calculating the Poisson brackets of V , D and A we get {V1 , A2 } = V1 V2 s12 V2−1 , X Eii ⊗ Eii V2−1 , {D1 , A2 } = −D1 V2

(4.1) (4.2)

i

where s12 = −

X i6=j

Di Fij ⊗ Eji . Di − Dj

The L-operator corresponding to the hyperbolic CM system is defined as the following function on the phase space: L = V −1 AV.

(4.3)

Calculation of the Poisson algebra of T ∗ G in terms of L, V, D results in {L1 , L2 } = [r˜12 , L1 ] − [r˜21 , L2 ], {V1 , L2 } = V1 s12 , X {D1 , L2 } = −D1 Eii ⊗ Eii ,

(4.4) (4.5) (4.6)

i

where we have introduced the matrix 1 r˜12 = −s12 + C. 2 The matrix r˜12 can be written in the following form: qij

r˜12

1X qij 1X e 2 1X =− cth Eij ⊗ Eji + Eii ⊗ Eii , (4.7) qij Eii ⊗ Eji + 2 2 2 2 i sinh 2 i6=j

i6=j

where qi = log Di . Now we clarify the connection of r˜12 with the dynamical r-matrix found in [1]. By 1 1 1 conjugating L-operator (4.3) with the matrix D 2 : L˜ = D 2 LD− 2 and calculating the Poisson bracket for L˜ with the help of (4.4) and (4.6), we arrive at {L˜ 1 , L˜ 2 } = [R˜ 12 , L˜ 1 ] − [R˜ 21 , L˜ 2 ],

(4.8)

where 1 1 1X −1 −1 Eii ⊗ Eii R˜ 12 = D12 D22 r˜12 D1 2 D2 2 − 2 i 1X qij 1X 1 =− cth Eij ⊗ Eji + q Eii ⊗ Eji . 2 2 2 sinh 2ij

i6=j

i6=j

(4.9)


27

It is important to note that R˜ differs from the matrix found in [1] by the term 1X 1 q Eii ⊗ (Eji + Eij ). 2 sinh 2ij i6=j

However, this term does not contribute to the bracket (4.8) if we take into account the ˜ representation of the L-operator of the hyperbolic CM system: L˜ =

X

pi Eii +

i

1X 1 q Eij , 2 sinh 2ij

(4.10)

i6=j

where (p, q) form a pair of canonically conjugated variables. Thus, on the reduced space ˜ these matrices define the same Poisson structure for L. ∗ Now we quantize T G in terms of A, V and D variables. We postulate the following commutation relations: [V1 , A2 ] = ~V1 V2 s12 V2−1 , X Eii ⊗ Eii V2−1 . [D1 , A2 ] = −~D1 V2

(4.11) (4.12)

i

One can verify that the compatibility of these relations with (3.2) follows from the following identity satisfied by s12 : X s12 − D1−1 s12 D1 + Eii ⊗ Eii = C. i

By using Eqs. (3.1), (4.11) and (4.12) one derives the commutation relations for the quantum L-operator: [L1 , L2 ] = ~[r˜12 , L1 ] − ~[r˜21 , L2 ] + ~2 [r˜12 , r˜21 ], X Eii ⊗ Eii , [D1 , L2 ] = −~D1

(4.13) (4.14)

i

[V1 , L2 ] = ~V1 s12 .

(4.15)

The relation (4.13) can be also written in the form (3.29): [L1 + ~r˜21 , L2 + ~r˜12 ] = 0.

(4.16)

One can check without problems that the L-operator (4.10) realizes the representation of the algebra (4.16). To complete our discussion let us show the existence of N mutually commuting operators in the algebra composed by the L-operator and the coordinates D. Obviously, In = trAn mutually commute. Applying the technique used in the previous section to derive the quantum integrals of motion, one can show that In can be expressed as the following function of L and D: t2 (L1 + ~st212 )n . In = trAn = trV Ln V −1 = tr12 C12

In the component form these integrals look as

28


In =

X

Lj 1 j 2 δj 1 m 1

j1 ,...,jn+1 m1 ,...,mn

L j2 j3 δm1 m2

δj 1 j 2 − 1 +~ δj 2 m 1 Dj1 Dj−1 −1 2

δ m j − δm j + ~ 1 3 −1 1 2 δj3 m2 D j2 Dj 3 − 1

Ljn jn+1 δmn−1 mn + ~

!

! ···

δmn−1 jn+1 − δmn−1 jn Djn Dj−1 −1 n+1

! δjn+1 mn

δjn+1 mn .

Let us note that In can not be expressed as a linear combination of trLn solely. 5. Conclusion The approach to R-matrix quantization of the RS models proposed in the paper seems to be general. The problem of real interest is to apply it to the trigonometric and elliptic cases. As was recently shown, the trigonometric and elliptic RS models are obtained from the cotangent bundles over the centrally extended loop group [19] and double loop group [17] respectively. A natural suggestion is to use for this purpose the above-mentioned phase spaces. It is known that the Heisenberg double [26] can be regarded as a natural deformation of the cotangent bundle T ∗ G. It seems to be interesting to investigate the Poisson structure of the Heisenberg double in the same parametrization. One could expect the appearance of another Poisson structure on the Frobenius group induced by the one on the dual Poisson-Lie group G∗ . The appearance of the quantum Frobenius group F states the problem of developing the corresponding representation theory. Owing to the method of orbits, one can suggest that irreducible representations of F should be in correspondence with the symplectic leaves of the Poisson-Lie structure. On the other hand, it is known [27] that the symplectic leaves of a Poisson-Lie structure are the orbits of the dressing transformation. Studying the orbits of F ∗ and corresponding representations of F , one can hope to obtain the quantum integrable systems being some “spin” generalizations of the RS model [28]. Another open problem related to the representation theory is to find the universal Frobenius R-matrix. As was shown in [29], the Ruijsenaars Hamiltonians can be related to the special L-operator satisfying the fundamental relation RLL = LLR with Belavin’s elliptic R-matrix. It would be interesting to clarify the relationship of this approach with our construction. Acknowledgement. The authors are grateful to L.O.Chekhov, P.B.Medvedev and N.A.Slavnov for valuable discussions. This work is supported in part by the RFFR grants N96-01-00608 and N96-01-00551 and by the ISF grant a96-1516.

References 1. 2. 3. 4.

Avan, J. and Talon, M.: Phys. Lett. B303, 33–37 (1993) Babelon, O. and Viallet, C.M.: Phys. Lett. B237, 411 (1989) Avan, J., Babelon, O. and Talon, M.. Alg. Anal. 6 (2), 67 (1994) Sklyanin, E.K.: Alg. Anal. 6 (2), 227 (1994)


29

5. Braden, H.W. and Suzuki, T.: Lett. Math. Phys. 30, 147 (1994) 6. Avan, J. and Rollet, G.: The classical r-matrix for the relativistic Ruijsenaars-Schneider system. preprint BROWN-HET-1014 (1995) 7. Suris, Yu.B.: Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-matrix as the Calogero-Moser ones ? hep-th/9602160 8. Nijhoff, F.W., Kuznetsov, V.B., Sklyanin, E.K. and Ragnisco, O.: Dynamical r-matrix for the elliptic Ruijsenaars-Schneider model. solv-int/9603006 9. Suris, Yu.B.: Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are governed by the same r-matrix. solv-int/9603011 10. Ruijsenaars, S.N.: Commun. Math. Phys. 110, 191 (1987) 11. Avan, J., Babelon, O. and Billey, E.: The Gervais-Neveu-Felder equation and the quantum CalogeroMoser systems. Preprint PAR LPTHE 95-25, May 1995; hep-th/9505091 (to appear in Commun. Math. Phys.) 12. Gervais, J.L. and Neveu, A.: Nucl. Phys. B238, 125 (1984) 13. Felder, G.: Conformal field theory and integrable systems associated to elliptic curves. hep-th/9407154 14. Babelon, O., Bernard, D. and Billey, E.: A quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations. Preprint PAR LPTHE 95-51, IHES/P/95/91, q-alg/9511019 15. Arutyunov, G.E. and Medvedev, P.B.: Phys. Lett. A223, 66–74 (1996) 16. Arutyunov, G.E., Frolov, S.A. and Medvedev, P.B.: Elliptic Ruijsenaars-Schneider model via the Poisson reduction of the affine Heisenberg double. hep-th/9607170 17. Arutyunov, G.E., Frolov, S.A. and Medvedev, P.B.: Elliptic Ruijsenaars-Schneider model from the cotangent bundle over the two-dimensional current group hep-th/9608013 18. Alekseev, A. and .Faddeev, L.D: Comm. Math. Phys. 141, 413 (1991) 19. Gorsky, A. and Nekrasov, N.: Nucl. Phys. B414, 213 (1994); Nucl.Phys. B436, 582 (1995); Gorsky,A.: Integrable many body systems in the field theories. Prep. UUITP-16/94, (1994) 20. Faddeev, L.D.: Integrable models in (1+1)-dimensional quantum field theory. In: Recent advances in field theory and statistical mechanics. Eds. Zuber, J.B., Stord, R. (Les Houches Summer School Proc. session XXXiX, 1982), Elsevier Sci.Publ., 1984 p. 561 21. Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method. Recent developments. In: Integrable quantum field theories. Eds. Hietarinta J., Montonen C., Lect .Not. Phys. 51, 1982 p. 61 22. Olshanetsky, M.A., Perelomov, A.M.: Phys. Reps. 71, 313 (1981) 23. Belavin, A.A. and Drinfel’d, V.G.: Funk. Anal. i ego pril. 16 (3), 1–29 (1982) 24. Chari, V. and Pressley, A.: A Guide to Quantum Groups. Cambridge: Cambridge University Press 25. Maillet, J.M.: Phys. Lett. B245, 480 (1990) 26. Semenov-Tian-Shansky, M.A.: Teor. Math. Phys. 93, 302 (1992)(in Russian) 27. Semenov-Tian-Shansky, M.A.: Publ. RIMS Kyoto Univ. 21 (6), 1237–1260 (1985) 28. Krichever, I.: A.Zabrodin, Spin generalizations of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra. hep-th/9505039 29. Hasegawa, K.: Ruijsenaars’ commuting difference operators as commuting transfer matrices qalg/9512029 Communicated by G. Felder

Commun. Math. Phys. 191, 31 – 60 (1998)

Communications in


Representation Theory of Lattice Current Algebras Anton Yu. Alekseev1,2,? , Ludwig D. Faddeev3 , ¨ Fröhlich2 , Volker Schomerus4,5 Jurg 1 Institute of Theoretical Physics, Uppsala University, Box 803 S-75108, Uppsala, Sweden. E-mail: [email protected] 2 Institut f¨ ur Theoretische Physik, ETH – Hönggerberg, CH-8093 Zürich, Switzerland 3 Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, Russia. E-mail: [email protected] 4 II. Institut f¨ ur Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany. E-mail: [email protected] 5 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan

Received: 25 April 1996 / Accepted: 14 April 1997

Abstract: Lattice current algebras were introduced as a regularization of the left- and right moving degrees of freedom in the WZNW model. They provide examples of lattice theories with a local quantum symmetry Uq (G ). Their representation theory is studied in detail. In particular, we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra. It is shown that for an arbitrary number of lattice sites, the representation categories of the lattice current algebras agree with their continuum counterparts. 1. Introduction Lattice current algebras were introduced and first studied several years ago (see [2, 13] and references therein). They were designed to provide a lattice regularization of the left- and right-moving degrees of freedom of the WZNW model [30] and gave a new appealing view on the quantum group structure of the model. In spite of many similarities between lattice and continuum theory, fundamental relations between them remain to be understood. In this paper we prove the conjecture of [2] that the representation categories of the lattice and continuum model agree. 1.1. Lattice current algebras. Lattice current algebras are defined over a discretized circle, i.e., their fundamental degrees of freedom are assigned to N vertices and N edges of a 1-dimensional periodic lattice. We enumerate vertices by integers n(modN ). Edges are oriented such that the nth edge points from the (n − 1)st to the nth vertex. Being defined over lattices of size N , the lattice current algebras come in families KN , N a positive integer. A precise definition of these (associative *-)algebras KN is given in the next section. We shall see that elements of KN can be assembled into (s × s)− matrices, Jn and Nn , n ∈ ZmodN , with KN -valued matrix elements such that ?

On leave of absence from Steklov Institute, Fontanka 27, St.Petersburg, Russia

32

A. Y. Alekseev, L. D. Faddeev, J. Fröhlich, V. Schomerus 1

2

2

1

1

1

2

2

2

1

R0 Jn Jn =Jn Jn R ,

Jn R Jn+1 =Jn+1 Jn , 2

1

R0 N n R N n = N n R0 N n R, 2

1

1

2

0 N n Jn =Jn R N n R ,

1

2

(1.1) 2

1

0 Jn N n−1 = R N n−1 R Jn .

elements in End(V ) ⊗ KN , where To explain notations we view the matrices Jn , Nn asP V is an s-dimensional vector space. In this way Jn = mn,ς ⊗ln,ς determines elements mn,ς ∈ End(V ) and ln,ς ∈ KN which are used to define X

1

Jn =

2

mn,ς ⊗ e ⊗ ln,ς , Jn =

X

e ⊗ mn,ς ⊗ ln,ς ,

where e is the unit in End(V ). Similar definitions apply to Nn . Throughout the paper we will use the symbol σ for the permutation map σ : End(V ) ⊗ End(V ) → End(V ) ⊗ End(V ), and application of σ to an object X ∈ End(V ) ⊗ End(V ) is usually abbreviated by putting a prime, i.e. X 0 ≡ σ(X). The matrix R = R(h) ∈ End(V ) ⊗ End(V ) which appears in Eqs. (1.1) is a one-parameter solution of the Yang Baxter Equation (YBE). Such solutions can be obtained from arbitrary simple Lie algebras. The lattice Kac-Moody algebra KN depends on a number of parameters, including the “Planck constant” h in the solution R(h) of the YBE and the “lattice spacing” 1 = 1/N . A first, nontrivial test for the algebraic relations (1.1) comes from the classical continuum limit, i.e., from the limit in which h and 1 = 1/N are sent to zero. Using the rules Nn ∼ 1 − 1η(x) , Jn ∼ 1 − 1j(x) , R ∼ 1 + iγhr with x = n/N , γ being a deformation parameter and the standard prescription {., .} = lim i h→0

[., .] h

to recover the Poisson brackets from the commutators, one finds that 1 2 γ [C, j (x)− j (y)]δ(x − y) + γCδ 0 (x − y) , 2 1 2 1 2 γ {η (x), η (y)} = [C, η (x)− η (y)]δ(x − y) , 2 2 1 2 1 γ {η (x), j (y)} = [C, j (x)− j (y)]δ(x − y) + γCδ 0 (x − y) . 2 1

2

{j (x), j (y)} =

Here C is the Casimir element C = r + r0 = r + σ(r). For clarity, let us rewrite these relations in terms of components. When we express C = ta ⊗ ta and j(x) = ja (x)ta in terms of generators ta of the classical Lie algebra, the relations become c jc (x)δ(x − y) + γδab δ 0 (x − y), {ja (x), jb (y)} = γfab c ηc (x)δ(x − y), {ηa (x), ηb (y)} = γfab c jc (x)δ(x − y) + γδab δ 0 (x − y). {ηa (x), jb (y)} = γfab c 0 c c s are the structure constants of the Lie algebra, i.e., [ta , tb ] = fab t . We easily The fab recognize the first equation as the classical Poisson bracket of the left currents in the WZNW model. Furthermore, the quantity j R (x) ≡ j L (x) − η(x) Poisson commutes

Representation Theory of Lattice Current Algebras

33

with j L (x) ≡ j(x) and satisfies the Poisson commutation relations of the right currents, i.e., 1 2 1 2 γ {j R (x), j R (y)} = − [C, j R (x)− j R (y)]δ(x − y) − γCδ 0 (x − y) , 2 1

2

{j R (x), j L (y)} = 0 .

(1.2)

Hence we conclude that the lattice current algebra as described in Eqs. (1.1) is the quantum lattice counterpart of the classical left and right currents. One would like to establish a close relationship between the lattice current algebra and its counterpart in the continuum model. A first step in this direction is described in this paper. We find that the representation categories of the lattice and the continuum theory coincide. For this to work, it is rather crucial to combine left- and right-moving degrees of freedom. For instance, the center of the lattice current algebra with only one chiral sector changes dramatically depending on whether the number of lattice sites is odd or even. The only *-operation known for such algebras [26] is constructed in the case of Uq (sl(2)) and does not admit straightforward generalizations. However, no such difficulties appear in the full theory. It therefore appears to be rather unnatural to constrain the discrete models to one chiral sector. 1.2. Main results. In the next section we use the methods developed in [4] to provide a precise definition of lattice current algebras. In contrast to the heuristic definition we use in this introduction, our precise formulation is applicable to general modular Hopf algebras G, in particular to Uq (G ), for an arbitrary semisimple Lie algebra G . The main result of Sect. 3 provides a complete list of irreducible representations for the lattice current algebras KN . Theorem A. (Representations of KN ) For every semisimple modular Hopf algebra G and every integer N ≥ 1, there exists a lattice current algebra KN which admits a family IJ on Hilbert spaces WNIJ . Here the labels I, J run of irreducible ∗-representations DN through classes of finite-dimensional, irreducible representations of the algebra G. The two labels I, J that are needed to specify a representation of KN correspond to the two chiralities in the theory of current algebras. In fact, the algebra K1 is isomorphic to the quantum double of the algebra G [21] and the pairs I, J label its representations. These results are in agreement with the investigation of related models in [22]. Next, we introduce an inductive limit K∞ of the family of finite dimensional algebras KN . It can be done using the block-spin transformation [13] KN → KN +1 . Under this embedding, every irreducible representation of KN +1 splits into a direct sum of IJ irreducible representations of KN . It appears that the representation DN +1 always splits IJ into several copies of the representation DN . Thus, representations of the inductive limit K∞ are in one to one correspondence with representations of K1 (or KN for arbitrary finite N ). In order to be able to take tensor products of representations of the lattice current algebras, we introduce a family of homomorphisms 3N,M : KN +M −1 → KN ⊗ KM , which satisfy the co-associativity condition

34

A. Y. Alekseev, L. D. Faddeev, J. Fröhlich, V. Schomerus

(id ⊗ 3M,L ) ◦ 3N,L+M −1 = (3N,M ⊗ id) ◦ 3M +N −1,L . Let us notice that the co-product 3N,M is supposed to provide a lattice counterpart of the co-product defined by the structure of superselection sectors in algebraic field theory [9, 10, 27, 28]. By combining the co-product 3N,M with the block-spin transformation we construct a new co-product 1N : KN → KN ⊗ KN which preserves the number of sites in the lattice (see Subsect. 4.3). This co-product is compatible with the block-spin transformation and, hence, it defines a co-product for the inductive limit K∞ : 1∞ : K ∞ → K ∞ ⊗ K ∞ . Our second result concerns tensor products of representations of the lattice current algebra K∞ . Theorem B (Representation category of the lattice current algebra). The braided tensor categories of representations of the lattice current algebra K∞ with the coproduct 1∞ and of the Hopf algebra K1 with the co-product 11 coincide. In principle, our theory must be modified to apply to Uq (G ), q p = 1. It is well known that Uq (G ) at roots of unity is not semi-simple. This can be cured by a process of truncation which retains only the “physical” part of the representation theory of quantized universal enveloping algebras. The algebraic implementation of this idea has been explained in [4] and can be transferred easily to the present situation. We plan to propose an alternative treatment in a forthcoming publication.

2. Definition of the Lattice Current Algebra Our goal is to assign a family of lattice current algebras (parametrized by the number N of lattice sites) to every modular Hopf-*-algebra G. Before we describe the details, we briefly recall some fundamental ingredients from the theory of Hopf algebras. 2.1. Semisimple, modular Hopf-algebras. By definition, a Hopf algebra is a quadruple (G, , 1, S) of an associative algebra G (the “symmetry algebra ”) with unit e ∈ G, a onedimensional representation : G → C (the “co-unit”), a homomorphism 1 : G → G ⊗G ( the “co-product”) and an anti-automorphism S : G → G (the “antipode”). These objects obey a set of basic axioms which can be found e.g. in [1]. The Hopf algebra (G, , 1, S) is called quasitriangular if there is an invertible element R ∈ G ⊗ G such that R 1(ξ) = 10 (ξ) R for all ξ ∈ G , (id ⊗ 1)(R) = R13 R12 ,

(1 ⊗ id)(R) = R13 R23 .

Here 10 = σ ◦ 1, with σ : G ⊗ G → G ⊗ G the permutation map, and we are using the standard notation for the elements Rij ∈ G ⊗ G ⊗ G. For a ribbon Hopf-algebra one postulates, in addition, the existence of a certain invertible central element v ∈ G (the “ribbon element”) which factorizes R0 R ∈ G ⊗ G (here R0 = σ(R)), in the sense that R0 R = (v ⊗ v)1(v −1 )


35

(see [24] for details). The ribbon element v and the element R allow us to construct a distinguished grouplike element g ∈ G by the formula X S(rς2 )rς1 , g −1 = v −1 P 1 rς ⊗ rς2 of R. The element g is where the elements rςi come from the expansion R = important in the definition of q-traces below. We want this structure to be consistent with a *-operation on G. To be more precise, we require that (2.1) R∗ = (R−1 )0 = σ(R−1 ) , 1(ξ)∗ = 10 (ξ ∗ ), and that v, g are unitary 1 . This structure is of particular interest, since it appears in the theory of the quantized universal enveloping algebras Uq (G ) when the complex parameter q has values on the unit circle [18]. At this point we assume that G is semisimple, so that every representation of G can be decomposed into a direct sum of finite-dimensional, irreducible representations. From every equivalence class [I] of irreducible representations of G, we may pick a representative τ I , i.e., an irreducible representation of G on a δI -dimensional Hilbert space V I . The quantum trace trqI is a linear functional acting on elements X ∈ End(V I ) by trqI (X) = T rI (Xτ I (g)) . Here T rI denotes the standard trace on End(V I ) with T rI (eI ) = δI and g ∈ G has been defined above. Evaluation of the unit element eI ∈ End(V I ) with trqI gives the quantum dimension of the representation τ I , dI ≡ trqI (eI ) . Furthermore, we assign a number SIJ to every pair of representations τ I , τ J , SIJ ≡ N (trqI ⊗ trqJ )(R0 R)IJ

with (R0 R)IJ = (τ I ⊗ τ J )(R0 R) ,

for a suitable, real normalization factor N . The numbers SIJ form the so-called S-matrix S. Modular Hopf algebras are ribbon Hopf algebras with an invertible S-matrix 2 . Let us finally recall that the tensor product, τ × τ 0 , of two representations τ, τ 0 of a Hopf algebra is defined by 0 0 (τ × τ )(ξ) = (τ ⊗ τ )1(ξ) for all ξ ∈ G .

In particular, one may construct the tensor product τ I × τ J of two irreducible representations. According to our assumption that G is semisimple, such tensor products of representations can be decomposed into a direct sum of irreducible representations, τ K . IJ in this Clebsch-Gordan decomposition of τ I × The multiplicities NK τ J are called fusion rules. Among all our assumptions on the structure of the Hopf-algebra (G, , 1, S) (quasitriangularity, existence of a ribbon element v, semisimplicity of G and invertibility of S), semisimplicity of G is the most problematic one. In fact it is violated by the algebras 1 Here we have fixed ∗ on G ⊗ G by (ξ ⊗ η)∗ = ξ ∗ ⊗ η ∗ . Following [18], we could define an alternative involution † on G ⊗ G which incorporates a permutation of components, i.e., (ξ ⊗ η)† = η † ⊗ ξ † and ξ † = ξ ∗ for all ξ, η ∈ G. With respect to †, 1 becomes an ordinary ∗ -homomorphism and R is unitary. 2 If a diagonal matrix T is introduced according to T 2 I IJ = $δI,J dI τ (v) (with an appropriate complex factor $), then S and T furnish a projective representation of the modular group SL(2, Z).

36


Uq (G ) when q is a root of unity. It is sketched in [4] how “truncation” can cure this problem, once the theory has been extended to weak quasi-Hopf algebras [18]. Example (Hopf-algebra Zq ). We wish to give one fairly trivial example for the algebraic structure discussed so far. Our example comes from the group Zp . To be more precise, we consider the associative algebra G generated by one element g subject to the relation g p = 1. On this algebra, a co-product, co-unit and an antipode can be defined by 1(g) = g ⊗ g , S(g) = g −1 , (g) = 1 . We observe that G is a commutative semisimple algebra. It has p one-dimensional representations τ r (g) = q r , r = 0, . . . , p − 1, where q is a root of unity, q = e2πi/p . We may construct characteristic projectors P r ∈ G for these representations according to Pr =

p−1

1 X −rs s q g for r = 0, . . . , p − 1 . p s=0

r

One can easily check that τ (P s ) = δr,s . The elements P r are employed to obtain a nontrivial R-matrix, X q rs P r ⊗ P s . R= r,s

When evaluated with a pair of representations τ r , τ s we find that (τ r ⊗ τ s )(R) = q rs . The R-matrix satisfies all the axioms stated above and thus turns G into a quasitriangular P −r2 Hopf-algebra. Moreover, a ribbon element is provided by v = q Pr . We can finally introduce a ∗-operation on G such that g ∗ = g −1 . The consistency relations 2.1 follow from the co-commutativity of 1, i.e. 10 = 1, and the property R = R0 . A direct computation shows that the S-matrix is invertible only for odd integers p. Summarizing all this, we have constructed a family of semisimple ribbon Hopf-*-algebras Zq , q = exp(2πi/p). They are modular Hopf-algebras for all odd integers p. 2.2. R-matrix formalism. Before we propose a definition of lattice current algebras, we mention that Hopf algebras G are intimately related to the objects Nn , n ∈ ZmodN, introduced in Eq. (1.1). To understand this relation, let us introduce another (auxiliary) copy, Ga , of G and let us consider the R-matrix as an object in Ga ⊗ G. To distinguish the latter clearly from the usual R, we denote it by N± , N− ≡ R−1 ∈ Ga ⊗ G , N+ ≡ R0 ∈ Ga ⊗ G . Quasi-triangularity of the R-matrix furnishes the relations 2

1

1a (N± ) =N ± N ± , 2

1

2

1

1

2

R N + N − =N − N + R , 1

(2.2)

2

R N ± N ± = N ± N ±R . Here we use the same notations as in the introduction, and 1a (N± ) = (1 ⊗ id)(N± ) ∈ Ga ⊗Ga ⊗G. The subscript a reminds us that 1a acts on the auxiliary (i.e. first) component of N± . To be perfectly consistent, the objects R in the preceding equations should all be equipped with a lower index a to show that R ∈ Ga ⊗ Ga etc. We hope that no confusion will arise from omitting this subscript on R. Equations (2.2) are somewhat redundant:


37

in fact, the exchange relations on the second line follow from the first equation in the first line. This underlines that the formula for 1a (N± ) encodes information about the product in G rather than the co-product 3 . More explanations of this point follow in Subsect. 2.3. Next, we combine N+ and N− into one element N ≡ N+ (N− )−1 = R0 R ∈ Ga ⊗ G . From the properties of N± we obtain an expression for the action of 1a on N , 2

1

1

2

1a (N ) = N + N + (N − )−1 (N − )−1 1

2

1

1

1

2

= R−1 N + N + R(N − )−1 (N − )−1 2

2

= R−1 N + (N − )−1 R N + (N − )−1 1

2

= R−1 N R N . As seen above, the formula for 1a (N ) encodes relations in the algebra G and implies, in particular, the following exchange relations for N : 1

2

R0 N R N = R0 R1a (N ) = R0 10a (N )R 2

1

= N R0 N R . This kind of relations first appeared in [23]. We used them in the introduction when describing the objects Nn assigned to the sites of the lattice. Thus we have shown that, for any modular Hopf-algebra G, one may construct objects N obeying the desired quadratic relations. The other direction, namely the problem of how to construct a modular Hopf-algebra G from an object N satisfying the exchange relations described above, is more subtle. To begin with, one has to choose linear maps π : Ga → C in the dual Ga0 of Ga . When such linear forms π ∈ Ga0 act on the first tensor factor of N ∈ Ga ⊗ G they produce elements in G: π(N ) ≡ (π ⊗ id)(N ) ∈ G for all π ∈ Ga0 . π(N ) ∈ G will be called the π-component of N or just component of N . Under certain technical assumptions it has been shown in [6] that the components of N generate the algebra G. In this sense one can reconstruct the modular Hopf-algebra G from the object N. Lemma 1 ([6]). Let Ga be a finite-dimensional, semisimple modular Hopf algebra and N be the algebra generated by components of N ∈ Ga ⊗ N subject to the relations 1

2

N R N = R1a (N ) , where we use the same notations as above. Then N can be decomposed into a product of elements N± ∈ Ga ⊗ N , 3 The co-product 1 of G acts on N according to 1(N ) = (id ⊗ 1)(N ) = N N ± ± ± ± ˜ ± ∈ Ga ⊗ G ⊗ G. Here N± [N˜ ± ] on the right hand side of the equation have the unit element e ∈ G in the third [second] tensor factor

38

A. Y. Alekseev, L. D. Faddeev, J. Fröhlich, V. Schomerus −1 N = N + N− such that −1 ∈ Ga ⊗ N ⊗ N , 1(N ) ≡ N+ N˜ N− −1 ∈ Ga ⊗ N (N ) ≡ e ∈ Ga , S(N± ) ≡ N±

define a Hopf-algebra structure on N . Here, the action of 1, , S on the second tensor −1 are supposed component of N, N± is understood. In the equation for 1(N ), N+ , N− P to have a trivial entry inPthe third component while N˜ = nς ⊗ e ⊗ Nς with e being the unit in N and N = nς ⊗ Nς ∈ Ga ⊗ N . As a Hopf algebra, N is isomorphic to Ga . Let us remark that the ∗-operation in G induces a ∗-operation in N which looks as follows: (2.3) N+∗ = N− . In our definition of the lattice current algebras below, we shall describe the degrees of freedom at the lattice sites directly in terms of elements ξ ∈ G, instead of working with N (as in the introduction). The δI -dimensional representations τ I of Ga ∼ = G furnish a δI × δI -matrix of linear forms on Ga . When these forms act on the first tensor factor of N , we obtain a matrix N I ∈ End(V I ) ⊗ G of elements in G, N I ≡ τ I (N ) = (τ I ⊗ id)(N ) . These matrices will turn out to be useful. us illustrate all these remarks on the examExample (R-matrix formalism for PZqrs). Let q P r ⊗ P s and that Zp has only one-dimensional ple of G = Zp . Recall that R = representations given by τ r (g) = q r . Evaluation of the objects N± in representations τ r r produces elements N± = (τ r ⊗ id)(N± ) ∈ C ⊗ Zq ∼ = Zq . Explicitly, they are given by X X r q rs Ps = g r and N− = q −rs Ps = g −r . N+r = Together with the property (τ r ⊗ τ s )1 ∼ = τ r+s the relations (2.2) become g ±(r+s) = g ±s g ±r , q rs g s g −r = g −r g s q rs . For N ∈ Zq ⊗ Zq we find N=

X

q 2rs Pr ⊗ Ps and N r = g 2r .

As predicted by the general theory, the elements N r ∈ Zq generate the algebra Zq when p is odd. 2.3. Definition of KN . Next, we turn to the definition of the lattice current algebras KN associated to a fixed modular Hopf algebra. Before entering the abstract formalism, it is useful to analyse the classical geometry of the discrete model. Our classical continuum theory contains two Lie-algebra valued fields, namely η(x) and j(x). To describe a configuration of η, for instance, we have to place a copy of the Lie-algebra at every point x on the circle. On the lattice, there are only N discrete points left and hence configurations of the lattice field η involve only N copies of the Lie algebra. When


39

passing from the continuum to the lattice, we encode the information about the field j(x) in the holonomies along links, Z jn = P exp( j(x)dx) . n

R

th

Here n denotes integration along the n link that connects the (n − 1)st with the nth site. The classical lattice field jn has values in the Lie group. Let us remark that, even at the level of Poisson brackets, the variables jn can not be easily included into the Poisson algebra. The reason is that jn ’s fail to be continuous functions of the currents. Therefore, we should regularize the Poisson brackets (or commutation relations) of the lattice currents. This regularization is done in the most elegant way with the help of Rmatrices. This consideration explains an immediate appearance of the quantum groups in the description of the lattice current algebras. In analogy to the classical description of the lattice field η, the lattice current algebras contain N commuting copies of the algebra G or, more precisely, KN contains an N -fold tensor product G ⊗N of G as a subalgebra. We denote by Gn the subalgebra G n = e ⊗ . . . ⊗ G ⊗ . . . ⊗ e ⊂ G ⊗N , where G appears in the nth position and all other entries in the tensor product are trivial. The canonical isomorphism of G and Gn ⊂ G ⊗N furnishes the homomorphisms ιn : G → G ⊗N for all n = 1, . . . , N. We think of the copies Gn of G as being placed at the N sites of a periodic lattice, with Gn assigned to the nth site. In addition, the definition of KN will involve generators Jn , n = 1, . . . , N . The generator Jn sits on the link connecting the (n − 1)st with the nth site. Definition 1. The lattice current algebra KN is generated by components 4 of Jn ∈ Ga ⊗ KN , n = 1, . . . , N, along with elements in G ⊗N . These generators are subject to three different types of relations. 1. Covariance properties express that the Jn are tensor operators transforming under the action of elements ξm ∈ Gm like holonomies in a gauge theory, i.e., ιn (ξ)Jn = Jn 1n (ξ) for all ξ ∈ G, 1n−1 (ξ)Jn = Jn ιn−1 (ξ), for all ξ ∈ G ιm (ξ)Jn = Jn ιm (ξ) for all ξ ∈ G, m 6= n, n − 1 modN .

(2.4)

The covariance relations (2.4) make sense as relations in Ga ⊗ KN , if ιn (ξ) ∈ Gn ⊂ KN is regarded as an element ιn (ξ) ∈ Ga ⊗ KN with trivial entry in the first tensor factor and 1n (ξ) ≡ (id ⊗ ιn )1(ξ) ∈ Ga ⊗ Gn ⊂ Ga ⊗ KN . 2. Functoriality for elements Jn on a fixed link means that 1

2

Jn Jn = R1a (Jn ).

(2.5)

This is to be understood as a relation in Ga ⊗ Ga ⊗ KN , where 1a : Ga ⊗ KN → Ga ⊗ Ga ⊗ KN acts trivially on the second tensor factor KN and R = R ⊗ e ∈ 4 Recall that a component of J is an element π(J ) ≡ (π ⊗ id)(J ) in the algebra K . Here π runs n n n N through the dual Ga0 of Ga .

40


Ga ⊗ Ga ⊗ KN . The other notations were explained in the introduction. We also require that the elements Jn possess an inverse Jn−1 ∈ Ga ⊗ KN such that Jn Jn−1 = e, Jn−1 Jn = e .

(2.6)

3. Braid relations between elements Jn , Jm assigned to different links have to respect the gauge symmetry and locality of the model. These principles require 1

2

2

1

if n 6= m, m ± 1modN,

Jn Jm =Jm Jn 1

2

2

1

Jn R Jn+1 = Jn+1 Jn .

(2.7)

R denotes the element R ⊗ e ∈ Ga ⊗ Ga ⊗ KN as before. The lattice current algebra KN contains a subalgebra JN generated by components of the Jn only. They are subject to functoriality (2) and braid relations (3). The subalgebra JN admits an action of G ⊗N (by generalized derivations) such that the full lattice current algebra KN can be regarded as a semi-direct product of JN and G ⊗N with respect to this action. Our covariance relations (1) give a precise definition of the semi-direct product. Let us briefly explain how Definition 1 is related to the description we used in the introduction. The relation of the Hopf algebras Gn and the objects Nn has already been discussed. Our covariance relations in Eq. (2.4) correspond to the exchange relations between N and J in the third line of Eq. (1.1). They can be related explicitly with the help of the quasi-triangularity of R, using the formula N = R0 R. We have, for instance, 2

1

1

1

0 0 N n Jn = (e ⊗ (R R)n ) Jn =Jn [(id ⊗ 1n )(R R)]213 1

1

2

= Jn R(e ⊗ (R0 R)n )R0 =Jn R N n R0 , where we use the notation (R0 R)n = (id ⊗ ιn )(R0 R) ∈ Ga ⊗ Gn and R = (R ⊗ e) ∈ Ga ⊗ Ga ⊗ KN as usual. [.]213 means that the first and the second tensor factors of the expression inside the brackets are exchanged. For finite-dimensional semisimple modular Hopf algebras G, Lemma 1 implies that one could define KN using the objects Nn ∈ Ga ⊗ Gn ⊂ Ga ⊗ KN instead of elements η ∈ G ⊗N . The generators Nn would have to obey the exchange relations stated in Eq. (1.1) and 1

2

N n R N n = R1a (Nn ) . The functoriality relation for Jn did not appear in the introduction. But we can use it now to derive quadratic relations for the Jn in much the same way as has been done for N , earlier in this section. Indeed we find 1

2

R0 Jn Jn = R0 R1a (Jn ) = R0 10a (Jn )R 2

1

= Jn Jn R . This exchange relation is the one used in the introduction to describe the lattice currents Jn . When the first two tensor factors in this equation are evaluated with representations of Ga ∼ = G one derives quadratic relations for the KN -valued matrices, JnI ≡ (τ I ⊗ id)(Jn ) ∈ End(V I ) ⊗ KN .


41

As we discussed earlier in this section, elements in the algebra KN can be obtained from Jn with the help of linear forms π ∈ Ga0 . To understand functoriality properly one must realize that it describes the “multiplication table” for elements π(Jn ) ∈ KN . If we pick two linear forms π1 , π2 ∈ Ga0 on Ga , the corresponding elements in KN satisfy π1 (Jn )π2 (Jn ) = (π1 ⊗ π2 )(R1(Jn )) . We can rewrite this equation by means of the (twisted) associative product ∗ for elements πi ∈ Ga0 , (π1 ∗ π2 )(ξ) ≡ (π1 ⊗ π2 )(R1(ξ)) . It allows us to express the product π1 (Jn )π2 (Jn ) in terms of the element π1 ∗ π2 ∈ Ga0 , π1 (Jn )π2 (Jn ) = (π1 ∗ π2 )(Jn ) for all π1 , π2 ∈ Ga0 . Remark. It may help here to invoke the analogy with a more familiar situation. In fact, the Hopf algebra structure of G induces the standard (non-twisted) product · on its dual G0, (π1 · π2 )(ξ) ≡ (π1 ⊗ π2 )(1(ξ)) for all ξ ∈ G . Let us identify π ∈ Ga0 with the image π(T ) = (π ⊗ id)(T ) of some universal object T ∈ Ga ⊗ G 0 and insert T into the definition of the product ·, π1 (T )π2 (T ) ≡ (π1 ⊗ π2 )(1a (T )) = (π1 · π2 )(T ) . Here and in the following we omit the · when multiplying elements π(T ). The derived multiplication rules for components of T ∈ Ga ⊗ G 0 are equivalent to the following functoriality: 1

2

T T = 1a (T ) 1

and imply RTT-relations:

2

2

1

R T T =T T R .

Such relations are known to define quantum groups, and hence our variables Jn describe some sort of twisted quantum groups. This fits nicely with the nature of the classical lattice field jn . As we have noted earlier, the latter takes values in a Lie group. We saw above that the definition of a product for components of Jn implies the desired quadratic relations. The converse is not true in general, i.e. functoriality is a stronger requirement than the quadratic relations. In the familiar case of Uq (sl2 ) for example, functoriality furnishes also the standard determinant relations which are usually “added by hand” when algebras are defined in terms of quadratic relations. Due to functoriality we are thus able to develop a universal theory which does not explicitly depend on the specific properties of the Hopf algebra G. We have shown that the mathematical definition of KN represented in this section agrees with the description used in the introduction. The algebra KN now appears as a special example of the “graph algebras” defined and studied in [4] to quantize ChernSimons theories. This observation will enable us to use some of the general properties established there. The most important one among such general properties is the existence of a *operation. On the copies Gn , a *-operation comes from the structure of the modular Hopf-*-algebra G. Its action can be extended to the algebra KN by the formula Jn∗ = Sn−1 Jn−1 Sn−1 ,

(2.8)

42


where Sn = (id⊗ιn )[1(κ−1 )(κ⊗κ)R−1 ] ∈ Ga ⊗Gn ⊂ Ga ⊗KN . Here ιn : G → Gn ⊂ KN is the canonical embedding, and the central element κ ∈ G is a certain square root of the ribbon element v, i.e. κ2 = v (cp. [4] for details). Let us note that the formula (2.8) can be rewritten using the elements Nn,± ∈ Ga ⊗ Gn constructed from the R-element according to Nn,+ ≡ (id ⊗ ιn )(R0 ) , Nn,− ≡ (id ⊗ ιn )(R−1 ) . The conjugate current is expressed as −1 −1 Jn Nn−1,− (κn−1 κn ) . Jn∗ = (κn−1 κn )−1 Nn,−

(2.9)

Here κn−1 = ιn−1 (κ), κn = ιn (κ). In order to verify the property (Jn∗ )∗ = Jn , one uses the following identities: −1 −1 vn−1 Jn vn−1 = va Nn−1 Jn , vn−1 Jn vn = va−1 Jn Nn ,

where va is the ribbon element in the auxilary Hopf algebra Ga . The ribbon elements v at different lattice sites generate automorphisms of the lattice current algebra which resemble the evolution automorphism in the quantum top [3]. Example (The U (1)-current algebra on the lattice). It is quite instructive to apply the general definition of lattice current algebras to the case G = Zq . Recall that Zq is generated by one unitary element g which satisfies g p = 1. Representations τ s of G were labeled by an integer s = 1, . . . , p − 1, and τ s (g) = q s with q = exp(2πi/p). We can apply the one-dimensional representations τ s to the current Jn ∈ Ga ⊗ KN to obtain elements Jns = τ s (Jn ) = (τ s ⊗ id)(Jn ) ∈ KN . Functoriality becomes Jns Jnt = q ts Jns+t , where we have used that (τ s ⊗ τ t )1a (ξ) = τ s+t (ξ) and (τ s ⊗ τ t )(R−1 ) = q −st (s + t is to be understood modulo p). The relation allows to generate the elements Jns from Jn1 ∈ KN . Observe that Jn0 is the unit element e in the algebra KN . From the previous equation we deduce that the pth power of the generator Jn1 is proportional to e, (Jn1 )p = q p(p−1)/2 Jnp = q p(p−1)/2 e . This motivates us to introduce the renormalized generators wn = q (1−p)/2 Jn1 ∈ Kn which obey wnp = e. In the following it suffices to specify relations for the generators wn and gn = ιn (g) of KN . The covariance equations (2.4) read gn wn = qwn gn , wn gn−1 = qgn−1 wn , since (τ 1 ⊗ id)1(g) = qg. The exchange relations for currents become wn wn+1 = q −1 wn+1 wn . The identity 1(κ−1 )(κ ⊗ κ)R−1 = e ⊗ e with κ =

P

q −r

2

/2

Pr finally furnishes

wn∗ = wn−1 . At this point one can easily recognize the algebra of wn ’s as the lattice U (1)-current algebra [12].


43

2.4. The right currents. Let us stress that there is a major ideological difference between our discussion of lattice current algebras and the work in [4]. In the context of Chern Simons theories, the graph algebras were introduced as auxiliary objects, and physical variables of the theory were to be constructed from objects assigned to the links, i.e. from the variables Jn . Here the Jn ’s represent only the left-currents, and we expect also right- currents to be present in the theory. They are constructed from the variables Jn and the elements ξn ∈ Gn and hence give a physical meaning to the graph algebras. We define a family of new variables JnR ∈ Ga ⊗ KN on the lattice by setting −1 −1 Jn Nn−1,+ . JnR = Nn,−

The JnR turn out to provide the right currents in our theory. For the rest of this section we will use the symbol JnL to denote the original left currents Jn . Proposition 2 (Right-currents on the lattice). With JnR ∈ Ga ⊗ KN defined as above, one finds that L commute for arbitrary n, m, 1. the elements JnR and Jm 1

2

2

1

R L L R Jn Jm =Jm Jn ;

2. the elements JnR satisfy the following exchange and functoriality relations: 2

1

1

2

R R R R Jn+1 R Jn = Jn Jn+1 , 2

(2.10)

1

R R R Jn Jn = R1a (Jn ) .

(2.11)

(Here we are using the same notations as in the definition of the lattice current algebra above.) If we denote the subalgebra in KN generated by the components of left currents JnL by JNL and, similarly, use JNR to denote the subalgebra generated by components of JnR , the result of this proposition can be summarized in the following statement: JNR and JNL form commuting subalgebras in KN , and JNR is isomorphic to (JNL )op . Here the subscript op means opposite multiplication. Both statements have their obvious counterparts in the continuum theory (cp. Eq. (1.2)). Example (The right U (1)-currents). We continue the discussion of the U (1)-current algebra on the lattice by constructing the right currents. Our general theory teaches us to consider wnR = gn wn−1 gn−1 . 1 1 = ιn (g) and gn−1 = Nn−1,− = ιn−1 (g). The reader is invited to check Here gn = Nn,+ L that these elements commute with wn = wn .

2.5. Monodromies. In the continuum R x theory one is particularly interested in the behavior of the chiral fields g C (x) = P exp( 0 j C (x)dx) under rotations by 2π, i.e. the monodromy of g C . Here and in the following, C stands for either L or R. The monodromy of g C is determined by the expression I C m = P exp( j C (x)dx).

44


Due to the regularizing effect of the lattice, left and right monodromies, M L , M R , are relatively easy to construct and control for our discrete current algebra. They are obtained as an ordered product of the chiral lattice holonomies JnL or JnR along the whole circle, i.e. L M L = va1−N J1L J2L · · · JN

and

R M R = va1−N JN · · · J2R J1R .

(2.12) (2.13)

When we derive relations for the monodromies, it is convenient to include the factors involving va = v ⊗ e ∈ Ga ⊗ KN . The definition in terms of left and right currents produces elements M L and M R in Ga ⊗KN . Their algebraic structure differs drastically from the properties of the currents Jn . We encourage the reader to verify the following list of equations: 1

2

L L L M R M = R1a (M ),

10 (ξ)M L = M L 10 (ξ),

2

1

R R R M R M = R1a (M ) ,

100 (ξ)M R = M R 100 (ξ)

for all ξ ∈ G and with 100 (ξ) = (id ⊗ ι0 )(10 (ξ)) = (id ⊗ ι0 )(σ ◦ 1(ξ)). The functoriality relations are familiar from Subsect. 2.2 and imply that the algebra generated by components of the monodromy M L or M R is isomorphic to G or Gop . There are several places throughout the paper where this observation becomes relevant for a better understanding of our results. As usual, we may evaluate the elements M C in irreducible representations of Ga . This results in a set of KN valued matrices M C,I ≡ (τ I ⊗ id)(M C ). Their quantum traces cIC ≡ tr Iq (M C,I ) are elements in the algebra KN which have a number of remarkable properties. First, they are central elements in the lattice current algebra KN , i.e. the cIC commute with all elements A ∈ KN . Even more important is that cIR , cIL ∈ KN generate two commuting copies of the Verlinde algebra [29]. Explicitly this means that X ¯ K IJ K ∗ NK cC and (cK cIC cJC = C ) = cC ¯ denotes the unique label such that N K K¯ = 1, for C = R, L. Here and in the following K 0 and 0 stands for the trivial representation τ 0 = . If the S-matrix SIJ = N (tr Iq ⊗ tr Jq )(R0 R) is invertible, and N is suitably chosen, the linear combinations χIC =

X

N dI SI J¯ cJC

J

provide a set of orthogonal central projectors in KN , for each chirality C = R, L, i.e., χIC χJC = δIJ χIC , (χIC )∗ = χIC . Proofs of all these statements can be found in [4]. We will see in the next section that products χIL χJR provide a complete set of minimal central projectors in the lattice current algebra or, in other words, they furnish a complete set of characteristic projectors for the irreducible representations of KN .


45

Example (The center of the lattice U (1) current algebra). In terms of the variable wn = q (1−p)/2 Jn1 = −q 1/2 Jn1 (cf. Subsect. 2.3 for notations), the definition of the monodromy M L,1 = (τ 1 ⊗ id)(M L ) becomes M L,1 = (−1)N q 3N/2−1 w1 w2 · · · wN ∈ KN . In this particular example, the quantum trace is trivial so that c1L = M L,1 . It is easily checked that c1L commutes with all the generators wn , gn ∈ KN and that it satisfies (c1L )p = c0L = e . Of course, this relation follows also from the formula crL csL = considerations apply to the right currents.

P

Ntrs ctL = cr+s L . Similar

2.6. The inductive limit K∞ . So far, the lattice current algebras KN depend on the number N of lattice sites, and one may ask what happens when N tends to infinity. A mathematically precise meaning to this question is provided by the notion of inductive limit. The latter requires an explicit choice of embeddings of lattice current algebras for different numbers of lattice sites. They will come from some kind of inverse block-spin transformation [13]. Suppose we are given two lattice current algebras KN and KN +1 with generators Jn , n = 1, . . . , N and Jˆm , m = 1, . . . , N + 1 respectively. The embeddings of G into KN or KN +1 will be denoted by ιn or ιˆm . An embedding γN : KN → KN +1 is furnished by γN (Jn ) = Jˆn for all n < N, γN (JN ) = va−1 JˆN JˆN +1 , and γN (ιn (ξ)) = ιˆn (ξ) for all n < N , γN (ιN (ξ)) = ιˆN +1 (ξ) . The intuitive idea behind γN is to pass from KN to KN +1 by dividing the N th link on the lattice of length N into two new links, so that we end up with a lattice of length ˆ ∈ Ga ⊗ KN +1 , and, N + 1. Observe that γN maps the monodromies M ∈ Ga ⊗ KN to M consequently, the same holds for our projectors χIL , χJR ∈ KN and χˆ IL , χˆ JR ∈ KN +1 , γN (χIL ) = χˆ IL , γN (χJR ) = χˆ JR .

(2.14)

Since the set of numbers N is directed, the collection of KN , together with the maps γN , forms a directed system, and we can define the inductive limit K∞ ≡ lim KN . N →∞

S

By definition, K∞ = N KN / ∼, where two elements AN ∈ KN and AN 0 ∈ KN 0 are equivalent, iff AN is mapped to AN 0 by a string of embeddings γM , i.e., AN 0 = γN 0 −1 ◦. . .◦γN +1 ◦γN (AN ). For the lattice U (1)-current algebra, a detailed investigation of this inductive limit was performed in [5]. We have chosen to define the block-spin transformation by dividing the N th link of the lattice. Now we introduce another block spin operation by dividing the 1st link of the lattice:

46


γ˜ N (Jn ) = Jˆn+1 for all n > 1, γ˜ N (J1 ) = va−1 Jˆ1 Jˆ2 , and γ˜ N (ιn (ξ)) = ιˆn+1 (ξ) for all n . Notice that the two block-spin operations ‘commute’ with each other: γ˜ N +1 ◦ γN = γN +1 ◦ γ˜ N . While the map γN is used in the definition of the inductive limit, we reserve γ˜ N for the definition of the co-product for lattice current algebras (see Sect. 4). 3. Representations of the Lattice Current Algebra The stage is now set to describe our main result on the representation theory of the lattice current algebras. We will begin with a much simpler problem of representing two important subalgebras of KN . Their representations will serve as building blocks for the representation theory of the lattice current algebra KN . 3.1. The algebra U. The lattice current algebras KN contain N − 1 non-commuting holonomies Uν , ν = 1, . . . , N − 1, Uν ≡ va1−ν J1 · · · Jν ∈ Ga ⊗ KN . The objects Uν commute with all elements in the symmetry algebras Gn , except the ones for n = 0 and n = ν. These properties of Uν remind us of holonomies in a gauge theory, which transform nontrivially only under gauge transformations acting at the endpoints of the paths. Together, the elements in G0 ⊗Gν ⊂ KN and the components of Uν generate a subalgebra, Uν , of the lattice current algebra KN . These subalgebras Uν ⊂ KN are all isomorphic to the algebra U which we investigate in this subsection. We begin with a definition of U . Definition 3. The algebra U is the *-algebra generated by components of elements U, U −1 ∈ Ga ⊗ U together with elements in G0 ⊗ G1 such that 1

2

U U = R1a (U ) , ι1 (ξ)U = U 11 (ξ),

10 (ξ)U = U ι0 (ξ),

and U −1 is the inverse of U . Here we use the same notations as in Subsect. 2.3. In particular, ι0,1 denote the canonical embeddings of G into G0 ⊗ G1 . The ∗-operation on U extends the ∗-operation on G0 ⊗ G1 ⊂ U, so that U ∗ = S1−1 U −1 S0 . Here Si = (id ⊗ ιi )(1(κ−1 )(κ ⊗ κ)R−1 ) ∈ Ga ⊗ U for i = 0, 1, and κ is a certain central square root of the ribbon element, as before. The algebra U admits a very nice irreducible representation, D, which is constructed by acting with elements in U on a “ground state” |0i. The state |0i may be characterized by the following (invariance-) properties: ιi (ξ)|0i = |0i(ξ) for all ξ ∈ G

i = 0, 1,


47

where is the trivial representation (co-unit) of G. Here and in the following we neglect to write the letter D when elements in U act on vectors. Since we are dealing with a unique representation of U, ambiguities are excluded. While the preceding formula means that |0i is invariant under the action of G0 ⊗ G1 , the components of U ∈ Ga ⊗ U create new states in the carrier space, 1/2 when the theory of pseudo-differential operators can be extensively used. Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

2

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

3

The Eikonal and Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4

Wave Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

5

The Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

6

The Diagonal Singularity of the Scattering Amplitude . . . . . . . . . . . . . . 204

7

Examples. Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

184

D. Yafaev

1. Introduction We consider the Schrödinger operator H = −1 + V (x) in the space L2 (Rd ) with a long-range potential V (x) satisfying a usual assumption |∂ α V (x)| ≤ Cα (1 + |x|)−ρ−|α| ,

ρ > 0.

(1.1)

Our goal is to develop the stationary scattering theory for the operator H and, in particular, to study the associated scattering matrix. We compare H with the operator H0 = −1 but consider the wave operators with a non-trivial identification J± (depending on the sign of t) (1.2) W± = W± (H, H0 ; J± ) = s − lim eiHt J± e−iH0 t . t→±∞

Such operators were introduced in [6], where the Enss method was used for the proof of their existence and completeness. Following [6], we construct J± as a Fourier integral operator with the phase function satisfying the eikonal equation and the amplitude satisfying the transport equation. We prefer to work with approximate but explicit solutions of these equations. Actually, J± enjoys all properties of pseudo-differential operators (pdo) since its phase function is sufficiently close to hx, ξi. The eikonal equation and the transport equations do not have global solutions so that we satisfy these equations outside of some cone where x is proportional to −ξ for the sign “+00 and to ξ for the sign “−00 . Neighbourhoods of these “bad” directions are cut off by some homogeneous function ζ± of degree 0. In contrast to [6] we use the stationary techniques for construction of wave operators (1.2) and show that long-range scattering fits into the theory of smooth (in the sense of Kato) perturbations. Indeed, the “effective” perturbation T± = HJ± − J± H0 is again a pdo with symbol vanishing as |x|−1 (because of the cut-off ζ± ) at infinity. Our proof of the existence and completeness of wave operators relies on the standard limiting absorption principle and the radiation conditions estimates introduced in [19] for the proof of asymptotic completeness for multi-particle systems. The radiation conditions estimates replace a rather tiresome construction of eigenfunctions of the operator H used (see e.g. [4]) in the first proof of the completeness in the long-range case. Once the existence and completeness of wave operators (1.2) are verified, it is easy to check (see the middle of Sect. 4) that they coincide with more traditional (see e.g. [4]) wave operators obtained by time-dependent modification of the free dynamics. Our basic goal is to study spectral properties of the corresponding scattering matrix S = S(λ), λ > 0, and the diagonal singularity of its kernel s(ω, ω 0 ) (the scattering amplitude). We stress that our paper is to a large extent motivated by [7]. Recall that in the short-range case (when V (x) = O(|x|−ρ ), ρ > 1, as |x| → ∞ and J± = I) the operator S − I is compact. Hence the spectrum of S consists of eigenvalues of finite multiplicity accumulating possibly at the point 1 only and the leading singularity of its kernel is the Dirac-function. As we shall see, the structure of S is completely different in the long-range case. Our study of the scattering matrix relies on its stationary representation. First, using the resolvent estimates, called usually microlocal or propagation estimates (see [14, 10, 9] or [5, 8]) we show, similarly to [7], that the part of S containing the resolvent of the operator H is an integral operator with smooth kernel. The remaining, singular, part S1 of S is quite explicit and can be obtained as the restriction of the operator J+∗ (HJ− −J− H0 ) on the sphere of radius λ1/2 in the momentum representation. It is convenient to treat S1 as a pdo (this view-point was adopted in [2] in the short-range case). We check that the

Scattering Amplitude for Schrödinger Equation with Long-Range Potential

185

principal symbol of S1 is an oscillating function determined by the asymptotics of V at infinity. In particular, this implies that the spectrum of the scattering matrix covers the whole unit circle. Note that in the short-range case the principal symbol of S equals 1 which corresponds to the Dirac-function in its kernel. Thus, in the long-range case this singularity disappears. The diagonal singularity of the scattering amplitude is given by the Fourier transform of the symbol of the operator S1 . It turns out that for an asymptotically homogeneous function V (x) of order −ρ, ρ < 1, X wj (ω, ω 0 ) exp iψj (ω, ω 0 ) , (1.3) s(ω, ω 0 ) = where the modules wj and the phases ψj are asymptotically homogeneous functions, as ω −ω 0 → 0, of orders −(d−1)(1+ρ−1 )/2 and 1−ρ−1 , respectively. The precise result is formulated in Theorem 6.7. Its proof relies on the stationary phase method applied to our integral representation of the scattering amplitude. In the case ρ = 1 (see Theorem 6.10) the sum (1.3) consists of one term only, the module w is asymptotically homogeneous of order −d + 1 and the phase ψ has a logarithmic singularity at the diagonal. We always assume that a potential V (x) is a C ∞ -function which satisfies the estimate (1.1) for all multi-indices α. Actually, only some finite number of derivatives is required but we imposed condition (1.1) for all α in order to use freely the calculus of pseudodifferential operators. We emphasize that the operator J± belongs to the Hörmander 0 with ρ determined by (1.1) and δ = 1 − ρ. The basic calculus of pseudoclass Sρ,δ differential operators is well developed for this class if ρ > 1/2 > δ. Thus we can lean directly on this theory only if (1.1) is fulfilled for ρ > 1/2. All our results remain valid for an arbitrary ρ > 0 but this requires a separate study of a special class of pdo with oscillating symbols. In this paper we concentrate on the case ρ > 1/2 in order to avoid additional technicalities. Let us mention that regularity of the scattering amplitude away from the diagonal was announced in [1] and its proof can be found in [7]. In the latter paper there is also an upper bound on |s(ω, ω 0 )| at the diagonal. However, as our asymptotic formula shows, this bound is not optimal. Let us say some words about terminology. In the short-range case the scattering amplitude f (ω, ω 0 ; λ) is defined (see e.g. [12]) as the coefficient at the outgoing spherical wave |x|−(d−1)/2 eik|x| in the asymptotics as |x| → ∞ of the scattering solution of the Schrödinger equation; the parameters ω 0 and ω = x/|x| are incident and outgoing directions, respectively. In terms of the scattering amplitude the kernel of the operator S(λ) − I equals (2π)−(d−1)/2 ieiπ(d−3)/4 λ(d−1)/4 f (ω, ω 0 ; λ). This relation between the scattering matrix and the scattering amplitude is preserved for ω 6= ω 0 in the long-range case although the plane and spherical waves are distorted. Thus, the scattering amplitude should be defined as f (ω, ω 0 ; λ) = −i(2π)(d−1)/2 e−iπ(d−3)/4 λ−(d−1)/4 s(ω, ω 0 ; λ),

(1.4)

0

but we often use this term with respect to the kernel s(ω, ω , λ) of the scattering matrix. According to (1.4) the scattering cross-section in an angle dω around ω is 6(ω, ω 0 , λ)dω, where X (1.5) (ω, ω 0 ; λ) = (2π)d−1 λ−(d−1)/2 |s(ω, ω 0 ; λ)|2 . Abusing somewhat terminology we call the function 6 itself the scattering cross-section. A choice of operators J± in (1.2) is, of course, not unique. Actually, let U± be multiplication by a function exp(i8± (ξ)) in the momentum representation and set J˜± =

186

D. Yafaev

J± U± . Clearly, the wave operators (1.2) and W± (H, H0 ; J˜± ) exist at the same time and the scattering amplitude s(ω, ˜ ω 0 ; λ) corresponding to J˜± is related to the original one by the equality s(ω, ˜ ω 0 ; λ) = exp(−i8+ (λ1/2 ω))s(ω, ω 0 ; λ) exp(i8− (λ1/2 ω 0 )).

(1.6)

Apparently, there is no preferable choice of identifications J± . Nevertheless, if the diagonal singularity of the function s(ω, ω 0 ; λ) is found for some operators J± , then, by (1.6), it is automatically known for all operators J˜± . In particular, the scattering cross-section 6(ω, ω 0 ; λ) does not depend at all on the choice of identifications. As shown by Gordon and Mott (see e.g. [12]), for the purely Coulomb potential (and d = 3) the quantum cross-section equals the classical one (its definition can be found in Sect. 7) for all scattering angles. According to our analysis, the quantum and classical cross-sections coincide in the limit of small scattering angles for all central potentials V (x) = v|x|−ρ , ρ < 1, and all dimensions d. On the other hand, for the Coulomb potential this is true in the case d = 3 only.

2. Preliminaries 1. We need some elementary facts about pseudo-differential operators (pdo) defined by the equality Z eihξ,xi a(x, ξ)fˆ(ξ)dξ, (2.1) (Af )(x) = (2π)−d/2 Rd

where fˆ(ξ) = (2π)−d/2

Z

e−ihξ,xi f (x)dx Rd

is the Fourier transform of f ∈ S(Rd ). We suppose that the symbol a ∈ C ∞ (Rd × Rd ) and (2.2) |∂xα ∂ξβ a(x, ξ)| ≤ C(1 + |x|)m−ρ|α|+δ|β| , C = Cα,β , for some numbers m ≤ 0, ρ > 1/2 > δ and all multi-indices α, β. Here and below C and c are different positive constants, whose precise values are of no importance. Moreover, we assume that a(x, ξ) = 0 for sufficiently large |ξ|. The class of symbols m . satisfying these conditions is usually denoted by Sρ,δ The adjoint operator A∗ to A is defined by Z ∗ d (A f )(ξ) = (2π)−d/2 e−ihξ,xi a(x, ξ)f (x)dx, Rd

and thus fits into a usual framework of the theory (see e.g. [3, 16]) of pseudo-differential operators (note that the roles of the variables x and ξ are interchanged). This allows to recover all results of the theory for the operator A itself. Necessary information on the calculus of pdo is collected in the four following assertions. Let us denote by X the operator of multiplication by (1 + |x|2 )1/2 . Proposition 2.1. The operator AX −m is bounded and AX −m1 , m1 > m, is compact in the space L2 (Rd ).


187 m

Proposition 2.2. Suppose that symbols aj of pdo Aj , j = 1, 2, belong to classes Sρ,δj . Then A∗2 A1 is the pdo with symbol a(x, ξ) which admits for any N the representation X a(x, ξ) = (−i)α ∂xα ((∂ξα a2 )(x, ξ)a1 (x, ξ))/α! + a˜ (N ) (x, ξ), |α|≤N m ˜ with m ˜ = m1 + m2 − (N + 1)(ρ − δ). where a˜ (N ) (x, ξ) ∈ Sρ,δ 0 and let A be the pdo Proposition 2.3. Let Aj , j = 1, 2, be pdo with symbols aj ∈ Sρ,δ ∗ with symbol a1 (x, ξ)a2 (x, ξ). Then the operator A − A1 A2 is compact. m Proposition 2.4. Suppose that A is a pdo with symbol a ∈ Sρ,δ and G is a pdo (actually, P a differential operator) with symbol g(x, ξ) = |s|≤s0 gs (x)ξ s , where gs ∈ C ∞ (Rd ) and ∂ α gs (x) = O(|x|n−|α| ) as |x| → ∞ for some n and all α. Let B be the pdo with symbol b(x, ξ) = |g(x, ξ)|2 a(x, ξ). Then the operator X p (G∗ AG − B)X p is bounded if 2p = ρ − m − 2n. m is invariant with respect to a change of variables (a In the case ρ > 1/2 the class Sρ,δ diffeomorphism in ξ). This allows to define pseudo-differential operators on manifolds. Moreover, the principal symbol of a pdo (its symbol modulo a term from the class m+1−2ρ ) is invariantly defined on the cotangent bundle of a manifold. We will need to Sρ,δ consider pdo on the unit sphere Sd−1 = {|ξ| = 1}.

2. Here we collect necessary estimates on the resolvent R(z) = (H − z)−1 (and its powers) of the Schrödinger operator H = −1 + V in the space H = L2 (Rd ). Various definitions of H-smoothness are discussed e.g. in [15] or [17]. We start with the limiting absorption principle. Proposition 2.5. Let assumption (1.1) be fulfilled, n = 1, 2, . . . and p > n − 1/2. Then the operator-function X −p Rn (z)X −p is continuous in norm with respect to the parameter z in the closed complex plane cut along [0, ∞), possibly with the exception of the point z = 0. In particular, the operator X −p is H-smooth for p > 1/2 on any interval 3 = (λ0 , λ1 ) for 0 < λ0 < λ1 < ∞. Proposition 2.5 implies, of course, that the positive spectrum of H is absolutely continuous. Its proof is naturally divided into two parts. The continuity of X −p Rn (z)X −p outside of the point spectrum of H is proven (for an arbitrary n), for example, in [9] and the absence of positive eigenvalues - in [15], v.4. The following resolvent estimates borrowed from [19, 20] are called radiation conditions-estimates there. Proposition 2.6. Let assumption (1.1) be fulfilled and p > 1/2. Set Gj u = X −1/2 (∂j u − |x|−2 h∇u, xixj ),

j = 1, . . . , d.

(2.3)

Then the operator-functions Gj R(z)G∗k , Gj R(z)X −p for all j, k = 1, . . . , d are weakly continuous with respect to the parameter z in the closed complex plane cut along [0, ∞), possibly with the exception of the point z = 0. strong operator In particular, the operator Gj is H-smooth on any interval 3 = (λ0 , λ1 ) for 0 < λ0 < λ1 < ∞. Propositions 2.5 for n = 1 and 2.6 are sufficient for the proof of existence and completeness of wave operators. However, our study of the scattering matrix requires additional resolvent estimates called usually microlocal or propagation estimates.

188

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Proposition 2.7. Let assumption (1.1) be fulfilled and n = 1, 2, . . .. Suppose that the m , a± (x, ξ) = 0 in some ball symbol a± (x, ξ) of a pdo A± belongs to some class Sρ,δ |ξ| ≤ c and that the support of a± (x, ξ) is contained in the cone ∓hξ, xi ≥ |x| |ξ|,

ξˆ = ξ/|ξ|, xˆ = x/|x|.

> 0,

Then the operator-functions X p−σ A∗+ Rn (z)X −p ,

X −p Rn (z)A− X p−σ ,

p > n − 1/2, σ > n + m,

are bounded and continuous in norm with respect to the parameter z in the region Re z ≥ λ0 > 0, Im z ≥ 0. Proposition 2.8. Under the assumptions of Proposition 2.7 for all p the operatorfunctions X p A∗+ Rn (z)A− X p are bounded and continuous in norm with respect to the parameter z in the region Re z ≥ λ0 > 0, Im z ≥ 0. Proofs of these assertions can be found either in [14, 10, 9] or in [5, 8]. The proof of [14, 10, 9] relies on the Mourre estimate and is easily accessible. 3. For construction of the scattering matrix we need to discuss restrictions of integral or pseudo-differential operators to the spheres ξ 2 = λ. Let (00 (λ)f )(ω) = 2−1/2 k (d−2)/2 fˆ(kω),

λ = k 2 > 0, ω ∈ Sd−1 ,

(2.4)

be (up to the numerical factor) the restriction of fˆ onto the sphere of radius k. Then the formally adjoint operator is defined by the equality Z eikhω,xi g(ω)dω. (2.5) (0∗0 (λ)g)(x) = 2−1/2 k (d−2)/2 (2π)−d/2 Sd−1

Denote N = L2 (Sd−1 ). The first of the following assertions is a direct consequence of the Sobolev trace theorem (or of Proposition 2.5). The second can be deduced from Proposition 2.6, see [20] for details. Proposition 2.9. For p > 1/2 the operator-function X −p 0∗0 (λ) : N → H is compact and continuous in norm with respect to the parameter λ > 0. Proposition 2.10. The operator-functions Gj 0∗0 (λ) : N → H are bounded and strongly continuous with respect to the parameter λ > 0. We call the operator A[ (λ), defined formally by the equality A[ (λ) = 00 (λ)A0∗0 (λ), the restriction of A to the sphere |ξ| = k. If A is a pdo and A[ (λ) is considered as an integral operator, then, by (2.4), (2.5), the kernel Z 0 [ 0 −1 −d d−2 eikhω −ω,xi a(x, kω 0 )dx (2.6) p (ω, ω ; λ) = 2 (2π) k Rd

(such integrals are understood, of course, in the sense of distributions) of the operator A[ (λ) is an infinitely differentiable function of ω and ω 0 (and λ > 0) for ω 6= ω 0 . However due to a possible strong singularity of function (2.6) at the diagonal ω = ω 0 a


189

precise definition of the restriction A[ (λ) requires some assumptions on the symbol of A. The operator A[ = A[ (λ) can also be considered as a pdo on Sd−1 . Actually, let κ be a local diffeomorphism of Sd−1 on an open set κ ⊂ Rd−1 and ωκ = κ(ω). Then Z Z 0 (A[ g)(ω) = (2π)−d+1 eihωκ −ωκ ,bi a[κ (ωκ , b)gκ (ωκ0 )dωκ0 db, gκ (ωκ ) = g(ω). κ

Rd−1

m , where 2ρ > 1, the symbols a[κ are connected for different κ by In the case a[κ ∈ Sρ,δ a usual formula of a change of variables for pdo. In particular, the principal symbol a[ m+1−2ρ (its symbol modulo a term from the class Sρ,δ ) of the pdo A[ is invariantly defined on the cotangent bundle of the unit sphere. The spectral family E0 (λ) of the operator H0 = −1 is described in terms of operators (2.4), (2.5). Actually, set δε (H0 − λ) = (2πi)−1 R0 (λ + iε) − R0 (λ − iε) ,

where R0 (z) = (H0 − z)−1 . Then for any λ > 0 and p > 1/2, lim X −p δε (H0 − λ)X −p = dX −p E0 (λ)X −p /dλ = X −p 0∗0 (λ) · 00 (λ)X −p .

ε→0

Let A be a pdo with symbol a(x, ξ) satisfying (2.2). If m < −1, then the operator A[ (λ) = 00 (λ)X −p X p AX p X −p 0∗0 (λ) , p = −m/2, is correctly defined as a bounded operator in N. In the case m = −1 a definition of the operator A[ (λ) requires vanishing of its symbol on the conormal bundle of the sphere |ξ| = k. −1 and a(tω, kω) = 0, ω ∈ Sd−1 , at least for Proposition 2.11. Suppose that a ∈ Sρ,δ sufficiently large |t|. Then for any functions f, g from the Schwartz class S(Rd ) the double limit

lim (Aδε (H0 − λ)f, δη (H0 − λ)g) =: (A[ (λ)00 (λ)f, 00 (λ)g)

ε,η→0

0 exists. The operator A[ (λ) is a pdo on Sd−1 with symbol from the class Sρ,δ so that [ [ A (λ) is a bounded operator in the space N. The principal symbol of A (λ) is given by the absolutely convergent integral Z ∞ a(zω + k −1 b, kω)dz, |ω| = 1, hω, bi = 0. a[ (ω, b; λ) = (4π)−1 k −1 −∞

This assertion was proved in [13], where it was supposed that ρ = 1, δ = 0. However, the proof of [13] extends automatically to the case ρ > 1/2 > δ (but not to the case ρ ≤ 1/2). Note that the existence of A[ can also be deduced from Proposition 2.6 (see Lemma 4.1 below). 4. Here we study the essential spectrum of some special class of pdo with oscillating symbols. In view of our applications we consider pdo on the unit sphere Sd−1 but the

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problem reduces by a diffeomorphism to operators acting in a domain ⊂ Rd−1 . We need the well-known formula for the action of a pdo A, Z Z 0 −d+1 0 eihω −ω,bi a(ω, b)f (ω 0 )dω 0 db, a ∈ Sρ,δ , (Af )(ω) = (2π)

Rd−1

on the exponential function. Actually, if f ∈ C0∞ (), b0 ∈ Rd−1 , b0 6= 0, and u,t (ω) = −(d−1)/2 f (−1 (ω − ω0 ))eithb0 ,ωi ,

(2.7)

then (Au,t )(ω) = a(ω, tb0 )u,t (ω) + r,t (ω), where kr,t kL2 ≤ C−1 t−ρ

(2.8)

as long as tρ → ∞. 0 Proposition 2.12. Let A be a pdo with symbol from the class Sρ,δ , where ρ > 1/2 > δ. Suppose that for some point (ω0 , b0 ), |ω0 | = 1, hω0 , b0 i = 0, b0 6= 0, the principal symbol a(ω, b) of A admits the representation

a(ω0 , tb0 ) = eiθ(t) (1 + o(1)),

t → ∞,

(2.9)

where θ(t) is a continuous function and lim sup θ(t) = ∞

t→∞

or

lim inf θ(t) = −∞.

t→∞

(2.10)

Then the spectrum of the operator A in the space N covers the unit circle. Proof. It suffices to construct for every point µ = eiϑ a sequence un such that ||un || = 1, un converges weakly to zero and lim ||Aun − µun || = 0.

n→∞

(2.11)

Consider a neighbourhood G ⊂ Sd−1 of the point ω0 , a diffeomorphism of G onto ⊂ Rd−1 and a unitary operator U : L2 (G) → L2 () corresponding to this diffeomorphism. Then the principal symbols of the operators U AU ∗ and A are the same. Since by the proof of (2.11) compact terms can be neglected, we may suppose that the symbol of U AU ∗ coincides with its principal symbol. Abusing somewhat notation we denote the operator U AU ∗ again by A and its symbol by a(ω, b), ω ∈ , b ∈ Rd−1 . This function 0 and satisfies condition (2.9). belongs to the class Sρ,δ 0 , we have that By the condition a ∈ Sρ,δ |a(ω, tb0 ) − a(ω0 , tb0 )| ≤ C|ω − ω0 | sup |(∇ω a)(ω, tb0 )| ≤ C|ω − ω0 | tδ . ω∈

Combining this estimate with (2.8) we obtain for functions (2.7) that ||Auε,t − a(ω0 , tb0 )uε,t || ≤ C(ε−1 t−ρ + εtδ ).

(2.12)

Set now ε = t−(ρ+δ)/2 . Then the right-hand side of (2.12) equals t−(ρ−δ)/2 and hence tends to zero as t → ∞. Let n → ∞ (or n → −∞) under the first (second) assumption (2.10). Choose a sequence tn → ∞ such that θ(tn ) = ϑ + 2πn and set un = uεn ,tn , −(ρ+δ)/2 . Then relation (2.11) follows from (2.9) and (2.12). where εn = tn


191

3. The Eikonal and Transport Equations In this section we collect standard material on approximate but explicit solutions of the eikonal and transport equations. These equations arise in an attempt to construct eigenfunctions of the Schrödinger operator in the form ψ = eiϕ a. 1. For proving existence and completeness of wave operators and a study of the scattering matrix up to compact terms it suffices to set a(x) = 1. Since (−1 + V )(eiϕ ) = eiϕ (|∇ϕ|2 + V − i1ϕ),

(3.1)

we require that ϕ be a special (approximate) solution of the eikonal equation |∇ϕ|2 + V (x) = λ,

λ > 0.

For an arbitrary unit vector ω ∈ Sd−1 we set ξ = λ1/2 ω and seek ϕ(x, ξ) in the form ϕ(x, ξ) = hξ, xi + 8(x, ξ),

(3.2)

where (∇8)(x, ξ) tends to zero as |x| → ∞ in some cone around ω. Let us first consider an auxiliary linear equation 2hξ, ∇8(x, ξ)i + V (x) = 0,

∇ = ∇x ,

(3.3)

ˆ xi and construct its solution 8(x, ξ) such that (∇8)(x, ξ) tends to zero in any cone hξ, ˆ ≥ κ. Here and below κ is an arbitrary number from the interval (−1, 1). For a given ξ = λ1/2 ω we decompose x = zω + b, where hω, bi = 0. In other words, we choose a coordinate system with the axis z directed along ω. Then (3.3) reduces to 2|ξ| ∂8(z, b)/∂z + V (z, b) = 0 and has a solution −1

Z

z

8(z, b) = −(2|ξ|)

V (z 0 , b)dz 0 + c(b),

(3.4)

(3.5)

0

where c(b) is an arbitrary function of b. According to (3.4) the function ∂8(z, b)/∂z vanishes as |x| → ∞ but the behaviour of Z z (∇b V )(z 0 , b)dz 0 + ∇c(b) (3.6) (∇b 8)(z, b) = −(2|ξ|)−1 0

is determined by a choice of the function c(b). Set Z ∞ c(b) = (2|ξ|)−1 (V (z 0 , b) − V (z 0 , 0))dz 0 .

(3.7)

0

Then, by (3.6), (∇b 8)(z, b) = (2|ξ|)−1

Z

∞

(∇b V )(z 0 , b)dz 0

z

tends to zero as |x| → ∞ in any cone z ≥ κ(z 2 + b2 )1/2 (or, equivalently, z ≥ κ(1 − κ2 )−1/2 |b|). Equalities (3.5), (3.7) for the function 8(x, ξ) =: 8+ (x, ξ) can be, of course, rewritten in an invariant (not depending on a coordinate system) form: Z ∞ V (x ± tξ) − V (±tξ) dt. (3.8) 8± (x, ξ) = ±2−1 0

The function 8− (x, ξ) = −8+ (x, −ξ) also satisfies Eq. (3.3) and (∇8− )(x, ξ) tends to ˆ xi zero in any cone hξ, ˆ ≤ κ. We always suppose that |ξ| ≥ c > 0.

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Proposition 3.1. Let assumption (1.1) hold. Define the function ϕ± (x, ξ) by equalities (3.2), (3.8). Then (0) , (3.9) (−1 + V − ξ 2 )(eiϕ± ) = eiϕ± q± where (0) = |∇8± |2 − i18± . (3.10) q± ˆ For all multi-indices α, β, any κ ∈ (−1, 1) and ±hξ, xi ˆ ≥ ±κ (or |x| ≤ c), |∂xα ∂ξβ 8± (x, ξ)| ≤ C|ξ|−1−|β| (1 + |x|)1−ρ−|α| , (0) |∂xα ∂ξβ q± (x, ξ)| ≤ C|ξ|−1−|β| (1 + |x|)−2ρ−|α| ,

|α| ≥ 1,

(3.11)

ρ ≤ 1.

(3.12)

Proof. Equalities (3.9), (3.10) follow directly from (3.1) and (3.2), (3.3). Differentiating (3.8) we find that Z ∞ t|β| (∂ α+β V )(x ± tξ)dt, |α| ≥ 1. ∂xα ∂ξβ 8± (x, ξ) = 2−1 (±1)1+|β| 0

Note that 2|x ± tξ|2 ≥ (1 − κ2 )(x2 + t2 ξ 2 ) Therefore (1.1) implies that

Z

|∂xα ∂ξβ 8± (x, ξ)| ≤ C

∞

if

± hξ, xi ≥ ±κ|x| |ξ|.

(3.13)

t|β| (1 + |x| + |ξ|t)−ρ−|α|−|β| dt.

0

Making the change of variables t = |ξ|−1 (1 + |x|)s in the last integral we see that it equals the right-hand side of (3.11). Estimate (3.12) for function (3.10) is a consequence of (3.11). Note that, in the case ρ < 1, inequality (3.11) for α = 0 follows from its validity for |α| = 1. 2. A description of the diagonal singularity of the scattering matrix requires a better approximation to eigenfunctions of H. It follows from (3.9) that (−1 + V − ξ 2 )(eiϕ± a± ) = eiϕ± q± ,

(3.14)

where

(0) a± − 2ih∇ϕ± , ∇a± i − 1a± . (3.15) q ± = q± The equality q± = 0 gives us the transport equation for a± . Its approximate solution can be constructed by a procedure similar to that of introduction. Define functions b(±) n (x, ξ) = 1, inductively by relations b(±) 0 (0) (±) (±) fn(±) = 2h∇8± , ∇b(±) n i + iq± bn − i1bn , Z ∞ −1 (x, ξ) = ±2 fn(±) (x ± tξ, ξ)dt, b(±) n+1

(3.16) (3.17)

0

and set ) a± (x, ξ) = a(N ± (x, ξ) =

N X

b(±) n (x, ξ).

(3.18)

n=0

Note that

(±) 2hξ, ∇b(±) n+1 i + fn = 0.

(3.19)


193

Proposition 3.2. Let assumption (1.1) with ρ ∈ (1/2, 1] hold. Then functions b(±) n , n≥ (N ) 1, and q± = q± defined by Eqs. (3.16), (3.17) and (3.15), (3.18) satisfy in the region ˆ xi ±hξ, ˆ ≥ ±κ (or |x| ≤ c) the estimates −n−|β| (1 + |x|)−ε0 n−|α| , |∂xα ∂ξβ b(±) n (x, ξ)| ≤ C|ξ|

and

ε0 = 2ρ − 1 > 0,

(N ) |∂xα ∂ξβ q± (x, ξ)| ≤ C|ξ|−N −|β| (1 + |x|)−1−ε0 (N +1)−|α| .

(3.20) (3.21)

Proof. Suppose that (3.20) holds for some n. Then it follows from (3.16) and Proposition 3.1 that |∂xα ∂ξβ fn(±) (x, ξ)| ≤ C|ξ|−n−|β| (1 + |x|)−2ρ−ε0 n−|α| .

(3.22)

Differentiating (3.17) we see that ∂xα ∂ξβ b(±) n+1 (x, ξ) consists of terms Z ∞ t|β1 | (∂xα+β1 ∂ξβ2 fn(±) )(x ± tξ, ξ)dt, β1 + β2 = β. 0

According to (3.22) this integral is bounded by Z ∞ t|β1 | (1 + |x| + |ξ|t)−2ρ−ε0 n−|α|−|β1 | dt. C|ξ|−n−|β2 | 0

Using inequality (3.13) and making the change of variables t = |ξ|−1 (1 + |x|)s we obtain bound (3.20) for n + 1. This proves inductively (3.20) for all n. It follows from (3.16), (3.19) that (0) (±) (±) (±) 2ihξ, ∇b(±) n+1 i + 2ih∇8± , ∇bn i − q± bn + 1bn = 0. (N ) Summing these equalities over n = 0, 1, . . . , N we find that q± = 2ihÊξ, ∇b(±) N +1 i. Thus, (3.21) is a consequence of (3.20) for n = N + 1.

Corollary 3.3. For functions ϕ± (x, ξ) and a± (x, ξ) constructed in Propositions 3.1 and 3.2, respectively, Eq. (3.14) holds with the function q± (x, ξ) satisfying estimates (3.21). Remark. If ρ > 1, then ρ should be replaced by 1 in estimates (3.12), (3.20) and (3.21). On the other hand, in this case one can set ϕ± (x, ξ) = hx, ξi and construct a± (x, ξ) as an approximate solution of the transport equation 2ihξ, ∇a± i + 1a± = 0. Then Eq. (3.14) remains true with a function q± (x, ξ) satisfying (3.21). 4. Wave Operators 1. Here we consider wave operators for the pair of Hamiltonians H0 = −1, H = −1+V and a suitable identification J± . To avoid unnecessary remarks we may suppose that (1.1) holds with ρ ∈ (1/2, 1). Let σ± ∈ C ∞ (−1, 1) be such that σ± (τ ) = 1 in a neighbourhood of the point ±1 and σ± (τ ) = 0 in a neighbourhood of the point ∓1 and let η ∈ C ∞ (Rd ) be such that η(x) = 0 in a neighbourhood of zero and η(x) = 1 for large |x|. Set ˆ xi) ζ± (x, ξ) = η(x)σ± (hξ, ˆ

(4.1)

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and choose some function ψ ∈ C0∞ (R+ \{0}). Due to the function ψ(ξ 2 ) all our considerations will be localized on a bounded energy interval disjoint from zero. The function η is introduced only to get rid of the singularity of the function |x|−1 x at the point x = 0. We construct J± as a pseudo-differential operator, Z (J± f )(x) = (2π)−d/2 eiϕ± (x,ξ) a± (x, ξ)ζ± (x, ξ)ψ(ξ 2 )fˆ(ξ)dξ, (4.2) Rd

where ϕ± and a± are the functions defined in Sect. 3. According to (3.11), the function exp(8± (x, ξ)) satisfies estimates (2.2) with m = 0, the same ρ as in (1.1) and δ = 1 − ρ. It follows that the symbol ei8± (x,ξ) a± (x, ξ)ζ± (x, ξ)ψ(ξ 2 ) 0 of pdo (4.2) belongs to the class Sρ,δ . Thus, by Proposition 2.1, J± is a bounded oper) ator in the space H. Moreover, by virtue of estimates (3.20), the operator J± (a(N ± )− (0) (0) J± (a± ), a± = 1, is compact. Therefore the wave operators (1.2) (and the corresponding scattering matrices) are the same for all N so that in this section we may set a± (x, ξ) = 1. We consider also “inverse” wave operators

W± (H0 , H; J ∗ ) = s − lim eiH0 t J ∗ e−iHt E(R+ ), t→±∞

where J = J+ or J = J− . Recall (see Proposition 2.5) that E(R+ ) coincides with the projection on the absolutely continuous subspace. Our goal in this section is to show that the triple H0 , H, J± fits into the framework of the theory of smooth perturbations so that wave operators (1.2) exist and are complete. To that end we need the following Lemma 4.1. Let T be a pdo with symbol 2 ˆ ), t(x, ξ) = g(x, ξ)w(hx, ˆ ξi)η(x)ψ(ξ −1 where g ∈ Sρ,δ , w ∈ C ∞ [−1, 1] and w(±1) = 0. Then T admits the representation

T =

d X

G∗j B (s) Gj + X −p B (r) X −p ,

(4.3)

j=1

where Gj are defined by (2.3), p = (1 + ρ)/2 and the operators B (s) , B (r) are bounded. Proof. Let us define B (s) as a pdo with symbol ˆ xi ˆ 2 )−1 t(x, ξ). b(s) (x, ξ) = (1 + |x|2 )1/2 |ξ|−2 (1 − hξ, 0 The function (1 − τ 2 )−1 w(τ ) is C ∞ on [−1, 1] so that b(s) ∈ Sρ,δ and the operator B (s) Pd is bounded. Set T0 = j=1 G∗j B (s) Gj . Since

(1 + |x|2 )−1/2

d X

(ξj − |x|−2 hξ, xixj )2 b(s) (x, ξ) = t(x, ξ),

j=1

Proposition 2.4 implies that the operator X p (T − T0 )X p is bounded. Let us calculate the perturbation


T± = HJ± − J± H0 . According to (3.9) and (4.2), we have that Z eiϕ± (x,ξ) τ± (x, ξ)ψ(ξ 2 )fˆ(ξ)dξ, (T± f )(x) = (2π)−d/2

195

(4.4)

(4.5)

Rd

where τ± (x, ξ) = q± (x, ξ)ζ± (x, ξ) − 2ih∇ϕ± (x, ξ), ∇ζ± (x, ξ)i − 1ζ± (x, ξ)

(4.6)

(0) with ∇ = ∇x , 1 = 1x and q± = q± . This expression is O(|x|−1 ) as |x| → ∞ because of the cut-off function (4.1). Let us single out the term (s) ˆ xi)i, (x, ξ) = −2iη(x)hξ, ∇σ± (hξ, ˆ τ±

(4.7)

which decays as |x|−1 only. Other terms in (4.6) decay more rapidly. Actually, the (r) , defined by the equality function τ± (s) (r) (x, ξ) + τ± (x, ξ), τ± (x, ξ) = τ±

(4.8)

satisfies, according to Proposition 3.1, the estimates (r) |∂xα ∂ξβ τ± (x, ξ)| ≤ C(1 + |x|)−2ρ−|α| ,

x ∈ Rd , 0 < c0 ≤ |ξ| ≤ c1 < ∞.

(4.9)

Let us introduce pdo T±(s) , T±(r) with symbols i8± (x,ξ) (s) τ± (x, ξ)ψ(ξ 2 ), t(s) ± (x, ξ) = e

respectively. Then T± = T±(s) + T±(r) .

i8± (x,ξ) (r) t(r) τ± (x, ξ)ψ(ξ 2 ), ± (x, ξ) = e

(4.10)

Proposition 4.2. Operator (4.4) admits representation (4.3) with p = ρ. −1 Proof. Let us consider the operators T±(s) and T±(r) separately. Since t(s) ± ∈ Sρ,1−ρ and 0 σ± (τ ) = 0 in neighbourhoods of points −1 and 1, Lemma 4.1 can be directly applied to −2ρ T±(s) . It follows from (4.9) that t(r) ± ∈ Sρ,1−ρ and consequently, by Proposition 2.1, the (r) operator X ρ T± X ρ is bounded.

Taking into account that, by Propositions 2.5 and 2.6, the operators X −ρ , ρ > 1/2, and Gj are H0 - and H- smooth on any bounded disjoint from zero interval we arrive (see e.g. [15] or [17]) at the following result. Theorem 4.3. Suppose that condition (1.1) is fulfilled for ρ > 1/2. Let J± = J± (ζ± , ψ) be defined by (4.2). Then all wave operators W± (H, H0 ; J± ),

∗ W± (H0 , H; J± )

(4.11)

W± (H, H0 ; J∓ ),

∗ W± (H0 , H; J∓ )

(4.12)

and

exist. Operators (4.11) as well as (4.12) are adjoint to each other.

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2. Let us show that the wave operators W± (H, H0 ; J± ) are isometric (on some subspace of H) and complete. Lemma 4.4. The following relations hold: ∗ J± − ψ 2 (H0 ))e−iH0 t = 0, s − lim (J±

(4.13)

∗ s − lim J± J± e−iH0 t = 0.

(4.14)

t→±∞

t→∓∞

∗ Proof. By Propositions 2.1 and 2.2 for N = 0, up to a compact term, J± J± is the pdo 2 2 2 (we denote it by A) with symbol ζ± (x, ξ)ψ (ξ ). Thus, Z 2 2 (Ae−iH0 t f )(x) = (2π)−d/2 eihξ,xi−iξ t ζ± (x, ξ)ψ 2 (ξ 2 )fˆ(ξ)dξ. (4.15) Rd

The stationary point ξ = x/(2t) of this integral does not belong to the support of the function ζ± (x, ξ) if t → ∓∞. Therefore supposing that f ∈ S(Rd ) and integrating by parts we estimate integral (4.15) by CN (1 + |x| + |t|)−N for an arbitrary N . This proves (4.14). To prove (4.13) we apply the same arguments to the pdo with symbol 2 (x, ξ) − 1)ψ 2 (ξ 2 ). (ζ± Below we consider an interval 3 = (λ0 , λ1 ), where 0 < λ0 < λ1 < ∞, and choose a function ψ ∈ C0∞ (R+ ) such that ψ(λ) = 1 on 3. Proposition 4.5. The operators W± (H, H0 ; J± ) are isometric on the subspace E0 (3)H and ∗ ) = 0. (4.16) W± (H, H0 ; J∓ ) = 0, W± (H0 , H; J∓ Proof. The results on the operators W± (H, H0 ; J± ) and W± (H, H0 ; J∓ ) are immediate consequences of (4.13) and (4.14), respectively. The second equality (4.16) is a ∗ consequence of the first because W± (H0 , H; J∓ ) = W± (H, H0 ; J∓ )∗ . Now it is easy to prove completeness of the operators W± (H, H0 ; J± ). Theorem 4.6. Suppose that condition (1.1) is fulfilled for ρ > 1/2. Then the asymptotic completeness holds: Ran (W± (H, H0 ; J± )E0 (3)) = E(3)H.

(4.17)

Proof. We have to check that for any f ∈ E(3)H there exists f0 ∈ E0 (3)H such that lim ||e−iHt f − J± e−iH0 t f0 || = 0.

t→±∞

(4.18)

∗ ∗ ) shows that for f0 = W± (H0 , H; J± )f , The existence of the operator W± (H0 , H; J± ∗ −iHt lim ||J± e f − e−iH0 t f0 || = 0,

t→±∞

and hence

∗ −iHt e f − J± e−iH0 t f0 || = 0. lim ||J± J±

t→±∞

(4.19)

∗ −iHt e f || = 0 and, consequently, The second equality (4.16) implies that limt→±∞ ||J∓ ∗ −iHt e f || = 0. lim ||J∓ J∓

t→±∞

(4.20)


197

∗ ∗ By Proposition 2.3, up to a compact term, J± J± + J∓ J∓ is the pdo with symbol 2 2 2 2 (ζ± (x, ξ) + ζ∓ (x, ξ))ψ (ξ ). Choosing the functions σ± (see (4.1)) in such a way that 2 (τ ) = 1, we deduce from (4.19), (4.20) that σ+2 (τ ) + σ−

lim ||ψ 2 (H0 )e−iHt f − J± e−iH0 t f0 || = 0.

t→±∞

Since the operator ψ 2 (H) − ψ 2 (H0 ) is compact, this yields (4.18).

∗ ) Remark. Equality (4.17) is equivalent to isometricity of the operator W± (H0 , H; J± on E(3)H.

3. Let us, finally, show that W± (H, H0 ; J± ) coincide with wave operators defined in terms of a time-dependent modification of e−iH0 t . Following [18], we choose a modification of this group in x-representation. First, we check Lemma 4.7. Let a family of unitary operators be defined by the equality (U0 (t)f )(x) = e∓πdi/4 eiΞ(x,t) (2|t|)−d/2 fˆ((2t)−1 x), where −1 2

Ξ(x, t) = (4t)

Z

±t > 0,

(4.21)

1

x −t

V (sx)ds.

(4.22)

0

Then for any function ζ± , lim ||J± (ψ, ζ± )e−iH0 t f − U0 (t)ψ(H0 )f || = 0,

t→±∞

∀f ∈ L2 (Rd ).

Proof. Consider the representation Z 2 −iH0 t −d/2 f )(x) = (2π) eiϕ± (x,ξ)−iξ t ζ± (x, ξ)ψ(ξ 2 )fˆ(ξ)dξ, (J± e Rd

f ∈ S(Rd ). (4.23)

Stationary points ξ± (x, t) of the phase function are determined by the equation x + (∇ξ 8± )(x, ξ± (x, t)) − 2ξ± (x, t)t = 0.

(4.24)

Due to the function ζ± (x, ξ)ψ(ξ 2 ) we are interested only in points ξ± (x, t) such that 2 ≤ C < ∞ and ±hξˆ± , xi ˆ ≥ ±κ for some κ ∈ (−1, 1). Using estimate 0 < c ≤ ξ± (3.11) on ∇ξ 8± we see that for large |t| Eq. (4.24) has a unique solution ξ± (x, t) and ξ± (x, t) = (2t)−1 x + (2t)−1 (∇ξ 8± )(x, (2t)−1 x) + O(|t|−2ρ ).

(4.25)

Applying the stationary phase method to integral (4.23) and taking into account that ζ± (x, ξ± ) = η(x) we find that 2 (x, t))fˆ(ξ± (x, t)), (J± e−iH0 t f )(x) ∼ e∓πid/4 eiΞ± (x,t) (2|t|)−d/2 ψ(ξ±

t → ±∞. (4.26) Here “ ∼00 means that the difference of the left- and right-hand sides tends to zero in L2 (Rd ) and 2 Ξ± (x, t) = hξ± (x, t), xi + 8± (x, ξ± (x, t)) − ξ± (x, t)t

= (4t)−1 x2 + 8± (x, ξ± (x, t)) − (4t)−1 |(∇ξ 8± )(x, ξ± (x, t))|2 ,

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where we have used Eq. (4.24). Formula (4.25) allows us to simplify this expression neglecting terms which in the case 2ρ > 1 tend to zero as t → ±∞: Ξ± (x, t) = (4t)−1 x2 + 8± (x, (2t)−1 x)). Using Eq. (3.8) we see that Ξ± (x, t) = Ξ(x, t) for ±t > 0. Finally, again by (4.25), we 2 ˆ )f (ξ± ) by ψ((2t)−2 x2 )fˆ((2t)−1 x) in (4.26). can replace ψ(ξ± Lemma 4.7 allows us to reformulate the results of this section as Proposition 4.8. Suppose that condition (1.1) is fulfilled for ρ > 1/2. Define U0 (t) by Eqs. (4.21), (4.22). Then the wave operators W± = s − lim eiHt U0 (t) t→±∞

exist, are isometric and Ran W± = E(R+ )H. Furthermore, for any ψ ∈ C0∞ (R+ \ {0}) and any function (4.1), W± ψ(H0 ) = W± (H, H0 ; J± (ψ, ζ± )). 4. Local singularities of a potential can be easily accommodated. Actually, suppose that V (x) satisfies condition (1.1) outside of some ball |x| ≤ r0 only. Let η0 ∈ C ∞ (Rd ) be such that η0 (x) = 0 for |x| ≤ r0 and η0 (x) = 1 for |x| ≥ R0 + 1. Set V0 (x) = η0 (x)V (x) and construct the phases 8± (x, ξ) by formula (3.8), where V is replaced by V0 . This gives an additional term (1 − η0 (x))V (x)ζ± (x, ξ) in the right-hand side of (4.6). However, it disappears if the function η(x) in (4.1) is chosen in such a way that η(x) = 0 for |x| ≤ r0 + 1. Then all proofs of this section work without any modification as long as singularities of V (x) inside the ball |x| ≤ r0 are not strong enough to violate estimates of Propositions 2.5 and 2.6. Quite similarly, all constructions are preserved if a potential is a sum V + Vs of a function V satisfying (1.1) and of a short-range term Vs (x) = O(|x|−ρs ), ρs > 1, as |x| → ∞. In this case an additional term Vs J± arising in (4.4) admits, according to Proposition 2.1, the factorization Vs J± = X −p B± X −p , p = ρs /2, with a bounded operator B± . 5. The Scattering Matrix 1. It follows from Theorems 4.3, 4.6 and Proposition 4.5 that the scattering operator S = W+∗ (H, H0 ; J+ )W− (H, H0 ; J− ) commutes with H0 and is unitary on the subspace E0 (3)H. Thus, in a diagonal representation of H0 it reduces to the operator of multiplication by the operator-function S(λ) : N → N called the scattering matrix. We consider the standard diagonal representation of H0 . Let the unitary operator U : H → L2 (R+ ; N) be defined by equalities (U f )(λ) = 00 (λ)f and (2.4). Since U H0 U ∗ acts in the space L2 (R+ ; N) as multiplication by the independent variable λ, the operator U SU ∗ acts as multiplication by the operator-function S(λ). It is unitary for λ ∈ 3.


199

We need a stationary formula for the scattering matrix in the case when identifications J+ and J− for t → +∞ and t → −∞ are different. Let us introduce auxiliary wave operators (5.1) Ω± = s − lim eiH0 t J+∗ J− e−iH0 t . t→±∞

The operator Ω± commutes with H0 and hence U Ω± U ∗ acts in the space L2 (R+ ; N) as multiplication by the operator-function ± (λ) : N → N. The scattering matrix admits two representations which can be formally written as (5.2) S(λ) = + (λ) − 2πi00 (λ) J+∗ T− − T+∗ R(λ + i0)T− 0∗0 (λ) and

S(λ) = − (λ) − 2πi00 (λ) T+∗ J− − T+∗ R(λ + i0)T− 0∗0 (λ)

with T± defined by (4.4). Both representations can be given a precise meaning in the framework of the smooth scattering theory. For definiteness, we use below representation (5.2). Let us formulate a precise result assuming that H0 = −1, H = −1 + V with a relatively compact perturbation V and some bounded operators J± . Note, however, that Proposition 5.1 has, actually, a sense in the abstract framework. Proposition 5.1. Suppose that T ± = K ∗ B± K

for both signs and

˜ J+∗ T− = K ∗ BK,

(5.3)

where B± , B˜ are bounded operators in some auxiliary Hilbert space G and K : H → G 1/2 is H0 -bounded. Assume that U0 (λ; K) = K0∗0 (λ) : N → G are bounded operators strongly continuous in λ > 0. Let, finally, the operator-function R(z; K) = KR(z)K ∗ : G → G be weakly continuous with respect to the parameter z in the closed complex plane cut along [0, ∞), possibly with the exception of the point z = 0. Then the wave operators W± (H, H0 ; J), W± (H0 , H; J ∗ ) (where J = J+ or J = J− ) and (5.1) exist and the scattering matrix S(λ) admits the representation ˜ 0 (λ; K) + 2πiU0∗ (λ; K)B+∗ R(λ + i0; K)B− U0 (λ; K). S(λ) = + (λ) − 2πiU0∗ (λ; K)BU (5.4) In particular, the operator-function S(λ) − + (λ) is weakly continuous in λ > 0. Equality (5.4) is formally the same as (5.2) but in the right-hand side of (5.4) we have a combination of bounded operators. This gives a correct meaning to (5.2). Below we usually write representation (5.2) keeping in mind that its precise form is given by (5.4). A proof of Proposition 5.1 is practically the same as the proof of the corresponding assertion in [17] in the case J+ = J− . Therefore we shall give only a sketch of the proof concentrating on formula representations and omitting details of their justification. Under assumptions of Proposition 5.1 wave operators W± (J) = W± (H, H0 ; J) admit stationary representations Z ∞ lim (T R0 (λ ± iε)f0 , δε (H − λ)f )dλ (5.5) (W± (J)f0 , f ) = (Jf0 , f ) + −∞ ε→0

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D. Yafaev

and Z (E(3)W± (J)f0 , f ) =

lim (δε (H0 − λ)f0 , J ∗ g − T ∗ R(λ ± iε)f )dλ,

3 ε→0

(5.6)

where J = J+ or J = J− and T = HJ − JH0 . Representation (5.6) ensures that lim (δε (H − λ)W+ (J)f0 , f ) = lim ((J − R(λ − iε)T )δε (H0 − λ)f0 , f ).

ε→0

ε→0

(5.7)

It follows from (5.5) for J = J− and f = W+ (J+ )g0 that (W+ (J− )f0 − W− (J− )f0 , W+ (J+ )g0 ) Z ∞ lim (T− δε (H0 − λ)f0 , δε (H − λ)W+ (J+ )g0 )dλ. = 2πi −∞ ε→0

Equality (5.7) implies (at least formally) that δε (H − λ)W+ (J+ )g0 in the last integral may be replaced by (J+ − R(λ − iε)T+ )δε (H0 − λ)g0 so that for any f0 , g0 ∈ H Z = 2πi

∞

(W+ (J− )f0 − W− (J− )f0 , W+ (J+ )g0 ) lim ((J+∗ − T+∗ R(λ + iε))T− δε (H0 − λ)f0 , δε (H0 − λ)g0 )dλ

−∞ ε→0 Z ∞

= 2πi −∞

((J+∗ − T+∗ R(λ + i0))T− dE0 (λ)f0 /dλ, dE0 (λ)g0 /dλ)dλ.

Since dE0 (λ)/dλ = 0∗0 (λ)00 (λ), this is equivalent to representation (5.2). 2. Let us apply Proposition 5.1 to our triple H0 , H and J± . Thus, we assume that V is multiplication by a function satisfying (1.1) for ρ ∈ (1/2, 1) and J± is given by (4.2), ) where a± = a(N ± for some N . Note, first that according to (4.14) s − lim J− exp(−iH0 t) = 0 t→+∞

so that Ω+ = 0 and hence + (λ) = 0 for all λ > 0. The operator T± is determined by (N ) . Properties of the operator J+∗ T− are summarized in Eqs. (4.5), (4.6), where q± = q± the following Lemma 5.2. The operator J+∗ T− is a pdo with symbol a(x, ξ) = eiφ(x,ξ) wN (x, ξ) + uN (x, ξ), where φ(x, ξ) = 2−1

Z

∞ −∞

V (tξ) − V (x + tξ) dt,

(5.8)

ˆ ˆ + wN (x, ξ) ˆ ξi)hξ, ∇σ− (hx, ˆ ξi)i wN (x, ξ) = −2iη 2 (x)ψ 2 (ξ 2 )σ+ (hx,

(5.9) (5.10)

and −2ρ wN ∈ S1,0 ,

−1−(N +1)ε0 uN ∈ Sρ,1−ρ ,

ε0 = 2ρ − 1.

(5.11)


201

Proof. Let us apply Proposition 2.2 to pdo A1 = T− and A2 = J+ with symbols t− (x, ξ) and j+ (x, ξ), respectively. It follows from (4.2), (4.5) that ∂xα ((∂ξα j+ )(x, ξ)t− (x, ξ)) = eiφ(x,ξ) vα (x, ξ). −1−ε0 |α|

Here φ(x, ξ) = 8− (x, ξ)−8+ (x, ξ) satisfies (5.9) according to (3.8) and vα ∈ S1,0 according to (3.11), (3.20), (3.21). In particular, ˆ ˆ v0 (x, ξ) = −2iη 2 (x)ψ 2 (ξ 2 )σ+ (hx, ˆ ξi)hξ, ∇σ− (hx, ˆ ξi)ia + (x, ξ)a− (x, ξ).

Equality (3.18) and estimate (3.20) allow us to replace a± by 1 here. This gives the representation (5.8)–(5.10) for a(x, ξ). ˆ 0 (x, ξ), where ˆ ξi)a Let A0 be pdo with symbol σ+ (hx, ˆ a0 (x, ξ) = −2ieiφ(x,ξ) η 2 (x)ψ 2 (ξ 2 )hξ, ∇σ− (hx, ˆ ξi)i.

(5.12)

It follows from Lemma 5.2 and Proposition 2.1 that J+∗ T− − A0 = X −ρ BX −ρ ,

(5.13)

where B is some bounded operator. Furthermore, by Lemma 4.1, the operator A0 admits representation (4.3) with p = ρ. Representations (4.3) for T± and J+∗ T− give factorizations (5.3) for these operators. Now the space G consists of several copies of H, K is a “vector” operator with components X −ρ , Gj , j = 1, . . . , d, and B± or B˜ are “matrix” operators constructed in terms of different operators B (r) and B (s) . Therefore weak continuity of R(z; K) and strong continuity of U0 (λ; K) are consequences of Propositions 2.5, 2.6 and 2.9, 2.10, respectively. Thus, all assumptions of Proposition 5.1 are fulfilled and we can reformulate it for our case. We also take into account that for unitary operator-functions the weak continuity implies the strong one. Theorem 5.3. Let condition (1.1) with ρ > 1/2 hold and let J± be defined by (4.2), ) where a± = a(N ± for some N . Then the scattering matrix S(λ) for the triple H0 , H and J± admits representation (5.2), where + (λ) = 0. In particular, S(λ) is strongly continuous in λ > 0. Remark. A representation of S(λ) to a large extent similar to (5.2) appeared first in [7]. However it seems to us that 00 (λ)J+∗ T− 0∗0 (λ) was not well defined there as a bounded operator in N. Indeed, its definition requires either Proposition 2.6 or 2.11 but assertions of such type were not used in [7]. 3. Now we can start our analysis of singularities of the scattering matrix. We may suppose that a± = 1 in (4.2) till the end of this section. Remark that the terms S1 (λ) = −2πi00 (λ)J+∗ T− 0∗0 (λ),

(5.14)

S2 (λ) = 2πi00 (λ)T+∗ R(λ + i0)T− 0∗0 (λ)

(5.15)

depend on the choice of the cut-off functions ζ± in definition (4.2) of the identifications J± , but their sum S(λ) = S1 (λ) + S2 (λ) does not depend on it. Below we always suppose that σ+ (τ ) = 1 for τ ∈ [−, 1] and σ− (τ ) = 1 for τ ∈ [−1, ], (5.16) where ∈ (0, 1). Then the term S2 (λ) is in some sense negligible.

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Lemma 5.4. With the above choice of J± the operator S2 (λ) is compact and normcontinuous in λ for λ > 0. Proof. Recall that T± = T±(r) + T±(s) , where T±(r) , T±(s) are pseudo-differential operators (s) with symbols t(r) ± , t± defined by (4.10). Let us write (5.15) as S2 (λ) = 2πi 00 (λ)X −p X p B(λ)X p X −p 0∗0 (λ) ,

(5.17)

where p ∈ (1/2, ρ], B(λ) = B1 (λ) + B2 (λ) + B3 (λ) : H → H

(5.18)

B1 (λ) = (T+(r) )∗ R(λ + i0)T−(r) ,

(5.19)

B2 (λ) = (T+(r) )∗ R(λ + i0)T−(s) + (T+(s) )∗ R(λ + i0)T−(r) ,

(5.20)

B3 (λ) = (T+(s) )∗ R(λ + i0)T−(s) .

(5.21)

and

In view of estimates (4.9) Proposition 2.1 implies that the operators X p T±(r) X p are bounded. Therefore, applying Proposition 2.5, we find that the operator X p B1 (λ)X p is bounded and norm-continuous in λ. According to (4.7), (4.10) the support of the ˆ ≤ − and, similarly, the support of the ˆ ξi symbol t(s) + (x, ξ) belongs to the cone hx, ˆ (x, ξ) belongs to the cone h x, ˆ ξi ≥ . Both functions t(s) symbol t(s) − ± (x, ξ) satisfy (2.2) with m = −1 and ρ > 1/2, δ = 1 − ρ. Therefore Proposition 2.7 can be applied to the operators X −p R(λ + i0)T−(s) X p , X p (T+(s) )∗ R(λ + i0)X −p , and Proposition 2.8 can be applied to the operator X p B3 (λ)X p . It follows that the operators X p Bj (λ)X p , j = 2, 3, and hence X p B(λ)X p are bounded and norm-continuous in λ. To conclude the proof we return to representation (5.17) and use that, by Proposition 2.9, the operator 00 (λ)X −p : H → N is compact and norm-continuous in λ. Let us now consider operator (5.14). Its genuinely non-compact part is determined by the operator A0 . Proposition 5.5. Let A0 be the pdo with symbol (5.12) and set S0 (λ) = −2πi00 (λ)A0 0∗0 (λ).

(5.22)

Then the operator S(λ) − S0 (λ) is compact and norm-continuous in λ for λ > 0. Proof. By virtue of (5.13), Proposition 2.9 implies that the operator 00 (λ)(J+∗ T− − A0 )0∗0 (λ) is compact and norm-continuous in λ > 0. Under assumption (5.16) the function ˆ xi) ˆ xi) σ+ (hξ, ˆ equals 1 on the support of ∇σ− (hξ, ˆ and hence it may be omitted in the symbol of A0 . So it remains to take Lemma 5.4 into account.


203

An obvious drawback of this assertion is that the operator A0 contains the cut-off function σ− although the scattering matrix does not depend on it. In the next part we shall show that the principal of the pdo (5.22) does not, actually, depend on this cut-off. ˆ ωi) = 0 in the cones hx, ˆ ωi ≥ κ for some κ ∈ (0, 1) and hx, ˆ ωi ≤ 0, 4. Since ∇σ− (hx, Proposition 2.11 can be applied to the operator A0 . Therefore the operator 00 (λ)A0 0∗0 (λ) (note that Proposition 2.11 gives also another proof of its existence) is a pdo on Sd−1 with the principal symbol Z

s(ω, b; λ) = (4πk)−1

∞ −∞

a0 (zω + k −1 b, kω)dz,

|ω| = 1,

hω, bi = 0.

(5.23)

To calculate this integral we remark that, by (5.9), φ(zω + k −1 b, kω) = 2−1 k −1 V(ω, k −1 b), where the function Z V(ω, b) =

∞ −∞

V (tω) − V (b + tω) dt,

|ω| = 1,

k = λ1/2 ,

hω, bi = 0,

(5.24)

(5.25)

ˆ xi ˆ = does not depend on z. Clearly, η(zω + k −1 b) = 1 for sufficiently large b and hξ, z(z 2 + k −2 b2 )−1/2 for ξ = kω, x = zω + k −1 b. Since σ− (1) = 0 and σ− (−1) = 1, we have that Z ∞ Z ∞ ˆ xi)idz hξ, ∇σ− (hξ, ˆ =k ∂σ− (z(z 2 + k −2 b2 )−1/2 )/∂z dz = −k. −∞

−∞

Taking into account that under assumption (5.16) the symbol of the pdo A0 equals a0 (x, ξ), ψ(λ) = 1 for λ considered and comparing (5.12), (5.23) and (5.24) we find that s(ω, b; λ) = exp i2−1 k −1 V(ω, k −1 b) ,

|ω| = 1,

hω, bi = 0.

(5.26)

This gives us Proposition 5.6. The operator S0 (λ) is a pdo on Sd−1 with the principal symbol 0 . s(ω, b; λ) defined by (5.25), (5.26). In particular, s ∈ Sρ,1−ρ Note that the operator S0 (λ) is determined by its principal symbol up to a compact term. Combining Propositions 5.5 and 5.6 we obtain Theorem 5.7. Let condition (1.1) with ρ > 1/2 hold. Then the scattering matrix S(λ) for the operators H0 , H, identifications J± and λ ∈ 3 admits the representation ˜ S(λ) = S0 (λ) + S(λ), where S0 (λ) is a pdo on Sd−1 with the principal symbol (5.25), (5.26) and the operator ˜ S(λ) is compact.

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According to (5.26), Proposition 2.12 can be applied to the operator S0 (λ) if we require that lim sup V(ω0 , tb0 ) = ∞

t→∞

or

lim inf V(ω0 , tb0 ) = −∞

t→∞

(5.27)

for some point ω0 , b0 6= 0, |ω0 | = 1, hω0 , b0 i = 0. Of course, this condition is satisfied for all points ω0 , b0 if V (x) is an asymptotically homogeneous function of order −ρ. Taking into account that the spectrum of a unitary operator belongs to the unit circle and that, by Weyl’s theorem, a compact operator can not change the essential spectrum we arrive at the following result. Theorem 5.8. Let condition (1.1) with ρ > 1/2 hold. Suppose that function (5.25) satisfies condition (5.27) for some point ω0 , b0 , |ω0 | = 1, hω0 , b0 i = 0. Then for all λ > 0 the spectrum of the scattering matrix S(λ) coincides with the unit circle. 5. Here we collect some additional remarks. 1. We do not have any information on the structure of the spectrum of the scattering matrix. Note, however, that in the radial case V (x) = V (|x|) it consists of eigenvalues. By Theorem 5.8, they are dense on the unit circle. 2. Let U be multiplication by a function exp(i8(ω)), where 8 ∈ C ∞ (Sd−1 ). Proposition 2.12 can be applied also to the operator U S0 (λ). It follows that, under assumptions of Theorem 5.8, the spectrum of S(λ) covers the unit circle for any choice of identifications J± . 3. Theorems 5.7 and 5.8 can be easily extended to more general classes of potentials discussed near the end of Sect. 4 provided the resolvent estimates of Propositions 2.5, 2.7 and 2.8 remain true. In particular, the spectrum of S(λ) coincides with the unit circle if a potential is a sum of a function obeying assumptions of Theorem 5.8 and of a short-range term. 4. If condition (1.1) is fulfilled for ρ > 1, then function (5.25) satisfies Z ∞ V (tω)dt. (5.28) V(ω, b) = c(ω) + O(|b|−ρ+1 ), where c(ω) = −∞

˜ Representation (5.2) remains, of course, true in this case and the operator S(λ) is compact. Conditions (5.27) and (5.28) are incompatible and S0 (λ) differs by a compact term from the operator of multiplication by exp(i2−1 k −1 c(ω)). On the other hand, in the case ρ > 1 one can set 8± = 0 (cf. the remark at the end of Sect. 3). Then representation (5.2) holds with + (λ) = I. In particular, one recovers the standard representation of the scattering matrix if a± = 1. 6. The Diagonal Singularity of the Scattering Amplitude 1. In this section we suppose that identifications J± = J± (a± ) are given by formula ) (4.2), where the function a± = a(N ± is constructed in the middle of Sect. 3 and N is sufficiently large. We proceed from the stationary representation of the scattering matrix S(λ) given by Theorem 5.3. Let us start with analysis of the regular part S2 (λ) = S2(N ) (λ) of the scattering matrix defined by formula (5.15). We shall show that S2 (λ) is an integral operator with smooth kernel s2 (ω, ω 0 ; λ) in variables ω, ω 0 ∈ Sd−1 and λ > 0. Actually,


205

) the function s2 = s(N is getting more and more regular as N increases. The proof of 2 the following assertion is similar to that of Lemma 5.4. ) Lemma 6.1. Suppose that condition (1.1) with ρ ∈ (1/2, 1) holds. Let J± = J± (a(N ± ) and let condition (5.16) be satisfied. Then the operator S2(N ) (λ) is an integral operator ) ∈ C M (Sd−1 × Sd−1 × R+ ), where an integer M satisfies with kernel s(N 2

M < (2ρ − 1)(N + 1) − d + 1.

(6.1)

Proof. According to (5.15) and (2.4), (2.5), the kernel s2 (ω, ω 0 ; λ) is formally given by the formula s2 (ω, ω 0 ; λ) = 2−1 ik d−2 (2π)−d+1 (T+∗ R(λ+i0)T− ψ0 (kω 0 ), ψ0 (kω)),

k = λ1/2 , (6.2)

) where ψ0 (x, ξ) = exp(ihξ, xi) and the operators T± = T± (a(N ± ) are determined by Eqs. (4.5), (4.6). Formula (6.2) is automatically justified if its right-hand side is, say, a continuous function of ω, ω 0 ∈ Sd−1 and λ > 0. Actually, we shall check that this function belongs to the class C M . To that end we differentiate (6.2) and remark that 0 (∂ωα ∂ωα0 ∂λm s2 )(ω, ω 0 ; λ) consists of terms 0

(T+∗ Rn (λ + i0)T− ψ0(β ) (kω 0 ), ψ0(β) (kω)),

(6.3)

where ψ0(β) (x, ξ) = xβ ψ0 (x, ξ), |β| + |β 0 | + n = |α| + |α0 | + m ≤ M

(6.4)

and |β| ≥ |α|, |β 0 | ≥ |α0 |, n ≤ m. Clearly, X −|β|−p ψ0(β) (kω) ∈ L2 (Rd ) if p > d/2. Therefore we need to check boundedness of the operator X |β|+p T+∗ Rn (λ + i0)T− X |β

0

|+p

,

p > d/2.

(6.5)

We decompose again the function (4.6) into a sum (4.8) but define now the regular (r) (r) (x, ξ) by the equality τ± = q± ζ± . Then T± = T±(r) + T±(s) , where T±(r) , T±(s) are part τ± pdo with symbols (4.10). By estimates (3.11), (3.21), m t(r) ± ∈ Sρ,1−ρ , where m = −1 − ε0 (N + 1),

and

−1 t(s) ± ∈ Sρ,1−ρ ,

ε0 = 2ρ − 1,

ˆ ≤ . t(s) ˆ ξi ± (x, ξ) = 0 if ∓ hx,

Therefore the operator B(λ) = T+∗ Rn (λ + i0)T− satisfies equalities (5.18) - (5.21) with 0 R(λ + i0) replaced by Rn (λ + i0). Set Gj (λ) = X |β|+p Bj (λ)X |β |+p , j = 1, 2, 3. By Proposition 2.5, the operator G1 is bounded if n + p + max{|β|, |β 0 |} < ε0 (N + 1) + 3/2.

(6.6)

By virtue of (6.4) this condition is, of course, satisfied under assumption (6.1). By Proposition 2.7, the operator G2 is bounded if, additionally to (6.6), n + 2p + |β| + |β 0 | < ε0 (N + 1) + 2. This condition is again satisfied under assumption (6.1). Finally, by Proposition 2.8, the operator G3 is bounded for all n, p and β, β 0 .

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2. Let us proceed to the analysis of the singular part S1 (λ) of the scattering matrix. Recall that S1 (λ) is defined by equality (5.14) and J+∗ T− is a pdo with symbol a(x, ξ), given −1 so that the kernel s1 (ω, ω 0 ; λ) of the operator by (5.8) - (5.10). In particular, a ∈ Sρ,1−ρ S1 (λ) is determined by formula (2.6): s1 (ω, ω 0 ; λ) = −i2−1 (2π)−d+1 k d−2

Z

0

eikhω −ω,xi a(x, kω 0 )dx,

k = λ1/2 ,

Rd

ω 6= ω 0 .

(6.7) Thus, s1 (ω, ω 0 ; λ) is an infinitely differentiable function of ω, ω 0 for ω 6= ω 0 and of λ > 0. Using now Lemma 6.1 and taking into account that M is arbitrary there, we arrive at the following assertion. Proposition 6.2. Under assumption (1.1), where ρ > 1/2, the kernel s(ω, ω 0 ; λ) of the scattering matrix S(λ) is an infinitely differentiable function of ω, ω 0 for ω 6= ω 0 and of λ > 0. By virtue of Lemma 6.1, formula (6.7) determines the singular part of s(ω, ω 0 ; λ). Moreover, the second inclusion (5.11) implies that the term uN in the right-hand side of (5.8) is negligible. Let us formulate the precise result. Proposition 6.3. Let assumption (1.1) with ρ ∈ (1/2, 1) hold. Set s0 (ω, ω 0 ; λ) = −i2−1 (2π)−d+1 k d−2

Z

0

0

eikhω −ω,xi eiφ(x,kω ) w(x, kω 0 )dx,

(6.8)

Rd

where φ and w = wN are defined by (5.9) and (5.10), respectively. Then s − s0 ∈ C M (Sd−1 × Sd−1 × R+ ), where M is the same as in Lemma 6.1. Remark. The function w in (6.8) is constructed in terms of approximate solutions a± = ) (N ) contains a(N ± of the transport equation and hence it is quite explicit. Thus s0 = s0 all diagonal singularities of the scattering matrix. This result remains meaningful in the case ρ > 1 when one can set 8± = 0. Our goal is to find explicitly the leading singularity of the kernel s(ω, ω 0 ; λ) at the diagonal ω = ω 0 . Below we fix the vector ω 0 =: ω0 and study s0 (ω, ω0 ; λ) as ω → ω0 . Set z = hω0 , xi and b = x − zω0 (so that b belongs to the hyperplane 3ω0 orthogonal to ω0 ). It is convenient to introduce the vector ϑ = ω − hω, ω0 iω0 ∈ 3ω0 .

(6.9)

Clearly, hω0 − ω, bi = −hϑ, bi,

hω0 − ω, ω0 i = 1 − (1 − |ϑ|2 )1/2 =: f (ϑ) = O(|ϑ|2 ) (6.10)

as ϑ → 0. Dependence of different expressions on ω0 and k will be often omitted. Let us formulate an intermediary result.


Lemma 6.4. Function (6.8) admits the representation Z −1 −1 −d+1 d−1 k e−ikhϑ,bi ei2 k V(ω0 ,b) p(b, ϑ)db, s0 (ω, ω0 ; λ) = (2π)

207

(6.11)

Rd−1

where V is given by (5.25) and p(b, ϑ) = −i2−1 k −1

Z

∞ −∞

w(zω0 + b, kω0 )eikf (ϑ)z dz.

(6.12)

Moreover, p(b, ϑ) = P0 (f (ϑ)b) + p1 (b, ϑ),

(6.13)

where P0 ∈ S(R), P (0) = 1, the function p1 (b, ϑ) is infinitely differentiable with respect to b and (6.14) |∂bα p1 (b, ϑ)| ≤ Cα (1 + |b|)−ε0 −|α| , ε0 = 2ρ − 1 ∈ (0, 1), with constants Cα not depending on ϑ. Proof. Equalities (6.11), (6.12) follow directly from (5.24) and the first equality (6.10). Consider function (5.10) for x = zω0 + b, ξ = kω0 . Remark that ψ(ξ 2 ) = 1 for λ ˆ = 1 on the support of ˆ ξi) considered, under assumption (5.16) the function σ+ (hx, ˆ and ∇σ− (hx, ˆ ξi) ˆ = −2ik∂σ− (z(z 2 + b2 )−1/2 )/∂z. ˆ ξi)i − 2ihξ, ∇σ− (hx,

(6.15)

Let p0 (b, ϑ) be defined by integral (6.12), where w(zω0 + b, kω0 ) is replaced by function (6.15). Changing the variable of integration z by z|b|, we see that p0 (b, ϑ) = P0 (f (ϑ)b), where Z ∞ P0 (t) = −

eiktz ∂σ− (z(z 2 + 1)−1/2 )/∂z dz.

−∞

−2ρ According to (5.10), estimates (6.14) on p1 = p−p0 follow from the inclusion w ∈ S1,0 .

3. Assume now for simplicity that V (x) is a homogeneous function for sufficiently large |x|: −ρ ˆ , v ∈ C ∞ (Sd−1 ), |x| ≥ r0 . (6.16) V (x) = V0 (x) := v(x)|x| Lemma 6.5. If (6.16) is fulfilled for some ρ < 1, then, up to some constant ν(ω), function (5.25) is homogeneous of degree 1 − ρ for sufficiently large |b|: ˆ 1−ρ + ν(ω), V(ω, b) = v(ω, b)|b| Z

where

∞

ˆ = v(ω, b)

−∞

Z

and

r

ν(ω) = −r

does not depend on b (and r).

V0 (tω) − V0 (bˆ + tω) dt

(6.17)

V (tω) − V0 (tω) dt,

|b| ≥ r0 ,

r ≥ r0 ,

(6.18)

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D. Yafaev

Proof. It suffices to make a change of variables s = |b|t in the representation Z ∞ V0 (sω) − V0 (b + sω) ds + ν(ω), |b| ≥ r0 . V(ω, b) = −∞

Since integral (6.11) over compact domain is bounded uniformly in ϑ, we may suppose that (6.17) holds for all b. Let us set γ = ρ−1 and make the change of variables b = λ−γ |ϑ|−γ y in (6.11). Using also (6.13), we see that Z ˆ −(d−1)γ 2γ−1 −d+1 (2πk ) µ(ω0 , k) eiLψ(ω0 ,ϑ,y) q(y, ϑ)dy, (6.19) s0 (ω, ω0 ; λ) = |ϑ| Rd−1

where µ(ω0 , k) = exp i(2k)−1 ν(ω0 ) , L = k 1−2γ |ϑ|1−γ , q(y, ϑ) = P0 (λ−γ |ϑ|−γ f (ϑ)y) + p1 (λ−γ |ϑ|−γ y, ϑ),

(6.20)

ˆ y) = −hϑ, ˆ yi + 2−1 v(ω0 , y)|y| ψ(ω0 , ϑ, ˆ 1−ρ .

(6.21)

It is important that L → ∞ as ϑ → 0 if ρ < 1. Therefore the asymptotics of integral ˆ y) = 0, or, according to (6.19) is determined by stationary points y, where ∇y ψ(ω0 , ϑ, (6.18), Z ∞ ˆ (∇V0 )(y + tω0 )dt = 0, y ∈ 3ω0 . (6.22) 2ϑ + −∞

Let us denote by H(ω0 , y) the Hessian of the function V(ω0 , y), i.e. H(ω0 , y) is the (d − 1) × (d − 1) - matrix with elements Z ∞ ∂ 2 V0 (y + tω0 )/∂yi ∂yj dt, y ∈ 3ω0 . Hij (ω0 , y) = − −∞

h(ω0 , y) = | det H(ω0 , y)|−1/2 exp iπ sgn H(ω0 , y)/4 .

Set also

(6.23)

ˆ y) equals 2−1 H(ω0 , y). By (6.21), the Hessian of ψ(ω0 , ϑ, ˆ there is a unique point Lemma 6.6. Let ρ ∈ (1/2, 1). Suppose that, for a given ϑ, ˆ 6= 0 satisfying (6.22) and that det H(y(ϑ)) ˆ 6= 0. Then y(ϑ) Z ˆ ˆ ϑ)) ˆ ˆ iLψ(ϑ,y( eiLψ(ϑ,y) q(y, ϑ)dy = (4π)(d−1)/2 L−(d−1)/2 h(y(ϑ))e (1 + O(|ϑ|ε )) Rd−1

(6.24)

for some ε > 0 as ϑ → 0.

Proof. According to Lemma 6.4, function (6.20) is bounded in a neighbourhood of ˆ uniformly in ϑ and its derivatives in y are even bounded by C|ϑ|2−γ . a point y(ϑ) Therefore, applying the stationary phase method we find that integral (6.24) over a ˆ equals neighbourhood of the point y(ϑ) ˆ iLψ(ϑ,y(ϑ)) q(y(ϑ), ˆ ϑ)(1 + O(L−1 )). (4π)(d−1)/2 L−(d−1)/2 h(y(ϑ))e ˆ

ˆ

(6.25)

ˆ ϑ) = 1 + O(|ϑ|2−γ ) and hence (6.25) equals the According again to Lemma 6.4, q(y(ϑ), right-hand side of (6.24).


209

ˆ can be estimated by direct inteIntegral (6.24) outside of a neighbourhood of y(ϑ) gration by parts (m times). Hereby one should take into account that the point y = 0 is singular for function (6.21) and use the bounds |∇ψ(y)| ≥ c max{1, |y|−ρ }

and

|∂ α ψ(y)| ≤ C|y|1−ρ−|α| ,

|α| ≥ 2.

Together with the estimate on q(y, ϑ) used above this shows that the integral considered is bounded by L−m , which is negligible if m > (d − 1)/2. Of course, Lemma 6.6 can be extended to the case of a finite number of stationary points. Combine now the results obtained. Theorem 6.7. Let assumptions (1.1) and (6.16) with ρ ∈ (1/2, 1) hold. Fix k > 0, ω0 ∈ Sd−1 and let ω and ϑ be related by (6.9). Suppose that for a given ϑˆ there is a finite ˆ . . . , yn (ϑ) ˆ satisfying (6.22) and that det H(ω0 , yj (ϑ)) ˆ 6= 0 for all number of points y1 (ϑ), j = 1, . . . , n. Define the functions ψ and h by equalities (6.21) and (6.23), respectively. Then the kernel of the scattering matrix admits as ω → ω0 or, equivalently, ϑ → 0 the representation s(ω, ω0 ; λ) = (πk 2γ−1 )−(d−1)/2 |ϑ|−(d−1)(1+γ)/2 µ(ω0 , k) n X ˆ exp ik 1−2γ |ϑ|1−γ ψ(ω0 , ϑ, ˆ yj (ϑ)) ˆ (1 + O(|ϑ|ε )), h(ω0 , yj (ϑ)) ×

(6.26)

j=1

where γ = ρ−1 , ε = ε(ρ) > 0. Remark. It is possible, of course, that for some ϑˆ there are no points y satisfying (6.22). In this case s0 (ω, ω0 ; λ) → 0 as ϑ → 0 quicker than any power of |ϑ| so that the kernel s(ω, ω0 ; λ) of the scattering matrix remains bounded. 4. The case ρ = 1 is essentially different. Lemma 6.8. If (6.16) is fulfilled for ρ = 1, then function (5.25) admits the representation ˆ + ν1 (ω), V(ω, b) = v(ω) ln |b| + ν(ω, b) where v(ω) = v(ω) + v(−ω), Z Z ˆ ˆ V0 (sω) − V0 (b + sω) ds − ν(ω, b) = |s|≥1

Z ν1 (ω) =

r −r

|s|≤1

V (tω)dt − v(ω) ln r,

|b| ≥ r0 ,

(6.27)

V0 (bˆ + sω)ds,

(6.28)

r ≥ r0 .

Proof. Replacing in (5.25) the potential V by its asymptotics V0 for |b| ≥ r or |t| ≥ r we see that Z r Z Z r V0 (tω) − V0 (b + tω) dt. (6.29) V (tω)dt − V0 (b + tω)dt + V(ω, b) = −r

−r

|t|≥r

Changing in the integral over |t| ≥ r the variable t = |b|s, we rewrite it as

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D. Yafaev

Z

Z V0 (sω)ds +

|s|≤r/|b|

r/|b|≤|s|≤1

ˆ V0 (bˆ + sω)ds + ν(ω, b),

(6.30)

ˆ is defined by (6.28). The first integral in (6.30) equals v(ω) ln(|b|/r). The where ν(ω, b) second integrals in (6.29) and (6.30) cancel each other. Therefore substituting (6.30) into (6.29) we arrive at (6.27). We emphasize that, in contrast to (6.17), the coefficient v(ω) in (6.27) does not ˆ depends on it. Note also that the expression depend on bˆ and, on the contrary, ν(ω, b) for ν1 (ω) does not depend on the parameter r ≥ r0 . Using (6.27) and changing the variable b = |θ|−1 y we rewrite (6.11) as Z ˆ −d+1 d−1 −d+1−iv(ω0 )/2k k µ1 |ϑ| eiΞ(y,ϑ,ω0 ,k) q(y, ϑ)dy, (6.31) s0 (ω, ω0 ; λ) = (2π)

Rd−1

where µ1 (ω0 , k) = exp i(2k)−1 ν1 (ω0 ) , ˆ yi + (2k)−1 v(ω0 ) ln |y| + (2k)−1 ν(ω0 , y) ˆ ω0 , k) = −khϑ, ˆ Ξ(y, ϑ,

(6.32)

and q(y, ϑ) = p(|ϑ|−1 y, ϑ). Let us show that the function q(y, ϑ) in integral (6.31) can be replaced by 1 in the limit ϑ → 0. Lemma 6.9. In the case ρ = 1 for any < 1 Z ˆ eiΞ(y,ϑ,ω0 ,k) (q(y, ϑ) − 1)dy = O(|ϑ| )

as

Rd−1

|ϑ| → 0.

Proof. Let η ∈ C ∞ be such that η(y) = 0 for |y| ≤ 1 and η(y) = 1 for |y| ≥ 2. We consider separately the integrals Z ˆ eiΞ(y,ϑ,ω0 ,k) (q(y, ϑ) − 1)(1 − η(y))dy I1 (ϑ) = Rd−1

Z

and I2 (ϑ) =

e−ikhϑ,yi r(y, ϑ)η(y)dy, ˆ

(6.33)

Rd−1

where r(y, ϑ) = G(y)(q(y, ϑ) − 1),

G(y) = |y|i(2k)

−1

v i(2k)−1 ν(y) ˆ

e

.

Below we use the bounds on p formulated in Lemma 6.4; note that (6.14) holds with any ε0 < 1. Let us estimate q(y, ϑ) − 1 = (P0 (f0 (ϑ)y) − 1) + p1 (|ϑ|−1 y, ϑ),

(6.34)

where f0 (ϑ) = |ϑ|−1 f (ϑ) = O(|ϑ|) as ϑ → 0. The first term in the right-hand side is bounded by C|ϑ||y| and the second - by (1 + |ϑ|−1 |y|)− . Therefore Z |q(y, ϑ) − 1|dy ≤ C1 |ϑ| . |I1 (ϑ)| ≤ C |y|≤2

To estimate integral (6.33) we integrate m times by parts. This gives


|I2 (ϑ)| ≤ C

X |α|+|β|=m

211

Z Rd−1

|∂ α r(y, ϑ)| |∂ β η(y)|dy.

(6.35)

Now we need estimates on derivatives of the function (6.34) for |y| ≥ 1. Clearly, |∂ α (P0 (f0 (ϑ)y) − 1)| = |f0 (ϑ)|α| P0(α) (f0 (ϑ)y)| ≤ C|ϑ| |y|−|α|+1

(6.36)

because |s||α|−1 P0(α) (s), |α| ≥ 1, is a bounded function. Inequality (6.36) holds also for α = 0. According to (6.14), −1 −|α|− , |∂yα p1 (|ϑ|−1 y, ϑ)| = |ϑ|−|α| |p(α) 1 (|ϑ| y, ϑ)| ≤ C|ϑ| |y|

|y| ≥ 1.

It follows that ∂ α (q(y, ϑ) − 1) is bounded by C|ϑ| |y|−|α|+1 for |y| ≥ 1. Since ∂ α G(y) = O(|y|−|α| ), the same bound holds for the function r(y, ϑ): |∂ α r(y, ϑ)| ≤ C|ϑ| |y|−|α|+1 ,

|y| ≥ 1,

∀α.

Choosing m = d + 1, we estimate the right-hand side of (6.35) by C|ϑ| .

Therefore representation (6.31) combined with Proposition 6.3 implies Theorem 6.10. Let assumptions (1.1) and (6.16) with ρ = 1 hold. Fix k > 0, ω0 ∈ Sd−1 and let ω and ϑ be related by (6.9). Set v(ω0 ) = v(ω0 ) + v(−ω0 ) and Z ˆ −d+1 d−1 ˆ k eiΞ(y,ϑ,ω0 ,k) dy, (6.37) c(ϑ, ω0 , k) = µ1 (ω0 , k)(2π) Rd−1

where the function Ξ is defined by (6.28) and (6.32). Then the kernel of the scattering matrix admits as ω → ω0 or, equivalently, ϑ → 0 the representation ˆ ω0 , k)|ϑ|−d+1−iv(ω0 )/2k (1 + O(|ϑ| )), s(ω, ω0 ; λ) = c(ϑ,

∀ < 1.

(6.38)

5. Let us make some comments on Theorems 6.7 and 6.10. 1. In formulas (6.26), (6.38) factors µ and µ1 which do not depend on ϑ and have modulus 1 are inessential because they can be changed by a choice of the operator J. 2. Condition (6.16) on V can be replaced by conditions (6.17) or (6.27) on V. Actually, for validity of (6.26) or (6.38) at some point ω0 it suffices to require (6.17) or (6.27) at this point ω0 only. 3. Theorems 6.7 and 6.10 can be extended to the case −ρ + V1 (x), V (x) = v(x)|x| ˆ

where V1 (x) satisfies (1.1) for some ρ1 > ρ (and |x| ≥ 1). 4. Local singularities of V are treated automatically as long as resolvent estimates of Propositions 2.5, 2.7 and 2.8 are preserved.

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5. If a potential is a sum V + Vs of a function V satisfying (1.1) and of a short-range function Vs (x) = O(|x|−ρs ) with ρs > d as |x| → ∞, then the additional term Vs (x)ζ± (x, ξ) appears in (4.6). Therefore we can not make (see Lemma 6.1) the ) 0 regular part s(N 2 (ω, ω ; λ) of the scattering amplitude as smooth as we want. To prove that this is a continuous function of ω, ω 0 ∈ Sd−1 (and λ > 0) we need to check boundedness of operator (6.5) for β = β 0 = 0 and n = 1. This requires the results of Proposition 2.7 for n = 1, m = 0 and some p > d/2 and the results of Proposition 2.8 for n = 1 and some p > d/2−1. We expect that this is true although we have not found the proper reference in the literature. Under the assumption ρs > d the contribution of Vs to the singular part (5.14) is also a continuous function of ω, ω 0 ∈ Sd−1 (and λ > 0). Thus, the results of Theorems 6.7 and 6.10 remain true. 7. Examples. Classical mechanics 1. Let us consider several examples. We start with asymptotically central potentials when a leading term of s(ω, ω0 ) as ω → ω0 is a function of |ϑ| only. Note that Z ∞ (1+t2 )−(ρ+2)/2 dt = 2ρ−1 I(ρ), where I(ρ) = π 1/2 0((1+ρ)/2)0(ρ/2)−1 (7.1) −∞

and 0 is the Gamma-function. Example 7.1. Suppose that v(x) ˆ = v = const in (6.16) and ρ ∈ (1/2, 1). Then as ϑ → 0, s(ω, ω0 ; λ) = µ0 w0 |ϑ|−(d−1)(1+γ)/2 eiψ0 |ϑ|

1−γ

(1 + O(|ϑ|ε ),

γ = ρ−1 , ε > 0,

(7.2)

where µ does not depend on ϑ, |µ| = 1, w0 = (2π)−(d−1)/2 ρ−1/2 k (1−2γ)(d−1)/2 (I(ρ)|v|)γ(d−1)/2 , ψ0 = ρ(1 − ρ)−1 k 1−2γ (I(ρ)|v|)γ sgn v. In particular, the scattering cross-section (1.5) :

6(ω, ω0 ; λ) = ρ−1 (I(ρ)|v|λ−1 )γ(d−1) |ϑ|−(d−1)(1+γ) 1 + O(|ϑ|ε ) .

(7.3)

Indeed, using (7.1), we obtain for function (5.25) and large |b| Z ∞ Z ∞ (∇V)(b) = − (∇V0 )(b + tω)dt = ρvb (b2 + t2 )−(ρ+2)/2 dt = 2vI(ρ)b|b|−ρ−1 . −∞

−∞

Therefore the coefficient v in (6.17) does not depend on ω and bˆ and it equals v = 2(1 − ρ)−1 I(ρ)v.

(7.4)

ˆ for any ϑ: ˆ Equation (6.22) reads now as I(ρ)v|y|−ρ yˆ = ϑˆ and has a unique solution y(ϑ) ˆ = Y ϑˆ sgn v, y(ϑ)

Y = (I(ρ)|v|)γ .

(7.5)

Calculating the Hessian of the function |y|1−ρ , y ∈ Rd−1 , we find that function (6.23) equals


where µ(0)

213

h(y) = µ0 (1 − ρ)−(d−1)/2 ρ−1/2 (|v|−1 |y|ρ+1 )(d−1)/2 , = exp iπ(d − 3)sgn v/4 . It follows from (7.4), (7.5) that ˆ = µ0 2−(d−1)/2 ρ−1/2 (I(ρ)|v|)γ(d−1)/2 . h(y(ϑ))

Similarly, we obtain for function (6.21) ˆ y(ϑ)) ˆ = −2hϑ, ˆ y(ϑ)i ˆ + vY 1−ρ = 2ρ(1 − ρ)−1 (I(ρ)|v|)γ sgn v. ψ(ϑ, Thus, formula (7.2) with µ0 = µ(0) µ is a consequence of (6.26). Consider now asymptotically Coulomb potentials. Example 7.2. Suppose that V (x) = v|x|−1 for sufficiently large |x|. Then formula (6.38) holds with v(ω0 ) = 2v and c = µ0 π −(d−1)/2 0((d − 1)/2 + iv/2k)0(−iv/2k)−1 ,

(7.6)

where µ0 does not depend on ϑ and |µ| = 1. In particular, the scattering cross-section 6(ω, ω0 ; λ) = 2d−1 λ−(d−1)/2 |0((d − 1)/2 + iv/2k)0(−iv/2k)−1 |2 |ϑ|−2d+2 (1 + O(|ϑ| )) (7.7) for any < 1 as ϑ → 0. Indeed, under the assumptions above equality (6.27) is fulfilled with v(ω) = 2v and ˆ By (6.32), integral (6.37) reduces to the coefficient ν(ω), which does not depend on b. calculated explicitly the Fourier transform of the function |y|iv/k and can be in terms of −1 the Gamma-function. This gives (7.6) with µ0 = µ1 exp −ivk ln(k/2) . In the case d = 3 (7.7) implies that 6(ω, ω0 ; λ) = v 2 λ−2 |ϑ|−4 (1 + O(|ϑ| )),

ϑ → 0.

(7.8)

Thus we recover (in the limit ϑ → 0) the celebrated formula of Gordon and Mott. Let us compare formulas (7.3) for ρ < 1 and (7.7) for ρ = 1. The right-hand side of (7.3) is well-defined also for ρ = 1 and contains the same power |ϑ|−2d+2 as the right-hand side of (7.7). On the other hand, the asymptotic coefficient (|v|λ−1 )d−1 in (7.3) for ρ = 1 coincides with that in (7.7) in the case d = 3 only. 2. Let us also consider an example of an essentially non-central potential. We choose ˆ = −hx, ˆ ni in a potential of a dypole type. Let n ∈ Rd be some given vector and v(x) (6.16). We compute all quantities in the right-hand side of (6.26) for this case. Calculating ˆ in (6.17) integral (6.18) and using (7.1) we find that the asymptotic coefficient v(ω0 , b) equals ˆ = πI(ρ)−1 hn, bi, ˆ b ∈ 3 ω0 . (7.9) v(ω0 , b) It is convenient to replace n here by its orthogonal projection m = m(ω0 ) = n−hn, ω0 iω0 on the hyperplane 3ω0 . We assume that m 6= 0. It follows from (7.9) that ˆ b)|b| ˆ −ρ . (∇b V)(ω0 , b) = πI(ρ)−1 (m − ρhm, bi ˆ y) = 0 for function (6.21) reads as The equation ∇y ψ(ϑ, ˆ m ˆ − ρhm, ˆ yi ˆ yˆ = M −1 |y|ρ ϑ,

where

M = π|m|(2I(ρ))−1 .

(7.10)

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D. Yafaev

Considering the scalar product of (7.10) with m ˆ we see that this equation may have ˆ ˆ solutions y = y(ϑ) only if p := hm, ˆ ϑi > 0. We seek the angular part yˆ of y in the form ˆ yˆ = αm ˆ + β ϑ.

(7.11)

Comparing coefficients at m ˆ in (7.10) we obtain that α and β are connected by the relation αρ(α + βp) = 1. (7.12) Solving this equation together with the normalization condition α2 + β 2 + 2αβp = 1 we get α4 − (q + 2γ)α2 + (q + 1)γ 2 = 0, q = p2 (1 − p2 )−1 , γ = ρ−1 , so that

1/2 α2 = 2−1 q + γ ± q 1/2 4−1 q + γ − γ 2 .

(7.13)

Thus, Eq. (7.10) has solutions only if q ≥ 4(γ 2 − γ), which gives the condition ˆ ≥ 2(1 − ρ)1/2 (2 − ρ)−1 . hm, ˆ ϑi

(7.14)

Under this assumption the right-hand side of (7.13) is non-negative for both signs so that there are four numbers ±αj , j = 1, 2, (we suppose that αj ≥ 0 and α1 corresponds to the sign “+00 ) satisfying (7.13). Now we obtain ±βj from (7.12) and then determine ±yˆj , j = 1, 2, by (7.11). To find |yj | we remark that, according to (7.12), ˆ − ρhm, ˆ = p − ρ(αj + βj p)(αj p + βj ) = −βj α−1 = p−1 (1 − γα−2 ). hm, ˆ ϑi ˆ yˆj ihyˆj , ϑi j j Hence it follows from Eq. (7.10) that |yj | = M γ p−γ (1 − γαj−2 )γ . This determines the ˆ := ψ(ϑ, ˆ yj ) = −ψ(ϑ, ˆ −yj ) : phase ψj (ϑ) ˆ = −hyj , ϑi ˆ + 2−1 πI(ρ)−1 hm, yj i|yj |−ρ . ψj (ϑ)

(7.15)

It remains only to find the Hessian H(ω0 , y) of the function V(ω0 , y) = πI(ρ)−1 hm, yi|y|−ρ ,

y ∈ 3 ω0 .

Suppose for simplicity that d = 3 and choose in 3ω0 ' R2 a system of orthogonal coordinates with the first axis directed along m. Denote by y (1) , y (2) the corresponding coordinates of y ∈ R2 . Then H(ω0 , y) = πρI(ρ)−1 |m||y|−1−ρ h(y), where h(y) is a 2 × 2 - matrix with elements h11 (y) = |y|−3 ((ρ − 1)(y (1) )3 − 3y (1) (y (2) )2 ), h22 (y) = |y|−3 ((ρ + 1)y (1) (y (2) )2 − (y (1) )3 ), h12 (y) = h21 (y) = |y|−3 ((ρ + 1)(y (1) )2 y (2) − (y (2) )3 ), Let us now calculate h(yj ). By (7.12), |y|−1 yj(1) = hyˆj , mi ˆ = αj + βj p = ρ−1 αj−1 =: tj , and hence

|y|−1 yj(2) = (1 − t2j )1/2 ,


215

h11 (yj ) = (ρ + 2)t3j − 3tj ,

h22 (yj ) = −(ρ + 2)t3j + (ρ + 1)tj , h12 (yj ) = h21 (yj ) = (1 − t2j )1/2 (ρ + 2)t2j − 1 .

Therefore tr h(yj ) = (1 − 2γ)αj−1 < 0 and det h(yj ) = (2 − ρ)ρ−2 αj−2 − 1. It follows from (7.13) that det h(y1 ) < 0 and det h(y2 ) > 0. Taking into account the equality H(ω0 , y) = −H(ω0 , −y), we see that sgn H(ω0 , ±y1 ) = 0, sgn H(ω0 , ±y2 ) = ∓2. Let us formulate the result obtained. Example 7.3. Let V be given by (6.16), where v(x) ˆ = −hx, ˆ ni, ρ ∈ (1/2, 1), and d = 3. Then s(ω, ω0 ; λ) remains bounded if ϑ → 0 in the cone ˆ mi hϑ, ˆ < 2(1 − ρ)1/2 (2 − ρ)−1 . If, on the contrary, the strict inequality (7.14) holds, then as ϑ → 0, ˆ ϑi−1−γ |m|γ k 1−2γ π γ−1 ρ−1 (2I(ρ))−γ s(ω, ω0 ; λ) = µhm, 1−γ 1−γ ˆ cos(k 1−2γ ψ1 (ϑ)|ϑ| ˆ ˆ sin(k 1−2γ ψ2 (ϑ)|ϑ| ˆ × w1 (ϑ) ) + w2 (ϑ) ) (1 + O(|ϑ|ε )), ˆ are defined by (7.15) and where ε > 0, µ does not depend on ϑ, |µ| = 1, ψj (ϑ) wj = [(2 − ρ)ρ−2 αj−2 − 1|−1/2 (1 − ρ−1 αj−2 )1+γ ,

ˆ αj = αj (ϑ).

Let us also write down a final answer in the case d = 2 when 3ω0 is a line. Example 7.4. Let V be given by (6.16), where v(x) ˆ = −hx, ˆ ni, ρ ∈ (1/2, 1), and d = 2. Then s(ω, ω0 ; λ) remains bounded if ϑ → 0 from the direction of −m and s(ω, ω0 ; λ) = µw0 |ϑ|−(1+γ)/2 cos(k 1−2γ ψ0 |ϑ|1−γ −π/4)(1+O(|ϑ|ε )),

ε > 0, (7.16)

as ϑ → 0 from the direction of m. Here µ does not depend on ϑ, |µ| = 1, w0 = π −1/2+γ/2 2−γ/2+1/2 ρ−1/2 (1 − ρ)γ/2 k −γ+1/2 (I(ρ)−1 |m|)γ/2 , ψ0 = ρ(1 − ρ)γ−1 (2I(ρ))−γ (π|m|)γ .

(7.17) (7.18)

Indeed, Eq. (7.10) reduces now to (1 − ρ)m ˆ = M −1 |y|ρ ϑˆ and has solutions if and only if ϑˆ = m. ˆ In this case there are two solutions ±y0 determined by y0 = (1 − ρ)γ (2I(ρ))−γ (π|m|)γ . ˆ ±y0 ) = ±ψ0 with ˆ ±y0 ) = ±ρ(1 − ρ)−1 y0 , so that ψ(ϑ, By virtue of (7.15), ψ(ϑ, ψ0 defined by (7.18). In the case 3ω0 ' R the Hessian of the function V(ω0 , y) = ±πI(ρ)−1 |m||y|1−ρ for ±y > 0 reduces to the second derivative, which allows us to calculate easily function (6.23). Therefore application of (6.26) leads to (7.16). Let us consider the exceptional case ρ = 1.

216

D. Yafaev

Example 7.5. Let V (x) = −hx, ˆ ni|x|−1 for sufficiently large |x| (and d is arbitrary). Then the asymptotics of s(ω, ω0 ; λ) is given by formulas (6.37), (6.38), where v(ω0 ) = 0 and ˆ = −khϑ, ˆ yi − π(2k)−1 hn, yi ˆ Ξ(y, ϑ) (note that this expression does not depend on ω0 ). ˆ = Indeed, v(ω0 ) = 0 since V0 (x) = −V0 (−x). Calculating (6.28) we find that ν(b) ˆ −πhn, bi. So the result formulated above is an immediate consequence of Theorem 6.10. 3. Let us compare asymptotic formula (7.3) for the quantum scattering cross-section with the corresponding classical result. Recall (see e.g. [11]) that for the classical motion in a central potential V (|x|) the scattering cross-section Σ(θ; λ) at the energy λ may be determined by the dependence of the scattering angle θ ∈ (0, π] on the impact parameter l (the minimal distance of a free particle from the coordinate center): Σ(θ; λ) = (sin θ)−d+2 l(θ; λ)d−2 |dl/dθ|.

(7.19)

We emphasize that, as in the quantum case, the physical differential cross-section in some angle equals the product of Σ and of the measure of this angle. The following result is known in the physics literature (see e.g. [11]) but we have not found its correct justification. Proposition 7.6. Suppose that V (x) = V (|x|) and V (r) = vr−ρ , where ρ is an arbitrary positive number, for sufficiently large r. Then in the limit θ → 0, (7.20) Σ(θ; λ) = ρ−1 (I(ρ)|v|λ−1 )(d−1)γ θ−(d−1)(1+γ) 1 + O(θ) , where the coefficient I(ρ) is defined by (7.1). Proof. Recall (see e.g. [11]) that the scattering angle θ is determined by the formula θ = |π − 2ϕ|, where Z ∞ −1/2 r−2 1 − λ−1 V (r) − l2 r−2 dr (7.21) ϕ=l rmin

and

rmin = sup{r : 1 − λ−1 V (r) − l2 r−2 < 0}.

Clearly, V (r) = vr−ρ for large l and r ≥ rmin . Changing in (7.21) the variable r = ly −1 we find that Z y0 −1/2 ϕ= 1 − y 2 − 2εy ρ dy, (7.22) 0

where ε = v(2λlρ )−1 and y0 = y0 (ε) is the positive root of the equation 1−y02 −2εy0ρ = 0. Our goal is to find the asymptotics of integral (7.22) as ε → 0. Fix some c ∈ (0, 1) and consider separately the intervals (0, c) and (c, y0 ). By the Taylor formula, Z c Z c Z c −1/2 −1/2 −3/2 2 ρ 2 1 − y − 2εy 1−y 1 − y2 dy = dy + ε y ρ dy + O(ε2 ). 0

0

In the integral over (c, y0 ) we put z = y 2 + 2εy ρ . Then

0

(7.23)


Z

y0 (ε)

1 − y 2 − 2εy ρ

2

−1/2

Z

1

dy = c2 +2εcρ

c

217

9ε (z)(1 − z)−1/2 dz,

(7.24)

where 9ε (z) = (y +ερy ρ−1 )−1 and y is considered as a function of z. An easy calculation shows that, away from the point z = 0, 9ε (z) = z −1/2 + ε(1 − ρ)z (ρ−3)/2 + O(ε2 ), and hence integral (7.24) equals Z 1 Z z −1/2 (1 − z)−1/2 dz + ε(1 − ρ) c2 +2εcρ

1

z (ρ−3)/2 (1 − z)−1/2 dz + O(ε2 ).

c2 +2εcρ

Comparing this result with (7.23) we find that ϕ(ε) = π/2 − I(ρ)ε + O(ε2 ),

(7.25)

where 2I(ρ) = 2c

ρ−1

2 −1/2

(1−c )

Z

c2

−

(1−z)−3/2 z (ρ−1)/2 dz−(1−ρ)

Z

1

(1−z)−1/2 z (ρ−3)/2 dz.

c2

0

Since I(ρ) does not depend on c, we can calculate it taking the limit c → 1. This gives Z 1 −1 (1 − z)−3/2 (z (ρ−1)/2 − 1)dz = π 1/2 0((1 + ρ)/2)0(ρ/2)−1 , I(ρ) = 1 − 2 0

so that the numbers I(ρ) and I(ρ) are the same. Formula (7.25) implies the following dependence of the scattering angle θ on the impact parameter l: θ(l) = I(ρ)|v|λ−1 l−ρ + O(l−2ρ ).

(7.26)

The same arguments as above show that this equality may be differentiated: dθ(l)/dl = −ρI(ρ)|v|λ−1 l−ρ−1 + O(l−2ρ−1 ). Solving Eq. (7.26) with respect to l we find that l(θ) = (I(ρ)|v|λ−1 )γ θ−γ 1 + O(θ) ,

γ = ρ−1 .

Combining the results obtained we get asymptotics (7.20) for function (7.19).

Comparing (7.3) and (7.20), we see that in the case ρ < 1 the quantum and classical cross-sections coincide in the limit of small scattering angles (when |ϑ| = sin θ → 0). According to (7.8) the same is true for the Coulomb potential if d = 3. Actually, in this case the formula of Gordon and Mott for the quantum scattering cross-section for an arbitrary angle is the same as the classical Rutherford formula. On the other hand, asymptotics (7.7) and (7.20) show that for the Coulomb potential both quantum and classical cross-sections increase as the same power θ−2d+2 of θ for any d, but the coefficients at this power are different if d 6= 3. It is curious that the classical coefficient is a limit of quantum coefficients for ρ < 1 as ρ → 1. We emphasize that the small angles asymptotics of the scattering cross-section is given in the classical mechanics by the same formula (7.20) for all ρ > 0. In the quantum case 6(ϑ) behaves as |θ|−2d+2ρ if ρ ∈ (1, d) and hence it grows less rapidly than the classical cross-section as ϑ → 0. If ρ > d, then 6(ϑ) has even a finite limit as ϑ → 0.

218

D. Yafaev

References 1. Agmon, S.: Some new results in spectral and scattering theory of differential operators in Rn . Seminaire Goulaouic Schwartz, Ecole Polytechnique, 1978 2. M. Birman, Sh. and Yafaev, D.R.: Asymptotics of the spectrum of the scattering matrix. J. Soviet Math. 25, no. 1, (1984) 3. Hörmander, L.: The Analysis of Linear Partial Differential Operators III, Berlin–Heidelberg–New York: Springer-Verlag, 1985 4. Hörmander, L.: The Analysis of Linear Partial Differential Operators IV, Berlin–Heidelberg–New York: Springer-Verlag, 1985 5. Isozaki, H., Kitada, H.: Micro-local resolvent estimates for 2-body Schrödinger operators. J. Funct. Anal. 57, 270–300 (1984) 6. Isozaki, H., Kitada, H.: Modified wave operators with time-independent modifies. J. Fac. Sci, Univ. Tokyo, 32, 77–104 (1985) 7. Isozaki, H., Kitada, H.: Scattering matrices for two-body Schrödinger operators. Sci. Papers College Arts and Sci., Univ. Tokyo, 35, 81–107 (1985) 8. Isozaki, H., Kitada, H.: A remark on the micro-local resolvent estimates for two-body Schrödinger operators. Publ. RIMS, Kyoto Univ., 21, 889–910 (1986) 9. Jensen, A.: Propagation estimates for Schrödinger-type operators. Trans. Am. Math. Soc. 291, 129–144 (1985) 10. Jensen, A., Mourre, E., Perry, P.: Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. Inst. Henri Poincaré, Phys. théor. 41, 207–225 (1984) 11. Landau, L.D. and Lifshitz, E.M.: Classical mechanics. London: Pergamon Press, 1960 12. Landau, L.D. and Lifshitz, E.M.: Quantum mechanics. London: Pergamon Press, 1965 13. Lerner, N., Yafaev, D.: Trace theorems for pseudo-differential operators. Université de Rennes, Prépublication 96–18, 1996 14. Mourre, E.: Opérateurs conjugués et propriétés de propagation. Comm. Math. Phys. 91, 279–300 (1983) 15. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics III, IV, New York: Academic Press, 1979, 1978 16. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Berlin–Heidelber–New York: Springer-Verlag, 1987 17. D. Yafaev, R.: Mathematical Scattering Theory, Am. Math. Soc., 1992 18. D. Yafaev, R.: Wave operators for the Schrödinger equation. Theor. Math. Phys. 45, 992–998 (1980) 19. Yafaev, D.R.: Radiation conditions and scattering theory for N -particle Hamiltonians. Comm. Math. Phys. 154, 523–554 (1993) 20. Yafaev, D.R.: Resolvent estimates and scattering matrix for N -particle Hamiltonians. Int. Eq. Op. Theory 21, 93–126 (1995) Communicated by B. Simon

Commun. Math. Phys. 191, 219 – 248 (1998)

Communications in


On Summability of Distributions and Spectral Geometry R. Estrada1 , J. M. Gracia-Bond´ıa2,? , J. C. Várilly3,? 1

P.O. Box 276, Tres R´ıos, Costa Rica. E-mail: [email protected] Departamento de F´ısica Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain 3 Centre de Physique Th´ eorique, CNRS–Luminy, Case 907, 13288 Marseille, France. E-mail: [email protected]

2

Received: 10 February 1997 / Accepted: 8 May 1997

Abstract: Modulo the moment asymptotic expansion, the Cesàro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities arising from elliptic pseudodifferential operators. We show how Cesàro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesàro asymptotic development.

1. Introduction Most approaches to spectral geometry rely on the asymptotic expansion of the heat kernel and Tauberian theorems. In this work, motivated by a string of recent papers by Connes, we develop spectral geometry from a more fundamental object. According to a deep statement by Connes [10], there is a one-to-one correspondence between Riemannian spin geometries and commutative real K-cycles, the dynamics of the latter being governed by the spectral properties of its defining Dirac operator. On ordinary manifolds, gravity (of the Einstein and the Weyl variety) is the only interaction naturally described by the K-cycle [1, 27, 28]. That is to say, in noncommutative geometry, existence of gauge fields requires the presence of a noncommutative manifold structure, whose “diffeomorphisms” incorporate the gauge transformations. Connes’ new gauge principle points thus to an intrinsic coupling between gravity and the other fundamental interactions. In a recent formulation [7], the Yang–Mills action functional is replaced by a “universal” bosonic functional of the form: Bφ [D] = Tr φ(D2 ), ?

On leave from Department of Mathematics, Universidad de Costa Rica, 2060 San Pedro, Costa Rica.

220

R. Estrada, J. M. Gracia-Bond´ıa, J. C. Várilly

with φ being an “arbitrary” positive function of the Dirac operator D. Chamseddine and Connes’ work on the universal bosonic functional has two main parts. In the first one, they argue that Bφ has the following asymptotic development: Bφ [D/3] ∼

∞ X

fn 34−2n an (D2 )

as 3 → ∞,

(1.1)

n=0 2 the an are the R ∞coefficients of the heat kernel0 expansion [19] for D and f0 = Rwhere ∞ xφ(x) dx, f1 = 0 φ(x) dx, f2 = φ(0), f3 = −φ (0), and so on. Then they proceed 0 to compute the development for the K-cycle currently [9, 32] associated to the Standard Model, indeed obtaining all terms in the bosonic part of the action for the Standard Model, plus gravity, plus some new ones. Their approach gives prima facie relations between the parameters of the Standard Model, in terms of the cutoff parameter 3, falling rather wide of the empirical mark. In the second part of their paper, they enterprise to improve the situation by use of the renormalization group flow equations [2]. This need not concern us here. Formula (1.1) can be given a quick derivation, by assuming that φ is a Laplace transform. This condition, however, will almost never be met in practice. In order to see that the asymptotic development of Bφ cannot be taken for granted, let us consider, as Kastler and coworkers have done [6, 26] the characteristic functions φ3 := χ[0,3] . This looks harmless enough, giving nothing but ND2 (32 ), the counting number of eigenvalues of D2 below the level 32 . However, it has been known for a long time —see for instance [24]— that there is no asymptotic development for the counting functional beyond the first term. Therefore Eq. (1.1), as it stands, is not applicable to that situation. One of our aims in this paper is to decrypt the meaning of “arbitrary functional”; a related one is to put on a firm footing the development (1.1). Our contribution turns around the Cesàro behaviour of distributions, and its relation with asymptotic analysis. Most results are new, or seem to be ignored in the literature; the paper is written with a pedagogical bent. The article is organized as follows. Section 2 is the backbone of the paper; there the Cesàro behaviour of distributions and Cesàro summability of evaluations are examined. The distributional theory of asymptotic expansions [15] is summarized. The latter is brought to bear by finding the essential equivalence between the Cesàro behaviour and the parametric behaviour of distributions at infinity. Also we prove that a distribution satisfies the moment asymptotic expansion iff it belongs to K0 , the dual of the space of Grossmann–Loupias–Stein operator symbols [20]. These results are new, having been obtained very recently by one of us [RE, 12]. We try to enliven this somewhat technical section with pertinent examples. Next we consider elliptic, positive pseudodifferential operators; let H be one of those; the functional calculus for H can be based on the spectral density, formally written as δ(λ − H). This is arguably a more basic object than the heat kernel, and its study is very rewarding. In Sect. 3, we show that δ(λ − H) is an operator-valued distribution in K0 . With that in hand, one can proceed to give a meaning to the universal bosonic action for a very wide class of functionals. Following some old ideas by Fulling [17], insufficiently exploited up to now, we emphasize that the Cesàro behaviour of the spectral density for differential operators is local, i.e., independent of the boundary conditions. This is practical for computational purposes, as it sometimes allows to replace an operator in question by a more convenient local model. In Sect. 4, we reach the heart of the matter: let dH (x, y; λ) denote the distributional kernel of δ(λ − H); a formula for dH is given and immediately applied to compute the

On Summability of Distributions and Spectral Geometry

221

coefficients of its asymptotic expansion on the diagonal, in terms of the noncommutative residues [38] of certain powers of H. We hope to have clarified in the paper that the identification of the higher Wodzicki terms is essentially a “finite-part” calculation. The spectral density is actually a less singular object for operators with continuous spectra than for operators with discrete spectra, and all of the above applies to operators associated to noncompact manifolds: for that purpose, taking account of locality, we work with densities of noncommutative residues throughout. We go on to extend Connes’ trace theorem [8] to noncompact K-cycles. The case of generalized Laplacians is then treated within our procedure. In the light of the preceding, the last two sections of the paper are concerned, respectively, with the counting number and the heat kernel expansions. The counting functional NH (λ) is treated mainly by way of example. Then we reexamine the status of arbitrary smoothing asymptotic expansions, in particular the Laplace-type expansions like the Chamseddine–Connes Ansatz. We point out conditions for the expansions to be valid without qualification, and to be valid only in the Cesàro sense. Also we exemplify circumstances under which the formal Laplace-type expansion does not say anything about the true asymptotic development. The Chamseddine–Connes expansion is derived and reinterpreted. 2. Cesàro Computability of Distributions Besides the standard spaces of test functions and distributions, the space K first introduced in [20] and its dual K0 play a central role in our considerations. Familiarity with the properties of K and K0 and with some of their elements will be convenient. For all general matters in distribution theory, we refer to [18]. As our interest is mainly in spectral theory, we consider Grossmann–Loupias– Stein symbols in one variable, almost exclusively. A smooth function φ of a real variable belongs to Kγ for a real constant γ if φ(k) (x) = O(|x|γ−k ) as |x| → ∞, for each k ∈ N. A topology for Kγ is generated by seminorms kφkk,γ = supx∈R { max(1, |x|k−γ ) |φ(k) (x)| }, and so Kγ ,→ Kγ 0 if γ ≤ γ 0 . Notice that φ(k) ∈ Kγ−k if φ ∈ Kγ . The space K is the inductive limit of the spaces Kγ as γ → ∞. Since every polynomial is in K, a distribution f ∈ K0 has moments µn := hf (x), xn i,

n∈N

of all orders; this is an indication that f decays rapidly at infinity in some sense. Denote by D00 (T) the space of periodic distributions with zero mean. They constitute a first class of examples: if f ∈ D00 (T), then, for n suitably large, the periodic primitive with zero mean fn of f of order n is continuous and defines the evaluation of f at φ ∈ K by a convergent integral: hf (x), φ(x)i = (−1)n hfn (x), φ(n) (x)i. Note that in this case all the moments are zero. The algebra K is normal (i.e., S is dense in K) and is a subalgebra of the multiplier algebras OM , M of S, respectively for the ordinary product and the Moyal star product [16]. Other properties of K and K0 will be invoked opportunely. The usefulness of K in phase-space Quantum Mechanics lies in the similitude of behaviour of the ordinary and the Moyal product, when applied to elements of K. The link between both appearances of K is still mysterious to us.

222


The natural method of studying generalized functions at infinity is by considering the parametric behaviour. The moment asymptotic expansion of a distribution [15] is given by ∞ X (−1)k µk δ (k) (x) as λ → ∞. (2.1) f (λx) ∼ k! λk+1 k=0

The interpretation of this formula is in the distributional sense, to wit hf (λx), φ(x)i =

N X µk φ(k) (0) k=0

k! λk+1

+O

1 λN +2

as λ → ∞,

for each φ in an appropriate space of test functions. Such an expansion holds only for distributions that decay rapidly at infinity, in a sense soon to be made completely precise; it certainly does not hold for all tempered distributions, as their moments do not generally exist. Distributions endowed with moment asymptotic expansions are said to be “distributionally small at infinity”. We are not happy with this terminology and invite suggestions to improve it. On the other hand, the classical analysis [23] notion of Cesàro or Riesz means of series and integrals admits a generalization to the theory of distributions, that we intend to exploit in this paper. It turns out that Cesàro limits and “distributional” ones are essentially equivalent; this will enable us to apply the simpler ideas of parametric analysis to complicated averaging schemes. We begin now in earnest by introducing the basic concept of Cesàro behaviour of the distributions; justification will follow shortly. Assume f ∈ D0 (R), β ∈ R \ {−1, −2, . . .}. Definition 2.1. We say that f is of order xβ at infinity, in the Cesàro sense, and write f (x) = O(xβ )

(C)

as x → ∞,

if there exists N ∈ N, a primitive fN of f of order N and a polynomial p of degree at most N − 1, such that fN is locally integrable for x large and the relation fN (x) = p(x) + O(xN +β )

as x → ∞

(2.2)

holds in the ordinary sense. The relation f (x) = o(xβ ) (C) is defined similarly. The notation (C, N ) can be used if one needs to be more specific; if an order relation holds (C, N ) for some N , it also holds (C, M ) for all M > N . The assumption β 6= −1, −2, . . . is provisionally made in order to avoid dealing with the primitives of x−1 , x−2 and such (see Sect. 6 for the general case). If β > −1, the polynomial p is arbitrary and thus irrelevant. We shall suppose when needed that our distributions have bounded support, say, on the left. In that case, we denote by I[f ] the first order primitive of f with support bounded on the left. When f is locally integrable, then, Z x f (t) dt. I[f ](x) = −∞

The notation

f (x) = o(x−∞ ) β

(C)

as x → ∞

will mean f (x) = O(x ) (C) for every β. For the proof of the following workhorse proposition we refer to [12].


223

Lemma 2.1. (a) Let f ∈ D0 such that f (x) = O(xβ )

as x → ∞.

(C, N )

Then for k = 1, 2, 3, . . . we have: f (k) (x) = O(xβ−k )

as x → ∞.

(C, N + k)

(b) Let f ∈ D0 such that f (x) = O(xβ )

(C)

as x → ∞,

and let α ∈ R. Provided that α + β is not a negative integer, we have: xα f (x) = O(xα+β )

(C)

as x → ∞.

Definition 2.2. We write limx→∞ f (x) = L (C) when f (x) = L + o(1) (C) as x → ∞. That is, limx→∞ f (x) = L (C, k) when fk (x) k!/xk = L + o(1), for fk a primitive of order k of f . For example, if f is periodic with zero mean value, there exists n ∈ N and a continuous (thus bounded) periodic function fn with zero mean value such that fn(n) = f ; then clearly as x → ∞, f (x) = o(x−∞ ) (C) a fact that yields, for f periodic with mean value a0 : lim f (x) = a0

x→∞

(C).

Let f ∈ D0 be a distribution with support bounded on the left and let φ be a smooth function. The following is a key concept of the theory. Definition 2.3. We say that the hf (x), φ(x)i has the value L in the Cesàro sense, and write hf (x), φ(x)i = L (C) if there is a primitive I[g] for the distribution g(x) = f (x)φ(x), satisfying lim I[g](x) = L

x→∞

(C)

as x → ∞.

A similar definition applies when f has support bounded on the right. If f is an arbitrary distribution, let f = f1 + f2 be a decomposition of f , where f1 has support bounded on the left and f2 has support bounded on the right. Then we say that hf (x), φ(x)i = L (C) if both hfi (x), φ(x)i = Li (C) exist for i = 1, 2 and L = L1 + L2 : this definition is seen to be independent of the decomposition. For instance, let f be a periodic distribution of zero mean and let f1 , f2 , . . . , fn+1 denote the periodic primitives with zero mean of f , up to the order n + 1. Then xn f1 (x) − nxn−1 f2 (x) + n(n − 1)xn−2 f3 (x) − · · · + (−1)n n! fn+1 (x) is a first order primitive of xn f (x), and since fi (x) = o(x−∞ ) (C) for i = 1, . . . , n as x → ∞, it follows that hf (x), xk i = 0 (C)

224


for all k ∈ N. To perceive the point of our hitherto abstract definitions, it is worthwhile to recall here briefly the classical theory [23]. Let {an }∞ n=1 be a sequence of real or complex numbers. Often it has no limit, but the sequence of averages Hn(1) := (a1 + · · · + an )/n does. Then people write lim an = L (C, 1). n→∞

Hn(1)

If still does not have a limit, then one may apply the averaging procedure again and again, hoping that eventually a limit will be obtained. There are two main procedures to perform such higher order averages: the Hölder means and the Cesàro means. The Hölder means are single-mindedly constructed as Hn(k) :=

H1(k−1) + · · · + Hn(k−1) , n

and limn→∞ Hn(k) = L is written lim an = L

n→∞

(H, k).

The properly named Cesàro means are defined as follows: let A(0) n := an and define (k−1) (k−1) (k) k recursively A(k) = A + · · · + A . If lim k! A /n = L, we write n→∞ n n n 1 lim an = L

n→∞

(C, k),

so that the (C, 1) and the (H, 1) notions are identical. The Cesàro limits have nicer analytical properties. The good news, at any rate, is that both procedures are equivalent: lim an = L

n→∞

(C, k) ⇐⇒ lim an = L n→∞

(H, k).

One uses the simpler notation limn→∞ an = L (C) if limn→∞ an = L (C, k) for some k ∈ N. A third averaging procedure is equivalent to Cesàro’s, the so-called Riesz typical means. For real µ, one writes lim an = L

n→∞

if

(R, k, n)

n k−1 1 X 1− an = L. µ→∞ µ µ lim

n≤µ

Riesz originally studied this formula for integral µ, but the means have more desirable P∞ properties with µ real. Now, one may study the summability P of a series n=1 an by ∞ studying the generalized function of a real variable f (x) = n=1 an δ(x − n). The definition of Ces` a ro limits of distributions is tailored in such a way that hf, 1i (C) P∞ P∞ a (C) coincide: a primitive of order k of a δ(x − n) is given by and n=1 n n n=1 P fk (x) = n≤x (x − n)k−1 an /(k − 1)! Note that one could consider distributions of the P∞ form h(x) = n=1 an δ(x − pn ), with pn ↑ ∞; this gives rise to the (R, k, pn ) means. In summary, we have demonstrated the following equivalence.


Theorem 2.2. The evaluation X ∞

225

an δ(x − n), φ(x)

=L

(C)

n=1

holds iff

P∞ n=1

an φ(n) = L in the Cesàro sense of the theory of summability of series.

In the same vein: Theorem 2.3. If f is locally integrable and supported in (a, ∞), then hf (x), φ(x)i = L if and only if

Z

(C)

∞

f (x)φ(x) dx = L a

in the Cesàro sense of the theory of summability of integrals. As shown below, if f ∈ K0 and φ ∈ K, then the evaluation hf (x), φ(x)i is always (C)-summable. We pause an instant to show by example just how useful is the concept of Ces`P aro computability of evaluations. An interesting periodic distribution is the Dirac ∞ comb n=−∞ δ(x − n). Its mean value is 1; therefore ∞ X

δ(x − n) = 1 + f (x),

(2.3)

n=−∞

with f ∈ D00 (T). The distributions ∞ X

δ(x − n) − H(x − 1),

n=1

∞ X

δ(x − n) − H(x),

n=1

where H is the Heaviside function, belong to K0 . In effect, take aPfunction φ1 ∈ K such ∞ that φ1 (x) = 1 for x > 1/2, φ1 (x) = 0 for x < 1/4. Then φ1 (x) n=−∞ δ(x − n) − 1 P∞ P∞ only differs from n=1 δ(x − n) − H(x − 1) or n=1 δ(x − n) − H(x) by a distribution of compact support. It follows that the evaluation X X Z ∞ ∞ ∞ δ(x − n) − H(x − 1), φ(x) = φ(n) − φ(x) dx n=1

1

n=1

is Cesàro summable whenever φ ∈ K. Now, xα does not belong to K unless α ∈ N, but the previous argument, using φα (x) = φ1 (x) xα , allows us to conclude that the evaluation Z(α) :=

X ∞

δ(x − n) − H(x − 1), x

α

n=1

is (C)-summable for any α ∈ C. Also, Z(α) is an entire function of α, since φα is. We find a formula for Z(α) by observing that if < α < −1, then the evaluation is given by the difference of a series and an integral, so that

226


Z(α) =

∞ X

Z

∞

nα −

xα = ζ(−α) +

1

n=1

1 , α+1

−1 then f (x) = O(|x|α )

as x → ±∞

(C)

if and only if

as λ → ∞

f (λx) = O(λα )

(2.4) (2.5)

in the topology of D0 . If −j − 1 > α > −j − 2 for some j ∈ N, then (2.4) holds if and only if there are constants µ0 , . . . , µj such that f (λx) =

j X (−1)k µk δ (k) (x) k=0

k! λk+1

+ O(λα )

in the topology of D0 as λ → ∞. Proof. We prove the theorem in the case f has support bounded on the left. The general case follows by using a decomposition f = f1 + f2 , where f1 has support bounded on the left and f2 has support bounded on the right. First we have to clarify the meaning of (2.5). It is a weak or distributional relation: we write f (x, λ) = O(λα ) as λ → ∞ whenever as λ → ∞, hf (x, λ), φ(x)i = O(λα ) for all φ ∈ D. Note that this yields ∂f (x, λ) , φ(x) = −hf (x, λ), φ0 (x)i = O(λα ). ∂x Now, if (2.5) holds, there exists N such that the primitive of order N of f (λx), with respect to x, exists and is bounded by M λα , say for |x| ≤ 1 and λ ≥ λ0 . We have then a primitive fN of order N of f (x), such that |fN (λx)| ≤ M λα+N ,

|x| ≤ 1, λ ≥ λ0 .

Taking x = 1 and replacing λ by x we obtain |fN (x)| ≤ M xα+N , and thus f (x) = O(xα )

(C, N ),

x ≥ λ0 , as x → ∞.

Reciprocally, assume α > −1 and f (x) = O(x ) (C, N ), as x → ∞. Then, if fN is the (locally integrable for x large) primitive of order N of f with support bounded on the α

228


left, an obvious estimate gives fN (λx) = O(λα+N ), as λ → ∞, and on differentiating N times with respect to x one obtains λN f (λx) = O(λα+N ), so that (2.5) follows. The case when α is nonintegral and less than −1 is more involved, as one has to deal with the polynomial p in (2.2). Then one shows that the moments hf (x), xk i = µk

(C)

up to a certain order exist, those being essentially the coefficients of p. For the gory details, we refer once again to [12]. A characterization of the distributions that have a moment asymptotic expansion follows. Theorem 2.5. Let f ∈ D0 . Then the following are equivalent: (a) f ∈ K0 . (b) f satisfies

f (x) = o(|x|−∞ )

as x → ±∞.

(C)

(c) There exist constants µ0 , µ1 , µ2 , . . . such that f (λx) ∼

µ0 δ(x) µ1 δ 0 (x) µ2 δ 00 (x) − + − ··· λ λ2 2! λ3

as λ → ∞

in the weak sense. Proof. It is proven in [15] that the elements of K0 satisfy the moment asymptotic expansion. For the converse, it is enough, as customary, to consider distributions with support bounded on one side. We show that if (b) holds, then f ∈ Kγ0 for all γ. From the hypothesis it follows that f (x) = O(x−γ−2 ) (C) as x → ∞. Thus, for a certain n, the nth order primitive fn of f with support bounded on one side is locally integrable and satisfies fn (x) = p(x) + O(x−γ−2+n ) as x → ∞, where the polynomial p has degree at most n − 1. We conjure up a compactly supported continuous function g whose moments of order up to n − 1 coincide with those of f . If gn is the primitive of order n of g −γ−2+n ). If φ ∈ Kγ−n , the with support R ∞ bounded on the left, then fn (x) − gn (x) = O(x integral −∞ (fn (x) − gn (x))φ(x) dx converges. Hence f = (fn − gn )(n) + g ∈ Kγ0 . The rest is clear. We get at once a powerful computational method for duality evaluations. Corollary 2.6. If f ∈ K0 and φ ∈ K, the evaluation hf (x), φ(x)i is Cesàro summable. Proof. It is enough to check for φ = 1. But, according to the previous theorem, if f ∈ K0 , then f (x) = o(x−∞ ) (C) as x → ∞. By the proof of Theorem 2.4, hf (x), 1i is (C)-summable. Fourier transforms are defined by duality and, in general, if f ∈ S 0 , we cannot make sense of fˆ(u) because the evaluation heixu , f (x)i is not defined. However, if φ ∈ K and u 6= 0, Corollary 2.6 guarantees that the Cesàro-sense evaluation heixu , φ(x)i (C) is well defined. Thus ˆ φ(u) = heixu , φ(x)i

(C)

when

φ ∈ K, u 6= 0.

b ⊂ K0 ; this follows also from Proposition 4 of [20]. It is clear that K Note as well that the moments of f ∈ K0 are (C)-summable. The converse is true:


229

Theorem 2.7. Let f ∈ D0 . If all the moments hf (x), xn i = µn (C) exist for n ∈ N, then f ∈ K0 . For the easy proof, we refer to [12]. It is clearly worthwhile to characterize spaces of distributions in terms of their Cesàro behaviour. Particularly important is the characterization of tempered distributions: Theorem 2.8. Let f ∈ D0 . Then the following statements are equivalent: (a) f is a tempered distribution. (b) There exists α ∈ R such that as λ → ∞

f (λx) = O(λα ), in the weak sense. (c) There exists α ∈ R and k ∈ N such that f (k) (x) = O(|x|α−k )

(C)

as x → ∞.

Proof. Again, it is enough to consider the case when f has support bounded on one side. It is well known that if f ∈ S 0 then there is a primitive F of some order N of slow growth at infinity; it follows that f (x) = O(|x|α ) (C). The rest is clear, in view of the equivalence theorem 2.4 and the fact that distributional order relations can be differentiated at will. We finish by giving several estimates that we will need later. The first one is just a rewording of the properties of the distribution (2.3). R∞ Lemma 2.9. If g ∈ K and if −∞ g(x) dx is defined, then ∞ X

1 g(nε) = ε n=−∞

Z

Lemma 2.10. If g ∈ K(Rn ) and if g(kε) = ε−n

R Rn

Z

Lemma 2.11. If g ∈ K and if

n=1

g(nε) =

1 ε

as ε ↓ 0.

g(x) dx is defined, then g(x) dx + o(ε∞ )

as ε ↓ 0.

Rn

k∈Zn

∞ X

g(x) dx + o(ε∞ )

−∞

By the same token:

X

∞

Z

R∞ 0

g(x) dx is defined, then

∞

g(x) dx + 0

∞ X ζ(−n)g (n) (0) n=0

n!

Proof. This follows from the zeta function example.

εn + o(ε∞ )

as ε ↓ 0.

(Results of this type were used to prove some formulas by Ramanujan in [15].)

230


3. Spectral Densities Let H be a concrete Hilbert space, the space of square integrable sections of an Euclidean vector bundle over a Riemannian manifold M , and let H be an elliptic positive selfadjoint pseudodifferential operator on H, with domain X . We consider the derivative, in the distributional sense, of the spectral family of projectors EH (λ) associated to H: dH (λ) :=

dEH (λ) . dλ

For instance, if H is defined on a compact manifold, and 0 < λ1 ≤ λ2 ≤ · · · is the complete set of its eigenvalues, with orthonormal basis of eigenfunctions uj , the kernel of the spectral family is given by [25]: X |uj )(uj |, EH (λ) := λj ≤λ

and the derivative is dH (λ) :=

X

|uj )(uj | δ(λ − λj ).

j

This spectral density is a distribution with values in L(X , H). The defining properties of E(λ): Z Z ∞

∞

dE(λ),

I=

H=

−∞

λ dE(λ) −∞

(in the weak sense) become, in the language of the previous section: I = hdH (λ), 1i,

H = hdH (λ), λi.

The spectral density is used to construct the functional calculus for H. Indeed, we can define φ(H) whenever f is a distribution such that the evaluation hdH (λ), f (λ)i makes sense, by φ(H) := hdH (λ), φ(λ)i, with domain the subspace of the x ∈ H for which the evaluation h(y | dH (λ)x), φ(λ)iλ is defined for all y ∈ H. Especially, one is able to deal with the “zeta operator": H −s := hdH (λ), λ−s i

(3.1)

(for 0 ∈ / sp H), the heat operator: e−tH := hdH (λ), e−tλ i,

t>0

(3.2)

and the unitary group of H, which is just the Fourier transform of the spectral density: UH (t) := hdH (λ), e−itλ i. The useful symbolic formula dH (λ) = δ(λ − H) recommends itself, and we shall employ it from now on.

(3.3)


231

We want to studyTthe asymptotic behaviour of δ(λ − H). Let Xn be the domain of ∞ H n and let X∞ := n=1 Xn . The fact that X∞ is dense has, in view of the theory of Sect. 2, momentous consequences. We have H n = hδ(λ − H), λn i in the space L(X∞ , H). Hence, δ(λ − H) belongs to the space K0 (R, L(X∞ , H)). Therefore the moment asymptotic expansion holds: δ(λσ − H) ∼

∞ X (−1)n H n δ (n) (λ) n=0

as σ → ∞,

n! σ n+1

and δ(λ − H) vanishes to infinite order at infinity in the Cesàro sense: δ(λ − H) = o(|λ|−∞ )

(C)

as |λ| → ∞.

Of course, the last formula is trivial when H is bounded. The space D(M ) of test functions is a subspace of X∞ . We can then realize the spectral density by an associated kernel dH (x, y; λ), an element of D0 (R, D0 (M × M )). Ellipticity actually implies that dH (x, y; λ) is smooth in (x, y). The expansion dH (x, y; λσ) ∼

∞ X (−1)n (H n δ)(x − y) δ (n) (λσ) n=0

as σ → ∞

n! σ n+1

holds in principle in the space D0 (R, D0 (M × M )). We also get dH (x, y; λ) = o(|λ|−∞ )

as |λ| → ∞

(C)

(3.4)

in the space D0 (M × M ). Eq. (3.4) is the mother of all incoherence principles. For instance, passing to the primitive with respect to λ, for an elliptic operator on a compact manifold with eigenfunctions ψn , n ∈ N, one concludes: X ψ¯ n (x)ψn (y) = o(|λ|−∞ ) (C) as |λ| → ∞, λn ≤λ

for x 6= y, which is Carleman’s incoherence relation [5]. It should be clear that the expansions cannot hold pointwise in both variables x and y, since we cannot set x = y in the distribution δ(x − y). In fact, our interest in this paper lies in the coincidence limit dH (x, x; λ), which is not distributionally small. However, it is proven in [13] that, away from the diagonal of M × M , the expansions are valid in the sense of uniform convergence of all derivatives on compacta. On the other hand, if H1 and H2 are two pseudodifferential operators whose difference over an open subset U of M is a smoothing operator, and if d1 (x, y; λ) and d2 (x, y; λ) are the corresponding spectral densities, then [13]: d1 (x, y; σλ) = d2 (x, y; σλ) + o(σ −∞ )

as σ → ∞

in D0 (U × U ). Also, it can be shown that d1 (x, y; λ) = d2 (x, y; λ) + o(λ−∞ )

(C)

uniformly on compacts of U × U , even at the diagonal.

as λ → ∞

232


We exemplify the reported behaviour with the simplest possible examples. Let H denote first the Laplacian on the real line. Its spectral density is dH (x, y; λ) =

√ 1 √ cos λ(x − y) , 2π λ

and therefore it is clear that dH (x, x; λ) is not distributionally small, but rather dH (x, x; λ) =

1 √ + o(λ−∞ ) 2π λ

(C)

as λ → ∞.

Let H denote now the Laplacian on the circle; the eigenvalues are λn = n2 , n = 0, 1, 2, . . ., with multiplicity 2 from n = 1 on, with normalized eigenfunctions ψn± (x) = (2π)−1/2 e±inx . Therefore X 1 δ(λ) + 2 cos n(x − y) δ(λ − n2 ) . 2π ∞

dH (x, y; λ) =

n=1

Then ∞ ∞ X X 1 δ (2j) (x − y) δ (j) (λ) δ(λσ) + 2 cos n(x − y) δ(λσ − n2 ) ∼ 2π j! σ j+1 n=1

as σ → ∞

j=0

in D0 (R, D0 (S1 × S1 )), while X 1 δ(λ) + 2 cos n(x − y) δ(λ − n2 ) = o(λ−∞ ) 2π ∞

(C)

as λ → ∞

n=1

if x and y are fixed, x 6= y. On the other hand, X 1 δ(λ) + 2 δ(λ − n2 ) 2π ∞

dH (x, x; λ) =

n=1

does not belong to K0 (R, C ∞ (S1 )). For the first time in this paper, but not the last, we have to find out what the Cesàro behaviour of a given spectral kernel is. We shall have recourse to a variety of tricks. For now, applying Lemma 2.11 to g(x) := φ(x2 ), for φ a Schwartz function, say, we get: Z ∞ ∞ X 1 φ(εn2 ) = √ x−1/2 φ(x) dx − 21 φ(0) + o(ε∞ ) as ε ↓ 0. 2 ε 0 n=1

It is then clear that dH (x, x; λ) =

1 √ + o(λ−∞ ) 2π λ

(C)

as λ → ∞,

and it is also immediately clear that the distributional and Cesàro behaviour of the spectral density and its kernel are exactly the same as in the previous example. That the manifold be compact or not and the spectrum be discrete or continuous is immaterial for that purpose. If we seek a boundary problem for the Laplacian, say on a bounded


233

interval of the line, we obtain still the same kind of behaviour (off the boundary, where a sharp change takes place). Note also the estimate: √ X λ (C) as λ → ∞. |ψn± (x)|2 ∼ π ±; λn ≤λ

As an aside, we turn before closing this section to the functional calculus formulas and compare (3.2) with (3.3). Obviously e−t(·) has an extension belonging to K, so there is no difficulty in giving a meaning to the heat operator. Also, as we shall see in Sect. 6, it is comparatively easy to study the asymptotic development of the corresponding Green function as t ↓ 0. One of the motivations of the present approach to spectral asymptotics is to define a sense for expansions of Schrödinger propagators and the like, that do not possess a “true” asymptotic expansion. Such an approach can be based in the following idea: Theorem 2.8 points to a rough duality between K0 and S 0 . Let g ∈ S 0 (R) and find α so that g(λx) = O(λα ) weakly as λ → ∞. For any φ ∈ S(R), the function 8 defined by 8(x) := hg(λx), φ(λ)iλ is smooth for x 6= 0 since 8(x) = |x|−1 hg(λ), φ(λx−1 )iλ , and satisfies 8(n) (x) = O(|x|α−n )

as |x| → ∞.

/ supp f , we can define hf (x), g(λx)ix as a tempered Therefore, if f ∈ K0 with 0 ∈ distribution. When 0 ∈ supp f , we need to ascertain independently smoothness of 8 at the origin. It turns out that, for this purpose, it is enough to demand distributional smoothness of g, i.e., the existence of the distributional values g (n) (0), in the sense of [31], for n = 0, 1, 2, . . .. Then g(tH) admits a distributional expansion in L(X∞ , H) as t ↓ 0. This can eventually lead to a proper treatment of some questions in quantum field theory. We say no more here and refer instead to the forthcoming [13]. In Sect. 6 of this paper, results will be stated for g belonging to S(R); for the rest of the paper we will venture outside safe territory only in examples.

4. The Cesàro Asymptotic Development of dH (x, x; λ) In this section we obtain the asymptotic expansion for the coincidence limits of spectral density kernels. We are fortified with the results of the previous section, implying that the Cesàro behaviour of the spectral density of pseudodifferential operators is a local matter. Let A be any pseudodifferential operator of order a positive integer d, with complete symbol σ(A), on the Riemannian manifold M . To simplify the discussion, we consider only operators acting on scalars; the treatment of matrix-valued symbols presents no further difficulty. The noncommutative or Wodzicki residue of A is defined by integrating (the trace of) the partial symbol σ−n (A)(x, ξ) of order −n over the cosphere bundle { (x, ξ) : |ξ| = 1 }: Z Z σ−n (A)(x, ξ) dξ dx. Wres A := M

Sn−1

234


Here dx denotes the canonical volume element on M . If M is notR compact, Wres A may not exist, but there always exists the local density of the residue Sn−1 σ−n (A)(x, ω) dω, that we denote by wres A(x). We recall that σ(AB) − σ(A)σ(B) ∼

X (−i)|α| ∂ξα σ(A)∂xα σ(B). α!

|α|>0

The kernel kA of A is by definition: kA (x, y) := (2π)−n hei(x−y)·ξ , σ(A)(x, ξ)iξ . In particular, on the diagonal: kA (x, x) := (2π)−n h1, σ(A)(x, ξ)iξ .

(4.1)

In order to figure out the symbol for a spectral density, we start by considering (the selfadjoint extension of) an elliptic operator H with constant coefficients. In this case σ(H n ) = σ(H)n and we assert: σ δ(λ − H) = δ(λ − σ(H)), justified by the identities: Z λn δ(λ − σ(H)) dλ = σ(H n ),

λ = 0, 1, 2, . . . .

In the general case of nonconstant coefficients, we make the Ansatz that: σ δ(λ−H) ∼ δ(λ−σ(H))−q1 δ 0 (λ−σ(H))+q2 δ 00 (λ−σ(H))−q3 δ 000 (λ−σ(H))+· · · (4.2) R in the Cesàro sense. Computation of λn σ(δ(λ − H)) dλ for λ = 0, 1, 2, . . . then gives q1 = 0; q2 = 21 σ(H 2 ) − σ(H)2 ; q3 = 16 σ(H 3 ) − 3σ(H 2 )σ(H) + 2σ(H)3 , (4.3) and so on. This development, it turns out, gives ever lower powers of λ in the asymptotic expansion of σ(δ(λ − H)). We are interested in explicit formulas for the Cesáro asymptotic development of the coincidence limit for the kernel of a positive operator H as λ → ∞. From (4.1) and (4.2) with p := σ(H), we get dH (x, x; λ) ∼ (2π)−n h1, δ(λ − p(x, ξ)) + q2 (x, ξ) δ 00 (λ − p(x, ξ)) − · · ·iξ

(C).

In polar coordinates on the cotangent fibres, ξ = |ξ|ω with |ω| = 1, this becomes Z dω h|ξ|n−1 , δ(λ − p(x, |ξ|ω)) + q2 (x, |ξ|ω) δ 00 (λ − p(x, |ξ|ω)) − · · ·i|ξ| . (2π)−n |ω|=1

Hence, if we denote by |ξ|(x, ω; λ) the positive solution of the equation p(x, |ξ|ω) = λ, we need to compute: Z ∂2 n−1 |ξ|n−1 (x, ω; λ) + ∂λ (x, ω; λ) − · · · 2 q2 (x, |ξ|(x, ω; λ)ω)|ξ| −n . dω (2π) p0 (x, |ξ|(x, ω; λ)ω) Sn−1 (4.4)


235

Write: p(x, |ξ|ω) ∼ pd (x, ω)|ξ|d + pd−1 (x, ω)|ξ|d−1 + pd−2 (x, ω)|ξ|d−2 · · · . To solve p(x, |ξ|ω) = λ amounts to a series reversion. In order to see how that is done, let us assume for a short while that H is a firstorder operator with constant coefficients —for instance, the absolute value of the Dirac operator on Rn . We then expect |ξ|(x, ω; λ) ∼

p0 (ω) 1 λ− − p−1 (ω) λ−1 + · · · . p1 (ω) p1 (ω)

Integration over |ω| = 1 gives dH (x, x; λ) ∼ (2π)−n a0 λn−1 + a1 λn−2 + a2 λn−3 + · · ·

(C),

where, clearly, a0 = wres H −n . To compute a1 , a2 , . . . we can as well assume that the development of p is analytic as |ξ| → ∞. Let ψ(z) := z n−1 /p0 (z), so that Z ψ(|ξ|(x, ω; λ)) dω. a0 λn−1 + a1 (x)λn−2 + a2 (x)λn−3 + · · · ∼ Sn−1

If 0 is a circle containing |ξ|(x, ω; λ), wound once around ∞, we have the Cauchy integral: I 1 ψ(z)p0 (z) dz ψ(|ξ|(x, ω; λ)) = ψ(p−1 (λ)) = − 2πi 0 p(z) − λ I I 1 1 ψ(ζ −1 )p0 (ζ −1 ) dζ dζ = . = 2 −1 n+1 2πi 0−1 ζ (p(ζ ) − λ) 2πi 0−1 ζ (p(ζ −1 ) − λ) R Thus aj (x) = Sn−1 cj (x, ω) dω, where I 1 cj (ω) = sn−j−2 ψ(p−1 (1/s)) ds 2πi |s|=ε I I dζ 1 n−j−2 s ds = n+1 (p(1/ζ) − 1/s) (2πi)2 |s|=ε ζ −1 0 I I dζ 1 sn−j−1 ds = 2 n+1 (2πi) 0−1 ζ |s|=ε s p(1/ζ) − 1 I 1 dζ , = 2πi 0−1 ζ n+1 p(1/ζ)n−j which is the coefficient of ζ n in the expansion of p(1/ζ)j−n . Integrating over |ω| = 1 yields thus aj = wres H j−n , so, finally: dH (x, x; λ) ∼

1 (wres H −n λn−1 + wres H −n+1 λn−2 + · · ·) (2π)n

(C),

236


where the densities of Wodzicki residues are constant for a constant-coefficient operator. It is amusing that we have arrived at a version of the classical Lagrange–Bürmann expansion [29], with Wodzicki residues in the place of ordinary residues. Notice that an = 0. This is a very simple “vanishing theorem” (see for instance [3]). Returning to the general case, if H is a positive pseudodifferential operator of order d, then A := H 1/d is a positive pseudodifferential operator of first order. Setting µ = λ1/d , we have δ(µ − A) δ(λ1/d − H 1/d ) = , δ(λ − H) = δ(µd − Ad ) = dµd−1 dλ(d−1)/d and so dH (x, x; λ) ∼

1 a0 (x)λ(n−d)/d + a1 (x)λ(n−d−1)/d d (2π)n + a2 (x)λ(n−d−2)/d + · · · (C).

(4.5)

Clearly, a0 = wres H −n/d . Now, the order of q2 is at most 2d − 1, therefore its higher order contribution to this development is in principle to a1 ; the order of q3 is at most 3d − 2, so it contributes to a2 at the earliest, and so on. Formula (4.5), obtained through fairly elementary manipulations, is the main result of this section. To illustrate its power, we show how to reap from it a rich harvest of classical results (with a little extra effort). Corollary 4.1 (Connes’ trace theorem). For positive elliptic pseudodifferential operators of order −n on a compact n-dimensional manifold, the Dixmier trace and the Wodzicki residue are proportional: Dtr H =

1 Wres H. n (2π)n

Proof. Let H be of order d = −n in (4.5). We get dH (x, x; λ) ∼ −

1 wres H(x) λ−2 + · · · n (2π)n

(C).

Assume the manifold is compact. We then know that H is a compact operator. Now, 0 (λ) ∼ −λ−2 , ergo NH (λ) ∼ λ−1 , ergo heuristically the argument goes as follows: NH λl (H) ∼ l−1 . A Tauberian argument can be used at this point [37] to ensure that the second asymptotic estimate is valid without the Cesàro condition; and then the result follows. But this is by no means necessary. One can steal a look at Sect. 6 and, by approaching step functions by elements of S, prove in an elementary way that for any given ε > 0 there is l(ε) such that C(1 + ε) C(1 − ε) < λl (H) < , l(ε) l(ε) where C = n−1 (2π)−n Wres H.

On a noncompact spin manifold, consider now the Dirac operator on the space of spinors L2 (S). The noncommutative integral of |D|−n does not exist. However, if R a(x) dx is defined, it is computable by a noncommutative integral:


237

Theorem 4.2. Let a be an integrable function with respect to the volume form on M . Then Z 1 a(x) dx = Wres(a|D|−n ), Cn n (2π)n M where on the right hand side A is seen as a multiplication operator on L2 (S). The constants are C2k = (2π)−k /k! and C2k+1 = π −k−1 /(2k + 1)!! Proof. That follows from Theorem 5.3 of [37] if a is a smooth function with compact support. For a positive and integrable, use monotone convergence on both sides; the general case follows at once. The former is a small step in the direction of a theory of K-cycles (or “spectral triples”, as they are nowadays called) over noncompact manifolds. Corollary 4.3 (Weyl’s estimate). Let NH (λ) denote the counting function of H, a Laplacian on a compact manifold or bounded region M acting on scalar functions. Then n vol M n/2 λ , NH (λ) ∼ n(2π)n where n is the surface area of the unit ball in Rn . Proof. The same type of arguments as in Corollary 4.1 work. Indeed, this estimate is a corollary of it [37]. Next consider the Schrödinger operators −1 + V (x), with symbol p(x, ξ) = |ξ|2 + V √(x). We can take a slightly different tack and solve the equation p(x, ξ) = λ by |ξ| = (λ − V (x))+ . Corollary 4.4 (The correspondence principle). For Schrödinger operators: Z n n/2 NH (λ) ∼ (λ − V (x))+ dx. n(2π)n See [22], for instance, for the reasons for the terminology. A word of caution is in order here. The development (4.5) cannot be integrated term by term in general. Consider, for instance, the harmonic oscillator hamiltonian H = 21 (−d2 /dx2 + x2 ) on R: according to the theory developed here, its spectral density √ behaves as 1/ λ. If ψn , n ∈ N denote the normalized wavefunctions, then indeed, like in Fourier series theory, √ X λ 2 ψn (x) ∼ π 1 n+ 2 ≤λ

is true and can be independently checked. But wres H√−1/2 is not integrable over the real line, λ. Actually, as we saw in Sect. 2, H (λ) behaves as P∞ so one cannot1 conclude that N−∞ δ(λ − (n + )) = H(λ) + o(λ ) (C), so N (λ) = λH(λ) + o(λ−∞ ) (C). Now, H n=0 2 Corollary 4.2 applies, so we have 2 NH (λ) ∼ 2π

Z

√

2λ

√ − 2λ

p 2λ − x2 dx = λH(λ)

238


precisely as it should. (See the discussion in [30].) Consider n-dimensional Schrödinger operators with (continuous) homogeneous potentials V (x) ≥ 0, V (ax) = ta V (x). The previous formula gives Z NH (λ) ∝ λn/2+n/a V (x)−n/a dx, Sn−1

and this means that if the cone { x ∈ R : V (x) = 0 } is too big, in the counting number estimate we are heading for trouble [36]. But the “nonstandard asymptotics” that might then intervene do not detract from the validity of the nonintegrated formula (4.5). In the remainder of the section, we focus on the computation of spectral densities for Laplacians. Nothing essential is won or lost by considering general vector bundles, so we work on scalars. The more general Laplacian operator on a Riemannian manifold is (minus) the Laplace–Beltrami operator 1 plus potential vector and scalar potential terms, with symbol n

p(x, ξ) = −g ij (x) ξi ξj + (i0kij (x)ξk + 2Ai (x)ξj )

+ (Ai (x)Aj (x) + i(0kij (x)Ak (x) − ∂i Aj (x))) + V (x)

=: −g ij (x)ξi ξj + B i (x)ξi + C(x). Formula (4.5) would seem to give for this case: dH (x, x; λ) ∼

1 a0 (x)λ(n−2)/2 + a1 (x)λ(n−3)/2 + a2 (x)λ(n−4)/2 + · · · (C). n 2 (2π)

In fact, it will be seen in a moment that a1 = a3 = · · · = 0. Also we know already that a0 (x) = Wres 1−n/2 = n . Our task is to compute the next coefficients; it is a rather exhausting one, whose results can be inferred from the extensive work already carried out [19] on heat kernel expansions (see Sect. 6), so we will limit ourselves to the computation of a2 (x) to illustrate the relative simplicity of our approach. Let n ≥ 3. Write a for g ij (x)ωi ωj , then b for B i (x)ωi and c for C(x). Our method calls for solving for the positive root of a|ξ|2 + b|ξ| + (c − λ) = 0 and substituting this in |ξ|n−1 /(2a|ξ| + b). In diminishing powers of λ, we obtain for the latter the development: n(n − 2)b2 (n − 2)c 1 (n−2)/2 (n − 1)b (n−3)/2 (n−4)/2 − λ − λ + λ +· · · . (4.6) 8a 2 2an/2 2a1/2 One sees that odd-numbered terms in this expansion contain odd powers of ω and thus give vanishing contributions, after the integration on the cosphere. Also, the contribution of the q2 term in (4.2) will start at order 21 n − 2 in λ, the contribution of q3 will start at order 21 n − 3 and so on: the terms in the asymptotic expansion of the density kernels of √ Laplacian operators differ by powers of λ, not of λ, as one would expect on general grounds. It is convenient now to use geodesic coordinates at each point; this is justified by the nature of the result. In these coordinates 0kij (x0 ) = 0 and we have the Taylor expansion X 1 (x − x0 )α ∂ α g(x0 ) gij (x) ∼ δij + Riklj (x0 ) (x−x0 )k (x−x0 )l + 3 α!

as x → x0 ,

|α|≥3

where RikljPdenotes the Riemann curvature tensor. Recall P that the Ricci tensor is given l by Rkj := l Rklj and the scalar curvature by R := kj g kj Rkj .


239

From (4.3) one obtains for q2 (x0 , ξ), 1 X i−|α| α ∂ξ −g ij (x0 )ξi ξj + B i (x0 )ξi ∂xα |x=x0 −g ij (x)ξi ξj + B i (x)ξi + C(x) . 2 α! |α|>0

(4.7) Let us take for a moment Ai = 0. Then in geodesic coordinates B i (x0 ) = 0 and it is not hard to see that the only surviving term in (4.7) is equal to 13 Rkj (x0 )ξ k ξ j . Also b = 0 in (4.6). So, in view of (4.4) we are left with two terms at order λ(n−4)/2 , to wit: Z (n − 2)C(x0 ) (n−4)/2 λ dω − 2 n−1 S that comes from the third term in (4.6), and the first order contribution of Z ∂2 n−1 (x, ω; λ) ∂λ2 q2 (x, |ξ|(x, ω; λ)ω)|ξ| . dω p0 (x, |ξ|(x, ω; λ)ω) Sn−1 In effect, q2 contributes here a factor of order λ, so the second derivative in the previous formula gives rise to a term of order λ(n−4)/2 also. To finish the computation, we use Z n ij g Aij , dω Aij ω i ω j = n Sn−1 to get

(n − 2)n 1 (4.8) 6 R(x0 ) − C(x0 ) . 2 Notice that for a pure Laplace–Beltrami operator, the contribution to a2 , when computed in geodesic coordinates, comes exclusively through the q2 term. It remains to convince ourselves that vector potentials give no contribution at this stage. On one hand, the c term in (4.6) would contribute now the extra terms a2 (x0 ) =

−

(n − 2)n j (A Aj + i∂j Aj ). 2

On the other, the term in b2 in the same formula would contribute a term of the form 1 j j 2 (n − 2)n A Aj , and in the computation of q2 there appears now a term (2i/n) ∂j B i j that contributes 2 (n − 2)n ∂j A and thereby cancels the rest. Therefore (4.8) stands also in that case. Actually the coefficients of the Cesàro asymptotic expansion of d(x, x; λ) are all (local densities of) Wodzicki residues for n odd: a2k (x) = wres 1−n/2+k (x), for k ∈ N. For n even we have a2k = wres 1−n/2+k only as long as −n/2 + k < 0 (the Wodzicki residues of nonnegative powers of a differential operator being of course zero); the following coefficients for the parametric expansion are, in our terminology of Sect. 2 (further explained in the next two sections), not “residues” but “moments”. Note that for n = 2, the coefficient a2 is already a “moment” and cannot be computed by a Cesàro development. This strikingly different behaviour of the odd-dimensional and the evendimensional cases is concealed in the uniformity of the usual heat kernel method, but it reflects itself in the fact that the corresponding zeta functions have an infinite number of poles, corresponding to the residues, in the odd-dimensional case; and a finite number in the even-dimensional case. One has [38]:

240


Res ζH (s) =

s=n/2−k

Z

where ζH (s) =

M

1 2

Wres H k−n/2 ,

hdH (x, x; λ), λ−s iλ dx

(< s 0)

is the kernel of the zeta operator (3.1). A direct, “elementary” proof of the essential identity between Wodzicki residues and residues of the poles of the zeta functions is obviously in the cards, but we will not go further afield here. For a nontrivial use of the noncommutative residue in zeta function theory, have a look at [11].

5. Cesàro Developments of Counting Functions We consider here operators on compact manifolds without boundary and look at the behaviour of the counting function X 1. N (λ) := λl ≤λ

In order to refresh our intuition, we shall follow a deliberately na¨ıve approach and temporarily forget some of what we learned at the end of last section. Envisage first the scalar Laplacian on T2 with the flat metric; then the counting function is given by the following table: λ 0 1 2

4

5

8

9 10 13 16 17 18 20 25 26 · · ·

N (λ ) 1 5 9 13 21 25 29 37 45 49 57 61 69 81 89 · · · +

No doubt, N (λ) ∼ πλ is a reasonable first approximation; but it is also plain that the remainder undergoes wild oscillations. The precise determination of this remainder is a difficult problem, not unlike the problem of determining the next-to-main term in the asymptotic development of prime numbers. An even simpler and more telling example is provided by the eigenvalues λl of the Laplacian on the n-dimensional sphere. They are given by l+n l+n−2 − , (5.1) λl = l(l +n−1) with respective multiplicities ml = n n for l ∈ N. For example, if n = 2, the eigenvalues are l(l + 1) and the multiplicities are (2l + 1). The leading term is N (λ) ∼

2 n/2 λ n!

as λ → ∞.

On the other hand, asymptotically: N (λ+ ) − N (λ− ) ∼ and so

2 ln−1 , (n − 1)!

λ(1−n)/2 N (λ+ ) − N (λ− ) ∼

2 . (n − 1)!


241

Plainly, we cannot find an asymptotic formula for N (λ) with error term o(λ(n−1)/2 ) and continuous main term. The example is taken from Hörmander’s work [24, 25]. The foregoing is a “Gibbs phenomenon” related to the lack of smoothness of the characteristic function. The problem is “solved” if one is prepared to look at the expansions in the Cesàro sense. The fact that higher order terms in the asymptotic expansion of the eigenvalues of the Laplacian were to be understood in an averaged sense was pointed out by Brownell [4] many years ago. Going back to tori, consider the distribution of nonvanishing eigenvalues {λl }∞ l=1 of the scalar Laplacian on an n-dimensional torus Tn , with the flat metric. The eigenfuncn tions {φl }∞ l=1 can be seen as nonzero smooth functions in R that satisfy 1φl + λl φl = 0 and the periodicity conditions φl (x1 + 2k1 π, . . . , xn + 2kn π) = φl (x1 , . . . , xn ), where the girths of the torus are taken to be 2π in all directions. Those eigenvalues are given by λk = k12 + · · · + kn2 for k = (k1 , . . . , kn ) ∈ Zn , with corresponding eigenfunctions φk (x1 , . . . , xn ) = eik·x . Thus the λl are the nonnegative integers ql that can be written as a sum of n squares. The multiplicity of each such value is the number of integral solutions of the Diophantine equation ql = k12 + · · · + kn2 . We wish to compute the terms in the parametric and Cesàro developments of N (λ) next to leading Weyl term (which in fact for this problem goes back to Gauss): N (λ) ∼

n n/2 λ n

as λ → ∞.

To do so, we start with the derivative N 0 (λ); this is nothing but (2π)n d(x, x; λ), but, as advertised, it is more instructive to forget for a while the discussion in Sect. 4. We have: ∞ X X δ(λ − λl ) = δ(λ − k12 − · · · − kn2 ). N 0 (λ) = k∈Zn

l=1

Let φ ∈ D(R), let σ be a large real parameter and set ε = 1/σ, so that ε ↓ 0. Then X φ(ε|k|2 ) hN 0 (σλ), φ(λ)iλ = εhN 0 (x), φ(ελ)iλ = ε k∈Zn

Z

φ(|x|2 ) dx + o(ε∞ ) Z ∞ r(n−2)/2 φ(r) dr + o(ε∞ ). = 21 n ε1−n/2

= ε1−n/2

Rn

0

The third equality is just Lemma 2.10. Hence, weakly: −1+n/2

N 0 (σλ) = 21 n σ −1+n/2 λ+

+ o(σ −∞ )

as σ → ∞,

and upon integration N (σλ) =

n n/2 n/2 λ+ σ + o(σ −∞ ) n

as σ → ∞.

242


Observe that the constant of integration µ0 vanishes, as do all the other moments. Then Theorem 2.4 yields: n n/2 λ + o(λ−∞ ) n

N (λ) =

as λ → ∞.

(C)

Hence the error term, although definitely not small in the ordinary sense, is of rapid decay in the (C) sense. We turn to examine P∞ some cases of spheres. The derivative of the counting function for S2 is N 0 (λ) = l=0 (2l + 1) δ(λ − l(l + 1)). To deal with this case, we need a heavier gun than Lemmata 2.9–2.11. This is provided by: Lemma 5.1. Let f ∈ K0 (Rn ), so that it satisfies the moment asymptotic expansion. If p is an elliptic polynomial and φ ∈ S, then hf (x), φ(tp(x))i ∼

∞ X hf (x), p(x)m i φ(m) (0)

m!

m=0

as t → 0.

tm

Proof. The proof consists in showing that the Taylor expansion φ(tp(x)) =

N X φ(m) (0)p(x)m

m!

m=0

tm + O(tN +1 )

holds not only pointwise, but also in the topology of K(Rn ). Consider now the distribution X ∞

f (λ) := (2λ + 1)

δ(λ − l) − H(λ) ,

l=1

that lies in K0 . Notice that hf (λ), φ(t(λ2 + λ))i =

∞ X

Z

l=1

=

∞ X

∞

(2l + 1) φ(t(l2 + l)) − Z

(2λ + 1)φ(t(λ2 + λ)) dλ

0 ∞

(2l + 1) φ(t(l2 + l)) −

φ(tµ) dµ. 0

l=1

From Lemma 5.1 we conclude that, for φ ∈ S, hN 0 (λ), φ(tλ)i =

∞ X

(2l + 1) φ(t(l2 + l))

l=0 ∞

Z ∼

φ(tµ) dµ + φ(0) + 0

∞ X hf (λ), (λ2 + λ)j i φ(j) (0)

j!

j=0

tj

The parametric expansion of N 0 (λ) is thus N 0 (λ/t) ∼ H(λ) + δ(λ)t +

∞ X (−1)j µj δ (j) (λ) j=0

j!

tj+1

as t ↓ 0,

as t ↓ 0.


243

where the “generalized moments” µj are given by µj = hf (λ), (λ2 + λ)j i =

∞ X

Z

∞

(2l + 1)(l2 + l)j −

(2λ + 1)(λ2 + λ)j dλ

(C).

0

l=1

It follows that N 0 (λ) ∼ H(λ) + o(λ−∞ ) (C) as λ → ∞. In view of our gymnastics with Riemann’s zeta function in Sect. 2, the computation of the µj presents no difficulties. We obtain 2 µ0 = 2ζ(−1) + ζ(0) = − , 3

µ2 = 2ζ(−5) + 4ζ(−3) =

1 , 15 and so on. On integrating, we get µ1 = 2ζ(−3) + ζ(−1) = −

N (λ/t) ∼

8 , 315

µ3 = 2ζ(−7) + 9ζ(−5) + ζ(−3) = −

1 1 4 0 λ H(λ) + H(λ) + δ(λ) t + δ (λ) t2 + · · · t 3 15 315

2 , 105

as t ↓ 0,

(5.2)

and N (λ) ∼ λ H(λ)+ 13 H(λ)+o(λ−∞ ) (C). Note that the λ0 th order term in the Cesáro development for N (λ) comes from the first moment. The curvature of a sphere Sn is given by R = n(n − 1), so the second term in the development is precisely what we had expected. We look now at the derivative of the counting function for the Laplace–Beltrami simpler to consider the operator 1 − 1, for which we have, operator on S3 . It is slightly P ∞ according to (5.1): N 0 (λ) = l=0 (l + 1)2 δ(λ − (l + 1)2 ). Consider the distribution X ∞ 2 f (λ) := (λ + 1) δ(λ − l) − H(λ + 1) , l=0

lying in K0 . We have: hf (λ), φ(t(λ + 1) )i = 2

∞ X

Z (l + 1) φ(t(l + 1) ) − 2

2

l=0

∞

(λ + 1)2 φ(t(λ + 1)2 ) dλ.

−1

One sees that the moments all cancel: hf (λ), (λ + 1)2j i = ζ(−2j − 2) = 0, for j ∈ N. Therefore we get simply Z ∞ √ 1 φ(u) u du as t ↓ 0, hN 0 (λ), φ(tλ)i ∼ 3/2 2t 0 and thus in this case we collect just the Weyl term N (λ) ∼

λ3/2 H(λ) 3

(C)

as λ → ∞.

(5.3)

We may reflect now that the counting number for these Laplacians on S2 , S3 behave in the expected way for even and odd dimensional cases, respectively. For a generalized Laplacian which is the square of a Dirac operator the qualitative picture is the same. In particular, the Chamseddine–Connes expansion corresponds to n = 4, whereupon the counting functional behaves in much the same way as the one for S2 . Therefore, formal application of the Chamseddine–Connes Ansatz to the characteristic function of the spectrum, as done in [6, 26] misses the terms involving δ and its derivatives —whose physical meaning, if any, is unclear to us.

244


6. Spectral Density and the Heat Kernel Now we tackle the issue of the small-t behaviour of the Green functions associated to an elliptic pseudodifferential operator H. These are the integral kernels of operator-valued functions of H, of the form G(t, x, y) = hdH (x, y; λ), g(tλ)iλ , where g, as already advertised, will in this section belong (or can be extended) to the Schwartz space S (i.e., we deal with the standard theory as opposed to the framework sketched at the end of Sect. 3). The basic question is whether G(t, x, y) has an asymptotic expansion as t ↓ 0. In effect, we shall see immediately how to obtain from the (C) asymptotic expansion for the spectral density an ordinary asymptotic expansion for Green functions. The emphasis in recent years has been on Abelian type expansions, the so-called heat kernel techniques [19]. It is common folklore that Cesàro summability implies Abel summability, but not conversely. As we just claimed, one can go from the Cesàro expansion to the heat kernel expansion. The reverse implication does not work quite the same. If we know the coefficients of the heat kernel expansion and we independently know that a Cesàro type expansion for the spectral density exists, we can infer the coefficients of the latter from the former. But it may happen that the formal Abel–Laplace type expansion does not say anything about the “true” asymptotic development. √ For instance, if f (λ) := sin λ e λ for λ > 0, then limλ→∞ f (λ) (C) does not exist, since no primitive of f can have polynomial order in λ. Even so, one can show that k(t) = hf (λ), e−tλ i has a Laplace expansion k(t) ∼ a−1 t−1 + a0 + a1 t + · · · as t ↓ 0, that is, limλ→∞ f (λ) = a−1 (A). To get an example of a bounded function with this behaviour, one uses the fact that fm (λ) = sin λ1/m obeys limλ→∞ fm (λ) = 0 (C, N ) only for N > m, together with Baire’s theorem, to construct a bounded function f (λ) = P −k fmk (λ) that does not have a Cesàro limit as λ → ∞, but for which f (λ) → 0 k≥1 2 in the Abel sense. In order to relate our Cesàro asymptotic expansions with heat kernel developments, we need to examine expansions of distributions f (λ) that may contain nonintegral powers of λ. Suppose that {αk }k≥1 is a decreasing sequence of real numbers, not including negative integers, and suppose further that f ∈ S 0 , supported in [0, ∞), has the Cesàro asymptotic expansion X X c k λα k + bj λ−j (C) as λ → ∞. f (λ) ∼ j≥1

k≥1

It follows from Theorem 32 of [15] and from Theorem 2.5 that f has the following parametric development: f (σλ) ∼

X k≥1

ck (σλ+ )αk +

X

bj Pf((σλ)−j H(λ)) +

j≥1

X (−1)m µm δ (m) (λ) m! σ m+1

as σ → ∞, where the “generalized moments” µm are given by X X k µm = hf (x) − c k xα bj Pf(x−j H(x)), xm i, + − k≥1

(6.1)

m≥0

(6.2)

j≥1

and where Pf denotes a “pseudofunction” R ∞ [14] obtained by taking the Hadamard finite part, that is: hPf(h(x)), g(x)i := F.p. 0 h(x)g(x) dx if supp h ⊆ [0, ∞). In particular,


hPf(x

−j

Z

∞

H(x)), g(x)i = F.p. 0

245

g(x) dx xj

j−1 (k) j−2 X X 1 g (0) k g (k) (0) x . g(x) − dx − j k! k!(j − k − 1) 1 0 x k=0 k=0 (6.3) Notice that taking the finite part involves dropping a logarithmic term proportional to g (j−1) (0). This has the consequence that Pf(x−j H(x)) fails to be homogeneous of degree −j by a logarithmic term; indeed, Z

∞

=

g(x) dx + xj

Z

1

Pf((σλ)−j H(σλ)) = σ −j Pf(λ−j H(λ)) +

(−1)j δ (j−1) (λ) log σ . (j − 1)! σ j

Consequently, hf (λ), g(tλ)iλ ∼

X k≥1

+

ck t

−αk −1

X

∞

F.p.

Z bj tj F.p.

λαk g(λ) dλ

0 ∞ 0

j≥1

+

Z

g (j−1) (0) g(λ) log t dλ − λj (j − 1)!

X µm g (m) (0) tm . m!

(6.4)

m≥0

−λ The heat kernel development R ∞may be recovered by taking g(λ) = e for λ ≥ 0. In that case, dαk is integral, F.p. 0 λαk g(λ) dλ = 0(αk + 1) and g (j−1) (0) = (−1)(j−1) . From this it is clear that the heat kernel of a pseudodifferential operator may generally contain logarithmic terms. Indeed, by harking back to (4.5), on using (6.4) we prove:

Corollary 6.1. The general form of the (coincidence limit of) the heat kernel for an elliptic pseudodifferential operator of order d on a compact manifold M of dimension n is given by X

K(t, x, x) ∼

γj−n (x)t(j−n)/d +

j−n∈dN / +

X

βj−n (x)t(j−n)/d log t +

γj−n (x) =

rm (x)tm

r=1

j−n∈dN+

as t ↓ 0, where

∞ X

0((n − j)/d) aj (x), d(2π)n

and similarly for the other coefficients. (See [21, Cor. 4.2.7].) Now suppose we know a priori that f (λ) has a Cesàro asymptotic expansion in falling powers of λ, and that we also know that 8(t) := hf (λ), e−tλ iλ has an asymptotic expansion as t ↓ 0 without log t terms. Then it follows that all bj = 0 in (6.1), i.e., there are no negative integral exponents in the Cesàro development of f , and consequently the constants µm are the moments of f . Thus (6.4) simplifies to 8(t) ∼

X k≥1

ck 0(αk + 1) t−αk −1 +

X (−1)m µm tm . m!

m≥0

246


This is precisely the case for a (generalized) Laplacian: if n is odd, only half-integer powers of λ appear in the spectral density and logarithmic terms in the heat kernel are thereby ruled out. Notice that the Cesàro development for an odd dimensional Laplacian need not terminate. For even dimensions, the term k = n/2 is proportional to wres H 0 λ−1 and later terms are proportional to wres H r λ−r−1 . However, since H r is a differential operator, its local Wodzicki residue vanishes for r ∈ N, and the Cesàro development terminates at the λ0 term. However, as we have seen, at this point the moments (6.2) enter the picture. It has become a habit to write the diagonal of the heat kernel for a Laplacian in the form ∞ X bk (x, x) tk/2 , K(t, x, x) ∼ (4πt)−n/2 k=0

where n is the dimension of the manifold and b0 (x, x) = 1. We see now that bk (x, x) = 0 for k odd, whereas 2k a2k (x) n (n − 2)(n − 4) . . . (n − 2k)

b2k (x, x) =

for k > 0.

A similar formula holds off-diagonal. As we have noted, these expansions are local in the sense that they do not distinguish between a finite and an infinite region of Rn , say. However, the smallness of the terms after the first is not uniform near the boundary, and hence the “partition function” Z

K(t, x, x) dx ∼ (4πt)−n/2

K(t) := M

∞ X

bk tk/2 ,

(6.5)

k=0

with b0 = vol(M ) for scalars, has an expansion with nontrivial boundary terms in general, starting to contribute in b2 [33]. As for the examples, the expansion (6.5) for S2 was first obtained as the partition function of a diatomic molecule [34] and is well known to physicists. On using vol(S2 ) = 4π, we read Mulholland’s expansion directly by looking at (5.2): KS2 (t) ∼

4 2 1 1 1 + + t+ t + ··· t 3 15 315

as t ↓ 0.

As for the SU (2) group manifold, from (5.3), on using vol(S3 ) = 2π 2 and et1 = et e−t(1−1) , the partition function is seen immediately to be √ π KS3 (t) ∼ 3/2 et . 4t We turn at last to the Chamseddine–Connes expansion. The theory of Cesàro and parametric expansions justifies (1.1), in the following way. We work in dimension n = 4 and take H = D2 , a generalized Laplacian, acting on a space of sections of a vector bundle E, over a manifold without boundary. The kernel of its spectral density satisfies dD2 (x, x; λ) ∼

rk E 1 λ+ wres D−2 (x) 16π 2 32π 4

(C)

as λ → ∞.

Integrating over M and using the formulas of this section with t = 3−2 , we then get


247

Z ∞ Z ∞ 1 4 2 2 Tr φ(D /3 ) ∼ λφ(λ) dλ + b2 (D )3 φ(λ) dλ rk E 3 16π 2 0 0 X + (−1)m φ(m) (0) b2m+4 (D2 ) 3−2m as t ↓ 0, 2

2

m≥0

where (−1)m b2m+4 (D2 ) = 16π 2 µm (D2 )/m! are suitably normalized, integrated moment terms of the spectral density of D2 . Thus, we arrive at (1.1). We finally take stock of the status of the Chamseddine–Connes development. If φ ∈ S, then the development becomes a bona fide asymptotic expansion. However, if one wishes to use (for instance) the counting function ND2 (λ ≤ 32 ), which does not lie in S, then the present formulae are not directly applicable and one must proceed like in Sect. 5; moreover the expansion beyond the first piece is only valid in the Cesàro sense. We close by noting that third piece of the Chamseddine–Connes Lagrangian has interesting conformal properties; this is better studied through the corresponding zeta function at the origin [35]. That term is definitely not a Wodzicki residue but a moment; as to whether this fact has any physical significance, the present authors are not of one mind. Acknowledgement. Heartfelt thanks to S. A. Fulling for sharing his ideas with us prior to the publication of [13]. We wish to thank M. Asorey, E. Elizalde, H. Figueroa, D. Kastler, F. Lizzi, C. P. Mart´ın, A. Rivero, T. Schücker and J. Sesma for fruitful discussions and G. Landi for a question that motivated the paragraph on harmonic oscillators in Sect. 4. JMGB and JCV acknowledge support from the Universidad de Costa Rica; JMGB also thanks the Departamento de F´ısica Teórica de la Universidad de Zaragoza and JCV the Centre de Physique Théorique (CNRS–Luminy) for their hospitality.

References

1. Ackermann, T.: A note on the Wodzicki residue. J. Geom. Phys. 20, 404–406 (1996) 2. Alvarez, E., Gracia-Bond´ıa, J. M., Mart´ın, C. P.: A renormalization group analysis of the NCG constraints mtop = 2 mW , mHiggs = 3.14 mW . Phys. Lett. B 329, 259–262 (1994) 3. Branson, T. P., Gilkey, P. B.: Residues of the eta function for an operator of the Dirac type. J. Func. Anal. 108, 47–87 (1992) 4. Brownell, F. H.: Extended asymptotic eigenvalue distributions for bounded domains in n-space. J. Math. Mech. 6, 119–166 (1957) ¨ 5. Carleman, T.: Uber die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen. Berichte Verhandl. Akad. Leipzig 88, 119–132 (1936) 6. Carminati, L., Iochum, B., Kastler, D., Schücker, T.: On Connes’ new gauge principle of general relativity. Can spinors hear the forces of spacetime? Preprint hep-th/9612228 7. Chamseddine, A. H., Connes, A.: Universal formula for noncommutative geometry actions: Unification of gravity and the Standard Model. Phys. Rev. Lett. 77, 4868–4871 (1996) 8. Connes, A.: The action functional in noncommutative geometry. Commun. Math. Phys. 117, 673–683 (1988) 9. Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995) 10. Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–175 (1996) 11. Elizalde, E., Vanzo, V., Zerbini, S.: Zeta-function regularization, the multiplicative anomaly and the Wodzicki residue. Preprint hep-th/9701160 12. Estrada, R.: The Cesàro behaviour of distributions. Preprint: San José 1996 13. Estrada, R., Fulling, S. A.: The asymptotic expansion of spectral functions. Preprint: College Station 1997

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14. Estrada, R., Kanwal, R. P.: Regularization, pseudofunction and Hadamard finite part. J. Math. Anal. Appl. 141, 195–207 (1989) 15. Estrada, R., Kanwal, R. P.: Asymptotic Analysis: a Distributional Approach. Boston: Birkhäuser 1994 16. Figueroa, H.: Function algebras under the twisted product. Bol. Soc. Paran. Mat. 11, 115–129 (1990) 17. Fulling, S. A.: The local geometric asymptotics of continuum eigenfunction expansions. I. Overview. SIAM J. Math. Anal. 13, 891–912 (1982) 18. Gelfand, I. M., Shilov, G. E.: Generalized Functions I. New York: Academic Press, 1964 19. Gilkey, P. B.: Invariance Theory, the Heat Equation and the Atiyah–Singer Theorem, 2nd edition. Boca Raton: CRC Press, 1995 20. Grossmann, A., Loupias, G., Stein, E. M.: An algebra of pseudodifferential operators and quantum mechanics in phase space. Ann. Inst. Fourier (Grenoble) 18, 343–368 (1968) 21. Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems. Boston: Birkhäuser, 1986 22. Gurarie, D.: The inverse spectral problem. In: Forty More Years of Ramifications: Spectral Asymptotics and its Applications, Fulling, S. A., Narcowich, F. J., eds., Texas A&M University, College Station (1991), pp. 77–99 23. Hardy, G. H.: Divergent Series. Oxford: Clarendon Press, 1949 24. Hörmander, L.: The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968) 25. Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Berlin: Springer, 1985 26. Iochum, B., Kastler, D., Schücker, T.: On the universal Chamseddine–Connes action. I. Details of the action computation. Preprint hep-th/9607158 27. Kalau, W., Walze, M.: Gravity, noncommutative geometry and the Wodzicki residue. J. Geom. Phys. 16, 327–344 (1995) 28. Kastler, D.: The Dirac operator and gravitation. Commun. Math. Phys. 166, 633–643 (1995) 29. Lagrange, J.-L.: Nouvelle méthode pour résoudre les e´ quations littérales par la moyen des séries. Mém. Acad. Royale des Sciences et Belles-lettres de Berlin 24, 251–326 (1770) 30. Landi, G.: An introduction to noncommutative spaces and their geometry. Preprint hep-th/9701078 31. Łojasiewicz, S.: Sur le valeur et la limite d’une distribution en un point. Studia Math. 16, 1–36 (1957) 32. Mart´ın, C. P., Gracia-Bond´ıa, J. M., Várilly, J. C.: The Standard Model as a noncommutative geometry: The low energy regime. Preprint hep-th/9605001, Phys. Rep., in press 33. McKean, H. P., Singer, I. M.: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom. 1, 43–69 (1967) P∞ 1 2 (2n + 1)e−σ(n+ 2 ) . Proc. Camb. Philos. Soc. 24, 34. Mulholland, H. P.: An asymptotic expansion for 0 280–289 (1928) 35. Rosenberg, S.: The Laplacian on a Riemannian manifold. Cambridge: Cambridge University Press 1997 36. Solomyak, M. Z.: Asymptotics of the spectrum of the Schrödinger operator with nonregular homogeneous potential. Math. USSR Sbornik 55, 19–37 (1986) 37. Várilly, J. C., Gracia-Bond´ıa, J. M.: Connes’ noncommutative differential geometry and the Standard Model. J. Geom. Phys. 12, 223–301 (1993) 38. Wodzicki, M.: Noncommutative residue I: Fundamentals. In Manin, Yu. I. (ed.) K-theory, Arithmetic and Geometry. Lecture Notes in Mathematics Vol. 1289, Berlin: Springer, 1987, pp. 320–399 Communicated by A. Connes

Commun. Math. Phys. 191, 249 – 264 (1998)

Communications in


Enriques Surfaces, Analytic Discriminants, and Borcherds’s 8 Function Jay Jorgenson1,? , Andrey Todorov2 1

School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA Department of Mathematics, University of California, Santa Cruz, CA 95064, USA, and Institute of Mathematics, Bulgarian Academy of Sciences

2

Received: 24 July 1995 / Accepted: 21 March 1997

Abstract: In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form 8 on the moduli space of marked, polarized Enriques surface of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. In [JT 94] and [JT 96], we defined and studied a meromorphic (multi-valued) modular form of weight 2, which we call the K3 analytic discriminant, on the moduli spaceQof marked, polarized, K3 surfaces of degree 2d; in certain cases, including when d = pk , where pk are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a determinant of the Laplacian on a polarized K3 surface with respect to the Calabi-Yau metric together with the L2 norm of the image of the period map with respect to a properly scaled holomorphic two form. Since the universal cover of any Enriques surface is a K3 surface, we can restrict the K3 analytic discriminant to the moduli space of degree 2 Enriques surfaces. The main result of this paper is the observation that the square of our degree 2 analytic discriminant, viewed as a function on the moduli space of degree 2 Enriques surfaces, is equal to the Borcherd’s 8 function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Moody algebras and elliptic curves, and suggests further connections with higher dimensional Calabi-Yau varieties, specifically those which can be realized as complete intersections in some, possibly weighted, projective space. 1. Introduction Let E be an elliptic curve over C, viewed as the complex plane C modulo the Z lattice generated by 1 and τ where τ = a + ib with b > 0. Let det ∆E be the zeta regularized product of the non-zero eigenvalues of the Laplacian relative to the unit area flat metric on E which acts on the space of smooth functions on E, and let dz be the canonical ?

Partially supported through a Sloan Fellowship.

250

J. Jorgenson, A. Todorov

holomorphic one form on E. By a direct calculation, which is essentially Kronecker’s first limit formula, it can be shown that det ∆E = |η(τ )|4 , kdzk2L2 where η(τ ) is the Dedekind eta function, defined by the infinite product η(τ )24 = qτ

∞ Y

1 − qτn

24

with qτ = e2πiτ .

n=1

One of the many fascinating features of this formula is the fact that the spectral invariant det ∆E and the L2 norm of the holomorphic one form dz can be used to obtain the Dedekind eta function η(τ ), which has numerous algebraic, among other, properties. In [JT 94] and [JT 96], we used the above realization of the Dedekind eta function as a guide in order to define and study a meromorphic form on the moduli space of marked, polarized K3 surfaces. Briefly, our construction is as follows. Let (X, e) be a polarized, algebraic K3 surface of degree 2d equipped with its canonical Calabi-Yau metric which is compatible with the polarization e and which gives X total volume one. Let {ω} be a meromorphic family of holomorphic two forms on MdK3,mpa , the moduli space of marked, polarized, algebraic K3 surfaces. Let det ∆(X,e) denote the zeta regularized product of the non-zero eigenvalues of the Laplacian which acts on the space of smooth functions on X. In [JT 94] it was shown that there exists a meromorphic (multi-valued) function fK3,ω,2d on MdK3,mpa such that (det ∆(X,e) )2 = |fK3,ω,2d |2 . kωk2L2 The moduli space MdK3,mpa is a Zariski open set in h2,19 , the symmetric space associated Nd to the group SO0 (2, 19), and there is an integer Nd such that the function fK3,ω,2d extends to a meromorphic form on h2,19 with respect to a certain arithmetic subgroup which depends on d (and the index of the abelianization of this arithmetic subgroup). Nd vanishes on h2,19 \ MdK3,mpa and possibly has additional zeros and The form fK3,ω,2d poles coming from the family of forms {ω}. In certain cases, incloding when d = 2, it is shown in [JT 94] that one can construct a normalized family of forms {ωnor } such that Nd has divisor whose support is contained in h2,19 \ MdK3,mpa . fK3,ω nor ,2d In this article, we shall prove that in the case d = 2, there is a particular choice of 2 when restricted to a certain subforms {ωnor } such that our analytic discriminant fK3,ω,4 2 variety of h2,19 satisfies a product formula. More specifically, we shall consider fK3,ω nor ,4 when restricted to the subvariety of marked, polarized, algebraic K3 surfaces of degree 2 which admit a fixed point free involution which is compatible with the polarization 2 as a function on the moduli space of and marking. Equivalently, we will view fK3,ω nor ,4 marked, polarized Enriques surfaces of degree 2. Using results from [JT 94] and [JT 95], 2 satisfies a few basic characterizing properties. In the recent we will show that fK3,ω nor ,4 article [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form 8 on the moduli space of marked, polarized Enriques surfaces of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. Borcherds’s 8

Enriques Surfaces, Analytic Discriminants, and Borcherds’s 8 Function

251

function is defined through a product formula which immediately extends to the symmetric space h2,10 associated to the group SO0 (2, 10), in which the moduli space of marked, polarized, Enriques surfaces of degree 2 is a Zariski dense subvariety. Our main 2 = c4 8. In other words, the result is that there exists a constant c4 such that fK3,ω nor ,4 holomorphic form which we construct by considering the spectral theory on marked, polarized Enriques surfaces of degree 2 coincides, up to a universal multiplicative constant, with the function constructed by Borcherds in his study of the fake monster Lie algebra. The contents of this article are as follows. In Sect. 2, we state the basic properties from the study of K3 surfaces and Enriques surfaces which we will need in our work. In Sect. 3, we will recall the construction of the analytic discriminant associated to a marked, polarized K3 surface of degree 2d, and then we show how the analytic discriminant can be used to construct a meromorphic modular form on the moduli space of marked, polarized Enriques surfaces of degree d. Finally, in Sect. 4, we shall recall results from [Bor 96] and prove that the Borcherds’s 8 function is related to our analytic discriminant in the case d = 2. In Sect. 5 we study our analytic discriminant for general more degree d polarized Enriques surfaces, and we relate different degree Enriques discriminants. 2. Basic Properties of K3 and Enriques Surfaces Let us review some basic properties of K3 surfaces and Enriques surfaces. For a more general and complete discussion, the reader is referred to [Ast 85]. A K3 surface X is a compact, complex two dimensional manifold with the following properties. a) There exists a non-zero holomorphic two form ω. b) H 1 (X, OX ) = 0. For the purposes of this article, we will assume that all surfaces are projective varieties. From the defining properties, one can prove that the canonical bundle on X is trivial. In [Sh 67], the following topological properties of K3 surfaces are proved. The surface X is simply connected, and the homology group H2 (X, Z) is a torsion free abelian group of rank 22. The intersection form h , i on H2 (X, Z) has the properties: a) hu, ui = 0 mod (2); b) det(hei , ej i) = −1, where {ei } is a basis of H2 (X, Z); c) the symmetric form h , i has signature (3,19). Theorem 5 from p. 54 of [Ser 73] implies that as an Euclidean lattice H2 (X, Z) is isomorphic to the K3 lattice 3K3 , where H2 (X, Z) ∼ = 3K3 = H3 ⊕ (−E8 )2 with

h H=

0 1

1 0

i3

being the hyperbolic lattice. Let α = {αi } be a basis of H2 (X, Z) with intersection matrix 3K3 . The pair (X, α) is called a marked K3 surface. Let e ∈ H 1,1 (X, R) be the class of a hyperplane section. The triple (X, α, e) is called a marked, polarized K3 surface. The degree of the polarization is the integer 2d such that he, ei = 2d. From [PSS 71] and [Ku 77] we have that the moduli space of isomorphism classes of marked,

252


polarized, algebraic K3 surfaces of a fixed degree is equal to an open dense set in the symmetric space h2,19 = SO0 (2, 19)/[SO(2) × SO(19)]. Let

0K3,d = {φ ∈ Aut(H 2 (X, Z)) | hφ(u), φ(v)i = hu, vi and φ(e) = e}.

The moduli space of isomorphism classes of polarized, algebraic K3 surfaces of a fixed degree 2d, which we denote by MdK3,pa , is isomorphic to a Zariski open set in the quasiprojective variety 0K3,d \h2,19 . If we allow our surfaces to have singularities which are at most double rational points, then the corresponding moduli space of isomorphism classes of marked, polarized, algebraic surfaces is equal to the entire symmetric space h2,19 . In other words, marked and polarized surfaces corresponding to points in the space h2,19 \ MdK3,mpa are those surfaces for which the projective embedding corresponding to any power of the polarization is singular with singularities which are double rational points. Specifically, the relation between MdK3,mpa and h2,19 is through the period map, which we now describe. The period map π for a marked K3 surface (X, α) is defined by integrating the holomorphic two form ω along the basis α of H2 (X, Z), meaning Z π(X, α) = (. . . , ω, . . .) ∈ P21 . αi

The Riemann bilinear relations hold for π(X, α), meaning hπ(X, α), π(X, α)i = 0 and hπ(X, α), π(X, α)i > 0. Choose a primitive vector e ∈ H2 (X, Z) such that he, ei = 2d, for some integer d. As in [PSS 71], one has the description of h2,19 as ¯ > 0 and hu, ei = 0}. h2,19 = {u ∈ P(H2 (X, Z) ⊗ C) : hu, ui = 0, hu, ui Results from [PSS 71] and [Ku 77] combine to prove that the period map is a surjection d d → MK3,pa be the natural map onto a Zariski open set in h2,19 . Let πmar,d : MK3,mpa which forgets the marking. From the surjectivity of the period map, it follows that πmar,d d coincides with the action of 0K3,d on MK3,mpa . d The set h2,19 \ MK3,mpa can be described as follows: Given a polarization class e ∈ 3K3 , set Te to be the orthogonal complement to e in 3K3 (i.e., Te is the transcendental lattice), so then we have the realization of h2,19 as one of the components of ¯ > 0}. {u ∈ P(Te ⊗ C) : hu, ui = 0 and hu, ui Define the set 1(e) = {δ ∈ 3K3 : he, δi = 0 and hδ, δi = −2}, and, for each δ ∈ 1(e), define the hyperplane H(δ) = {u ∈ P(Te ⊗ C) : hu, δi = 0}. Let

He =

[ δ∈1(e)

H(δ) ∩ h2,19 ,


and set

253

De = 0K3,d \He .

Results from [Ma 72], [Si 83], [Si 84] and [To 80], to name a few references, combine to give the relation DK3,e = 0K3,d \h2,19 \ MdK3,pa . Geometrically, points in the space DK3,e correspond to algebraic K3 surfaces with a class e which is pseudo-ample, meaning a divisor class for which a sufficently large multiple of the class yields a projective embedding of X with double rational singularities (see [Ma 72]). In Sect. 3, we shall show that for certain d, the divisor DK3,e is an ample divisor on the quasi-projective variety 0K3,d \h2,19 . An Enriques surface Y is a compact, complex surface for which the canonical class KY is not trivial but KY⊗2 is trivial, and H 1 (Y, OY ) = 0. Using the exponential cohomology sequence together with topological considerations, it can be shown that all Enriques surfaces are projective (in contrast to the case of K3 surfaces). In particular, as stated on p. 270 of [BPV 84], one can show that the map Pic(Y ) → H2 (Y, Z) is an isomorphism. As in the case of K3 surfaces, the structure of the homology group H2 (Y, Z) can be canonically determined. The intersection form h , i on H2 (Y, Z) is such that as an Euclidean lattice H2 (Y, Z) satisfies the (non-canonical) isomorphism H2 (Y, Z) ∼ = 3Enr ⊕ Z/2Z, where 3Enr is the Enriques lattice 3Enr = H ⊕ (−E8 ). Define H2 (Y, Z)f = H2 (Y, Z)/Tor(H2 (Y, Z)), where Tor(H2 (Y, Z)) denotes the torsion subgroup of H2 (Y, Z), so then we have the (non-canonical) isomorphism H2 (Y, Z)f ∼ = 3Enr . In particular, H2 (Y, Z)f is a torsion free abelian group of rank 10 and signature (1, 9). Let α = {αi } be a basis of H2 (Y, Z)f with intersection matrix 3Enr . The pair (Y, α) is called a marked Enriques surface. An H-marked Enriques surface is a pair (Y, η), where Y is an Enriques surface and η is an embedding of the lattice H into 3Enr . As shown in [Dol 84], every Enriques surface admits an H-marking. The two-torsion subgroup of H2 (Y, Z) implies the existence of an unramified degree two cover of Y which, by Noether’s formula, can be shown to be a K3 surface. Therefore, many aspects in the study of Enriques surfaces can be derived from the study of K3 surfaces. In particular, following the discussion in [BPV 84] and [Na 85], we have the following Torelli theorem for Enriques surfaces. Let % be the involution on the K3 lattice 3K3 defined by %(z1 ⊕ z2 ⊕ z3 ⊕ x ⊕ y) = (−z1 ⊕ z3 ⊕ z2 ⊕ y ⊕ x). Let 3+K3 and 3− K3 be the %-invariant and %-anti-invariant subspaces, respectively. Observe that the unimodular lattice 21 3+K3 is isometric to the Enriques lattice 3Enr . The class represented by the holomorphic two-form ω is anti-invariant; this follows from the fact that the canonical class of an Enriques surface is not trivial. Therefore,

254


if γ ∈ 3+K3 then h[ω], γi = 0. Let X be a K3 surface which admits a fixed point free involution σ, and let Y = X/hσi be the quotient Enriques surface. Lemma 19.1 of [BPV 84] proves the existence of an isometry φ : H2 (X, Z) → 3K3 such that φ ◦ σ ∗ = % ◦ φ. In other words, one can choose a marking of the K3 surface X which is consistent with a marking of the underlying Enriques surface Y . Therefore, there is a one to one correspondence between marked Enriques surfaces and pairs (Y, φ), where Y is an Enriques surface and φ is as above. Since every Enriques surface is projective, we define a marked, polarized Enriques surface of degree d to be the triple (Y, φ, e), where (Y, φ) is a marked Enriques surface and e is the class of an ample divisor with he, eiY = d. Equivalently, one can choose the class of an ample divisor e˜ on X which is invariant under the involution σ and with he, ˜ ei ˜ X = 2d, since σ induces a map σ ∗ : H2 (Y, Z)f → H2 (X, Z) which satisfies the property hσ ∗ (x), σ ∗ (y)iX = 2hx, yiY (see p. 203 of [Na 85]). There is a one to one correspondence between marked, polarized Enriques surfaces of degree d and marked, polarized K3 surfaces of degree 2d with fixed point free involution and whose polarization class lies in 3+K3 and is invariant under the involution. For simplicity, we shall say that a marked, polarized K3 surface with the above restrictions on marking and polarization covers the marked, polarized Enriques surface. Define the space Enr = P(3− K3 ⊗ C) ∩ K3 . From p. 282 of [BPV 84], it can be shown that Enr is isomorphic to two copies of h2,10 , the symmetric space associated to the group SO0 (2, 10). Let 0Enr be the discrete group defined by 0Enr = restr3− {g ∈ Aut(3K3 ) : g ◦ % = % ◦ g}. K3

We define the period map of marked, polarized Enriques surfaces to be the restriction of the period map for marked, polarized K3 surfaces which cover marked, polarized Enriques surface with image into 0Enr \Enr . As with the study of K3 surfaces, one can consider a Torelli theorem for marked, polarized Enriques surfaces. For the current considerations, it suffices to study Enriques surfaces with an almost polarization of degree 2, by which we mean a class of a line bundle of self-intersection 2 which has non-negative intersection with the class of any hyperplane section. It is stated on p. 5 of [St 91] that the results from [Ho 78a] and [Ho 78b] imply that any Enriques surface can be endowed with an almost polarization of degree 2. Following [Na 85], one has the following result. Let l ∈ 3− K3 be such that hl, li = −2. It is shown on p. 283 of [BPV 84] such that no point of the hyperplane Hl = {p ∈ Enr : hp, liY = 0} can be the period point of a marked Enriques surface. Further, Theorem 21.4 on p. 286 of [BPV 84] asserts that all points in the variety


0Enr \Enr \

0Enr \

[

255

! Hl

l

occur as period points of Enriques surfaces (with an almost polarization of degree 2). However, as stated on p. 168 of [Dol 84], the period map is a finite to one map, hence additional data associated to any Enriques surface is necessary. Let us denote the above variety by M4K3,% , since the above variety is indeed the moduli space of degree 4 K3 covers of Enriques surfaces, which are necessarily of degree 2. In order to obtain a moduli space of Enriques surfaces, with possible additional data thus yielding a variety which covers M2K3,% , we utilize the following result from [Dol 84]: Let M2Enr,H denote the moduli space of H-marked Enriques surfaces with an almost polarization of degree 2. Then there is a finite index subgroup 0Enr,H of 0Enr such that we have the isomorphism ! [ 2 ∼ M = 0Enr,H \Enr \ 0Enr,H \ Hl . Enr,H

l

Finally, let us note that the set DEnr = 0Enr \

[

! Hl

l

is an irreducible divisor on M2K3,% (see also p. 702 of [Bor 96]). When descending from the moduli space of polarized, algebraic K3 surfaces of degree 4, one can describe the moduli space M2Enr,H as follows. ˜ Enr denote the subgroup of 0Enr which is defined as follows. Let e1 and e2 be Let 0 two vectors which generate a copy of H in 3K3 , i.e. e21 = e22 = 0 and e1 e2 = 1. Set e = e1 + e2 and take the polarization on the K3 surface to be given by two distinct, orthogonal embeddings of e into 3K3 , meaning in the isomorphism 3K3 ∼ = H3 ⊕ (−E8 )2 the polarization is given by the vector E = (0, e, e, 0, 0). Note that the polarization is invariant, meaning ρ(E) = E. Define ˜ Enr = restr − {g ∈ Aut(3K3 ) : g ◦ % = % ◦ g and g(E) = E}. 0 3K3

Then we claim that one has the isomorphism M2Enr,H

∼ ˜ Enr \ Enr \ =0

[

! Hl

,

l

˜ Enr ∼ i.e. 0 = 0Enr,H (see p. 40 of [St 88]). We prefer this description of M2Enr,H since the discussion readily generalizes to the setting of general degree polarization. Equivalently, − the subgroup 0Enr,H of 0Enr can be described as follows. Let 3−,∗ K3 be the dual to 3K3 , so then we have an isomorphism 1 − ∼ 1 H(2)/H(2) ⊕ E8 (−2)/E8 (−2). 3−,∗ K3 /3K3 = 2 2 Then 0K3,H is the subgroup of 0K3 corresponding to those elements which respect the above decomposition into direct summands (see p. 41 of [St 88]). For more discussion of this point, see [Dol 84] or [St 88]. Finally, we note that M2Enr,H is a finite cover of − M2K3,% with covering group 3−,∗ K3 /3K3 .

256


3. Analytic Discriminants for Enriques Surfaces In this section we shall recall the definition of the analytic discriminant associated to any marked, polarized, algebraic K3 surface. The results in this section are proved in complete detail in [JT 94] and [JT 95]. When the given marked, polarized, algebraic K3 surface covers a marked, polarized Enriques surface, we shall relate our K3 discriminant to information constructed from the Enriques surface. From this, we shall define an analytic discriminant for Enriques surfaces. To begin, let us recall the definition of the Weil-Petersson metric on the moduli space MdK3,mp . Let (X, e) be a polarized K3 surface of degree 2d, and let T(X,e) be the sheaf of holomorphic vector fields on (X, e). From Kodaira-Spencer deformation theory, we can of the moduli space of MdK3,mp at the point (X, e) identify the tangent space TMd K3,mp

with H 1 (X, T(X,e) ). The existence of the non-vanishing holomorphic two-form ω on X implies that we can identify H 1 (X, T(X,e) ) with H 1 (X, ), where is the sheaf of to holomorphic one forms on X. One can then deduce that the tangent space TMd K3,mpa

the moduli space MdK3,mpa at the point (X, α, e) can be identified with the space H 1 (X, )0 = {u ∈ H 1 (X, ) | hu, ei = 0}.

We view any φ ∈ H 1 (X, T(X,e) ) as a linear map from 1,0 to 0,1 pointwise on X. Given φ1 and φ2 in H 1 (X, T(X,e) ), the trace of the map φ1 φ2 : 0,1 → 0,1 at a point x ∈ X with respect to the unit volume Calabi-Yau metric g (meaning a Kähler-Einstein metric compatible with the given polarization class e) is simply X nl¯ (φ1 )kl¯ ((φ2 )m Tr(φ1 φ2 )(x) = ¯ . n ¯ )g gkm k,l,m,n

The existence of a Calabi-Yau metric on X compatible with the polarization e is guaranteed by Yau’s theorem [Y 78]. We define the Weil-Petersson metric on MdK3,mpa via the inner product Z hφ1 , φ2 i =

Tr(φ1 φ2 )volg X

d on the tangent space of MK3,mpa at (X, α, e). It is shown in [To 89] that the Weil-Petersson metric on MX,α,e is equal to the restriction of the Bergman metric on h2,19 . Therefore, the Weil-Petersson metric is a Kähler metric with Kähler form µWP . Since h2,19 is simply connected, there exists a non-vanishing holomorphically varying family of holomorphic two-forms over MdK3,mpa . For any such family {ω}, consider the function on MdK3,mpa defined by Z 2 ¯ kωkL2 = hω, ωi = ω ∧ ω. X

In [To 89] and [Ti 88] it was proved that log kωk2L2 is a potential for the Weil-Petersson metric. The following result from [JT 94] proves the existence of a second potential for the Weil-Petersson metric.


257

Proposition 1. Let (X, e) be a polarized, algebraic K3 surface of degree 2d, and let µ denote the unit volume Kähler-Einstein form on X which is compatible with the polarization e. Let {ω} be a non-vanishing, holomorphically varying family of holomorphic two forms on h2,19 . a) Let det ∆(X,e) denote the zeta regularized product of the non-zero eigenvalues of the Laplacian which acts on the space of smooth functions on X. Then (det 1(X,e) )2 c = 0, dd log kωk2L2 or, equivalently −ddc log(det 1(X,e) )2 = −ddc log kωk2L2 = µWP . In other words, − log(det 1(X,e) )2 is a potential for the Weil-Petersson metric on MdK3,mpa . b) There is a holomorphic function (possibly multi-valued) fK3,ω,d on h2,19 which vanishes on the codimension one set h2,19 \ MdK3,mpa such that |fK3,ω,d ([X, α, e])| =

(det 1(X,e) )2 kωk2L2

;

whence fK3,d does not vanish on MdK3,mpa . The reader is referred to [JT 94] and [JT 95] for complete details of the proof of Proposition 1. Important Note: The function fK3,ω,d constructed in Proposition 1(b) is a possibly multi-valued function with divisor contained in h2,19 \ MdK3,mpa . At this point, we do not assert any behavior of fK3,ω,d with respect to the discrete group 0K3,d . For this, one would need to impose further restrictions on the family of forms {ω}, which may require the family of forms to be a meromorphic family, so then fK3,ω,d would have additional zeros and poles corresponding to the divisor of {ω} (see Remark 3). Remark 2. Let ∆(0,1) (X,e) denote the Laplacian which acts on the space of smooth (0, 1) forms relative to the unit volume Calabi-Yau metric which is compatible with the polar2 ization e. Using the Hodge decomposition, it is argued in [JT 95] that ∆(0,1) (X,e) = (∆(X,e) ) . With this in mind, our variational formula [JT 95] proves the following result, which is an analogue of Mumford’s theorem for curves. Let L0 = π∗ KX /Mdmpa and L1 = det R1 π∗ . NK3,d If NK3,d = 0K3,d / 0K3,d , 0K3,d , then the sheaf L20 ⊗ L−1 is trivial. 1 Remark 3. Proposition 1 is a generalization of a known result which exists in the setting of elliptic curves. An analogous result in the setting of Calabi-Yau manifolds was established in [JT 95]. For general degree K3 varieties, one can choose a family of NK3,d is a forms {ωd }, which possibly has zeros or poles, for which the function fK3,ω d ,d meromorphic form of weight 2NK3,d with respect to the action by 0K3,d . The ability NK3,d to choose such an integer NK3,d so that the function fK3,ω is single-valued on h2,19 d ,d follows from the Kodaira embedding theorem as applied to the bundle corresponding to automorphic forms of a particular weight, which is an ample bundle (see [Ba 70] and

258


p. 191 of [GH 78]). Note that the family of forms {ωd } may have additional zeros or NK3,d poles, which will contribute to the divisor of fK3,ω . By Proposition 1(b), we can view d ,d N

K3,d as a section of the line sheaf (π∗ KX /Md )NK3,d . The holomorphic function fK3,ω d ,d K3,pa fK3,ωd ,d will be called the analytic discriminant for the marked, polarized, algebraic K3 surfaces of degree 2d, relative to the family of forms {ωd }. Finally, if one picks N a different family of forms {ω˜ d }, the modular form fK3,K3,d ω ˜ d ,d will differ from the form

N

K3,d by a multiplicative factor which is a rational function on 0K3,d \h2,19 . fK3,ω d ,d

Remark 4. For certain degree d, one can construct a family of forms {ωd } such that NK3,d d the associated analytic discriminant fK3,ω has divisor which is supported on DK3 = d ,d d h2,19 \ MK3,mpa . For example, in the case 0K3,d has one zero dimensional cusp in BailyQ Borel compactification of 0K3,d \h2,19 , which occurs when d = pj , where {pj } is a set of distinct primes (see [Sc 87]), then one can construct a normalized family of forms {ωd } as follows: 1. Any marked, polarized, algebraic K3 surface is an element of a family of K3 surfaces E → D, where D is the unit disc, such that the monodromy has a Jordan cell of dimension 3; i.e., if T is the monodromy operator, on H2 (X, Z), then (T − id)3 = 0 and (T − id)2 6= 0 (see [To 76] and [JT 94] for details). 2. On the generic fibre Xt of this family, we have, up to sign, a unique cycle γ such that T γ = γ and any other T invariant cycle is an integer multiple of γ. Further, there exists a cycle µ such that T µ = γ + µ. 3. Since h2,19 is contractible, there exists a globally defined, non-vanishing, holomorphically varying family of holomorphic two forms, say ωt ∈ H 0 (h2,19 , π∗ KE d

K3,mpa

where KE d

K3,mpa

/h2,19

/h2,19 ),

is the relative canonical sheaf.

4. In [JT 94], it is shown that the function

Z ωt

φ(t) = γ

is non-vanishing on h2,19 . The normalized family of holomorphic two-forms is defined by {ωnor } = {ωt /φ(t)}. We remark that when following the identical steps in the case of elliptic curves, one constructs the family of holomorphic one forms {dz}. When the family {ωnor } exists, it is immediate that the function |fK3,d ([X, α, e])| · kωnor k2L2 = (det ∆(X,e) )2 is 0d invariant since det ∆(X,e) is independent of the marking of (X, e). Also, we follow NK3,d the standard definition of automorphic form, meaning that fK3,d satisfies a multiplicative transformation law with respect to 0d which involves a power of the functional determinant j(Z, g) appearing as a multiplicative factor (see, for example, [Ba 70] or NK3,d is 0d invariant. [Ba 72]). Equivalently, the function fK3,d


259

Let fK3,2 denote the K3 analytic discrminant associated to the normalized family of forms {ωnor } constructed in Remark 4. We now use the function fK3,2 to define a function on M2Enr,mp , hence an analytic discriminant associated to any marked Enriques surface with almost polarization of degree 2. Let (Y, φ, eY ) be any marked, polarized Enriques surface with almost polarization of degree 2, and let (X, α, eX ) be the marked, polarized K3 surface which covers (Y, φ, eY ). We define the degree 2 Enriques analytic discriminant fEnr,2 by fEnr,2 ([Y, φ, eY ]) = fK3,2 ([X, α, eX ]). By Proposition 1(b) and Remark 4, the function fEnr,2 extends to a (multi-valued) holomorphic function on h2,10 (which is one component of Enr which vanishes on DEnr . Remark 5. Using the methods of [Ko 88], one can prove 0Enr,2 / 0Enr,2 , 0Enr,2 has order 4. Indeed, since the Enriques lattice has roots of 2 different norms, its abelianization has at least 2 different elements of order 2, when we conclude 0Enr,2 / 0Enr,2 , 0Enr,2 ∼ = (Z/2Z)2 . 2 Therefore, the function fEnr,2 is a single-valued holomorphic modular form of weight 4 on the quasi-projective variety 0Enr,2 \Enr whose divisor is a positive multiple of the irreducible divisor DEnr .

4. Tube Domains and Borcherds’s 8 Function In this section we shall recall the construction presented in [Bor 96] of a holomorphic modular form on h2,10 relative to 0Enr,E , which is defined to be the image in O(3− K3 ) of the subgroup of O(3K3 ) which consists of those isometries of 3K3 that commute with the involution ρ and leave the degree 4 polarization class E invariant. To begin, we follow the tube domain construction of h2,10 as given in Sect. 1 of [Bor 96]. Let L be the lattice L = −2E8 ⊕ H, so the vector (v, m, n) with v ∈ −2E8 and n, m ∈ Z has norm v 2 + 2mn. In particular, the vectors ρ = (0, 0, 1) where ρ0 = (0, 1, 0) have norm zero. Let M be the lattice M = L ⊕ 2H, and let OM (Z) be the automorphism group of M . Let ¯ M > 0}. M = {w ∈ P(M ⊗ C) : hw, wiM = 0 and hw, wi + The space M has two connected components, and we let OM (Z) be the subgroup of OM (Z) which does not interchange the components. The positive norm vectors of L⊗R form two open cones; choose one cone, √ to be called CL . One of the two components of M can be identified with L ⊗ R + −1CL by the map √ L ⊗ R + −1CL → M ⊗ C

defined by

v 7→ (v, 1, −v 2 /2).

The identification depends on a choice of the cusp, which is equivalent to a choice of a point in the boundary of the moduli space of K3 surfaces corresponding to the

260


deepest degeneration of polarized K3 surfaces. Any automorphism of M ⊗ R induces the components an automorphism of M , and if the automorphism does not interchange √ of M , then there is an induced automorphism of L ⊗ R + −1CL . Therefore, as in Theorem 5.3 of [JT 96], we have an isomorphism √ + (Z)\(L ⊗ R + −1CL ) → 0Enr,E \h2,10 . π : OM If d is a vector of M of norm −2, let Hd be the set of points of M which are orthogonal to d. Then results from [BPV 84] and [Na 85] show that ! [ −1 ∼ Hd . π (DEnr ) = OM (Z)\ d

With this, we now have an explicit construction of the moduli space of Enriques surfaces as a tube domain, and we have identified the divisor DEnr in this tube domain. We shall view our Enriques discriminant as a function on this tube domain. We now recall Borcherds’s construction of an automorphic form 8 on the moduli space of Enriques surfaces. Definition 6. Let W be the reflection group √ of the lattice L generated by the reflections of vectors of norm −2. For y ∈ L ⊗ R + −1CL , define (−1)n 8 Y X 1 − e2πinhρ,w(y)iL det(w)e2πihρ,w(y)iL . 8(y) = n>0

w∈W

As stated in [Bor 96], results from [Bor 92] show that the function 8(y) satisfies the following product formula. Let 1+ be the set of vectors of L which have positive inner product with ρ or are positive multiples of ρ. Let {c(n)} be the coefficients of the function ∞ X c(n)qτn = η(τ )−8 η(2τ )8 η(4τ )−8 . f (τ ) = n=−1

Then for all y with Im(y) sufficiently large, we have 8(y) = e2πihρ,yiL

Y

1 − e2πihr,yiL

(−1)hr,ρ+ρ0 iL ·c(hr,riL /2)

.

r∈1+

In other language, Borcherds studies the above functions by considering the twisted denominator formula for an automorphism of the monster Lie algebra coming from an involution of the Leech lattice with an 8-dimensional fixed subspace. The group W is the Weyl group of the fake monster Lie superalgebra, ρ is the Weyl vector, and 1+ is the set of positive roots. We refer the reader to [Bor 92] and [Bor 96] for further discussion of these notions. For now, let us quote the following result from [Bor 96]. Proposition 7. The holomorphic function 8(y) is an automorphic form of weight 4 on √ + (Z). the tube domain L ⊗ R + −1CL with respect to a finite index subgroup 0 of OM The function 8 vanishes to first order along the divisor ! [ Hd D8 = 0\ d

and is non-zero elsewhere on the quasi-projective variety 0\(L ⊗ R +

√

−1CL ).


261

The main result of this paper is the following theorem which relates the Borcherds’s 8 function to the analytic discriminant for Enriques surfaces. Theorem 8. Let fEnr,2 be the analytic discriminant for marked, polarized Enriques surfaces with an almost polarization of degree 2, and let 8 be the Borcherds’s 8 function. 2 . Then there is a constant c2 such that 8 = c2 fEnr,2 Proof. Let N be the order of the finite group 0Enr,E / 0Enr,E , 0Enr,E N 2 and consider the quotient fEnr,2 /8 . By Proposition 1 and Proposition 7, the quotient √ is a modular function on the quasi-projective variety 0\(L ⊗ R + −1CL ). Indeed, both forms are sections of the bundle π∗ KM2Enr /h2,10 ⊗ Lχ , where Lχ is the flat bundle corresponding χ ∈ H 1 (M2Enr , Z) which is a finite group isomorphic to the character to 0Enr,E / 0Enr,E , 0Enr,E . Furthermore, both sections vanish on the same irreducible divisor (see [Na 85] and [Bor 96]), so the only possible pole of the quotient lies in the cusp of the Baily-Borel compactification. By Koecher’s principle, any such modular function must be constant, thus concluding the proof. Remark 9. Borcherds used the 8 function to show that the moduli space M2Enr is a quasi-affine variety (see [Bor 96]). We remark here that the realization of M2Enr as a quasi-affine variety also follows from our study of analytic K3 discriminant fK3,2 (see [JT 94] and Sect. 4 of [JT 95]). Specifically, the existence of the normalized family of forms {ωnor } on M2K3,pa which varies holomorpically and does not vanish then implies 2 . that the associated K3 analytic discriminant has divisor which is supported on DK3 2 Therefore, by arguing as in [Bor 96], one concludes that the moduli space MEnr is quasi-affine. 5. Degree d Analytic Discriminants for K3 Covers of Enriques Surfaces In this section we shall compare degree d analytic discriminants when restricted to covers of Enriques surfaces, i.e. marked K3 covers of Enriques surfaces. To compare with the notation of previous sections, we are considering the restriction of the analytic K3 discriminants to the analogue of the space M2K3,% , which is the moduli of degree 2 polarized K3 covers of Enriques surfaces, rather than the space M2Enr,H , which is the moduli of H-marked degree 2 Enriques surfaces. Consider the moduli space of marked, polarized K3 surfaces of degree 2d which admit a fixed point free involution % which preserves the polarization class, i.e., the moduli space of marked, degree d covers of Enriques surfaces. Denote this space by MdK3,%,m . The argument on p. 283 of [BPV 84] applies to show that for any l ∈ 3− K3 such that hl, liY = −2, then no point of the hyperplane Hl = {p ∈ Enr : hp, liY = 0} can be the period point of an Enriques surface of any degree. Therefore, the Torelli theorem for K3 surfaces of degree 2d asserts that one has the isomorphism [ MdK3,%,m = h2,10 \ Hl . l

262


Let GdK3,%,e be the image in O(3− K3 ) of the subgroup of O(3K3 ) which consists of isometries of 3K3 which commute with the involution ρ and which leave the polarization class e invariant. Then MdK3,% = GdEnr,%,e \MdK3,%,m . Let fK3,ω,d be a K3 analytic discriminant associated to the family of forms {ω} such that fK3,ω,d is a meromorphicmodular form with respect to same finite index subgroup of GK3,%,e (see Remark 3). Let fEnr,ω,d be the restriction of the K3 analytic discriminant to the subvariety of K3 surfaces which cover Enriques surfaces, and view fEnr,ω,d as a function on MdK3,% . With this, we have the following result. Theorem 10. Let fEnr,ωd1 ,d1 and fEnr,ωd2 ,d2 be analytic discriminants for marked, polarized Enriques surfaces of degrees d1 and d2 , respectively, viewed as meromorphic functions on h2,10 . Then there is a quasi-projective variety Z which is a finite degree 1 2 and MdK3,% , and integer N , and a rational function h on Z such cover of both MdK3,% that N

fEnr,ωd1 ,d1 /fEnr,ωd2 ,d2

= h.

Proof. Both analytic discriminants fEnr,ωd1 ,d1 and fEnr,ωd2 ,d2 can be viewed as meromor1 2 phic functions on h2,10 . Since the groups GdK3,% and GdK3,% are arithmetic, the subgroup 1 2 ∩ GdK3,% 0K3,%,d1 ,d2 = GdK3,% 1 2 is of finite index in both GdK3,% and GdK3,% . Therefore, we can view fEnr,ωd1 ,d1 and fEnr,ωd2 ,d2 as meromorphic forms of the same weight on the quotient space

Z = 0K3,%,d1 ,d2 \h2,10 . Indeed, both discriminants are sections of the bundle π∗ KZ/h2,10 ⊗ Lχ , where Lχ is the flat bundle corresponding to the character χ ∈ H 1 (Z, Z) which is a finite group isomorphic to 0K3,%,d1 ,d2 /[0K3,%,d1 ,d2 , 0K3,%,d1 ,d2 ]. By choosing N so that χN is the trivial character, the proof of the theorem is complete. 6. Concluding remarks One of the interesting aspects of Theorem 8 is that we have now connected two seemingly different fields of investigation: One field being the study of analytic discriminants and the construction of holomorphic functions via spectral invariants; the other field being Borcherds’s theory of fake monster Lie superalgebras and “monstrous moonshine”. Further study of this connection, as well as related questions, is certainly warranted. Along this line, we offer the following speculation which extends the above observations into the setting of K3 surfaces. Let (X, e) be a polarized, algebraic K3 surface. In [JT 94], we show the existence of three cycles γ0 , γ1 and γ2 such that the following holds. If (X, e) is a family of degenerating K3 surfaces with monodromy operator T for which (T − id)2 6= 0 yet (T − id)3 = 0, then


263

1 T (γ0 ) = γ0 T (γ1 + γ0 ) = γ1 + γ0 T (γ2 + γ1 + γ0 ) = γ2 + γ1 + γ0 . 2 Define He = {u ∈ H2 (X, Z) : hu, ei = 0} and set 3K3,e = {u ∈ He : hu, γ0 i = hu, γ2 i = 0}. One can show that the cup product restricts to 3K3,e to an inner product of signature (1, 18). Let 1K3,e = {` ∈ 3K3,e : h`, ì = −2}, and write 1K3,e = 1+K3,e ∪ (−1+K3,e ) (see [JT 96]). Define B(3K3,e ) as the Lie algebra associated to the Cartan matrix formed from 1+K3 . With all this, we speculate the existence of a constant cd and integer Nd such Nd is equal to the denominator of the Weyl character formula associated to the that cd fK3,d algebra B(3K3 ). Given the results from [JT 97] and Theorem 8 above, we can now speculate the existence of generalized Kac-Moody algebras associated to any polarized Calabi-Yau variety such that an associated projective embedding realizes the variety as a complete intersection. With such a construction, one can then study the generalization of Borcherds’s 8 function and the possibility of extending Theorem 8 to the setting of general complete intersections. Acknowledgement. Both authors benefited greatly through support from NSF grants and from the Institute for Advanced Study. Both authors thank the referee for comments which improved the exposition of the paper and for pointing out a gap in a previous version. We are grateful to V. Nikulin for writing the article [Ni 96], which pointed out that we omitted an important point in our definition of the K3 analytic discriminant, thus leading to a correction in the case when the degree of the K3 surface is such that the corresponding moduli space has more than a single zero dimensional boundary component in its Baily-Borel compactification (see [Sc 87]). Further ongoing work due to Gritsenko and Nikulin are related to results in this and a subsequent article by the authors.

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[JT 94]

Communicated by S.-T. Yau

Commun. Math. Phys. 191, 265 – 282 (1998)

Communications in


Associativity of Quantum Multiplication Gang Liu Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA Received: 17 November 1994 / Accepted: 8 May 1997

Abstract: We proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold V for which c1 (A) > 1 for any effective class A ∈ H2 (V ). The same proof also works for any positive compact symplectic manifold with c1 (A) > 1.

1. Introduction This paper is a short version of my dissertation on Quantum cohomology. Quantum cohomology was introduced by the physicist C.Vafa for compact Kahler manifolds (cf. [V]). Recently several efforts have been made to build up a mathematical foundation for it in the setting of symplectic geometry by using the Gromov-Witten invariant (cf. [MS1, RT1, RT2]). The purpose of this paper is to give a proof of the associativity of this new multiplication for a monotonic compact symplectic manifold V for which, c1 (A) > 1 for every effective class A ∈ H2 (V ). A symplectic manifold (V, ω) is called monotonic, if there exists a constant λ > 0 such that for any f ∈ π2 (V ), ω(f ) = λc1 (f ∗ T V ). By rescaling the symplectic form ω, we may assume that λ = 1. Quantum cohomology QH ∗ (V ) of a monotonic symplectic manifold is additively just the usual cohomology of V with coefficients in R[q], the polynomial ring in q. However, its multiplicative structure is a certain deformation of the ordinary cup-product which can be described as follows. If a01 , a02 , a03 are three cocycles in V with Poincaré duals a1 , a2 , a3 of cycles respectively, then the quantum multiplication a01 ∗a02 can be defined by specifying with all a03 . This triple index can be defined P its pairing 0 0 0 ω(f ) sign(f )q , where f is running through all discrete as follows: < a1 ∗ a2 , a3 >= f

J-holomorphic sphere f : (S 2 , 0, 1, ∞) → (V, a1 , a2 , a3 ). Intuitively this means that we are counting discrete J-holomorphic spheres with three marked points 0, 1, ∞ mapping to the three cycles a1 , a2 , a3 respectively. In order to justify this definition and to see

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what is involved in order to prove the associativity of this multiplication, we need to introduce some basic notations first. Let J (ω) be the set of all ω-compactible almost complex structures for a given symplectic manifold (V, ω). It is well-known that J (ω) is a non-empty contractible set. For a given almost complex structure J ∈ J (ω) and A ∈ H2 (V ), we define the moduli space M(A, J) = {f |f : S 2 −→ V is J − holomorphic and simple, [f ] = A}. It is proved, for example in [G] and [M], that there exists a dense set Jreg of the second category in J (ω) such that for any J ∈ Jreg and A ∈ H2 (V ) the moduli space M(A, J) is a smooth manifold of dimension 2c1 (A) + dim(V ). The p-fold evaluation map eA,J : M(A, J) × (S 2 )p−3 −→ V p is given by: (f, z1 , · · · , zp ) 7→ (f (0), f (1), f (∞), f (z1 ), · · · , f (zp−3 )). Now we can give a formal definition of the quantum multiplication for monotonic symplectic manifolds. Assume that dim(V ) = 2n and that a1 , a2 and a3 are cycles of codimension 2α1 , 2α2 and 2α3 respectively, which are represented by submanifolds of V and which are Poincaré dual to a01 , a02 and a03 of cocycles in H ∗ (V, R) respectively. Fix a generic J. Then the triple index is defined as follows: Definition 1.1. < a01 ∗ a02 , a03 >=

X

ω(A) #(e−1 , A,J (a1 × a2 × a3 ))q

A∈H2 (V )

where the sum is running over all such A that c1 (A) + n = α1 + α2 + α3 . Here eA,J is the 3-fold evaluation map and #(e−1 A,J (a1 × a2 × a3 )) is the oriented intersection number of eA,J and a1 × a2 × a3 . (This makes sense since MA,J , V 3 and a1 × a2 × a3 are oriented). Remark 1.2. (1) Using the Gromov compactness theorem and the monotonicity asω(A) is sumption, one can show that the coefficient #e−1 A,J (a1 × a2 × a3 ) before q finite for generic J. (2) Since α1 + α2 + α3 ≤ 3n, we have ω(A) = c1 (A) ≤ 2n. Therefore the triple index < a01 ∗ a02 , a03 > is in R[q]. (3) The zero order term of a01 ∗ a02 is just the ordinary cup product a01 ∪ a02 . This can be seen as follows: The condition that ω(A) P = c1 (A) = 0 implies that f is a constant map for any f ∈ MA,J . Therefore A,ω(A)=c1 =0 #e−1 A,J (a1 × a2 × a3 ) is nothing but the triple intersection number of a1 , a2 and a3 . According to our definition, in order to know (a01 ∗ a02 ) ∗ a03 one needs first to know what the Poincaré dual (a01 ∗ a02 )0 is. For generic J, consider the 2-fold evaluation map eA,J : M(A, J) → V 2 given by f 7→ (f (0), f (1)). Let M(A, J; a1 , a2 ) = e−1 A,J (a1 × a2 ), which is a smooth manifold of dimension 2(c1 (A) + n − α1 − α2 ). If we define ‘singular cycle’ 8A,J (a1 , a2 ) : M(A, J; a1 , a2 ) → V given by f 7→ f (∞), then it is easy to see that

Associativity of Quantum Multiplication

Lemma 1.3.

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(a01 ∗ a02 )0 =

X

8A,J (a1 , a2 )q ω(A) .

A∈H2 (V )

Let

eA,B,J : M(A, J) × M(B, J) → V 6

be the evaluation map, given by (f, g) 7→ (f (0), f (1), g(0), g(1), f (∞), g(∞)). For two spherical classes A, B ∈ H2 (V ) and cohomology class a0i ∈ H ∗ (V ) with its Poincaré dual ai of codimension 2αi , i = 1, · · · , 4, we define the moduli space of (A, B)-cuspcurves to be M(A, B, J; a1 , a2 ; a3 , a4 ) = e−1 A,B,J (a1 × a2 × a3 × a4 × 1), where 1 is the diagonal. This moduli space is a smooth manifold of dimension 2(c1 (A + B) + n − α1 − α2 − α3 − α4 ) for generic J. It is easy to see that when c1 (A + B) + n = α1 + α2 + α3 + α4 , #M(A, B, J; a1 , a2 ; a3 , a4 ) = #(8A,J (a1 , a2 ) ∩ 8B,J (a3 , a4 )). Lemma 1.4. X

=

< (a01 ∗ a02 ) ∗ a03 , a04 > #(8A,J (a1 , a2 ) ∩ 8B,J (a3 , a4 ))q ω(A+B) ,

A,B∈H2 (V )

the sum being taken over c1 (A + B) + n =

P4

i=1

αi .

Proof. By definition, X

< (a01 ∗ a02 ) ∗ a03 , a04 >=

sign(f )q ω(B) · q ω(A) ,

A,B∈H2 (V ), f ∈M(B,J)

with 8A,J (a1 , a2 ) of (a01 ∗ a02 )0 which is equal to P f (0) being in the component ω(A) , f (1) being in a3 , f (∞) being in a4 and A∈H2 (V ) 8A,J (a1 , a2 )q 2(c1 (B) + n) = codim8A,J (a1 , a2 ) + 2α3 + 2α4 . Geometrically, 8A,J (a1 , a2 ) is just the collection of A-curves intersecting with a1 and a2 at 0 and 1 respectively, and its dimension is 2(c1 (A) + n − α1 − α2 ). Therefore, each such B-curve f which is counted above, determines an A-curve g, so gives rise to an (A, B)-cusp-curve (f, g), which intersects the given 4-cycles and satisfies the condition that c1 (A + B) + n = α1 + α2 + α3 + α4 . In other words, each such f determines an element of M(A, B, J; a1 , a2 ; a3 , a4 ). The conclusion follows immediately. Now for any P ∈ H2 (V ) with c1 (P ) + n = α1 + α2 + α3 + α4 we define the extended Gromov-Witten invariant X 0 #(8A,J (a1 , a2 ) ∩ 8B,J (a3 , a4 )), 9P,J (a1 , a2 ; a3 , a4 ) = A,B∈H2 (V )

the sum being taken over A + B = P. Then Lemma 1.4 can be restated as

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X

< (a01 ∗ a02 ) ∗ a03 , a04 >=

90P,J (a1 , a2 ; a3 , a4 )q ω(P ) ,

P ∈H2 (V )

the sum being taken over c1 (P ) + n =

P4

i=1

αi . Similarly,

< a01 ∗ (a02 ∗ a03 ), a04 >= (−1)2α1 (2α2 +2α3 ) < (a02 ∗ a03 ) ∗ a01 , a04 > X = (−1)2α1 (2α2 +2α3 ) 90 (P, J)(a2 , a3 ; a1 , a4 )q ω(P ) , P ∈H2 (V )

the sum running over such P that c1 (P ) + n = α1 + α2 + α3 + α4 . Therefore, associativity will follow if we can prove that 90P,J (a1 , a2 ; a3 , a4 ) = (−1)2α1 (2α2 +2α3 ) 80P,J (a2 , a3 ; a1 , a4 ). This is equivalent to the fact that 90P,J is graded-commutative. To this end, we will relate 90P,J to the Gromov-Witten invariant 9P,J , which is graded-commutative by its definition. It can be defined as follows: Fix P, ai , i = 1, · · · , 4 as above with c1 (P ) + n = α1 + α2 + α3 + α4 . Consider the 4-fold evaluation map eP,J : M(P, J) × (S 2 − {0, 1, ∞}) → V 4 . Let M(P, J; a1 , a2 , a3 , a4 ) be e−1 P,J (a1 ×a2 ×a3 ×a4 ) which is a smooth submanifold of M(P, J)×(S 2 −{0, 1, ∞}) of dimension 2. Consider the restriction of the projection π2 : M(P, J) × (S 2 − {0, 1, ∞}) → S 2 − {0, 1, ∞} to M(P, J; a1 , a2 , a3 , a4 ) and denote it by π. Picking up a generic point z ∈ S 2 − {0, 1, ∞}, we define 9P,J (a1 , a2 , a3 , a4 ) = #(π −1 (z)). We will use Mz (P, J; a1 , a2 , a3 , a4 ) to denote π −1 (z). It is proved in [MS1] that when c1 (A) > 1, π −1 (z) is finite for generic J and that the Gromov-Witten invariant 9P is well defined, independent of the particular choices of z, J and representatives ai of cycles. It is also easy to see that 9P is graded-commutative. Therefore the associativity will follow if one can prove the following special decomposition rule: Theorem 1.5. If V is monotonic with c1 (A) > 1 for any effective class A ∈ H2 (V ), then we have 9P,J (a1 , a2 , a3 , a4 ) = 90P,J (a1 , a2 ; a3 , a4 ). To prove this, we will construct a family of gluing maps with a gluing parameter z ∈ C∗ , a M(A, B, J; a1 , a2 ; a3 , a4 ) → Mz (P, J; a1 , a2 , a3 , a4 ) #z : A,B∈H2 (V )

when |z| is large enough, where the disjoint union is taken over A+B = P and c1 (P )+n = α1 + α2 + α3 + α4 . The existence of these maps is established in Sect. 3 by using the gluing technique there. We can prove the special decomposition rule by showing that #z is an orientation-preserving bijection when |z| large enough. The injectivity of #z is


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more or less obvious since the domain of #z is a finite set. The surjectivity will follow from the uniqueness part of Lemma 3.9 of the Picard method if one can prove that when |z| is large enough, for any f ∈ Mz (P, J; a1 , a2 , a3 , a4 ) there exists an approximate J-holomorphic curve g = g1 χz g2 ,which is made from a cusp-curve, (g1 , g2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ), such that the C 0 − distance of f and g is small. This can be shown by analyzing what could happen for a sequence fn ∈ Mzn (P, J; a1 , a2 , a3 , a4 ) when |zn | tends to infinity, which is done in Sect. 4. The proof given in this paper was carried out during the fall of 1993 and completed in early 1994 as my dissertation. During the preparation of this paper, we learned that Y. Ruan and G. Tian proved the general decomposition rule, therefore the associativity in [RT1, RT2]. A different proof was given by D. McDuff and D.Salamon in [MS]. Our proof is independent of the above two proofs. 2. Transversality We start with a discussion about the smoothing of S 2 ∨ S 2 and pre-gluing for cusp curves. Let a 1 Si2 = ((C, wi ) (C, wi0 ))/(wi = 0 ), wi i = 1, 2, where w and w0 are the complex coordinates. Then S12 ∨ S22 is given by identifying w10 = 0 in S12 with w20 = 0 in S22 . Let y be the cuspidal point of S12 ∨ S22 , 0L , 1L and 0R , 1R be the points of w1 = 0, 1 in S12 and w2 = 0, 1 in S22 respectively. When |z| ˜ is small enough, the complex sphere S12 #z S22 with gluing parameter z = 2 ∗ z˜ 2 ∈ C and four ‘marked’ points 0L ,1L ,0R ,1R can be constructed as follows: ˜ |z| ˜ 0 2 2 One first cuts off |w10 | ≤ |z| 2 and |w2 | ≤ 2 from S1 and S2 respectively, then glues 2 |z| ˜ | z| ˜ the two remains along 2 < |w10 | < |z| ˜ and 2 < |w20 | < |z| ˜ by the formula w10 ·w20 = z˜2 . It has a ‘left’ and a ‘right’ complex coordinate w1 , w2 respectively with the relation w1 ·w2 = z˜22 . In w1 -coordinate, the points 0L ,1L ,0R ,1R have coordinates w1 = 0, 1, ∞, z˜22 respectively. Therefore the cross-ratio of these four points is z˜22 . Since the cross-ratio is the only invariant for 4-tuples in S 2 under the P SL(2, C) action, we may consider the moduli space Mz (P, J; a1 , a2 , a3 , a4 ) equally as Mz˜ (P, J; a1 , a2 , a3 , a4 ) f is J − holomorphic and simple, = f : S12 #z S22 → V | , f (0L ) ∈ a1 , f (1L ) ∈ a2 , f (0R ) ∈ a3 , f (1R ) ∈ a4 where z = z˜22 . There is another description of S12 #z S22 given by using a cylindrical coordinate, which is what we need in order to do gluing in Sec. 3. We may think of S 2 − {y} as a union of a half sphere Hi and a half infinite cylinder R+ × S 1 with cylindrical

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coordinate (τi , ti ), i = 1, 2, with ∂Hi identified with {0} × S 1 . In this coordinate the previous construction S12 #z S22 will become the following: ˜ + log 2 is cut The part of Si2 − {y}, i = 1, 2, with cylindrical coordinate τ > − log |z| off, and the rest of them are glued along the collars of the length log 2 of the cylinders twisted with an angle arg z˜ 2 . In a cylindrical coordinate, the pre-gluing f1 χz f2 of cusp curve (f1 , f2 ) is defined as follows: Let β be the ‘bump’ function supported in [0, log2 2 ] with β(τ ) being equal to 1 when τ < 0 and β(τ ) being equal to 0 when τ > log2 2 , and βz be the shifting of β by the amount −log|z|, ˜ where z = z˜22 , i.e. βz (τ ) = β(τ − log |z|), ˜ ˜ i = 1, 2. fi (w) if τi (w) < − log |z|, f1 χz f2 (w) = expy (βz (τ1 (w))f˜1 (w) + βz (τ2 (w))f˜2 (w)) otherwise, where (τi , ti ) is the cylindrical coordinate of Si2 − {y} and f˜i is the lifting of fi under the exponential map expy for i = 1, 2. Now we are ready to give the basic analytic setup. Fix p > 2 and A ∈ H2 (V ). Definition 2.1. We define the mapping space p = {f |f : S 2 → V, [f ] = A, kf k1,p < ∞}, B1,A

where kf k1,p is measured by the standard metric on S 2 and some fixed metric on V . By making a suitable choice of a metric on V such that ai is a geodesic submanifold of V for i = 1, · · · , 4, we can similarly define p (z; a1 , a2 , a3 , a4 ) B1,A f : S12 #z S22 → V, kf k1,p < ∞, = f| . f (0L ) ∈ a1 , f (1L ) ∈ a2 , f (0R ) ∈ a3 , f (1R ) ∈ a4 p ’s. To simplify our notation, We will omit the subscript A in B1,A

Definition 2.2. For any f in B1p , the fiber of the tangent bundles T B1p = W1p of B1p at f is W1p (f ) = {ξ|ξ ∈ Lp1 (f ∗ T V )}. Similarly for f in B1p (z; a1 , a2 , a3 , a4 ), the fiber of the tangent bundles of T B1p (z; a1 , a2 , a3 , a4 ) = W1p (z; a1 , a2 , a3 , a4 ) of B1p (z; a1 , a2 , a3 , a4 ) at f is p W 1 (f, z; a1 , a2 , a3 , a4 ) ξ(0L ∈ Tf (0L ) a1 , ξ(1L ) ∈ Tf (1L ) a2 , p ∗ = ξ|ξ ∈ L1 (f T V ), . ξ(0R ) ∈ Tf (0R ) a3 , ξ(1R ∈ Tf (1R ) a4 p p p The bundle L over B1 or B1 (z; a1 , a2 , a3 , a4 ) is defined to be the bundle whose fiber over f is Lp (0,1 (f ∗ T V )). Note that when p > 2, Lp1 (E) ,→ C 0 (E) for any vector bundle E over S 2 . Since in the marked point case the formulae of local charts and trivializations for these Banach manifolds and bundles are similar to the non-marked point case, we will only deal with the later case. Let ι be the injective radius of a fixed metric on V . Consider Uf = {ξ|ξ ∈ W1p (f ), kξk∞ < ι}. Then the maps Expf : Uf → Expf (Uf ) ,→ B1p given by ξ(τ, t) 7→ expf (τ,t) (ξ(τ, t)) form smooth local charts for B1p . Their derivatives DExpf : Uf × W1p → W1p given by


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(Exp(ξ, η))(τ, t) = Dexpf (τ,t) (ξ(τ, t))(η(τ, t)) will give local trivializations for W1p . The local trivializations for Lp1 can be obtained by using a J-invariant parallel transformation coming from a corresponding J-invariant connection which is described in detail, for instance, in [MS], Sect. 3.3. There they also showed that the connection ∇ can be chosen in such a way that T or = 41 N , where N is the O’Neil tensor of J. Now the ∂ J -operator can be thought of as a section of the bundle Lp over Bbp1 given by f 7→ df + J(f ) ◦ df ◦ i. Let ∂ J,f : Uf → Lp be the corresponding non-linear map under the above local charts Expf of W1p and the local trivializations of Lp over Uf , then we have Lemma 2.3. ∂ J,f has the following Taylor expression: ∂ J,f (ξ) = ∂ J (f ) + D∂ J,f (0)ξ + N (ξ), where (1) the first order term E = D∂ J,f (0) : Lp1 (f ∗ T V )) → Lp (0,1 (f ∗ T V ) is given by: 1 E(ξ) = ∇ξ + J(f ) ◦ ∇ξ ◦ i + NJ (∂J (f ), ξ), 4 where the connection ∇ is a J-invariant connection with its torsion proportional to the Nijenhuis tensor NJ . (2) the non-linear part is of the form: N (ξ) = L1 (ξ) ◦ ∇ξ + L2 (ξ) ◦ ∇ξ ◦ i + Q1 (ξ) ◦ du + Q2 (ξ) ◦ du ◦ i, where Li and Qi are linear and quadratic respectively in the sense that there exists a constant C(f ) depending only on f and the ‘geometry’ of V such that kLi (ξ)k∞ ≤ C(f )kξk∞ and kQi (ξ)k∞ ≤ C(f )kξk2∞ respectively for kξk∞ < ι when i = 1, 2. Proof. See [MS], Sect. 4 and [F1], Sect. 2 for the proof of (1) and (2) respectively.

Our next goal in this section is to state results on transversality. In this aspect, the result of [M] about deformation of J-holomorphic curves plays a fundamental role. Following [MS1], we will state it in its linearized form. Lemma 2.4. Given J ∈ J (V, ω), and a J-holomorphic sphere f : S 2 → V, there exists a constant δ such that for every v ∈ Tf (z0 ) V and every pair 0 < ρ < r < δ there exists a smooth vector field ξ(z) ∈ Tf (z) V along f and an infinitesimal almost complex structure Y ∈ C ∞ (End(T V, J, ω)) such that the following hold: (1) Df (ξ) + Y (f ) ◦ du ◦ i = 0, (2) ξ(z0 ) = v, (3) ξ is supported in Br (z0 ) and Y is supported in an arbitrary small neighborhood of f (Br (z0 ) − Bρ (z0 )). Proof. See [MS], Chapter 6.

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In order to state the transversality we need to give some formal definitions about cusp-curves and their evaluation maps. Let Ai ∈ H2 (V ), i = 1, · · · , n, jν ∈ {1, · · · , n − 1}, ν = 2, · · · , n, with jν < ν and D = {A1 , · · · , AN , j1 , · · · , jn }. Then the moduli space of simple cusp-curves of type D is defined as the following: M(D, J) = ( (f, w, z)|f ∈

n Y

) M (Ai , J), w, z ∈ CP n−1 , fjν (wν ) = fν (zν ), ν = 2, · · · , n ,

i=1

where w = (w2 , · · · , wn ) and z = (z2 , · · · , zn ). Note that if Ai = Aj , for some i 6= j in the above definition, we require that fi 6= fj ◦φ for any φ ∈ P SL(2, C). The p-fold evaluation map eD,T,J : M(D, J) × CP p → M p is defined as follows: eD,T,J (f, w, z, m) = (fT (1) (m1 ), · · · , fT (p) (mp )), where T : {1, · · · , p} → {1, · · · , n}. Now we are ready to state the main result on the transversality of eD,T,J . Lemma 2.5. Given P ∈ H2 (V ) and submanifolds a1 , · · · , ap of V in a general position, there exists a dense subset Jreg (ω) of the second category of J (ω) such that for any J ∈ Jreg (ω), eD,T,J is transversal to a1 × · · · × ap for all (D, T ) with c1 (D) = c1 (A1 + A2 + · · · + AN ) ≤ c1 (P ) when restricted to the set of all (f, w, z, m) ∈ M(D, J) × CP p which satisfy the conditions that for any i ∈ {1, · · · , p}, (1) fT (i) (mi ) 6= fT (i) (zT (i) ) and (2) If T (i) = jν for some ν ∈ {2, · · · , n}, fT (i) (mi ) 6= fT (i) (wν ). Proof. Let M(Ai , J ) =

a

M(Ai , J)

J∈J

denote the universal moduli space of class Ai , which is a Banach manifold (see, for example, [MS], Chapter 3). Consider the evaluation map eD,T :

n Y

M(Ai , J ) × CP 2(n−2) × CP p → V 2n−2 × V p

i=1

given by: (f, w, z, m) 7→ (fj2 (w2 ), f2 (z2 ), · · · fjn (wn ), fN (zn ), fT (1) (m1 ), · · · fT (p) (mp )) and the two relevant evaluation maps π1 ◦eD,T and π2 ◦eD,T , where π1 : V 2n−2 ×V p → V 2n−2 and π2 : V 2n−2 × V p → V p are the two projections. From Lemma 3.4, arguing similarly to [MS], we conclude that eD,T is transversal to 1n−1 × (a1 × · · · × ap ) when restricted to the corresponding domain with the same conditions (1) and (2) in the lemma for fixed J, and π1 ◦eD,T and π2 ◦eD,T are transversal to 1n−1 and a1 × · · · × ap respectively. By taking the inverse images of the above three


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submanifolds under the corresponding evaluation maps, which are transversal to them, and then projecting to J , we find a dense subset Jreg (ω) of second category of J (ω) , which has the that for any J ∈ Jreg (ω), the above three evaluation maps, when Qproperty n restricted to 1 M(Ai , J)×CP 2(n−2) ×CP p , are also transversal to the corresponding submanifolds as before. Now M(D, J) = (π1 ◦ eD,T,J )−1 (1n−1 ), and the claim of this lemma is just that π2 ◦ eD,T,J is transversal to a1 × · · · × ap when restricted to the open subset of M(D, J) in the lemma. This follows from the following elementary fact: Let M1 , M2 and M be three manifolds and fi : M → Mi ,i = 1, 2, be smooth maps. Assume that Ni is a submanifold of Mi such that fi is transversal to Ni , i = 1, 2 and (f1 , f2 ) is transversal to N1 × N2 . Let Wi = fi−1 (Ni ) i = 1, 2, and gj = fj |Wi , j 6= i, then gj is transversal to Nj , j = 1, 2. The proof of this fact follows from the corresponding linear algebra lemma by replacing everything above with a linear one. Remark 2.6. For those ‘bad’ points (f, w, z, m) in M(D, J) × CP p , at which at least one of the conditions in the lemma is not satisfied, the above argument is not applicable. Since the two bad cases we need to deal with are similar, we only consider the case where the condition (1) is violated, i.e., we need to consider those points (f, w, z, m) at which fT (i) (zT (i) ) = fT (i) (mi ) for some i ∈ {1, · · · , p}. In this case if we assume, without loss of generality, that i = 1, T (i) = 1, then we can form the (p − 1)-fold evaluation map eD,T 0 ,J from eD,T,J by deleting the T (1) factor, where T 0 : {2, · · · , p} → {2, · · · , n} given by T 0 (i) = T (i), i = 2, · · · , p. Now we can require that for generic J, eD,T 0 ,J is transversal to 1a1 × 1n−1 × (a2 × · · · × ap ) in the proper domain similar to the one in the lemma above. Here 1a1 = 1 ∩ (a1 × a1 ). Proceeding in this way inductively, we can form (p − i)-fold evaluation maps,i = 1, · · · , p, and achieve transversality for generic J. It is easy to see that all the ‘bad ’ points could be covered by an inverse image at some stage. This is what we need in Sect. 4. Now we have special cases of the above lemma. Lemma 2.7. Given P ∈ H2 (V ) and submanifolds ai of V , i = 1, 2, 3, 4, consider all evaluation maps eA,B,J : M(A, J) × M(B, J) → V 6 with A + B = P given by: eA,B,J (φ, ψ) = (φ(0), φ(1), ψ(0), ψ(1), φ(∞), ψ(∞)). Then for generic J, every eA,B,J is transversal to a1 × a2 × a3 × a4 × 1. As its corollary we have Corollary 2.8. If f = (f1 , f2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ), then for generic J Df = (Df1 , Df2 ) : W1p (f, a1 , a2 ; a3 , a4 ) → Lp (0,1 (f ∗ (T V ))) is surjective with kernel of dimension 2(n + c1 (A + B) − α1 − α2 − α3 − α4 ). Here as in Definition 2.2, we write   ξ ∈ Lp1 (f ∗ T V ),     ξ(0 ) ∈ Tf1 (0) a1 , ξ(1L ) ∈ Tf1 (1) a2 , W1p (f, a1 , a2 ; a3 , a4 ) = ξ| L .    ξ(0R ) ∈ Tf2 (0) a3 , ξ(1R ) ∈ Tf2 (1) a4 ,  ξ(∞L ) = ξ(∞R ) ∈ Tf1 (∞) V = Tf2 (∞) V Proof. The proof of the surjectivity of Df is basically the same as the one for Dg when g ∈ M(P, J), which can be found, for example, in [MS], chapter 3. The formula for the dimension of the kernel comes from Lemma 3.7 and the fact that KerDf = Tf M(A, B, J; a1 , a2 , a3 , a4 ). This is what we need in order to do gluing in the next section.

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3. Gluing In this section we will construct a gluing map a M(A, B, J; a1 , a2 ; a3 , a4 ) → Mz (P, J; a1 , a2 , a3 , a4 ) #z : A+B=P

when |z| is large enough, J generic, and c1 (P ) + n = α1 + α2 + α3 + α4 . Given a cusp-curve f = (f1 , f2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ) and zα ∈ C ∗ we will use the following short notations: Sα2 = S12 #zα S22 , fα = f1 χzα f2 , Dfα = Dα = D∂ J,fα (0), Df = D∞ = (Df1 , Df2 ). We will use wα (s, t) or (s, t) with (s, t) ∈ R×S 1 to denote the cylindrical coordinate starting from the ‘middle’ of Sα2 and (τi , ti ) ∈ R+ × S 1 , i = 1, 2 to denote the one with τi = 0 at the boundary of the hemisphere ∂Hi . Now we define the following norms on Lp (fα ) and Lp1 (fα , a1 , a2 , a3 , a4 ), which is essentially due to Floer. R If ξα is in Lp (fα ) or in Lp1 (fα , a1 , a2 ; a3 , a4 ), then we define ξ˜α0 = S 1 ξα ◦ wα (0, t)dt ∈ Tx V , where x = f1 (∞) = f2 (∞). Now we switch to the (τ, t)-coordinate. Fix a bump function β on Sα2 such that β(τi ) = 1 when τi > 1 and β(τi = 0 when τi < 1/2, i = 1, 2. Define ξα0 (τ, t) = Dexpx (f˜α (τ, t))(β(τ )ξ˜α0 ) and ξα1 = ξα − ξα0 , where f˜α is the lifting of fα under expx and ξ˜α0 is considered as a vector field along f˜α . Definition 3.1. Let 0 < ε < 1. For any ηα in Lp (0,1 (fα∗ (T V )) and ξα in W1p (fα , a1 , a2 , a3 , a4 ), we define kηα kχ,0 = kηα k0,p;ε = keετ ηα k0,p , kξα kχ,1 = kξα1 k1,p;ε + |ξα0 | = keετ ξα1 k1,p + |ξα0 |, where |ξα0 | = |ξ˜α0 |. Note that the metric on Sα2 we used in the above definition of k · k’s is the metric induced from the cylindrical coordinate wα (s, t). The main estimate in this section is Lemma 3.2. Given a generic J and a cusp-curve f ∈ M(A, B, J, a1 , a2 ; a3 , a4 ) with c1 (A + B) + n = α1 + α2 + α3 + α4 such that for this J, Lemma 2.7 and its corollary hold so that Df : W1p (f, a1 , a2 ; a3 , a4 ) → Lp (0,1 (f ∗ (T V )) is isomorphic in spherical coordinate. Then there exists a constant C independent of zα such that for |zα | large enough, Dα : W1p (fα , a1 , a2 , a3 , a4 ) → Lp (0,1 (fα∗ (T V )) with the norms above has a uniform inverse Gα such that kGα (η)kχ,1 ≤ Ckηkχ,0 for any η ∈ Lp (0,1 (fα∗ (T V ). Proof. We only need to prove that when |zα | is large enough, there exists a constant C such that kξα kχ,1 ≤ CkDα ξα kχ,0 for any ξα ∈ W1p (fα , a1 , a2 , a3 , a4 ). If this is not true, then there exists a sequence {ξα } ∈ W1p (fα , a1 , a2 , a3 , a4 ) with |zα | → ∞ such that


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(i) kξα kχ,1 = kξα1 k1,p;ε + |ξα0 | = 1, (ii) kDα ξα kχ,0 = kDα ξα k0,p;ε → 0 when α → ∞. We will prove that (i) and (ii) contradict each other. In the proof we will repeatedly use the following fact: Lemma 3.3. Let B be a Banach space with a norm k.kB and f : B → R+ be a convex continuous function. If {xi } is a sequence in B such that xi → x weakly for some x in B, then f (x) ≤ lim inf f (xi ). i

Lemma 3.4. (i) and (ii) above imply that there exists a subsequence {ξα0 } such that |ξα0 | → 0, when α → ∞. Proof. By definition, |ξα0 | = |ξ˜α0 |. From (i) we know that |ξ˜α0 | ≤ 1. This implies that 0 0 there exists a subsequence {ξ˜α0 } → ξ˜∞ , for some ξ˜∞ ∈ Tx V . Therefore we only need 0 ˜ to prove that ξ∞ = 0. The idea of the proof is to construct a ξ = (ξ1 , ξ2 ) ∈ W1p (f, a1 , a2 ; a3 , a4 ) such that 0 in the spherical coordinate. The assumption of Df ξ = 0, and that ξ1 (∞) = ξ2 (∞) = ξ˜∞ 0 = 0. Lemma 4.2 implies that ξ = 0. Therefore ξ˜∞ 0 0 = Dexpx (f˜(τ, t))(β(τ )ξ˜∞ ) in the same way as defining To this end, we define ξ∞ 0 0 ∈ 0(f ∗ (T V )). It is easy to see that ξα0 is locally C ∞ -convergent to ξ∞ . ξα0 . Then ξ∞ 2 2 2 Let DR be the domain in Sα or in S1 ∨ S2 of the union of the two half spheres at two ends plus the cylindrical part up to τ = R. From (1) we know that kξα1 k1,p;ε ≤ 1. This implies that for any R > 0, there exists a constant C(R) depending on R such that kξα1 k1,p ≤ C(R) for all α. Note that when R is 1 weakly fixed, all these ξα1 |DR ’s live in same space for large α. Therefore ξα1 |DR → ξ∞;R p p 1 in L1 - space for some ξ∞;R ∈ L1 (f |DR ). By letting R → ∞ and taking a subsequence 1 ’s of the diagonal, we conclude, by the Sobolev embedding argument, that all these ξ∞;R p 1 can be pasted together to give rise to a single section ξ∞ ∈ L1,loc (f, a1 , a2 , a3 , a4 ) such 1 1 |DR = ξ∞;R weakly in Lp1 -space. that ξα1 |DR → ξ∞ 0 1 + ξ∞ . Then ξα |DR → ξ∞ |DR weakly in Lp1 - space. Therefore Let ξ∞ = ξ∞ Dα ξα |DR → D∞ ξ∞ |DR weakly in Lp -space. Here we have used the fact that Dα = D∞ on DR when α is large enough. Our assumption (ii), which says that kDα ξα k0,p;ε → 0 as α → ∞, implies that k(Dα ξα )|DR k0,p → 0 as α → ∞ for any fixed R. From Lemma 3.3 we conclude that k(D∞ ξ∞ )|DR k0,p ≤ lim inf k(Dα ξα )|DR |0,p = 0. α

This implies that (D∞ ξ∞ )|DR = 0 for any R, therefore D∞ ξ∞ = 0. In spherical coordinate, this gives us a solution of D∞ ξ = 0 with a singularity at the cuspidal point y. Since the L2 -norm of ξ∞ is bounded, this singularity is removable. 0 is already smooth. We conclude that as τ → ∞, all these three sections, ξ∞ , Note that ξ∞ 0 1 ξ∞ and ξ∞ , are convergent uniformly with respect to t under the trivialization Dexpx of T V near x. Combining this and the fact that Z 1 p 1 epετ |ξ∞ | dτ dt ≤ kξ∞ k0,p;ε ≤ lim inf kξα1 k0,p;ε ≤ 1, R+ ×S 1

α

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1 we conclude that lim ξ∞ = 0. Therefore τ →∞

0 0 lim ξ∞ (τ, t) = lim ξ∞ (τ, t) = ξ˜∞ .

τ →∞

τ →∞

This proves that in the spherical coordinate ξ∞ has the required properties, and therefore finishes the proof of the lemma. Using Lemma 3.4, by a simple calculation one can show that Lemma 3.5. (i) and (ii) also imply that lim kDα ξα0 k0,p;ε = 0. α→∞

From Lemmas 3.4 and 3.5, we conclude that (I) kξα1 k1,p;ε = 1, (II) kDα ξα1 k0,p;ε → 0 when α → ∞. We will prove that (I) and (II) contradict each other. To do this, we need to have an estimate on the middle part of ξα1 . Let β2 be a ‘bump’ function on Sα2 which is supported in −2 < s < +2 and equal to 1 on − log 2 < s < + log 2, where (s, t) or wα (s, t) is the cylindrical coordinate of Sα2 starting from the middle. Lemma 3.6. lim kβ2 ξα1 k1,p;ε = 0. α→∞

Proof. Let ρα = − log z˜α + log 2, which is the length of the cylindrical coordinate wα (s, t) along the t-direction. Define ηα : [−ρα , +ρα ] × S 1 → Tx V by Dexpx (f˜α ◦ wα (s, t))(ηα (s, t)) = e|ρα |ε · ξα1 (wα (s, t)). Extend ηα trivially over the whole cylinder. Then from (I) there exists a constant C such that (1) ke−ε|s| · ηα kp ≤ C for all α. Let ηα;R be the restriction of ηα to the domain ZR = [−R, +R] × S 1 , then from (I) again there exists a constant C(R) depending on R such that kηα;R k1,p ≤ C(R) for all α. Therefore as α → ∞, (2) ηα;R → η∞,R weakly in Lp1 (ZR , Tx V ) for some η∞;R ∈ Lp1 (ZR , Tx V ). The same argument as in the proof of Lemma 3.4 will prove that when R → ∞, all these ηα;R ’s agree with each other on their overlaps to form a single element η∞ ∈ Lp1,loc (R × S 1 , Tx V ) such that η∞ |ZR = η∞;R . Now (2) implies that when α → ∞, ∂ J0 ηα;R → ∂ J0 η∞;R weakly in Lp (ZR , Tx V ), where ∂ J0 is the standard Cauchy-Riemann operator. Let E˜α be the lifting of Eα under expx as before. Then it is of the form ∂ηα;R ∂ηα;R ∂ηα;R E˜α ηα;R = + J0 + (J˜ − J0 )(f˜α ) + Aα;R · ηα;R , ∂s ∂t ∂t

(3)


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where Aα;R is the restriction to ZR of some zero order operator Aα . It is easy to see that when R is fixed, lim |Aα;R | = 0 and lim |(J˜ − J0 )|f˜α;R | = 0. α→∞ α→∞ From (II) we have lim kE˜α ηα;R kp = 0. Therefore α→∞

lim k∂ J0 ηα;R kp n o ≤ lim kE˜α ηα;R kp + |Aα;R |kηα;R kp + |(J˜ − J0 )|f˜α;R |kηα;R k1,p = 0.

α→∞

(4)

α→∞

From this, (3) and Lemma 3.3, we have k∂ J0 η∞;R kp ≤ lim inf k∂ J0 ηα;R kp = 0. α→∞

This implies that ∂ J0 η∞;R = 0 for any R > 0. Therefore ∂ J0 η ∞ = 0

(5)

From (1) and Lemma 3.3, we have that kη∞;R k0,p;(−ε) ≤ lim inf ke−ε|s| ηα kp is α→∞

bounded not depending on R. This implies that kη∞ k0,p;(−ε) < ∞. This together with (5) and the fact that the constant Fourier component of η∞ |{0}×S 1 is zero imply that η∞ = 0. By a Sobolev embedding argument we conclude that for any fixed R > 0, ηα;R is uniformly C 0 -convergent to zero. Therefore, when α → ∞, kβ2 ηα k1,p ≤ Ck∂ J0 (β2 ηα )kp ≤ C(kβ20 ηα kp + kβ2 ∂ J0 ηα kp ) → 0. This implies that lim eερα kβ2 ξα1 k1,p = 0. Hence α→∞

lim kβ2 ξα1 k1,p;ε = 0.

α→∞

p (f, a1 , a2 , a3 , a4 ) be the weighted Sobelev Finishing the proof of Lemma 3.2. Let W1;ε 2 2 space of sections over S ∪ S − {y} with cylindrical coordinate, which consists of ξ with kξk1,p;ε < ∞ and obvious constraints at the four marked points. p It is proved in [F1] and [LM] that when ε < 1, Df : W1;ε (f, a1 , a2 , a3 , a4 ) → p 0,1 ∗ L0,ε; ( (f (T V ))) is Fredholm. A removing singularity argument will show that any element ξ in the kernel of the above operator will be in W1p (f, a1 , a2 , a3 , a4 ). But by the assumption of this lemma this is impossible unless ξ = 0. p (f, a1 , a2 , a3 , a4 ). Therefore there exists a constant C Now (1 − β2 )ξα1 is in W1,ε independent of α such that

k(1 − β2 )ξα1 k1,p;ε ≤ CkDf {(1 − β2 )ξα1 }k0,p;ε = CkDα {(1 − β2 )ξα1 }k0,p;ε ≤ C kDα ξα1 k0,p;ε + kDα (β2 ξα1 )k0,p;ε ≤ C 2kDα ξα1 k0,p;ε + kβ20 ξα1 k0,p;ε → 0

when α → ∞.

Therefore kξα1 k1,p;ε ≤ k(1−β2 )ξα1 k1,p;ε +kβ2 ξα1 k1,p;ε → 0 when α → 0. This contradicts with (I).

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Lemma 3.7. lim k∂ J fα kχ,0 = 0. α→∞

Proof. If we use ∼ to denote the corresponding lifting maps and operators under expx , then f˜α = β1 f˜1 + β2 f˜2 , where β1 and β2 are supported in the middle part of Sα2 of length log 2. Since ∂˜ J f˜i = 0, |f˜i (τ, t)| ∼ e−τ when τ large, we have k∂ J fα kχ,0 ≤ Ck∂˜ J f˜α kχ,0 ≤ C(kβ10 f˜1 kχ,0 + kβ20 f˜2 kχ,0 ) ∼ e(ε−1)ρα → 0 when α → ∞.

By using Lemma 2.3 together with a simple calculation, one can show that Lemma 3.8. There exists a constant C1 only depending on f such that for any ξα , ηα ∈ W1p (fα , a1 , a2 , a3 , a4 ), kN (ξα )kχ,0 ≤ C1 kξα k∞ kξα kχ,1 , kN (ξα ) − N (ηα )kχ,0 ≤ C1 (kξα kχ,1 + kηα kχ,1 )kξα − ηα kχ,1 . Lemma 3.9 (Picard method). Assume that a smooth map f : E → F from Banach spaces (E, k·k) to F has a Taylor expansion f (ξ) = f (0)+Df (0)ξ+N (ξ) such that Df (0) has a finite dimensional kernel and a right inverse G satisfying kGN (ξ) − GN (η)k ≤ 1 . If kG ◦ f (0)k ≤ δ2 , then the zero C(kξk +kηk)kξ − ηk for some constant C. Let δ = 8C set of f in Bδ = {ξ|kξk < δ} is a smooth manifold of dimension equal to the dimension of ker Df (0). In fact, if Kδ = {ξ|ξ ∈ ker Df (0), kξk < δ} and K ⊥ = G(F ), then there exists a smooth function φ : Kδ → K ⊥ such that f (ξ + φ(ξ)) = 0 and all zeros of f in Bδ are of the form ξ + φ(ξ). The proof of this lemma is an elementary application of Banach ’s fixed point theorem. Applying this to our case we have Lemma 3.10. If A, B ∈ H2 (V ) with A + B = P , c1 (P ) + n = α1 + α2 + α3 + α4 , and f = (f1 , f2 ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ), then for generic J and a parameter z = z˜22 ∈ C ∗ with |z| large enough, there exists a gluing map #z : M(A, B, J, a1 , a2 ; a3 , a4 ) → Mz˜ (P, J, a1 , a2 , a3 , a4 ) with f = (f1 , f2 ) 7→ f1 #z f2 . Moreover, if gz is another element in Mz˜ (P, a1 , a2 , a3 , a4 ) ‘close’ to the pre-gluing 1 f1 χz f2 in the sense that kg˜z kχ,0 ≤ δ = 8CC , then gz = f1 #z f2 . Here g˜z is a vector field 1 along fz = f1 χz f2 defined by Dexp(fz (τ, t))(g˜ z (τ, t)) = gz (τ, t), and C and C1 are the constants which appeared in Lemma 3.2 and 3.8. Proof. By Lemma 3.2 and Lemma 3.8 we have kGN (ξ) − GN (η)kχ,1 ≤ CkN (ξ) − N (η)kχ,0 ≤ CC1 kξ − ηkχ,1 (kξkχ,1 + kηkχ,1 ), for any ξ and η over fz . Let δ =

1 8CC1 .

Then by Lemma 3.7 we have

kG(∂ J fz )kχ,1 ≤ Ck∂ J fz kχ,0
1 for every effective class A ∈ H2 (V ) for generic J. Here ‘generic’ means that all transversality about cusp-curves stated in Sect. 3 hold. Fix a generic J and a class P ∈ H2 (V ) with c1 (P ) + n = α1 + α2 + α3 + α4 . Since the only J-holomorphic sphere of class zero are the constant maps, we may assume that P is not zero. Assume that the given four cycles a1 , a2 , a3 , a4 , which have been assumed to be submanifolds of V as we remarked in Sect. 1, are put in a general position in V so that all possible intersections among them are still submanifolds of V . 2 → Lemma 4.1. Consider a sequence {fn } ∈ Mzñ (P, J; a1 , a2 , a3 , a4 ) with |z| = |z| ˜ 2 ∞. Each such fn gives rise to the two J-holomorphic spheres fL,n and fR,n under the ‘left’ and ‘right’ coordinates of S 2 #zn S 2 respectively, both mapping the ‘standard’ sphere S 2 to V . Then there are two possibilities:

(1) a1 ∩ a2 is not empty and {fR,n } is C ∞ -convergent to some fR ∈ M(P, J, a1 ∩ a2 , a3 , a4 ); or a3 ∩ a4 is not empty and {fL,n } is C ∞ -convergent to some fL ∈ M(P, J, a1 , a2 , a3 ∩ a4 ). (2) There exists a parametrized cusp-curve (fL , fR ) ∈ M(A, B, J; a1 , a2 ; a3 , a4 ) for some A + B = P with fL not equal to fR as unparametrized curves such that {(fL,n , fR,n )} is locally C ∞ - convergent to (fL , fR ) as parametrized curves. Proof. We will use ˆ· to denote the corresponding unparametrized curve and moduli ˆ space. By the Gromov compactness theorem, we have that {fˆn } → fˆ∞ = ∪m i=1 fi,∞ Pm 0 ˆ ˆ ˆ weakly with P = [fn ] = i=1 [fi,∞ ], and that for any C -neighborhood U of f∞ , fˆn is contained in U when n is large enough. The last statement implies that fˆ∞ has a non-empty intersection with any of the a0i s, i = 1, 2, 3, 4. By a detailed combinatorial

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analysis of the intersection pattern of fî,∞ ’s and a dimension counting argument, we conclude that m = 1 or 2 . The cases that m = 2, fˆ1,∞ = fˆ2,∞ and that m = 1 but the curve is multiplycovered do not occur. To rule out the first possibility, let [fî,∞ ] = A, i = 1, 2. Then ˆ fî,∞ ∈ M(A, J; a1 , a2 , a3 , a4 ), whose dimension is 2(c1 (A) + n − α1 − α2 − α3 − α4 ) = 2(c1 (P ) + n − α1 − α2 − α3 − α4 ) − 2(c1 (A)) which is less than zero by our assumption. It is similar for the second case. Now we can use Wolfson’s version of the Gromov compactness theorem to analyze the limit behavior of the sequence {fn }, as parametrized curves. Let lim fn (0L ) = l1 ∈ a1 , lim fn (1L ) = l2 ∈ a2 , n

n

lim fn (0R ) = l3 ∈ a3 , lim fn (1R ) = l4 ∈ a4 . n

n

(1) If l1 = l2 ∈ a1 ∩ a2 or l3 = l4 ∈ a3 ∩ a4 , then we have m = 1. Otherwise, for example, in the case l1 = l2 , fˆ∞ = (fˆ1,∞ , fˆ2,∞ ) is in the moduli space of (A, B)-cusp-curves with A + B = P , which intersects with a1 ∩ a2 ,a3 and a4 . A dimension counting argument shows that the dimension of this moduli space is less than zero. In the case l1 = l2 , we claim that {fR,n } is C ∞ - convergent to fR . If this is not true, then we have only one bubble at some point x1 in S 2 , and {fR,n }|S 2 −{x1 } is locally convergent to a constant map. x1 must be 0R , or 1R . Otherwise l3 = lim fn (0R ) = n ˆ lim fn (1R ) = l4 . This implies that fˆ∞ is in M(P, J; a1 ∩ a2 , a3 ∩ a4 ), which is empty n

for dimension reason. If x1 , for example, is 0R , then a similar argument show that ˆ l1 = l2 = l4 . Therefore fˆ∞ is in M(P, J; a1 ∩ a2 ∩ a4 , a3 ), which is empty again. Similarly, in the case l3 = l4 , we have that {fL,n } is C ∞ -convergent to fL . This gives the possibility (1) of the lemma. (2) Now we can assume that l1 6= l2 , and l3 6= l4 . Consider the sequence {fL,n }. When n tends to ∞, zn tends to the point ∞ in S 2 . But lim fL,n (zn ) = l3 6= l4 = lim fL,n (∞). n→∞ n→∞ This implies that the derivative of fL,n at ∞ blows up when n tends to ∞. Therefore we have one bubble at ∞. We claim that this is the only bubble {fL,n } could have, therefore that {fL,n } is locally C ∞ -convergent. Suppose that this is not true. Let x1 be another bubble point. Then x1 must be 0L or 1L . Otherwise since fL,n |S 2 −{x2 ,∞} is locally convergent to a constant map, we have l1 = lim fL,n (0L ) = lim fL,n (1L ) = l2 , which contradicts the assumption. n→∞

n→∞

If x1 , for example, is 0L , then lim fL,n |S 2 −{0,∞} = lim fL,n (1L ) = l2 locally. n→∞ n→∞ This implies that the limit curve fL is a union of the constant curve l1 with two bubbles f1,∞ and f2,∞ . It follows from the proof of Gromov’s compactness theorem that f1,∞ and f2,∞ are “lying” on l1 in the sense that f1,∞ (∞) = f2,∞ (∞) = l1 (see, for example, [P] or [PW]). This implies that for A = [f1,∞ ], B = [f2,∞ ], the limit cusp-curve is in the moduli space of such (A, B)-cusp-curves that intersect with the given four cycles and that their cuspidal points lie on a2 . But for generic J, this moduli space is empty for the dimension reason. We get a contradiction again. The proof of the other case of (2) is similar. Lemma 4.2. Assume that {fn } is weakly convergent to (fL , fR ) as in the case (2) of Lemma 4.1. Then for any given δ > 0, we have d(fn , fL χzn fR ) < 2δ when |zn | is large enough. Here d(fn , fL χzn fR ) = maxx∈S12 #zn S22 d(fn (x), fL χzn fR (x)) measured by a metric on V .


281

Proof. We first prove that fL (∞) = fR (∞). Note that if we view fL as the “base” curve of the cusp-curve (fL , fR ), then fR becomes the bubble. But its parametrization may differ from that coming from the bubbling procedure described in [PW]. Therefore the statement does not immediately follow from that part of the compactness theorem concerning bubble intersections. Assume that d(fL (∞), fR (∞)) = 5δ > 0. Let B1 (R) and B2 (R) be the open balls of radius R in the ‘left’ and ‘right’ coordinates of S12 #zn S22 respectively. Note that for a fixed R, B1 (R) does not intersect with B2 (R) when |zn | big enough. Denote B1 (r) ∪ B2 (R) by B(R) and ∂Bi (R) by Ci (R), i = 1, 2. By using the fact that fn is locally convergent to (fL , fR ) and that the area A(fn ) = A(fL ) + A(fR ), it is easy to see that when R and |zn | large enough, (a) d(fn (C1 (R)), fL (∞)) < δ, d(fn (C2 (R)), fR (∞)) < δ, (b) A(fn |S 2 −B(R) ) < Cδ 2 for some fixed constant C which we will specify soon. Now fn : (S 2 − B(R)) → V is a minimal surface with respect to the metric gJ . Its two boundaries lie on the two disjoint balls BfL (∞) (δ) and BfR (∞) (δ) respectively. Since BfL (∞) (2δ) ∩ fn (S 2 − B(R)) and BfR (∞) (2δ) ∩ fn (S 2 − B(R)) are two disjoint open subsets of the connected surface fn (S 2 − B(R), there exists a point x1 in fn (S 2 − B(R)) such that the distance between x1 and fL (∞) or fR (∞) is larger than 2δ. Therefore Bx1 (δ) does not intersect with the two boundary components of fn (S 2 − B(R)). Now we can apply the monotonicity for minimal surface to conclude that A(fn |S 2 −B(R) ) > A(Bx1 (δ) ∩ fn (S 2 − B(R))) > Cδ 2 for some constant C, which only depends on the geometry of (V, J, gJ ). If we choose the constant C which appeared in (b) above to be the same as the one here, then we get a contradiction. This proves that fL (∞) = fR (∞) = x, the cuspidal point of f . A similar argument, by using monotonicity for the minimal surface again, will show that when R and |zn | are large enough, fn (S 2 − B(R)) is contained in Bx ( δ4 ). From this the conclusion of the lemma follows immediately. Proof of Theorem 1.5. Let P ∈ H2 (V ) with c1 (P ) + n = α1 + α2 + α3 + α4 . By Lemma 3.10, when |zn | is big enough, there exists a gluing map a #zn : M(A, B, J, a1 , a2 ; a3 , a4 ) → Mzñ (P, J; a1 , a2 , a3 , a4 ). A+B=P

Note that for any cusp-curve f = (fL , fR ), fL #zn fR is locally convergent to f . This plus that the domain of #zn is a finite set implies that #zn is injective. The surjectivity of #zn is a consequence of Lemma (4.2) and Corollary 3.11. As we remarked before, #zn is also preserving the orientation. This proves the special decomposition rule, therefore the associativity of quantum multiplication. Acknowledgement. I would like to express my hearty thanks to my advisor Professor Dusa McDuff for introducing me to the subject and for her encouragement and various kinds of help. This work is indebted to her works and her general ideas about the subject very much. I also wish to acknowledge my debt to Floer, via his unpublished notes. In [F] he tried to established the key technique of this paper, the gluing of J-holomorphic curves, by imitating the corresponding construction in Floer homology. Because the gluing used in this paper is quite different from the one in Floer homology, his effort did not succeed . The key ingredients, Lemmas 3.4 and 3.5, are missing there. However, his general idea on gluing has much influenced my work in Sect. 3.

282

G. Liu

References [F1] [F2] [F] [G] [LM] [M] [MS] [PW] [RT1] [RT2] [V] [W]

Floer, A.: The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41, (775–813) (1988) Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120, (575–611) (1989) Floer, A.: Lecture notes, unpublished Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, (307–347) (1985) Lockhard, R.B. and McOwen, R.C.: Elliptic operators on noncompact manifolds. Ann. Sci. Norm. Sup. Pisa IV-12, (409–446) (1985) Mcduff, D.: Examples of symplectic structures. Invent. Math. 89, (13–36) (1987) Mcduff, D. and Salamon, D.A.: J-holomorphic curves and Quantum Cohomology. In print (1994) Parker, T.H. and Wolfson, J.G.: Pseudo-holomorphic maps and bubble trees. J. Geom.Anal. 3, 63–98 (1993) Ruan, Y. and Tian, G.: A mathematical theory of quantum cohomology(announcement). Math. Res. Lett. Vol 1 no 1, 269–278 (1994) Ruan, Y. and Tian, G.: A mathematical theory of quantum cohomology. Preprint (1994) Vafa, C.: Topological mirrors and quantum rings. In: Essays on Mirrors Manifolds, edited by S.-T. Yau, Hong Kong: International Press, 1992 Wolfson, J.G.: Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry. J. Diff. Geom. 28, 383–405 (1988)

Communicated by S.-T. Yau

Commun. Math. Phys. 191, 283 – 298 (1998)

Communications in


Complex Matrix Models and Statistics of Branched Coverings of 2D Surfaces Ivan K. Kostov1,? , Matthias Staudacher2 , Thomas Wynter1 1 Service de Physique Th´ eorique, C.E.A. - Saclay, F-91191 Gif-Sur-Yvette, France. E-mail: [email protected], [email protected] 2 CERN, Theory Division, CH-1211 Geneva 23, Switzerland. E-mail: [email protected]

Received: 11 April 1997 / Accepted: 9 May 1997

Abstract: We present a complex matrix gauge model defined on an arbitrary twodimensional orientable lattice. We rewrite the model’s partition function in terms of a sum over representations of the group U (N ). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum limit allowing the branch points to freely float over the surface. The simplest such limit reproduces two-dimensional chiral U (N ) Yang-Mills theory and its string description due to Gross and Taylor. 1. Introduction Recently, two of the authors have considered a complex matrix model which describes the ensemble of branched coverings of a two-dimensional manifold [1]. The model has been interpreted as a string theory invariant with respect to area-preserving diffeomorphisms of the target space. In this paper, we continue the investigation of this model. The geometrical problem we solve consists in the enumeration of the branched coverings of a two-dimensional manifold with a given number of punctures. The target manifold is characterized by its topology, the number of punctures, and its total area. All structures we are considering are invariant under area-preserving diffeomorphisms of the targ et manifold. We assume that the covering surfaces can have branch points located at the punctures. Two covering surfaces related by an area-preserving diffeomorphisn are considered identical. The problem will be reformulated in terms of a lattice gauge theory of N ×N complex matrices defined on a lattice representing a cell decomposition of the target manifold. The vertices of the lattice are the punctures of the target surface. The N1 perturbative expansion of this model generates the covering surfaces with the corresponding combinatorial ?

Member of CNRS

284

I. K. Kostov, M. Staudacher, T. Wynter

factors. These discrete surfaces can be interpreted as cell decompositions of continuum surfaces covering the target space. It sh ould be intuitively clear that the combinatorics of these surfaces should not depend on the cellular decomposition of the target manifold, but only on global features like the topology of the manifold, the topology of the covering Riemann surface, and the number and types of branch points allowed. Our solution below will confirm this intuition. The solution of the model is given in terms of a sum over polynomial representations of the group U (N ). This sum is, by construction, a generating function for the combinatorics of covering maps. The order of the representation gives the degree of the (connected and disconnected) coverings. The sum depends on the genus G of the target manifold, and a number of variables tied to geometrical data of the covering maps: N −2 is the genus expansion parameter of the covering surfaces, and we associate, for each cell corner p, a weight t(p) k corresponding to a branch point at p of order k. For some applications, one would consider a slightly more general problem of counting coverings with branch points that can occur anywhere on the smooth target manifold. In order to allow for this possibility, we are led to take a continuum limit: We simply cover the target manifold by a microscopically small cell decomposition, with the weights of these branch points tuned correspondingly. We will find that the combinatorics of the resulting statistics of movable branch points actually simplifies significantly over the general case of fixed branch points: Many special configurations of enhanced symmetry, corresponding to coalescing branch points, are scaled away. The weights associated with the punctures survive scaling; this makes the difference between the punctures and the rest of the points of the cell decomposition. Apart from the obvious mathematical interest of our approach, we are able to connect our results to recent work on the QCD string in two dimensions. In the case of a target space with nonnegative global curvature, the chiral (i.e., orientation preserving) sector of the Gross and Taylor [2] string theory describing two dimensional Yang-Mills theory, is a particular case of the string theory defined by our matrix model. We discuss the problem of the “ factors” in the string interpretation of the Yang-Mills theory and interpret the “-points” as punctures in the target space. 2. Definition of the Model Consider a smooth, two dimensional, compact, closed and orientable manifold MG of genus G with N0 marked points (punctures), which we denote by p = 1, ..., N0 . The manifold is compact in the followingR sense: We assume that there is a volume form dA on MG and the total area AT = dA is finite. We will consider the ensemble of nonfolding surfaces covering MG and allowed to have branch points at the punctures. These surfaces are smooth everywhere on MG and are given a volume form inherited from the embedding. In this way, the area of a surface covering n times the target manifold is equal to nAT . We will resolve the problem by discretizing the target manifold. Introduce a cell decomposition of the original target manifold MG such that each cell is a polyhedron homeomorphic to a disc. The vertices of the cell decomposition are by construction the N0 punctures of MG . The resulting polyhedral surface (cellular complex) MG is thus characterized by its genus G, and by its set of points p, links ` and cells c. The numbers of points, links and cells which we denote correspondingly by N0 , N1 and N2 , are related by the Euler formula (2.1) N0 − N1 + N2 = 2 − 2G.

Complex Matrix Models and Statistics of Branched Coverings

285

Each cell P contributes a fraction Ac to the total area AT of the target manifold, so that AT = c Ac . A given manifold can be discretized in many different ways, but the choice of discretization is irrelevant for our problem. A branched covering of MG of degree n is a surface Σ obtained by taking n copies of each of the polygons of MG and identifying pairwise the edges of the n polygons on either side of each link. The Riemann surface obtained in this way can have branch points of order k (k = 1, 2, ..., n) representing cyclic contractions of edges. They are located at the Ppoints p ∈ MG . The discretized surface Σ has nN1 links, nN2 polygons and nN0 − p bp points (with bp the winding number minus one at point p). Its total area is nAT . Its genus g is given by the Riemann-Hurwitz formula: X bp . (2.2) 2g − 2 = n(2G − 2) + p

The partition function is defined as the sum over all possible coverings Σ → MG conserving the orientation. A factor N 2−2g is assigned to the genus g of the covering surface. Furthermore, we introduce Boltzmann weights associated with its branch points. The weight of a branch point of order k is t(p) k , where k ≥ 2. A regular (analytic) point . gets a weight t(p) 1 The partition sum is now defined as a sum over all (not necessarily connected) coverings Σ → MG : Z=

X

e

−nAT

N

2−2g

N0 Y Y

nk (t(p) k ) ,

(2.3)

p=1 k≥1

Σ→MG

where, associated with the point p ∈ MG , nk (p) (with k ≥ 2) is the number of the branch points of order k of Σ and n1 (p) the number of regular points. The symmetry factor of the map is understood in the sum. Now we introduce a matrix model whose perturbative expansion coincides with (2.3). To each link ` = hpp0 i we associate a field variable 8` representing an N × N matrix with complex elements. By definition 8 = 8† . In order to be able to associate arbitrary weights to the branch points we will associate an external matrix field with the corners of the cells. Let us denote by (c, p > the corner of the cell c associated with the point p. The corresponding matrix will be denoted by B(c,p> . The partition function of the matrix model is defined as Z Y Y [D8` ] exp(e−Ac N Tr8c ), (2.4) Z= c

`

where 8c denotes the ordered product of link and corner variables along the oriented boundary ∂c of the cell c Y 8` B(c,p> , (2.5) 8c = `,p∈∂c

and the integration over the link variables is performed with the Gaussian measure [D8` ] = (N/π)N

2

N Y i,j=1

d(8` )ij d(8` )∗ij e−N

Tr 8` 8†`

.

(2.6)

286


3 2

Φ31 Φ13

Φ12 Φ21

B312 B413

1

B214

Φ41 Φ14

4

Fig. 1. The matrices associated with the links < 12 >, < 13 >, < 14 > and the corners (214 >, (312 >, (413 > associated with the point 1

The perturbative expansion of (2.4) gives exactly the partition function (2.3) of branched surfaces covering MG . The weight t(p) k of a branch point of order k (k ≥ 2) or regular point (k = 1) associated with the vertex p ∈ MG equals t(p) k =

1 Tr[Bpk ], N

where we have defined the matrices Bp as the ordered product Y Bp = B(c,p>

(2.7)

(2.8)

c

of the B-matrices around the vertex p.

3. Exact Solution by the Character Expansion Method The method consists in replacing the integration over complex matrices by a sum over polynomial representations of U (N ). Applying to (2.4) the same strategy as in ref.[3], we expand the exponential of the action for each cell c as a sum over the Weyl characters χh of these representations: exp(e−Ac N Tr8c ) =

X 1h h

h

χh (8c ) e−Ac |h| .

(3.1)

The representations are parametrized by the shifted weights h = {h1 , h2 , . . . , hN }, where hi are related to the lengths m1 , ..., mN of the rows of the Young tableau by hi = N − i + mi and are therefore subjected to the constraint h1 > h2 > . . . > hN ≥ 0. Here i = 1, 2, . . . denotes the first, second, . . . row of the tableau as counted down from


287

the top. We will denote by |h| = Σi mi the total number of boxes of the Young tableau. The dimension 1h of the representation h is given by 1h =

Y hi − hj

,

(3.2)

hi ! . (N − i)!

(3.3)

j−i

i<j

and the “Omega factor” h by h = N −|h|

N Y i=1

An explicit representation of the Weyl characters χh is only needed for the derivation of two essential integration formulas (fission and fusion rule, respectively): Z h χ (81 ) χh (82 ), [D8] χh (81 8 82 8† ) = 1h h Z (3.4) h χh (81 82 ). × [D8] χh (81 8) χh0 (8† 82 ) = δh,h0 1h For a simple proof of these identities, see [1]. It is now possible to exactly perform the integration with respect to the gaussian measure (2.6) at each link. Using the fusion rule one progressively eliminates links between adjoining cells until one is left with a single cell. The remaining links along that plaquette are integrated out by applying the fission rule. A similar procedure was first used in the exact solution of two-dimensional Yang-Mills theory [4, 5]. Employing the very useful relation χh (A1 ) =

1h h

with

1 TrAk1 = δk,1 , N

(3.5)

one finds the exact solution of the matrix model (2.4): Z=

N0 X 1h 2−2G Y χh (Bp ) −|h|AT e . h χh (A1 ) h

(3.6)

p=1

The characters χh (Bp ) are related to the branch point weights t(p) k at the site p through the Frobenius formula (we omit the index p for clarity): X chh (1n1 , 2n2 , 3n3 , . . .) χh (B) = n1 +2n2 +3n3 ...=|h|

|h|! N n1 +n2 +n3 +... tn1 tn2 tn3 . . . . 1n1 n1 ! 2n2 n2 ! 3n3 n3 ! . . . 1 2 3

(3.7)

The sum is over all partitions of the covering number |h|, and the chh (1n1 , 2n2 , 3n3 , . . .) are the characters of the symmetric group S|h| corresponding to the representation h and the partition class with cycle structure (1n1 , 2n2 , 3n3 , . . .) . Let us emphasize that (3.6), together with (3.7), is the complete solution of the combinatorial problem of counting branched covers (2.3). In Appendix A, we have given the first few terms of the expansion (3.6), in the special case where the branch point weights are identical on all N0 sites. We have also given the first few terms of the free energy F = log Z, corresponding to connected surfaces.

288


A special case of (3.6) is obtained when no branch points are allowed on the target manifold. This is done by choosing all matrices Bp = A1 . Then the solution (3.6) becomes completely independent of the cell decomposition of the manifold: The cell corners are indistinguishable from any other point on the surface. We obtain X h 2G−2 e−|h|AT . Z= 1h

(3.8)

h

This is completely trivial for the sphere and very simple (see [1],[2]) for the torus, but highly non-trivial for higher target space genus G ≥ 2:  2 −A for G = 0;  N eP T , ∞ 1 −kmAT 0 for G = 1; (3.9) log Z = N k=1,m=1 k e  2−2G −AT N e + (N 2−2G )2 21 (4G − 1)e−2AT + . . . for G ≥ 2. For G ≥ 2, log Z counts the number of smooth and locally invertible maps of a surface of genus g onto a surface of genus G. It would be interesting to find an analytic expression for log Z for all G ≥ 2.

4. Continuum Limit: Statistics of Movable Branch Points In the preceding section we have presented the full solution to the problem of counting the branched covers of a smooth, closed and orientable target manifold of any genus G and N0 punctures. By construction, the solution (3.6) depended, aside from G and the genus g of the branched cover, on the number and types of branch points at the punctures of MG : At each puncture p we are to specify a set of weights t(p) k associated with the winding numbers of the covering surface. It is natural to consider a related, but different combinatorial problem: Allow the covering surfaces to have branch points anywhere on the target manifold. In this case we have to sum over the positions of the branch points with an appropriate integral measure. The solution of this problem is actually contained in the solution of the previous one. Since branching in our model is constrained to occur at the sites of the cell decomposition, we are led to consider a continuum limit: The cell decomposition has to densely cover the manifold. The simplest choice is to assign identical weights at each branching sites, that T is, choose Bp = B everywhere, and assume that all cells have the same area Ac = A N0 . Set 1 t1 = TrB = 1, N (4.1) 1 τk for k ≥ 2, tk = TrB k = N N0 and take N0 → ∞, while holding AT and the continuum couplings τk fixed. Thus the probability for branching at a specific site p goes to zero in a prescribed way. In this continuum limit the configuration space of the branch points becomes infinite and their statistics drastically simplifies due to the fact that the probability to have more than one branch point associated with the same point of MG tends to zero. Mathematically, this simplification is immediately seen from the partition function (3.6), and the Frobenius formula (3.7). The product of quotients of characters exponentiates, and (3.6) becomes:


Z=

X 1h 2−2G h

h

exp

X ∞

289

τk N 1−k ξkh e−|h|AT .

(4.2)

k=2

Here, in view of (3.7), the tableau dependent numbers ξkh are given by |h|! chh (1|h|−k , k 1 ) . k(|h| − k)! chh (1|h| )

(4.3)

N N Y k 1X 1− , hi (hi − 1) . . . (hi − k + 1) k hi − hj j=1

(4.4)

ξkh = An explicit formula is ξkh =

i=1

j6=i

which is quickly derived from (3.5),(4.1), and the identity (coming from the Schur definition of the characters, see [4]) N N X χh˜ k (B) ∂ log χh (B) = ∂tk k χh (B)

with

k h˜ i = hi − δki k.

(4.5)

i=1

The ξkh are actually symmetric polynomials of degree k in the weights hi (see Appendix B, where we have also listed a few examples). They are N independent as long as |h| < N . In Appendix B, we have given the first few terms of the expansion (4.2), as well as the first few terms of the free energy F = log Z, corresponding to connected surfaces. Finally, let us mention that we can keep the weights associated with a given number of points unscaled. Then we obtain the partition function of the ensemble of branched coverings of a target manifold of area AT , genus G and N0 punctures. The evident generalization of Eqs. (3.6) and (4.2) is X N0 ∞ X 1h 2−2G Y χh (Bp ) 1−k h exp τk N ξk e−|h|AT . Z= h χh (A1 ) h

p=1

(4.6)

k=2

5. Relation to Chiral 2D Yang-Mills Theory The solution of the map counting problem with movable branch points considered in the last section has an interesting physical interpretation. Let us restrict ourselves to the case of only simple branch points: τk = 0 for k ≥ 3. Then, using the explicit result for ξ2h (see Appendix B), and choosing 1 τ2 = − AT , 2 we can rewrite the partition function (4.2) as X 1h 2−2G AT Z= e− N h h

(5.1)

C2h

,

(5.2)

290


where C2h is the second Casimir of the group U (N ). This is very nearly the partition function of two-dimensional U (N ) Yang-Mills theory. This theory was shown to be solvable on any target manifold in [4, 5]. Recently, it was demonstrated by Gross and Taylor [2], that Y M2 can be interpreted as a string theory. This was achieved by the tour de force approach of expanding the exact solution in N1 and interpreting the terms as string maps. Here we are inverting the Gross-Taylor program: we start with a theory whose interpretation as a string theory generating covering maps from a worldsheet to the target manifold is manifest, and aim to derive Y M2 . There remain, however, two subtle differences between (5.2) and Y M2 : (1) In U (N ) Y M2 the sum over polynomial representations h is extended to a sum over coupled (non-polynomial) representations. This difference is easy to understand: The missing representations clearly correspond to orientation reversing maps, which have been eliminated from the start by our chiral matrix model. Indeed it became evident from the work of [2], that Y M2 factorizes into two rather weakly interacting chiral sectors. Our class of models furnishes a precise realization of one such sector. , absent (2) Our partition function contains, for genus G 6= 1, the extra factor 2G−2 h in Y M2 . This factor eliminates the so-called “ points” and “−1 points” introduced originally by Gross and Taylor in order to sustain the string picture for G 6= 1. For G = 0, we can add two punctures and associate weight 1 with the windings around them (i.e. take a matrix source B = 1). The answer is given by (4.6) with N0 = 2 and B1 = B2 = 1 and coincide with the partition function of the Yang-Mills theory on the sphere. The two punctures are the two “ points”. In summary, in the case of a target space representing a sphere with two punctures our theory is exactly equivalent1 to chiral Y M2 . For G ≥ 2 however, it is not possible to eliminate the extra factor ( the “−1 points”) by adding punctures. On the other hand, the “−1 factors” have been given an interesting re-interpretation in the work of Cordes, Moore and Ramgoolam [5]. These authors showed that the Y M2 partition function could be rewritten as a sum over covering maps without non-movable, special singularities, but weighted instead with special topological invariants (the “Euler character of Hurwitz moduli space”). It would be interesting if this result could be reproduced by a simple matrix model, similar in spirit to (2.4) and possessing an equally clear surface interpretation.

6. Sphere-to-Sphere Maps: Saddle Point Analysis In the case of a target space of spherical topology G = 0 we can apply a saddle point technique to the sum over representations to extract the contribution from coverings with spherical topology (i.e. g = 0). This was explained in detail in [3, 6], and leads in the case of the model with fixed branch points (3.6) to the general Riemann-Hilbert problem discussed in [4]. Fortunately, the case of movable branch points (4.2) is much simpler. Introducing a continuous coordinate h = N1 hi and a density ρ(h) = −∂i/∂hi , (remember our convention hi = N − i + mi ) one finds, with the help of (4.4), Z a 1 1 ρ(h0 ) = log(h − b) + AT + V˜ 0 (h), − dh0 0 h − h 2 2 b

(6.1)

1 Adding a puncture to the target space of the Yang-Mills theory does not change anything. For example, the gauge theory defined on a cylinder with Dirichlet boundaries is identical to the gauge theory on the sphere.


291

where the effective potential V˜ 0 (h) is, for a finite number of non-zero τn ’s, a polynomial in h whose coefficients are selfconsistently dependent on the first few moments Hn = Ra P ρ(h0 ) N −1−n hni of the resolvent H(h) = 0 dh0 h−h 0 . It is found as follows. Define the potential ∞ X 1 τ k hk . (6.2) V (h) = k k=2

Then

1 V˜ 0 (h) = 2πi

I

ds V (s e−H(s) ). (h − s)2

(6.3)

Here the contour surrounds the cut [b, a] of H(h). The simplest non-trivial example consists of taking τk = 0 for k ≥ 3. The analysis then becomes very similar to the one for chiral Y M2 , obtained in [6]. Equation (6.1) then reads: Z a A T τ2 dh0 ρ(h0 ) + (1 − h). (6.4) = ln(h − b) + − 0 h − h 2 2 b The term log(h − b) is a consequence2 of the fact that the hi are a set of ordered integers. The density is therefore constrained to have a maximum value of 1 (see [4]). This equation is solved in the standard way by a contour integral Z a dh0 ρ(h0 ) H(h) = h − h0 0 (6.5) I h p ds ln(s − b) + A2T + τ22 (1 − s) √ . + (h − a)(h − b) = ln h−b 2πi (h − s) (s − a)(s − b) Expanding the contour out to infinity we catch a pole at s = h, the discontinuity across the cut of the logarithm, and a contribution from the contour at infinity. The final result is p τ2 τ2 AT − + (h − (h − b)(h − b)) H(h) = ln h + 2 2 √2 √ (6.6) h−a+ h−b √ . − 2 ln a−b The density ρ(h) is as given by the discontinuity of H(h) across its cut: 3 r τ2 p 2 h−b −1 − ρ(h) = cos (a − h)(h − b), π a−b 2

(6.7)

and the cut points a and b are determined from the behaviour of H(h) for large h: H(h) =

1 1 +O 2 . h h

(6.8)

Imposing this asymptotic behaviour on (6.6) leads to the two equations 2 A brief argument shows that for the phase corresponding to the sum over surfaces, a part of the density, starting at the origin and finishing at a point b, is saturated at its maximum value. The sum over P coverings is −A

an expansion in powers of e−AT and τ2 . For small τ2 and small e−AT , i.e. large AT , the e T in (4.2) attracts all the hi towards the origin saturating the constraint hi+1 < hi , ρ(h) ≤ 1. 3 Specifically ρ(h) = i (H(h + i) − H(h − i)), where is a small positive real number. 2π

i

hi

term

292


χ = τ22 e−AT eχ and η − τ2 e−AT +χ = 1,

(6.9)

where we have defined χ = AT + 2 log((a − b)/4) and η = (a + b)/2. Differentiating ∂ F = − < h > + 21 , whereRthe free energy F is defined (4.2) w.r.t. to AT leads to ∂A a 2 here by F = 1/N ln Z. The expectation value < h >= 0 dhρ(h)h can be calculated from the expansion for H(h) (6.6). Using (6.9) it is possible to integrate up to obtain an explicit expression for the spherical contribution to F , 1 3 F = e−AT +χ 1 − χ + χ2 . 4 6

(6.10)

From (6.9) it is clear that χ is a power series in τ22 e−AT . We can thus perform a standard Lagrange inversion on [9, 8], and obtain after a short calculation: F=

∞ X nn−3 n=1

n!

τ22n−2 e−nAT .

(6.11)

This result was first obtained in [7],[6]. To discuss the convergence properties of (6.11), it is natural to take the continuum coupling proportional to AT , since the branch points can be located anywhere on the manifold (see also (5.1)): τ2 = tAT . Note that the series is only convergent for t2 A2T e−AT < e−1 . Beyond this point the boundary conditions (6.9) lead to a non-physical complex value for χ. We see that the sum over branched coverings is convergent for both large and small areas. For large enough t and intermediate values of the area AT , however, the entropy of the branch points is sufficient to cause the sum to diverge. It is interesting to understand this divergence in terms of the Young tableau density ρ(h). Along the critical line (τ2 , AT ) and τ2 > 0 the density becomes flat at its upper end point a, i.e. the singularity at the end point changes from ρ(h) ∼ (a − h)1/2 to ρ(h) ∼ (a − h)3/2 . Along the critical line (τ2 , AT ) and τ2 < 0 it is the singularity of the density at the point b that changes from 1/2 to 3/2. This is just as occurs in matrix models of pure 2D gravity, and indeed for t2 A2 e−AT ∼ e−1 the free energy (6.11) behaves as F ∼ (e−1 − t2 A2T e−AT )5/2 . (6.12) So far in our analysis we have ignored the constraint that we are summing over positive Young tableaux, i.e. that b > 0. Since b = 0 is not a singular point in the boundary conditions (6.9) it does not correspond to a singularity in the sum over surfaces. If, however, we take the sum over representations (4.2) as fundamental (as would be the case for YM2 ) then one should take this constraint into account. In full YM2 , the inclusion of this constraint triggers the Douglas-Kazakov phase transition [8]. Unlike our case, the phase transition in YM2 can be related to the proliferation of branch points in the sum over surfaces. Setting b = 0 in (6.9) leads to the pair of equations √ (6.13) t2 A2 e−AT = χe−χ and AT = χ + 2 ln(2 − χ). which determine t and AT parametrically in terms of χ. This curve is plotted in Fig. 2. It connects the dot on the left to the shaded area on the right. Immediately below this line the support of the density is entirely positive, i.e. it starts at a positive value of h. In addition it is less than one for the full range of its support, see Fig. 2. The phase transition across this line is the analogue of the large N phase transition that occurs in the one-plaquette Wilson lattice gauge theory (see [11, 12]). There there is a phase


ρ ’(b)=0

1 0 0 1 0 1

d=0

0110 11001100 b

d c b a

d c

ρmax =1

b a

0110 10 -2

AT 6

a

5

4

3

b=0 2

1

b

a

-1

293

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 ρ’(a)=0

1

2

t

Fig. 2. Phase diagram in the t, AT plane. A typical density, ρ(h), is sketched in each phase

transition separating a strong coupling regime from a weak coupling phase, and as is the case here, there is no singularity in either phase indicating the transition point. Indeed it is possible to formulate the Gross-Witten [9] and Brézin-Gross [10] models in terms of a sum over representations [11] and the large N phase transition is precisely the point at which the density just begins to touch (or pull away from) the origin. Note, however, that in our case the phase transition is second order, while it is third order in Gross-Witten. Further analysis shows that there are two further phases for the model, as first observed in [6]. There is one phase where the density has an entirely positive support but attains its maximal value ρ(h) = 1 over a single finite interval, and another phase where the density starts at the origin and has two separated intervals where ρ(h) = 1. Both of these phases involve two separated nontrivial cuts and can be calculated in terms of elliptic functions. We do not present here explicit results for any of these extra phases. The complete phase diagram for the partition function is shown in Fig. 2. In each phase is sketched a typical density. Along the transition lines are indicated the corresponding critical behaviours of the density. The exact position of the line along which the point d equals zero has not been calculated. We indicate this by using a dotted line. The sum over representations is thus seen to have a much richer phase structure than the perturbative sum over surfaces it generates. 7. Concluding remarks We have introduced a new class of matrix gauge models which solve the general problem of counting branched covers of orientable two-dimensional manifolds with specified branch point structure. The result is given as a weighted sum over polynomial representations of U (N ). We conjecture that a similar methodology could be applied to non-orientable surfaces by replacing the complex matrices by real matrices and the group U (N ) by O(N ).

294


Our approach actually treats also the case of manifolds with boundaries. Indeed, due to the invariance with respect to area-preserving diffeomorphisms, the problem does not depend on the length of the boundaries and the latter can be considered as punctures. As an interesting by-product of our investigation, we have derived in a constructive and transparent fashion important features of the original Gross-Taylor interpretation of Y M2 as a string theory. An interesting problem which we did not consider in this paper is the calculation of Wilson loops. The Wilson loop functional defined as the sum of all nonfolding branched surfaces spanning a closed contour C ∈ MG should satisfy a set of loop equations with a contact term similar to those considered in [12]. Appendix A. Free energies for N0 fixed, identical branch points In this appendix we give some examples of how to extract the combinatorics of connected coverings of a smooth manifold with N0 fixed, identical branch points. The partition sum (3.6) for this case becomes X 1h 2−2G χ (B) N0 h e−|h|AT . Z= h χh (A1 )

(A.1)

h

Define Z =1+

∞ X

Zn e−nAT .

(A.2)

n=1

Using the Frobenius formula for characters, the first few Zn ’s, up to order four, are found to be: 0 Z1 = N 2−2G tN 1 1 Z2 = ( N 2 )2−2G 2 1 Z3 = ( N 3 )2−2G 6 1 +( N 3 )2−2G 3 1 4 2−2G Z4 = ( N ) 24

,

1 N0 1 t2 ) + (t21 − t2 )N0 , N N 3 3 2 3 2 N0 (t1 + t1 t2 + 2 t3 ) + (t31 − t1 t2 + 2 t3 )N0 + N N N N 3 1 N0 (t1 − 2 t3 ) , N 4 6 2 8 3 6 (t1 + t1 t2 + 2 t1 t3 + 2 t22 + 3 t4 )N0 + N N N N 6 8 3 6 + (t41 − t21 t2 + 2 t1 t3 + 2 t22 − 3 t4 )N0 + N N N N h 2 1 2 1 +( N 4 )2−2G (t41 + t21 t2 − 2 t22 − 3 t4 )N0 8 N N N i 2 1 2 + (t41 − t21 t2 − 2 t22 + 3 t4 )N0 + N N N 4 3 2 N0 1 4 2−2G 4 (t1 − 2 t1 t3 + 2 t2 ) . +( N ) 12 N N (t21 +

(A.3)

These expressions allow us to obtain explicit results for the map counting problem. The free energy counting connected surfaces is


F = log Z =

∞ X

Fn e−nAT .

295

(A.4)

n=1

Let us present some explicit low order results: G = 0: 0 F1 =N 2 tN 1 ,

[ 21 N0 ]−1

F2 =

X

N 2−2g

g=0

1 N0 2 N0 −2g−2 2g+2 (t ) t2 2 2g + 2 1

1 N0 3N0 −6 2 N0 3N0 −8 4 N0 3N0 −7 2 t t t F3 =N 4 t2 + t2 t 3 + t3 + 4 1 2, 1 1 3 2 1 N0 3N0 −12 6 3 N0 3N0 −11 4 + N 0 40 t t t2 + t2 t3 + 6 1 2 2, 2, 1 1 N 1 N0 3N0 −9 3 0 −2 0 −10 2 2 +2 t3N t t t + t ), 2 3 3 + O(N 3 3 1 2, 2 1 N 5 N 5 N N 1 2N0 4N0 −12 6 0 0 0 0 2 F4 =N + 13 + + − t1 124 t2 + 6 1, 4 4 2, 2 16 3 16 6 N N0 0 0 −11 4 + 27 +3 t4N t2 t 3 + 1 1, 4 1, 1, 2 N0 N0 4N0 −10 2 2 + 6 + t1 t2 t3 + 2, 2 2, 1 N N 1 N0 4N0 −8 2 0 0 0 −9 0 −9 3 t4N t4N t2 t 3 t 4 + t t3 + t4 + + 1 1 1, 1, 1 3 4 2 1 N0 1 N0 4N0 −10 3 + 4 + t1 t2 t4 + O(N 0 ). 3, 1 2 1, 1, 1 (A.5) G = 1: 2

0 F1 =tN 1 ,

[ N0 ]+1 N 3 2N0 2X 0 F2 = t1 + (t2 )N0 +2−2g t2g−2 N 2−2g 2 , 2 2 2g − 2 1 g=2 (A.6) 4 0 N0 3N0 −4 2 3N0 −3 −2 −4 F3 = t3N t + N t + 3N t t ), 16 + O(N 0 1 3 2 3 1 2 1 N 7 4N0 0 4N0 −4 2 4N0 −3 −2 F 4 = t1 + N t1 7N0 + 60 t2 + 9N0 t1 t3 + O(N −4 ). 4 2 1

296


G = 2: 0 F1 =N −2 tN 1 ,

[ 2 N0 ]+3 N X 0 −4 15 2N0 F2 =N t1 + (t2 )N0 +6−2g t2g−6 N 2−2g 8 , 2 2 2g − 6 1 g=4 N 0 3N0 −4 2 3N0 −3 −6 220 3N0 −8 F3 =N t t 640 +N t2 + 135N0 t1 t3 + O(N −10 ), 3 1 2 1 N 5275 4N0 0 4N0 −3 0 −4 2 F4 =N −8 t1 +N −10 3760N0 +41280 t4N t + 8505N t t 0 1 3 + 2 1 4 2 1

+ O(N −12 ). (A.7) Here have employed the standard notation for binomial and multinomial coefficients. It is instructive to draw the Riemann surfaces corresponding to the various terms in (A.5),(A.6),(A.7). As will be seen, the combinatorics involved increases rapidly in complexity. The above examples are easily checked to be in agreement with the Riemann-Hurwitz formula. Of course, this formula only gives a necessary condition for the existence of a Riemann surface. The above results can be used to decide whether a Riemann surface 2 N3 of given G,g and branch point structure tN 2 t3 . . . actually exists. For example, we see from F4 in (A.5) that there exists a fourfold cover of the sphere by a sphere with exactly six simple branch points, where two branchpoints are located at each of three locations 5 ). As a second example, we of the target manifold (the associated symmetry factor is 16 see from F3 in (A.6) that there exists a triple cover of the torus by a double torus with exactly one branch point of order 2 (the associated symmetry factor is 3).

Appendix B. Free Energies for an Arbitrary Number of Movable Branch Points In this appendix we give some concrete examples of how the maps are counted in the continuum limit we defined above. Define Z =1+

∞ X

Zn e−nAT .

(B.1)

n=1

The first few auxiliary ξkh (see (4.4)) are: X

1 hi − N (N − 1), 2 i X 1 2N − 1 X N (N − 1)(2N − 1) ξ2h = , h2i − hi + 2 i 2 6 i 1X 3 1X X 2N − 1 X 2 ξ3h = hi − hi hj − hi + 3 i 2 i 2 j i ξ1h =

X 1 9 9 N (N − 1)(3N 2 − 3N + 2) + ( N 2 − N + 2) . hi − 3 2 2 8 i

(B.2)


297

They are easily computed from the following representation of Eq. (4.4): ξnh =

1 n2

I

∞

X (−n)p ∂ p−1 dh h(h − 1) . . . (h − n + 1) exp H(h) , 2πi p! ∂h

(B.3)

p=1

where we have introduced H(h) =

PN

1 i=1 h−hi .

The first few Zn ’s, up to order four, are:

Z1 =N 2−2G , 1 1 1 Z2 =( N 2 )2−2G e N τ2 + e− N τ2 , 2 3 2 3 2 1 Z3 =( N 3 )2−2G e N τ2 + N 2 τ3 + e− N τ2 + N 2 τ3 + 6 1 1 + ( N 3 )2−2G e− N 2 τ3 3 6 8 6 1 4 2−2G N6 τ2 + 82 τ3 + 63 τ4 N N Z4 =( N ) + e − N τ2 + N 2 τ3 − N 3 τ4 + e 24 2 2 2 2 1 + ( N 4 )2−2G e N τ2 − N 3 τ4 + e− N τ2 + N 3 τ4 + 8 4 1 + ( N 4 )2−2G e− N 2 τ3 . 12

(B.4)

This gives the following continuum free energies (as can be checked easily from the results of Appendix A): G = 0: F1 =N 2 , ∞ X 1 1 τ 2g+2 , N 2−2g F2 = 2 (2g + 2)! 2 g=0 h1 1 1 i (B.5) F3 =N 2 τ24 + τ22 τ3 + τ32 + 6 2 6 h1 3 1 1 i τ26 + τ24 τ3 + τ22 τ32 + τ33 + O(N 0 ), + 18 8 2 18 h1 27 3 1 2 1 i F4 =N 2 τ26 + τ24 τ3 + τ22 τ32 + τ2 τ3 τ4 + τ33 + + τ23 τ4 + τ42 + O(N 0 ). 6 24 2 6 3 8 G = 1:

F1 =1, ∞

3 X 2−2g 2 τ 2g−2 , N F2 = + 2 (2g − 2)! 2 g=2 h i 4 F3 = + N −2 8τ22 + 3τ3 + O(N −4 ), 3 h i 7 F4 = + N −2 30τ22 + 9τ3 + O(N −4 ). 4 G = 2:

(B.6)

298


F1 =N −2 , ∞

15 X 2−2g 8 + τ 2g−6 , N 2 (2g − 6)! 2 g=4 h i 220 F3 =N −6 + N −8 320τ22 + 135τ3 + O(N −10 ), 3 h i −8 5275 + N −10 20640τ22 + 8505τ3 + O(N −12 ). F4 =N 4

F2 =N −4

(B.7)

References 1. Kostov, I. and Staudacher, M.: Phys. Lett. B 394, 75 (1997) 2. Gross, D.: Nucl. Phys. B 400, 161 (1993); Gross, D. and Taylor, W.: Nucl. Phys. B 400, 181 (1993); Nucl. Phys. B 403, 395 (1993) 3. Kazakov, V.A., Staudacher, M. and Wynter, T.: Advances in Large N Group Theory and the Solution of Two-Dimensional R2 Gravity. hep-th/9601153, 1995 Cargèse Proceedings 4. Migdal, A.A.: Zh. Eksp. Teor. Fiz. 69, 810 (1975) (Sov. Phys. JETP 42, (A 75), 413) 5. Rusakov, B.: Mod. Phys. Lett. A 5, 693 (1990) 6. Kazakov, V.A., Staudacher, M. and Wynter, T.: Commun. Math. Phys. 177, 451 (1996); Commun. Math. Phys. 179, 235 (1996); Nucl. Phys. B 471, 309 (1996) 7. Cordes, S., Moore, G. and Ramgoolam, S.: Large N 2-D Yang-Mills Theory and Topological String Theory. hep-th/9402107, and Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories. hep-th/9411210, 1993 Les Houches and Trieste Proceedings; Moore, G.: 2-D Yang-Mills Theory and Topological Field Theory. hep-th/9409044 8. Taylor, W. and Crescimanno, M.: Nucl. Phys. B 437, 3 (1995) 9. Taylor, W.. MIT-CTP-2297 hep-th/9404175 (1994) 10. Douglas, M.R. and Kazakov, V.A.: Phys. Lett. B 312, 219 (1993) 11. Gross, D. and Witten, E.: Phys.Rev. D 21, 446 (1980) 12. Gross, D. and Brézin, E.: Phys. Lett. B 97, 120 (1980) 13. Staudacher, M. and Wynter, T.: Unpublished (1996) 14. Kazakov, V.A. and Kostov, I.: Nucl. Phys. B 176, 199 (1980) Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 191, 299 – 323 (1998)

Communications in


On GNS Representations on Inner Product Spaces I. The Structure of the Representation Space Gerald Hofmann Universität Leipzig, NTZ, Augustusplatz, D-04109 Leipzig, Germany. E-mail:[email protected] Received: 21 May 1996 / Accepted: 12 May 1997

Abstract: A generalization of the GNS construction to hermitian linear functionals W defined on a unital *-algebra A is considered. Along these lines, a continuity condition (H) upon W is introduced such that (H) proves to be necessary and sufficient for the existence of a J-representation x → πW (x), x ∈ A, on a Krein space H. The property whether or not the Gram operator J leaves the (common and invariant) domain D of the representation invariant is characterized as well by properties of the functional W as by those of D. Furthermore, the interesting class of positively dominated functionals is introduced and investigated. Some applications to tensor algebras are finally discussed.

1. Introduction One of the most powerful theorems frequently used in mathematical physics is that about the GNS representation (Gelfand, Neumark, Segal) stating that for every i) positive functional W defined on ii) a unital C ∗ -algebra there is a cyclic *-representation by bounded operators on a Hilbert space ([26, Chap. 17.4, Theorem 2]). In order to apply that theorem to General (axiomatic) QFT, the following generalizations were considered about 30 years ago. While a generalization of ii) to topological *-algebras leading to unbounded *-representations was given by Borchers ([7]), Uhlmann ([34]) and Powers ([33]), a generalization of i) to hermitian linear functionals yielding representations on (possibly indefinite) inner product spaces ([4, 5]) was studied by Scheibe ([29], cf. Proposition 1, below). The last is of increasing interest because the investigations of some models (e.g., gauge fields ([6, Chap. 10], [21, 25, 14, 15]), massless fields ([31, 24])) considered within the Borchers-Uhlmann approach to General QFT lead to GNS representations on indefinite inner product spaces. Starting with a unital *-algebra A and an hermitian linear functional W defined on A, the following new features enter the theory in contrast to the well-known reconstruction theorem for positive functionals.

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– A priori, the (common and invariant) domain D of the operators of the representation πW (.) does not carry any scalar product [., .] such that the indefinite metric (., .) inherited by W on D satisfies (., .) = [., J.], where J is a bounded Gram-operator (cf. (1), (2), below). However, for as well mathematically (see ad 1) and ad 2) in Chap. 2) as physically (see [19, ch. 3]) motivated reasons , it is desirable to introduce a Hilbert space structure (H, J) on D. – If there are Hilbert-space structures on D, then even the maximal one (cf. Proposition 2) is not uniquely defined in general. Let us mention that a whole family of nonequivalent maximal Hilbert-space structures was explicitly constructed by Araki ([2]) in the case of A = (C2 )⊗ (tensor algebra over C2 ). – Furthermore, if there is a Hilbert-space structure on D, then in general, the representation πW is not involution preserving with respect to the Hilbert-space adjoint operators πW (.)[∗] , i.e., πW is not any *-representation ([30]). However, x → πW (x), x ∈ A, is a J-representation in the sense of Neumark (cf. [26, §41.6], [32], Definition 1). – In contrast to the case of *-representations it was observed by Yngvason that D 6⊂ D[∗] = JD (domain of the adjoint operators π(.)[∗] ) is possible. For applications, both cases i) D ⊂ D[∗] and ii) D 6⊂ D[∗] are of interest since there are models in General QFT such that respectively, i) (see [17, 22, 18, 12]) and ii) (see [13]) apply. Along these lines and guided by the idea that the whole theory is encoded in the functional W , the following questions arise. (Q1) Which conditions must the functional W satisfy such that a Hilbert space structure exists on D? (Q2) Under which conditions does the Gram operator J satisfy J : D → D ? (Q3) Under which conditions does exactly one maximal Hilbert space structure exist on D ? In the case of the tensor algebra S⊗ (field algebra of axiomatic QFT for one hermitian scalar field), (Q1) was answered by Yngvason in [35]. For the general case of the above *algebra A, question (Q1) will be answered in Theorem 3. More precisely, it is shown that a Hilbert space structure exists on D, if and only if W satisfies the Hilbert-space structure condition (H) introduced in Chap. 2. In Theorem 1 and Proposition 3 the structure of D and of its completion is considered as both an inner product space and a locally convex vector space. Using these results, a generalization of the GNS representation to Jrepresentations on Krein-spaces is given in Theorem 2. Since the proofs are constructive, there are given two possibilities for a construction of the corresponding Krein space (see Remark 4). (Q2) was also posed and discussed by Yngvason in [35, chapter 4]. In Theorem 4 an answer to (Q2) will be given in the settings of respectively, the inner product space D and the functional W . Concerning (Q3) it was mentioned by Araki (cf. [2]) that while the indefinite metric (., .) on D is supposed to be relevant to physical interpretation the positive definite metric [., .] is not intrinsic. However, if there is an affirmative answer to (Q2), then sufficient conditions for an affirmative answer to (Q3) are given in Proposition 4 and Corollary 2. Guided by the significance of positive functionals for the investigations on *algebras, the interesting class of positively dominated hermitian functionals is introduced. Among others, it is proven in Theorem 5 that every positively dominated hermitian functional satisfies (H) and consequently leads to a J-representation on a Krein space. It is further shown by two counter-examples (Examples 1, 2) that there are as

On GNS Representations

301

well hermitian linear functionals W not satisfying (H) as such functionals W satisfying (H) but not being positively dominated. On the other hand, for the interesting case of tensor algebras having a countable (algebraic) basis it is shown that the class of positively dominated hermitian functionals coincides with that of hermitian linear functionals (Proposition 5). The pattern of the present paper is as follows. The prerequisites for the following considerations are briefly recalled in Chap. 2. After introducing the Hilbert-space condition (H), quadratic majorants are constructed in Chap. 3. In Chap. 4 there is investigated the structure of the state space obtained by GNS representation. A discussion of (H) including answers to (Q1), (Q2), and (Q3) is given in Chap. 5. While the class of positively dominated functionals is introduced and studied in Chap. 6, some applications to tensor algebras are finally given in Chap. 7. Let us mention that an application of condition (H) to QFT with indefinite metric and transformations of linear functionals including truncation is given in [16]. Further, continuity properties of the representation x → πW (x) will be studied in a subsequent paper.

2. Preliminaries Throughout the present paper let A denote an (associative) *-algebra with unity 1 and W a linear and hermitian functional on A satisfying W (1) = 1. Recall the following GNS-like reconstruction theorem due to Scheibe. Proposition 1. Under the above assumptions there are (i) a vector space D with an inner product (., .), (ii) a vector ψ0 ∈ D satisfying (ψ0 , ψ0 ) = 1, (iii) a representation f 7→ πW (f ) of A by linear operators on D such that W (f ) = (ψ0 , πW (f )ψ0 ), D = span {πW (f )ψ0 ; f ∈ A}, cyclicity of ψ0 , (φ, πW (f )ψ) = (πW (f ∗ )φ, ψ), f ∈ A, φ, ψ ∈ D. Furthermore, D, ψ0 , and πW (.) are uniquely defined by (i),. . .,(iii) up to linear isomorphisms. Proof. See [29, 35].

In order to make the theory mathematically manageable (see ad 2), below) one has to define a Hilbert space structure on D, i.e., there is a positive definite inner product e k.k , k.k = √[., .], such that [., .] and a linear operator J = J ∗ (Gram operator) on H = D (f, g) = [f, Jg], kJk := sup kJxk < ∞,

(1) (2)

kxk≤1

where the continuously extended inner products are also denoted by (., .), [., .] on H. (Here and in the following, lete· k.k (resp.e· τ ) denote the completed hull of a set · relative

302

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to the (locally convex) topology defined by k.k (resp. τ ).) Let the above Hilbert space structure be denoted by (H, J). Recall further that two Hilbert space structures (Hj , Jj ), j = 1, 2, are called equivalent, if the two positive definite inner products [., .]j satisfy (f, g) = [f, J1 g]1 = [f, J2 g]2 , p and both norms k.kj = [., .]j , (j = 1, 2) are equivalent on D, thus H1 = H2 . Remember also that if the inverse operator J −1 exists as a bounded linear operator on H, then there is an equivalent Hilbert space structure (H0 , J 0 ) given by [x, y]0 = [x, |J|y], J0 =

J , |J|

x, y ∈ H ([9, §2.7(3)]). Noticing that J 0 is a symmetry on H0 (i.e., J 0 = J 0∗ = J 0−1 ), (H0 , J 0 ) is called a Krein space structure. The existence of a Hilbert space structure has the following consequences: 1) there is a maximal Hilbert space structure, 2) the theory of unbounded representations applies, and consequently, the theory is well-understood and mathematically manageable (see [30]). ad 1): A Hilbert space structure (H, J) on D is called maximal if there is no other ⊂

one (H1 , J1 ) satisfying H 6= H1 . Recall the following. Proposition 2. The following are equivalent. (i) (H, J) is a maximal Hilbert space structure, (ii) J −1 is a bounded operator on H, (iii) H is a Krein space. Proof. (i) ⇔ (ii) : [23, Theorem 5], (ii) ⇔ (iii) : [5, V.1.3].

Remark 1. In General QFT there are considered both a) a representation π of a unital *-algebra A and b) representations ρ of certain groups G (gauge groups, Poincaré group) on some inner product space H, (., .). While the present paper is concerned with a), a detailed analysis of indecomposable representations ρ of G with invariant inner product such as it is found, e.g., in the situation of Gupta-Bleuler QED was given by Araki in [3]. There were considered subspaces H1 ⊂ H such that ρ|H1 (restriction of ρ to H1 ) is irreducible and H1 = H1⊥⊥ (bi-orthogonal complement in H, (., .)), see [3, Theorem 1]. Since H, (., .) is given by π considered in a), there is the following interplay between a) and b). Recalling that if H, (., .) is a Krein space then H1 = H1⊥⊥ holds for every subspace H1 closed relative to the Krein-space topology, Araki’s theory immedeately applies to every closed subspace of Krein space H and topological complications are avoided. ad 2): If a Hilbert space structure is introduced on D, the theory of unbounded representations on Hilbert spaces applies. Let us recall some notions and results from the theory of unbounded representations of *-algebras on Hilbert spaces. Let A be an (associative) *-algebra with unity 1. A (closable) representation π of A on a Hilbert space H, [., .] with domain D(π) ⊂ H is a linear mapping π of A into the set of closable linear operators defined on D(π) such that


303

a) π(1) = I (identity operator on H), b) π(x)π(y)ξ = π(xy)ξ for all ξ ∈ D(π), x, y ∈ A, c) D(π) is dense in H, and π(x)ξ ∈ D(π) for all ξ ∈ D(π), x ∈ A. In order to obtain involution-preserving representations on Hilbert spaces, the concept of *-representations was introduced (see [30]). If one considers representations on inner product spaces, there is the concept of J-representation introduced in the case of algebras of bounded operators in [26, §41.6] and generalized to the case of unbounded ˆ ([32]). Following the idea that while the inner product (., .) is intrinsic operators by Ota the (positive definite) scalar product [., .] is only auxiliary, let us give the following reformulation of [32, Def. 2.3]. Definition 1. A representation π of a unital *-algebra A with domain D(π) is called a J-representation, if there is a Krein space (H, J) such that (i) π is a representation on (H, J), (ii) (1) holds, (iii) π ⊂ π (∗) (for an explanation see the following remarks). Remark 2. a) Assuming that (i), (ii) of Definition 1 apply, for every densely defined operator T there are the both adjoint operators T (∗) and T [∗] defined by (T ξ, ζ) = (ξ, T (∗) ζ), [T ξ, ζ] = [ξ, T [∗] ζ], ξ ∈ dom(T ), ζ ∈ dom(T (∗) ) resp. ∈ dom(T [∗] ), and T [∗] = JT (∗) J is satisfied. Setting D(π (∗) ) =

\

dom(π(x)(∗) ),

x∈A

π (∗) (x) = π(x∗ )(∗) |D(π(∗) ) , the restriction of π(x∗ )(∗) to D(π (∗) ), (iii) reads as π(x)ξ = π (∗) (x)ξ for all x ∈ A, ξ ∈ D. Hence, if π is a J-representation, then (ξ, π(x)ζ) = (π(x∗ )ξ, ζ), ξ, ζ ∈ D, and π(x∗ ) = π(x)(∗) , x ∈ A on D. b) Assume that Definition 1 applies. Considering the algebra of operators {π(a); a ∈ A} defined on D and endowed with the *-operation π(a)+ = π(a)(∗) |D , π becomes a *homomorphism between *-algebras A and {π(a); a ∈ A}. c) Continuity properties of π will be considered in a subsequent paper. In order to investigate J-representations the concept of P-functionals was introduced ˆ ([1]), and further investigated in [12, 13]. Setting by Antoine and Ota NW = {f ∈ A; W (g ∗ f ) = 0 for all g ∈ A}, let us consider the following generalization of that concept for answering (Q2) in Chap. 5.

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Definition 2. An hermitian linear functional W defined on a unital *-algebra A is called a generalized P-functional, if W (1) = 1 and there is a linear mapping α : A → A such that α2 αn α1 W (xy) W (α(x∗ )) W ((αx)∗ x)

= id (identity mapping on A), = n for n ∈ NW , = 1, = W (α(x)α(y)), = W ((αx)∗ ), ≥0

for all x, y ∈ A. Remark 3. Considering the projection P =

1 (α + id), 2

it was shown in [1] that P : A → B is an abstract conditional expectation ([27]) of A onto some *-subalgebra B. (Let us mention that the notation "P-functional" refers to the above projection P .) At the end of this section let us recall some notions from the geometry of inner product spaces frequently used in the following ([5]). An inner product space E, (., .) is called decomposable if it admits a fundamental decomposition, i.e., there are positive and negative definite (linear) subspaces E + and E − such that .

.

E = E + (+)E − (+)E (0) , where (x+ , x− ) = 0 for every x± ∈ E ± , E (0) := E ∩ E ⊥ (set of isotropic vectors of E). Further, E, (., .) is called non-degenerate, if E (0) = {0}. Letting E, (., .) be nondegenerate and decomposable, every fundamental decomposition of E is of the form .

E = E + (+)E − , where E ± are as above. Fixing such a fundamental decomposition of E and letting P ± : E → E ± denote the fundamental projectors belonging to it, define the fundamental symmetry J := P + − P − relative to the fundamental decomposition chosen above. Noticing now that [., .]J := (., J.) is a positive definite inner√product (scalar product) on E, consider the topology τJ defined by the norm k.kJ := [., .]J . Since |(x, y)|2 ≤ kxkJ kykJ , x, y ∈ E, the inner product is (jointly) continuous relative to τJ , and thus τJ is called a decomposition majorant assigned to J.


305

3. Condition (H) and Quadratic Majorants Let us be given a *-algebra A with unity 1 and an hermitian linear functional W on A satisfying W (1) = 1. Let us consider the following Hilbert-space structure condition. (H) There is a quadratic seminorm p on A such that for each g ∈ A there is a constant Cg ≥ 0 and |W (g ∗ f )| ≤ Cg p(f )

(3)

(f, g) = W (g ∗ f )

(4)

is satisfied for all f ∈ A. Noticing that defines an inner product on A, (H) means that p defines a quadratic and partial majorant on A. Let us also consider the isotropic part A(0) = NW . Let us further introduce the topologies τn (n = 1, 2, . . .) defined by the seminorms P(τn ) = {p(n) }, where p(1) = p, p(n+1) (f ) = sup{|(f, g)|; p(n) (g) ≤ 1}

(5)

(the existence will follow from Lemma 1). Notice that the topologies τn are not separated in general, and furthermore, τn+1 is the polar topology of τn with respect to (4). For any seminorm q, let us put ker(q) = {f ∈ A; q(f ) = 0}. Some of the properties of the topologies τn and the seminorms p(n) used extensively afterwards are collected in Lemma 1. Lemma 1. Assuming that (H) applies, the following are implied. i)

It holds ∞ > p(1) (f ) ≥ p(3) (f ) = p(5) (f ) = p(7) (f ) = . . . , p (f ) = p(4) (f ) = p(6) (f ) = . . . < ∞ (2)

for each f ∈ A. ii) The seminorms p(n) are quadratic on A, n = 1, 2, . . . iii) τn are partial majorants on A, and |(f, g)| ≤ p(n) (f )p(n+1) (g) hold for each f, g ∈ A, n = 1, 2, . . . iv) ker(p(1) ) ⊂ A(0) , ker(p(m) ) = A(0) for m = 2, 3, . . .

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Proof. The proof uses some ideas taken from [5, Lemmata III.4.1-3] i), iii): Using (H), there is some constant Cg(1) < ∞ such that |(f, g)| ≤ Cg(1) p(1) (f ) for each f ∈ A. Hence, p(2) (g) = sup{|(f, g)|; p(1) (f ) ≤ 1} ≤ Cg(1) < ∞ for each g ∈ A. Further,

|(f, g)| ≤ p(1) (f )p(2) (g)

(6)

for each f, g ∈ A.Using |(f, g)| = |(g, f )|, interchanging f and g, and setting Cg(2) = p(1) (g) < ∞, (6) implies

|(f, g)| ≤ Cg(2) p(2) (f ), p(3) (g) = sup{|(f, g)|; p(2) (f ) ≤ 1} ≤ Cg(2) = p(1) (g), |(f, g)| ≤ p(2) (f )p(3) (g)

for each f, g ∈ A. Arguing as above, p(n+2) (g) ≤ p(n) (g), |(f, g)| ≤ p(n) (f )p(n+1) (g)

(7) (8)

follow for n = 1, 2, . . .. (8) proves iii). Noticing that (5) and (7) also yield p(n+3) (g) ≥ p(n+1) (g), g ∈ A, n = 1, 2, . . . , i) follows. iv): Using iii), for each g ∈ A there are Cg(n) < ∞ such that |(f, g)| ≤ Cg(n) p(n) (f ) for all f ∈ A, n = 1, 2, . . .. Hence, f ∈ ker(p(n) ) implies (f, g) = 0 for all g ∈ A, i.e., f ∈ A(0) . Thus, (9) ker(p(n) ) ⊂ A(0) . Conversely, letting y ∈ A(0) , (x, y) = 0 holds for all x ∈ A, and (3) yields p(n+1) (y) = sup{|(x, y)|; p(n) (x) ≤ 1} = 0

(10)

for n = 1, 2, . . . . Hence A(0) ⊂ ker(p(m) ) for m = 2, 3, . . . . Now, iv) follows from (9), (10). ii): Assuming that there is an s ∈ N such that p(s) is quadratic, there is a positive inner product (., .)s on A such that p p(s) (f ) = (f, f )s , f ∈ A. Consider the quotient space E = A/ker(p(s) ) and note that a positive definite inner product is defined by (fˆ, g) ˆ (s) = (f, g)s


307

on E, where f, g ∈ A and fˆ, gˆ denote the residue classes containing f, g, respectively. Noticing that g ∈ ker(p(s) ) ⊂ A(0) implies W (g ∗ f ) = 0, W (f ∗ g) = W (g ∗ f ) = 0 for each f ∈ A, an inner product (x, ˆ y) ˆ E = W (y ∗ x), x, y ∈ A, is defined on E. Consider the linear functional ˆ = (x, ˆ y) Ê Lyˆ (x) on E. Using (8), the k.k(s) -continuity of Lyˆ (.) follows from ˆ = |W (y ∗ x)| = |(x, y)| ≤ Cy(s) p(s) (x) = Cy(s) kxk ˆ (s) , |Lyˆ (x)| p ˆ x) ˆ (s) , xˆ ∈ E, Cy(s) := p(s+1) (y). Hence there is a unique extension where kxk ˆ (s) = (x, ˜ where E˜ denotes the completed hull of E with respect to the norm k.k(s) . L˜ yˆ to E, Applying the Riesz representation theorem, there is zˆ ∈ E˜ such that L˜ yˆ (x) ˆ = (x, ˆ z) ˆ (s) . Moreover, kzk ˆ (s) = kL˜ yˆ k = sup{|W (y ∗ x)|; p(s) (x) ≤ 1} = p(s+1) (y).

(11)

Since the above mapping y 7→ zˆ is linear and k.k(s) is quadratic, (11) implies that the parallelogram-identity applies to p(s+1) . Hence, p(s+1) is quadratic, too. Recalling that p(1) is quadratic due to (H), ii) follows. Using the seminorms p(n) introduced above there are two further interesting topologies ρj on A defined by the seminorms r 1 (j) 2 (j) q = ((p ) + (p(j+1) )2 ), 2 j = 1, 2, respectively. Lemma 1 readily implies the following. Corollary 1. ρj are quadratic majorants on A, and ker(q (1) ) ⊂ ker(q (2) ) = A(0) hold. Furthermore, ρ2 ≤ ρ1 . In order to show that there are hermitian linear functionals not satisfying (H) let us consider the following example. Example 1. Letting B denote the set of all sequences x = (xn )∞ n=0 , xn ∈ C, and introducing algebraic operations by setting (x + y)n = xn + yn , (xy)n = xn yn , (x∗ )n = x¯ n , n = 0, 1, 2, . . . , B becomes a *-algebra. Let

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G.Hofmann

A = {(λ, x); λ ∈ C, x ∈ B} be the *-algebra with unity (1, 0) obtained from B by adjunction of unity ([26, §7.2]). Consider now the subspace L = {x ∈ B; there exists nx ∈ N with xν = 0 for ν = nx , nx + 1, . . .} . ˇ where (L) ˇ ∗ = L. ˇ Define then an and an (algebraic) direct decomposition A = L + L, hermitian linear functional W on A by setting P∞ λ + j=0 xj for x ∈ L W ((λ, x)) = . λ for x ∈ Lˇ

Statement. W does not satisfy condition (H). Proof. Assuming that W satisfies (H), there is a norm k.k on A such that for each a ∈ A there is a constant Ca < ∞ with |W (a∗ b)| ≤ Ca kbk

(12)

(n) for all b ∈ A. Choosing e(n) = (δn,j )∞ = (0, e(n) ) ∈ A, n = 0, 1, 2, . . . , x = j=0 ∈ B, y ∞ (xm )m=0 ∈ B with xm = mCy(m) , m = 0, 1, 2, . . . , z = (0, x) ∈ A, where Cy(m) is taken from (12). Noticing e(n)∗ x ∈ L,

W ((0, e(n)∗ x)) = xn = n Cy(n) , and setting a = y (n) , b = z in (12), it follows n Cy(n) ≤ Cy(n) kzk,

n = 0, 1, 2, . . . , which is impossible.

4. GNS Representation on Krein Spaces Throughout the present chapter let us assume that (H) applies. In order to apply the well-developed theory of non-degenerate inner product spaces (see [5, ch. IV.4]), let us consider the vector space D = A/A(0) endowed with the non-degenerate inner product (x, ˆ y) ˆ D = W (y ∗ x), where x ∈ x, ˆ y ∈ y, ˆ and ˆ. denotes the residue class of . in D. Furthermore, q (2) inherits a normed and quadratic majorant also denoted by q (2) on D. Setting kf k(1) = q (2) (f ), f ∈ D, let us recursively define kf k(n+1) =

r

1 ((kf k(n) )2 + (kf k0(n) )2 ), 2

where k.k0(.) = sup{|(., g)|; kgk(.) ≤ 1} denotes the polar norm.

(13)


309

Lemma 2. k.k(n) , n = 1, 2, . . . , are quadratic majorants on D satisfying (i) (ii)

|(f, g)| ≤ kf k(n) kgk(n) , khk(n) ≥ khk(n+1) ≥ √12 khk(n) ≥

√1 khk0 , (1) 2

f, g ∈ D, 0 6= h ∈ D, n = 1, 2, . . . . Proof. Recall first that every k.k(n) , n = 1, 2, . . . , defines a majorant satisfying (i) on D, see [5, Lemma IV.4.1]. Applying now [5, Lemma III.4.3], Corollary 1 and (13) imply that k.k(n) , n = 1, 2, . . . , are quadratic. Noticing that (i) readily implies khk0(n) ≤ khk(n) , Eq. (13) yields the first two inequalities of (ii). Noting now that khk(n) ≥ khk(n+1) implies khk0(n+1) ≥ khk0(n) , it follows khk(n) ≥ khk0(n) ≥ khk0(1) proving the remainder of (ii).

Applying Lemma 2, let us consider kf k(∞) = lim kf k(n) , n→∞

f ∈ D. Noting that k.k(∞) is a quadratic norm on D, the following hold. Lemma 3. It holds kf k(∞) = limn→∞ kf k0(n) , f ∈ D, and k.k(∞) is a quadratic, minimal and self-polar (i.e., k.k(∞) = k.k0(∞) ) majorant on D. Proof. See [5, Lemma IV.4.1, Theorem IV.4.2].

Noticing that Lemmata 2, 3 imply that the 0-neighborhood U = {f ∈ D; kf k(1) ≤ 1} is k.k(∞) -closed, D ⊂ H(1) ⊂ H(∞)

(14)

follow, where e k.k(1) , H(1) = D e k.k(∞) , H(∞) = D (see [20, §18.4(4)]). Using Lemmata 2, 3, let us k.k(1) − (resp. k.k∞ −) continuously extend the inner product (.,.) of D onto H(1) (resp. H(∞) ), and let us also denote these extensions by (.,.). Let further τ∞ denote the topology defined by k.k(∞) on H(∞) . For the inner product spaces so obtained, the following hold. Theorem 1. The inner product spaces H(∞) and H(1) are i) decomposable and ii) nondegenerate. Furthermore, iii) H(∞) is a Krein space, and iv) there is a fundamental . decomposition H(∞) = H+ (+)H− such that 1ˆ ∈ H+ .

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Proof. i): Noticing that Lemmata 2, 3 imply that k.k∞ (resp. k.k(1) ) are Hilbert majorants on H(∞) (resp. H(1) ), the first assertion to be shown follows from [5, Theorem IV.5.2]. ii): For the following let H and k.k stand for one of the Hilbert spaces under consideration and its norm, respectively. In order to show ii), let us consider f ∈ H and a sequence {fm }∞ m=1 , fm ∈ D, such that lim kf − fm k = 0

m→∞

(15)

and (f, g) = 0

(16)

for all g ∈ H. Using (15), (16) and Lemmata 2, 3, it follows that |(fm , g)| = |(f − fm , g)| ≤ kf − fm kkgk → 0

(17)

for each g ∈ H as m → ∞. Since k.k(1) ≥ k.k(∞) , lim kf − fm k(∞) = 0

m→∞

(18)

applies to both cases under consideration. Considering the seminorms Pm (h) = |(fm , h)|, h ∈ H, m = 1, 2, . . . , and noticing that for each fixed h ∈ H, {Pm (h)}∞ m=1 is bounded (since limm→∞ Pm (h) = 0, see (17)), the uniform boundedness principle applies. Hence for each > 0 there is some k.k(∞) −neighborhood V of 0 such that |Pm (v)| < , for all v ∈ V, m = 1, 2, . . . , i.e., kfm k0(∞) = sup |(fm , g)| → 0 g∈V

(19)

as m → ∞. Using k.k0(∞) = k.k(∞) (see Lemma 3), f = 0 now follows from (18) and (19). iii): Since k.k(∞) defines a minimal Hilbert majorant on H(∞) , iii) follows from ˆ is an ortho-complemented Proposition 2 (i) ⇒ (iii). iv): Noticing that L = span{1} (∞) and positive definite subspace of H , there is a fundamental decomposition H(∞) = . H+ (+)H− such that L ⊂ H+ (see [5, Theorems V.3.4, V.3.5]), and 1ˆ ∈ H+ . Recalling that in Krein spaces all the interesting l.c. topologies such as all the decomposition majorants and the Mackey topology τM (H(∞) , H(∞) ) of the duality (H(∞) , H(∞) ) defined by the inner product (., .) coincide there is as well a natural notion of continuity in H(∞) as further equivalent descriptions of the Krein space topology τ∞ . Along these lines, the following considerations are aimed at further descriptions of the Krein-space norm k.k(∞) . Let us consider the positive definite inner products [., .](n)

(resp. [., .]0(n) , [., .](∞) )

which are defined by k.k(n) (resp. k.k0(n) ,k.k(∞) ) on H(1) , n = 1, 2, . . . . Lemma 2 implies that there is an hermitian and bounded operator G (Gram operator) such that (x, y) = [Gx, y](1) ,

(20)


311

kGk(1) = sup |[Gx, x](1) | = sup |(x, x)| ≤ sup kxk2(1) = 1, kxk(1) =1

kxk(1) =1

kxk(1) =1

x, y ∈ H . A relation between [., .](∞) and [., .](1) is given in the following. (1)

Lemma 4. It holds [x, y](∞) = [|G|x, y](1) , x, y ∈ H(1) , where G is taken from (20). Proof. Let us introduce hermitian and bounded operators Hn,n−1 , Hn by [x, y](n) = [Hn,n−1 x, y](n−1) , [x, y](n) = [Hn x, y](1) , Hn = H1,2 H3,2 . . . Hn,n−1 , x, y ∈ H(1) , n = 2, 3, . . .. Notice that the inverse operators (Hn )−1 , (Hn,n−1 )−1 exist due to Lemma 2. Since kxk0(1) =

sup |(x, y)| =

kyk(1) ≤1

it follows that

sup |[Gx, y](1) | = kGxk(1) ,

kyk(1) ≤1

[x, y]0(1) = [G2 x, y](1) , 1 [x, y](2) = [ (I + G2 )x, y](1) , 2

x, y ∈ H(1) . Hence, H2,1 =

1 (I + G2 ). 2

Then, kxk0(2) = =

sup |(x, y)| =

kyk(2) ≤1

sup |[Gx, y](1) |

kyk(2) ≤1

sup |[(H2,1 )−1 Gx, y](2) | = k(H2,1 )−1 Gxk(2) ,

kyk(2) ≤1

[x, y]0(2) = [(H2,1 )−2 G2 x, y](2) = [(H2,1 )−1 G2 x, y](1) , 1 [x, y](3) = [ (I + (H2,1 )−2 G2 )x, y](2) . 2 Hence, 1 (I + (H2,1 )−2 G2 ), 2 1 H3 = H2,1 H3,2 = (H2,1 + (H2,1 )−1 G2 ) 2

H3,2 =

follow. Arguing as above it follows that [x, y]0(n) = [(Hn )−2 G2 x, y](n) , 1 Hn+1,n = (I + (Hn )−2 G2 ), 2 1 Hn+1 = (Hn + (Hn )−1 G2 ), 2

(21)

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where the operators Hn and G are commuting, and Hn , Hn−1 are strictly positive due to mn+1 := m1 =

inf [Hn+1 x, x](1) ≥

kxk(1) =1

1 1 inf [Hn x, x](1) =: mn , 2 kxk(1) =1 2

inf [x, x](1) = 1,

kxk(1) =1

R1 n = 1, 2, . . . . Using the spectral resolution G = −1 λdEλ , it follows straightforwardly that G2 ≤ (H2,1 )2 and (22) 0 ≤ G2 ≤ Hn+1 ≤ Hn . √ 2 2 Hence, H = limn→∞ Hn exists, and (21) yields H = G , and also H = |G| (=: G2 ) due to (22). The proof of the lemma under consideration is completed. Starting with k.k(1) and using Theorem 1i), ii), there is a special fundamental symmetry G (23) J˜ = |G| satisfying J˜ = J˜∗ = J˜−1 on H(1) , where G is taken from (20). Considering the positive definite inner product ˜ (24) [x, y](J)˜ = (x, Jy), the decomposition majorant τJ˜ is defined by the norm q kxk(J)˜ = [x, x](J)˜ , x, y ∈ H(1) . Furthermore, let τJ denote any decomposition majorant on H(1) . τ

τ

J J˜ g g (1) (1) , and ii) τ = τ = τ (H(∞) , H(∞) ) = Proposition 3. It holds i) H(∞) = H =H J M J˜ τ∞ on H(∞) , (where τJ , τJ˜ also denote the l.c. topologies on H(∞) obtained by continuous extension of the corresponding norms).

Proof. i): Noticing that (20), (23) and (24) imply ˜ = [x, GJy] ˜ (1) = [x, |G|y](1) , [x, y](J)˜ = (x, Jy) x, y ∈ H(1) , Lemma 4 yields τJ˜ = τ∞ on H(1) . Recalling also that every fundamental decomposition of H(1) leads to a decomposition majorant τJ which is equivalent to τJ˜ (see [5, IV.6.4]), i) follows from [5, V.2.1]. ii): Since H(∞) is a Hilbert space with scalar product [., .](J)˜ , (24) implies that τM (H(∞) , H(∞) ) is defined by the norm ˜ (J)˜ , p(x) = kJxk x ∈ H(∞) (cf. [5, ch. IV.8]). Since J˜ is a symmetry q q ˜ J˜2 x) = (Jx, ˜ x) = kxk(J)˜ ˜ (J)˜ = (Jx, kJxk follows. Hence, τJ˜ = τM (H(∞) , H(∞) ) on H(∞) completing the proof of ii).

Remark 4 (to Proposition 3). Assuming that (H) applies and noticing that the foregoing considerations are constructive, there are the following two constructions leading to equivalent Krein space topologies.


313

1. Using Lemmata 1, 2, a Krein space topology is given by kf k(∞) = lim kf k(n) = lim kf k0(n) . n→∞

n→∞

2. Using the Gram operator (see (20)), Lemma 4 yields p kf k(∞) = k |G|f k(1) , f ∈ D. Combining Proposition 1 and Theorem 1, the following generalization of the GNS construction to J-representations on indefinite inner product spaces is implied. Theorem 2. Let us be given A and W as above. a) If W satisfies condition (H), then there is a J-representation π with an algebraically cyclic vector ψ0 ∈ D(π) such that W (f ) = (ψ0 , π(f )ψ0 ), f ∈ A, and Jψ0 = ψ0 . b) Conversely, if π is a J-representation on D, then the hermitian linear functional T (f ) = (ψ, π(f )ψ), f ∈ A, satisfies (H) for each ψ ∈ D. ˆ Theorem 1 implies now that Proof. a) Using Proposition 1, set π = πW , ψ0 = 1. x, a ∈ A, xˆ ∈ D, Definition 1 Definition 1 (i), (ii) apply. Recalling that πW (a)xˆ = ac (iii) follows from ∗ x, y) ˆ = (x, ˆ a cy) = W ((ay)∗ x) = W (y ∗ (a∗ x)) = (ad ˆ = (πW (a∗ )x, ˆ y), ˆ (x, ˆ πW (a)y)

a, x, y ∈ A. Further, Theorem 1 iv) implies Jψ0 = ψ0 . b) The linearity of T is obvious. Applying Definition 1 (iii), the hermiticity of T follows from T (x∗ ) = (ψ, π(x∗ )ψ) = (π(x)ψ, ψ) = (ψ, π(x)ψ) = T (x). Note further that |T (x∗ y)| = |(ψ, π(x∗ )π(y)ψ)| = |(π(x)ψ, π(y)ψ)| = |[Jπ(x)ψ, π(y)ψ]| ≤ kJπ(x)ψk kπ(y)ψk = Cx kπ(y)ψk, where Cauchy-Schwarz inequality was applied to the scalar product [., .], and k.k = √ [., .], Cx := kJπ(x)ψk < ∞. Noticing finally that kπ(.)ψk is a quadratic seminorm on A, it follows that (H) applies to T . Remark 5 (to Theorem 2). a) Noting that every P-functional on A (for definition see [1]) obviously satisfies all the assumptions of Theorem 2a), the main result of [1, Theorem 3] is readily implied by our Theorem 2. b) In contrast to the well-known case of a positive functional W , a crucial distinction enters the theory concerning the uniqueness of the "reconstructed GNS-data" (π(.), ψ0 , D, H). While for positive W the GNS-data are uniquely defined up to a unitary intertwiner U , the situation is quite different in the case at hand in the following way.

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1) The Krein-space structure is not uniquely determined in general, (see also Proposition 5 and Corollary 2, below). 2) If (π1 (.), ψ1 , D1 , H1 ) is a second set of GNS-data also satisfying Theorem 2a), then due to Proposition 1 there is a linear and invertible operator T : D → D1 such that T ψ0 = ψ1 , π1 (.) = T π(.)T −1 , T −1 = T (∗) (pseudo-unitary), and however, T is not extendable to H as a bounded operator in general. Notice also that T is unitary with respect to [., .], if and only if T J = JT . c) Using the constructions explained in Remark 4, there is a further distinction in contrast to the well-known case of a positive functional W . While if the functional W under consideration is positive, the scalar product < ., . > of the state space obtained by GNS construction satisfies ∗ h >, < fcg, hˆ >=< g, ˆ fd

f, g, h ∈ A, in the case at hand [., .] does not satisfy such a condition in general. 5. Some Further Properties of the Representation Space There is the following answer to (Q1). Theorem 3. Considering (i) (ii) (iii) (iv) (v)

Condition (H) is satisfied, there is a quadratic majorant on D, there exists a Hilbert-space structure (H, J) on D, there exists a Krein-space structure (H0 , J 0 ) on D, e τJ , there is a non-degenerate and decomposable subspace E ⊂ D such that D ⊂ E where τJ refers to the corresponding decomposition majorant on E, (vi) statement (iv) applies and D ∩ J 0 D is τJ 0 -dense in D,

the following implications are implied: (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) ⇐ (v) ⇔ (vi). Proof. (i) ⇒ (ii) : follows from Lemma 2. (ii) ⇒ (iii) : Corollary 1 implies that q (2) inherits a positive definite inner product [., .] on D. Since ρ2 is a majorant on A, there is a Gram operator J such that (1), (2) apply. (iii) ⇒ (iv) : Assuming (iii), Eq. (1) and (2) yield the spectral resolution Z m λ dEλ , J= −m

0 < m < ∞, in Hilbert space H, [., .]. Considering a second scalar product [x, y]0 := [x, |J|y], x, y ∈ H, it follows

Setting P + :=

Rm 0

kxk0 :=

p p p √ [x, |J|x] ≤ kJxk kxk ≤ m kxk.

dEλ and J 0 := 2P + − I, it follows (x, y) = [x, Jy] = [x, |J|(2P + − I)y] = [x, J 0 y]0 .


315

e k.k0 , J 0 ) is a Krein-space structure Noticing J 0 = J 0∗ = J 0−1 , it follows that (H0 = H on D. Hence (iv) applies. (iv) ⇒ (i) : Since a Krein space structure is a special Hilbert space structure (1), (2) apply. Then, ˆ = |[fˆ, J g]| ˆ ≤ kfˆkkJ gk ˆ = Cg p(f ) |W (f ∗ g)| = |(fˆ, g)| q ˆ < ∞, p(f ) = [fˆ, fˆ]. (v) ⇒ (i) : If E, (., .) is a nonyield (H), where Cg = kJ gk degenerate. and decomposable inner product space with fundamental decomposition E = E + (+)E − and fundamental projections P ± : E → E ± , then the corresponding decomposition majorant τJ is given by p kxkJ = (x, Jx), J = P + − P − , x ∈ E. Let k.kJ be continuously extended onto E˜ τJ , and this extended quadratic norm also be denoted by k.kJ . Using [5, Lemma II.11.4], the proposition under consideration now follows from ˆ y)| ˆ ≤ kxk ˆ J kyk ˆ J = Cy p(x), |W (y ∗ x)| = |(x, where Cy = kyk ˆ J , p(x) = kxk ˆ J , x ∈ x, ˆ y ∈ y, ˆ x, ˆ yˆ ∈ D. (v) ⇒ (vi) : Assuming (v), . consider a fundamental decomposition E = E + (+)E − , the corresponding fundamental symmetry J = P + − P − and decomposition majorant τJ . Since J : E → E, E = E ∩ JE ⊂ D ∩ JD e τJ , it follows that D∩JD is τJ -dense in D. Considering H = E e τJ , follows. Due to D ⊂ E let the τJ -continuous extensions of respectively, J and the inner product (., .) from E and E × E onto H and H × H be denoted by J 0 and (., .)0 . Noticing J 0 = J 0∗ = J −1 , it follows that (H, [., .]J 0 = (., J 0 .)0 ) is a Krein space structure on D. (vi) ⇒ (v) : Assuming (vi) and setting E = D ∩ JD, (v) follows. Remark 6. a) If Theorem 3 (iv) applies, then Theorem 1 (iv) implies that there is an ˆ ˜ satisfying J˜1ˆ = 1. equivalent Krein-space structure (D, J) b) Let us mention that Theorem 3 (v) ⇒ (i) yields a sufficient criterion useful for constructing non-decomposable inner product spaces. More precisely, if an inner product space does not satisfy (H), then necessarily, it is non-decomposable. Hence, the inner product space considered in Example 1 is non-decomposable. c) The proof of Theorem 3 (iii) ⇒ (iv) is based on the idea of introducing the new scalar product [., .]0 on H which is due to Ginsburg and Iokhvidov (see [9, §2 7(3)] and [23, Remark to Theorem 5]). More precisely, if the inverse J −1 of the Gram operator J considered in (iii) ⇒ (iv) of the above proof is unbounded in H, [., .], then both scalar products [., .] and [., .]0 are inequivalent, and thus the Hilbert-space structure (H, J) is inequivalent to the Krein-space structure (H0 , J 0 ). An answer to (Q2) is given in the following. Theorem 4. The following are equivalent: (i) there is a Krein-space structure (H, J) on D such that J : D → D, (ii) D, (., .) is a decomposable inner product space, (iii) W is a generalized P-functional (see Def. 2).

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Proof. (i) ⇒ (ii) : Assuming (i), consider D± := {x ± Jx; x ∈ D} ⊂ D. .

Then, D = D+ (+)D− yields a fundamental decomposition of the inner product space D, (., .), and hence (ii) is satisfied. (ii) ⇒ (iii) : Assuming (ii), there is a fundamental decomposition .

D = D+ (+)D− with 1ˆ ∈ D+ . The corresponding fundamental symmetry J = P + − P − then satisfies ˆ Considering now a direct decomposition J 1ˆ = 1. .

A = NW + B with 1 ∈ B, there is a linear bijection η : B → D. Define now a linear mapping α : A → A by setting x for x ∈ NW . α(x) = η −1 Jη(x) for x ∈ B It straightforwardly follows that Definition 2 applies to W and α. (iii) ⇒ (i) : Applying [1, Lemma 2, Theorem 3], the implication under consideration follows. The following is concerned with (Q3). Recalling the definition of intrinsic topology τint on definite subspaces of inner product spaces (see, e.g., [5, ch. III.3]), there is the following supplement to Theorem 3 (v) ⇒ (i). Proposition 4. Assume that there is a decomposable subspace E ⊂ D with fundamental . decomposition E = E + (+)E − such that D ⊂ E˜ τJ . If E + or E − are intrinsically complete, then up to equivalence, there exists exactly one Krein-space structure on D. ˜ on D. Hence, Proof. Theorem 3 implies that there is some Krein-space structure (H, J) τJ˜ is a minimal majorant on E. Assume now that there is a further Krein-space structure (H0 , J 0 ) on D being non-equivalent to the above. Consequently, τJ˜ |D 6= τJ 0 |D. However, recalling that there is only one minimal majorant on E ([5, Theorem IV.6.2]), τJ˜ |E = τJ 0 |E follows. Now D ⊂ E˜ τJ implies τJ˜ |D = τJ 0 |D which is a contradiction to the above. For the important class of quasi-positive (resp. quasi-negative) inner product spaces, i.e., D does not contain any negative definite (resp. positive definite) subspace of infinite dimension, the following corollary readily follows from Proposition 4 and the fact that every quasi-positive (resp. quasi-negative) inner product space is decomposable. Corollary 2. If D is quasi-positive or quasi-negative, then there exists exactly one Kreinspace structure on D. There is the following interesting application of Corollary 2 to axiomatic quantum field theory. Remark 7. Recalling that the one-particle space D1 of the free massless quantum field of space-time dimension 2 is quasi-positive ([8]) there is exactly one Krein-space structure on D1 . Hence, all the n-particle spaces of the corresponding Fock-space are uniquely determined.


317

6. On Positively Dominated Functionals Introducing the cone of positive elements ( A+ =

M X

) f (i)∗ f (i) ; f (i) ∈ A, M ∈ N ,

i=1

a linear functional T on A is called positive if T (f ) ≥ 0, f ∈ A+ . Inspired by the significance of positive functionals within the investigations on *-algebras (see [26, 30] and references cited there), let us consider an interesting subclass of hermitian functionals leading to Krein-space structures. Definition 3. A hermitian linear functional W on A is called positively dominated, if there is some positive functional T on A such that |W (g ∗ f )|2 ≤ T (g ∗ g)T (f ∗ f ), g, f ∈ A. A characterization of positively dominated functionals is given in the following. Theorem 5. Considering (i) W is positively dominated, (ii) there exist a Krein-space structure on D and a positive functional T on A such that kfˆk(∞) ≤

p T (f ∗ f ), f ∈ fˆ,

q where kfˆk(∞) = (J fˆ, fˆ), fˆ ∈ D, (iii) there are a quadratic majorant k.k on D and a positive functional T on A such that p kfˆk ≤ T (f ∗ f ), f ∈ fˆ, (iv) there are positive functionals T (j) on A, j = 1, 2, such that W = T (1) − T (2) , (v) there is a locally convex topology τ on A such that W is continuous and A+ is normal, the following implications are implied: (v) ⇒ (i) ⇔ (ii) ⇔ (iii) ⇔ (iv). Remark 8. Concerning (v) let us mention that in [11] there are explicitly constructed some families of normal topologies in the case of tensor algebras A = E⊗ .

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√ Proof. (i) ⇒ (ii) : Noticing that p(f ) := T (f ∗ f ), f ∈ A, is a quadratic seminorm satisfying |W (g ∗ f )| ≤ Cg p(f ), where Cg = p(g), (H) follows. Hence, there exists a Krein-space structure on D due to Theorem 3. In order to show the remainder of (ii), notice that the polar seminorm p0 satisfies p0 (f ) ≤ p(f ) because of |W (g ∗ f )| ≤ p(g)p(f ), g, f ∈ A. Hence, r 1 00 2 ˆ ˆ kf k(∞) ≤ kf k(1) = p (f ) + p0 (f )2 2 r r 1 1 2 0 2 ≤ p(f ) + p (f ) ≤ p(f )2 + p(f )2 = p(f ) 2 2 p = T (f ∗ f ) completes the proof of (ii). (ii) ⇒ (iii): Lemma 3 yields (iii) with k.k = k.k(∞) . (iii) ⇒ (iv): Assuming that (iii) apply, |W (f ∗ f )| ≤ kfˆk2 ≤ T (f ∗ f ), (1) (2) f ∈ A, follows. p Setting T = Tp+ W, T = T , (iv) is implied. (iv) ⇒ (i): Assuming (j) ∗ (j) ∗ (iv), set aj = T (f f ), bj = T (g g), j = 1, 2, and consider the positive linear functional T = T (1) + T (2) .

Now, (i) follows from |W (g ∗ f )|2 = |T (1) (g ∗ f ) − T (2) (g ∗ f )|2 ≤ ≤ a21 b21 + a22 b22 + 2|T (1) (g ∗ f )T (2) (g ∗ f )| ≤ ≤ a21 b21 + a22 b22 + 2a1 b1 a2 b2 ≤ (a21 + a22 )(b21 + b22 ) = = T (f ∗ f )T (g ∗ g), f, g ∈ A. (v) ⇒ (iv) : follows from [28, V.3.3 Cor. 3].

In Proposition 5 it will be shown that in the case of tensor algebras having a countable algebraic basis every hermitian linear functional is positively dominated. On the other hand, the existence of non-positively dominated functionals follows from the following. Example 2. Consider

B = {1, e, f, ef, f e, ee}

and the vector space A = span (B) of dimension 6. Defining b = b∗ , 1 · b = b, b ∈ B, and e · f = ef, f · e = f e, e · e = −f · f = ee, ef · b0 = b0 · ef = f e · b0 = b0 · f e = ee · b0 = b0 · ee = 0, b0 ∈ B \ {1}, A becomes a unital *-algebra. Consider further the hermitian linear functional W defined by W (b) = 1, b ∈ B. Noticing A+ = {λ1 + µee; λ ≥ 0, µ ∈ R}, it follows that every positive functional T on A satisfies T (ee) = 0,


319

and thus Theorem 5 (iv) does not apply to W . Furthermore, the inner product (f, g) = W (g ∗ f ), g, f ∈ A, satisfies (f, g) = [f, Ag], here [., .] denotes the Euclidean metric and the transformation A is given by the matrix A = (aij )6i,j=1 with respect to the basis B, where a1j = aj1 = a22 = a23 = a32 = −a33 = 1, j = 1, . . . , 6, and aik = 0 otherwise. Introducing an eigenbasis of A, it follows (f, g) = [V f, DV g], where D = diag {λ1 , . . . , λ6 }, λ1 ≈ 3.07, λ2 ≈ −1.84, λ3 ≈ −1, 15, λ4 ≈ 0.92, λ5 = λ6 = 0, and V describes the corresponding transformation of coordinates. Hence, A(0) = {x ∈ A; (V x)j = 0, j = 1, . . . , 4} and D = A/A(0) is a Krein space, where [x, y]0 =

4 X

|λi |(V x)i (V y)i

i=1

and J = diag {1, −1, −1, 1} is the corresponding fundamental symmetry.

7. Applications to Tensor Algebras Let us apply the preceding to the special case of tensor algebras A = E⊗ := C ⊕ E1 ⊕ E2 ⊕ . . . , where En = E ⊗ E ⊗ . . . ⊗ E ( n copies). (For definitions and notions the reader is referred to [10, 11].) There is the following immediate consequence of Theorem 3. Corollary 3. If A = E⊗ , then the following are equivalent. (i) Condition (H) applies, (ii) there is a sequence of quadratic semi norms pn on En such that for each gm ∈ Em there are constants Cgm < ∞ with ∗ |Wn+m (gm ⊗ fn )| ≤ Cgm pn (fn ),

fn ∈ En (m, n = 0, 1, 2, . . .).

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Proof. (i) ⇒ (ii) : Using the assertion of Theorem 3 (iii), ∗ |Wn+m (gm ⊗ fn )| = |[fˆn , J gˆm ]| ≤ kJ gˆm kkfˆn k = Cgm pn (fn ),

where P Cgm = kJ gˆm k, pn (fn ) = kfˆn k. (ii) ⇒ (i) : Considering any g ∈ E⊗ , put ∞ Cg := n=0 Cgn and notice Cgn < ∞. Defining the quadratic semi norm v u∞ uX p(f ) = t 2n pn (fn )2 n=0

on E⊗ , |W (g ∗ f )| ≤

X

|Wm+n (gn∗ ⊗ fm )| ≤

m,n

X

Cgn pm (fm ) ≤

m,n

v uX ∞ X √ u∞ ≤ Cg t 2−j 2m pm (fm )2 = 2Cg p(f ) j=0

yields (H).

m=0

Since there are no non-trivial relations between the elements of E⊗ , the following being of some interest with respect to Theorem 5 is implied. Proposition 5. Let us be given a tensor algebra E⊗ having a countable (algebraic) basis. The following are equivalent. (i) W = (1, W1 , W2 , . . .) is an hermitian linear functional on E⊗ , (ii) W is positively dominated. Proof. (ii) ⇒ (i) is obvious. (i) ⇒ (ii) : Let {b(n) }∞ n=1 denote a basis of the basic space E. Setting [n1 , n2 , . . . , nm ] := b(n1 ) ⊗ b(n2 ) ⊗ . . . ⊗ b(nm ) , a basis of E⊗ is given by B = {1} ∪ {∪∞ m=1 [n1 , . . . , nm ]}, nj ∈ N, (j = 1, . . . , m). Let us also use the abbreviations ˜ := [nm , . . . , n1 ], [n] := [n1 , . . . , nm ], [n] [n] ∪ [r] := [n1 , . . . , nm , r1 , . . . , rs ], where [r] := [r1 , . . . , rs ], ri ∈ N, (i = 1, . . . , s). Setting ˜ wn1 ,...,nm := w[n] = |W ([n])| = |W ([n])|, the following is aimed at a recursive definition of constants cn1 ,...,nm := c[n] > 0 such that the following two systems of inequalities are satisfied: |w[s]∪[t] |2 ≤ c[s] c[t] ,

(25)

|c[s]∪[t] |2 ≤ c[s] c[t]∪[˜t]∪[˜s] ,

(26)

for all [s], [t] ∈ B. In order to introduce an ordering in B \ {1} let us consider the decomposition B = ∪∞ m=0 Dm


321

(m) into nonintersecting classes, where D0 = {1}, Dm = {∪m l=1 1l },

1(m) m = {[n1 , . . . , nm ]; nj ≤ m (j = 1, . . . , m)}, 1(m) = {[n1 , . . . , nl ]; nj ≤ m (j = 1, . . . , l), there is a l j0 ∈ {1, . . . , l} with nj0 = m}, (mj )

(j) l = 1, 2, . . . , m − 1. Considering any [n(j) 1 , . . . , nlj ] ∈ 1lj ordering " ≺ " by setting

, j = 1, 2, define an

(1) (2) (2) [n(1) 1 , . . . , nl1 ] ≺ [n1 , . . . , nl2 ] (1) if i) m1 < m2 , or ii) m1 = m2 and l1 < l2 , or iii) m1 = m2 , l1 = l2 and [n(1) 1 , . . . , n l1 ] (2) stands before [n(2) 1 , . . . , nl2 ] with respect to lexicographic ordering. Define now c[1] = max{1, |w[1] |2 }. Let us then consider some [n] = [n1 , . . . , ns ] ∈ B \ {1} and assume that c[m] are defined for all [m] ∈ B \ {1} with [m] ≺ [n]. Let us put

˜ Z[n] := {([r], [b]) ∈ B × B; [r] ≺ [n], [b] ≺ [n], [r] ∪ [n] = [b] ∪ [b]}, 2 |c[b] | ; ([r], [b]) ∈ Z[n] , d[n] := max c[r] |w[m]∪[n] |2 e[n] := max ; [m] ≺ [n] , c[m]

(27) (28)

and notice that the max in (27), (28) is only taken over finitely many items. Define then c[n] = max{1, d[n] , e[n] }, and note that (25), (26) are satisfied. Setting now T0 = 1, T2s−1 = 0, T2s ([m1 , . . . , ms ]∗ ⊗ [n1 , . . . , ns ]) := δn1 m1 δn2 m2 . . . δns ms c[n1 ,...,ns ] , s = 1, 2, . . . , (δ.. denotes Kronecker’s δ) (28), (29) imply after linear extension of T2s , |Wν+µ (fν∗ ⊗ gµ )|2 ≤ T2ν (fν∗ ⊗ fν )T2µ (gµ∗ ⊗ gµ ) , |Tν+µ (fν∗

⊗ gµ )| ≤ 2

T2ν (fν∗

⊗

fν )T2µ (gµ∗

⊗ gµ ) ,

(29) (30)

for all fν ∈ Eν , gµ ∈ Eµ (ν, µ ∈ N). Recall that there are sequences {αn }∞ n=0 , αn ≥ 0, α0 > 0, α2s+1 = 0 (s = 0, 1, . . .), such that the inequality of matices G ≥ E (unity matrix) is satisfied, where  α0 0 −α2 0 −α4 0 . . .  0 α2 0 −α4 0 −α6 . . . G= −α2 0 α4 0 −α6 0 . . . ........................... 

(see [11, Construction 3.6]). Thus,

322

G.Hofmann ∞ X

T2n (fn∗ ⊗ fn ) ≤

∞ X

n=0

α2n T2n (fn∗ ⊗ fn )

n=0 ∞ X

−

n=0

≤

∞ X

X p T2r (fr∗ ⊗ fr )T2s (fs∗ ⊗ fs )

α2n

r+s=2n r6=s

α2n T2n (fn∗ ⊗ fn ) +

n=0

=

∞ X

∞ X

α2n

n=0

α2n T2n (

n=0

X

X

Tr+s (fr∗ ⊗ fs )

r+s=2n r6=s

fr∗ ⊗ fs ),

(31)

r+s=2n

f = (f0 , f1 , . . . , fN , 0, 0, . . .) ∈ E⊗ , are implied. Considering T˜ = (T˜0 , T˜1 , . . .) with Tñ = αn Tn , n = 0, 1, . . ., (29), (31) imply |W (f ∗ g)|2 ≤

∞ X X

|Wµ+ν (fµ∗ ⊗ gν )|

n=0 µ+ν=n

≤

∞ X X

T2µ (fµ∗ ⊗ fµ )T2ν (gν∗ × gν )

n=0 µ+ν=n

≤

∞ X

T2µ (fµ∗ ⊗ fµ )

µ=0

∞ X

T2ν (gν∗ × gν )

ν=0

  ! ∞ ∞ X X X X ∗ ∗ α2µ T2µ ( fr ⊗ fs ) α2ν T2ν ( fr ⊗ fs ) ≤ µ=0 ∗

r+s=2µ

ν=0

r+s=2ν

∗

= T˜ (f f )T˜ (g g) for all f, g ∈ E⊗ . Hence W is positively dominated by T˜ . The proof is completed.

Acknowledgement. The author is indebted to Professor H. Araki for helpful and stimulating discussions. The suggestions of the referee are gratefully acknowledged.

References ˆ S.: Unbounded GNS representations of a *-algebra in a Krein space. Lett. Math. 1. Antoine, J.-P., Ota, Phys. 18, 267–274 (1989) 2. Araki, H.: On a pathology in indefinite inner product spaces. Commun. Math. Phys. 85, 121–128 (1982) 3. Araki, H.: Indecomposable representations with invariant inner product. A theory of the Gupta-Bleuler triplet. Commun. Math. Phys. 97, 149–159 (1985) 4. Azizov, T.Y., Iokhvidov, I.S.: Linear operators in spaces with indefinite metric. New York: John Wiley & Sons Inc., 1989 5. Bognár, J.: Indefinite inner product spaces. Berlin: Springer-Verlag, 1974 6. Bogoljubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T.: General principles of quantum field theory. Dordrecht: Kluwer, 1990


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7. Borchers, H.J.: On the structure of the algebra of field operators. Nuovo Cimento 24, 214–236 (1962) 8. Dubin, D.A., Tarski, J.: Indefinite metric resulting from regularization in the infrared region. J. Math. Phys. 7, 574–577 (1966) 9. Ginsburg, Ju.P., Iokhvidov, I.S.: Investigations on the geometry of infinite-dimensional vector spaces with bilinear metric. Uspechi Matem. Nauk 17, no.4, 3–56 (1962) (in Russian) 10. Hofmann, G.: On topological tensor algebras. Wiss. Z. Univ. Leipzig, Math.-Naturw. Reihe 39, 598–622 (1990) 11. Hofmann, G.: On algebraic #-cones in topological tensor algebras, I. Basic properties and normality. Publ. RIMS, Kyoto University 28, 455–494 (1992) 12. Hofmann, G.: An explicite realization of a GNS representation in a Krein-space. Publ. RIMS, Kyoto University 29, 267–287 (1993) 13. Hofmann, G.: On the cones of α-positivity and generalized α-positivity for quantum field theories with indefinite metric. Publ. RIMS, Kyoto Univ. 30, 641–670 (1994) 14. Hofmann, G.: On the GNS representation of generalized free fields with indefinite metric. Rep. Math. Phys. 38, 67–83 (1996) 15. Hofmann, G.: Generalized free field like U (1)-gauge theories within the Wightman framework. Rep. Math. Phys. 38, 85–103 (1996) 16. Hofmann, G.: The Hilbert space structure condition for Quantum Field Theories with indefinite metric and transformations with linear functionals. To appear in Lett. Math. Phys. 17. Ito, K.R.: Canonical linear transformation on Fock space with an indefinte metric. Publ. RIMS, Kyoto Univ. 14, 503–556 (1978) 18. Jakóbczyk, L.: Borchers algebra formulation of an indefinite inner product quantum field theory. J. Math. Phys. 29, 617–622 (1984) 19. Jakóbczyk, L., Strocchi, F.: Euclidean formulation of quantum field theory without positivity. Commun. Math. Phys. 119, 529–541 (1988) 20. Köthe,G.: Topological vector spaces, I. Berlin, New York, Heidelberg: Springer-Verlag, 1984 21. Kugo, T., Ojima, I.: Local covariant operator formalism of non-abelian gauge theories and quark confinement problem. Suppl. of the Progr. of Theor. Phys. 66, 1–130 (1979) 22. Mintchev, M.: Quantization in indefinite metric. J. Phys. A: Math. Gen. 13, 1841–1859 (1980) 23. Morchio, G., Strocchi, F.: Infrared singularities, vacuum structure and pure phases in local quantum field theory. Ann. Inst. Henri Poincaré 33, 251–282 (1980) 24. Morchio, G., Pierotti, D., Strocchi, F.: Infrared and vacuum structure in two-dimensional local quantum field theory models. The massless scalar field. J. Math. Phys. 31, 1467–1477 (1990) 25. Nakanishi, N., Ojima, I.: Covariant operator formalism of gauge theories and quantum gravity. Singapore: World Scientific, 1990 26. Neumark, M.A.: Normierte Algebren. Thun, Frankfurt am Main: Verlag Harri Deutsch, 1990 27. Sakai, S.: C ∗ -Algebras and W ∗ -Algebras. Berlin: Springer-Verlag, 1971 28. Schaefer, H.H.: Topological vector spaces. London: Collier-Macmillan Limited, 1966 ¨ 29. Scheibe, E.: Uber Feldtheorien in Zustandsräumen mit indefiniter Metrik. Mimeographed notes of the Max-Planck Institut für Physik und Astrophysik in München, München, 1960 30. Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Berlin: Akademie-Verlag, 1990 31. Strocchi, F.: Selected Topics on the General Properties of QFT. Lecture Notes in Physics Vol. 51, Singapore, New Jersey, Hong Kong: World Scientific, 1993 ˆ S.: Unbounded representation of a *-algebra on indefinite metric space. Ann. Inst. Henri Poincaré 32. Ota, 48, 333–353 (1988) 33. Powers, R.T.: Self-adjoint algebras of unbounded operators, I. Commun. Math. Phys. 21, 261–293 (1971), II. Trans. Am. Math. Soc. 167, 85 (1974) ¨ 34. Uhlmann, A.: Uber die Definition der Quantenfelder nach Wightman und Haag. Wiss. Zeitschr. d. Univ. Leipzig 11, 213–217 (1962) 35. Yngvason, J.: Remarks on the reconstruction theorem for field theories with indefinite metric. Rep. Math. Physics 12, 57–64 (1977) Communicated by H. Araki

This article was processed by the author using the LaTEX style file pljour1 from Springer-Verlag.

Commun. Math. Phys. 191, 325 – 395 (1998)

Communications in


Rogers–Schur–Ramanujan Type Identities for the M (p, p0 ) Minimal Models of Conformal Field Theory Alexander Berkovich1 , Barry M. McCoy2 , Anne Schilling2 1

Physikalisches Institut der Rheinischen Friedrich-Wilhelms Universität Bonn, Nussallee 12, D-53115 Bonn, Germany. E-mail: berkov [email protected] 2 Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA. E-mail: [email protected]; [email protected] Received: 23 July 1996 / Accepted: 15 May 1997

Dedicated to the memory of Poline Gorkova

Abstract: We present and prove Rogers–Schur–Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model M (p, p0 ). The proof uses the continued fraction decomposition of p0 /p introduced by Takahashi and Suzuki for the study of the Bethe’s Ansatz equations of the XXZ model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of Forrester and Baxter. (p,p0 ) for many We use this method to get fermionic representations of the characters χr,s classes of r and s.

1. Introduction Rogers–Schur–Ramanujan type identities is the generic mathematical name given to the identities which have been developed in the last 100 years from the work of Rogers [0], Schur [1] and Ramanujan [2] who proved, among other things, that for a = 0, 1, 2 ∞ X q j +aj

j=0

(q)j

=

∞ Y j=1

1 (1 − q 5j−1−a )(1 − q 5j−4+a )

∞ 1 X j(10j+1+2a) = (q − q (5j+2−a)(2j+1) ), (q)∞ j=−∞

(1.1)

where (q)k =

k Y j=1

(1 − q j ), k > 0; (q)0 = 1.

(1.2)

326

A. Berkovich, B. M. McCoy, A. Schilling

These identities are of the greatest importance in the theory of partitions [4–5] and number theory and by the early 50’s at least 130 of them were known [5]. The emergence of these identities in physics is much more recent, starting with the work of Baxter [6], Andrews, Baxter and Forrester [7] and the Kyoto group [8] in the 80’s on the order parameters of solvable statistical mechanical models. An even more recent relation to physics is the application to conformal field theories invented by Belavin, Polyakov and Zamolodchikov [9] in 1984. Here the left hand side is obtained by using a fermionic basis and the right hand side is obtained by using a bosonic basis for the Fock space. The bosonic constructions are done in a universal fashion using the methods of Feigin and Fuchs [11, 12]. The construction of the fermionic basis is more involved. The earliest example of such a fermionic representation is for modules of the affine Lie algebra A(1) 1 [12], but the general theory of this application of the identities has only been explicitly developed in the last several years [13]-[17]. However some of the mathematics of these constructions is already present in the original identities (1.1) which are now recognized as being the fermi/bose identities for the conformal field theory M (2, 5). Thus in some sense one might say that the 1894 paper of Rogers [0] is one of the first mathematical contributions to conformal field theory even though conformal field theory as a physical theory was invented only in 1984 [9]. The theory of bosonic representations of conformal field theory characters is well developed. In particular the characters of all M (p, p0 ) minimal models are given by the formula [18] (p,p0 ) (p,p0 ) (q) = q 1r,s −c/24 Br,s (q), (1.3) χˆ r,s where Br,s (q) =

∞ 0 1 X j(jpp0 +rp0 −sp) (q − q (jp +s)(jp+r) ), (q)∞ j=−∞

(1.4)

with conformal dimensions 0

(p,p ) 1r,s =

(rp0 − sp)2 − (p − p0 )2 (1 ≤ r ≤ p − 1, 1 ≤ s ≤ p0 − 1), 4pp0

and central charge c=1−

6(p − p0 )2 . pp0

(1.5)

(1.6)

p and p0 are relatively prime and we note the symmetry property Br,s (q) = Bp−r,p0 −s (q). It is obvious that (1.4) generalizes the sum on the right-hand side of (1.1). The generalization of the q-series on the left-hand side of (1.1) has a longer history. The first major advance was made in the 70’s when Andrews realized [19] that there were generalizations of (1.1) in terms of multiple sums of the form X m1 ,···,mk−1

1

q2m

T

Bm+AT m

k−1 Y i=1

1 (q)mi

(k ≥ 2)

(1.7)

where B is a (k − 1) by (k − 1) matrix, A is a (k − 1)–dimensional vector and the summation variables mi run over positive integers. These results are now recognized as the characters of the M (2, 2k + 1) models. In the interpretation of [14–18] we say that each mi represents the number of fermionic quasi-particles of type i. A second generalization of the form (1.7) is that the summation variables may obey restrictions such as mi being even or odd or having linear combinations being congruent to some

Rogers–Schur–Ramanujan Identities for Minimal Models of CFT

327

value Q (mod N ) (where N is some integer). The odd/even restriction is present in the original work of Rogers [0] and the (mod N ) restrictions were first found by Lepowsky and Primc [12]. Further generalizations of (1.7) are needed to represent the most general character. In particular we need the general form of what we call the “fundamental fermionic form” which was first found in [14] (and generalizes the special case (5.5) of [20] which in retrospect is M (5, 5k + 2)) fr,s (u, q) =

X

q

T 1 T 2 m Bm+A m

m,restr.

k−1 Y i=1

((Ik−1 − B)m + u)i mi

,

(1.8)

q

where Ik−1 is the (k−1) by (k−1) dimensional unit matrix and u is a (k−1)–dimensional vector with components (u)i . (In general we impose the notation that the components of a vector u are either denoted by (u)i or ui . ui would denote a vector labeled by i and not its ith component). We define the q-binomial coefficients for nonnegative m and n as (q)m+n m+n m+n (1.9) = = (q)m (q)n if m, n ≥ 0, n q m q 0 otherwise. There exist generalizations of (1.9) to negative n, and their use in the context of fermionic characters was first found in [21]. We also note that using the property 1 m+n = (1.10) lim n→∞ m q (q)m the general form (1.8) reduces to (1.7) when ui → ∞ for all 1 ≤ i ≤ k − 1. Then in terms of these fundamental fermionic forms the generic form of the generalization of (1.1) is now given as the linear combination X q ci fr,s (ui ; q) = q Nr,s Br,s (q), (1.11) Fr,s (q) = i

where Nr,s is a normalization constant. Character identities of this form which generalize the results of [19] were conjectured for some special cases of M (p, p0 ) in [16] including p0 = p + 1. Proofs of the identities for M (p, p+1) are given in [23–25]. Several other special cases of p and p0 and particular values of r and s are proven in [25]. In a previous letter [26] two of the present authors gave results for the case of arbitrary p and p0 for certain selected values of r and s. Here we generalize and prove the results of that letter. Our method of proof is to generalize the infinite series for the bosonic and fermionic forms of the characters in (1.11) to polynomials Br,s (L, q) and Fr,s (L, q) whose order depends on an integer L and then to prove that both Br,s (L, q) and Fr,s (L, q) satisfy the same difference equations in L with the same boundary conditions. The generalization from infinite series to a set of polynomials is referred to as “finitization” [27]. For the proof of the L–difference equations we utilize the technique of telescopic expansion first introduced in [22] to prove the conjecture of [17, 18] for the M (p, p + 1) model and subsequently used to prove identities for the N = 1 supersymmetric model SM (2, 4ν) [21] and for general series of the A(1) 1 coset models with integer levels [28]. This method is the extension to many quasi particles of the recursive proof of (1.1) given by [2, 4 and 30]. (Somewhat different methods have been used to prove polynomial analogues of the Andrews–Gordon identities in [31–33].)

328


There are many ways to finitize the fermionic and bosonic forms of the characters. For example the bosonic character (1.4) has a polynomial generalization in terms of q-binomial coefficients Br(b),s (L, b; q) ∞ X 0 0 L L (jp+r(b))(jp0 +s) q j(jpp +r(b)p −sp) L+s−b , − q = L−s−b − jp0 q − jp0 q 2 2 j=−∞ (1.12) where L + s − b is even and r(b) is a prescribed function of b with 1 ≤ b ≤ p0 − 1 (see (3.4) below). This generalization first appeared in the work of Andrews, Baxter and Forrester [7] for p0 = p + 1 and for general p p0 in the work of Forrester and Baxter [33]. The bosonic polynomials in (1.12) have the symmetry Br(b),s (L, b; q) = Bp−r(b),p0 −s (L, p0 − b; q).

(1.13)

Thanks to (1.10), Eq. (1.12) tends to (1.4) as L → ∞. Notice that in this limit the dependence on b drops out and hence for each character identity there are several different polynomial identities with the same limit. The polynomials (1.12) generalize the polynomials used by Schur [1] in connection with difference two partitions and is the finitization we use in this paper. They satisfy a simple recursion relation in L and can be interpreted as the generating functions for partitions with prescribed hook differences [34]. However, there are several other known polynomial finitizations [22, 25, 33, 38] which satisfy other L difference equations and prove to be useful in other contexts. In this paper we will present a method which allows the construction of fermionic polynomials which satisfy the identities Fr(b),s (L, b; q) = q Nr(b),s Br(b),s (L, b; q)

(1.14)

for the general minimal model M (p, p0 ) in principle for all b and s. It is however difficult to find a notation which allows for a compact treatment of all values of b and s at the same time. Consequently even though our methods in this paper are general, we present results only for certain classes of b and s. However, we emphasize that all cases can be treated by the same methods. Additional results will be presented elsewhere. The polynomials appearing on the left-hand side of (1.14) generalize the polynomials originally used by MacMahon [3] in his analysis of (1.1). In contrast to the bosonic polynomials (1.12) the form of the fermionic polynomials depends on the values of b and s. This is because the fermionic polynomials depend on the continued fraction decomposition of p0 /p. We will present the formalism of this decomposition in Sect. 2 and defer the presentation of our results to Sect. 3. The proof of our results for p0 > 2p is given in Sects. 4–11. The case p < p0 < 2p is obtained from the case p0 > 2p in Sect. 12 by the method of the dual transformation discussed in [26]. We close in Sect. 13 with a discussion of several ways our results can be extended and with an interpretation of our polynomial identities in terms of new Bailey pairs.


329

2. Summary of the Formalism of Takahashi and Suzuki We begin with the observation that the models M (p, p0 ) are obtained as a reduction of the XXZ spin chain X y x z (σkx σk+1 + σky σk+1 + 1σkz σk+1 ), (2.1) HXXZ = − k

where σki (i = x, y, z) are the Pauli spin matrices and 1 = − cos π

p . p0

(2.2)

Consequently we may use the results of the classic study of the thermodynamics of the XXZ chain made by Takahashi and Suzuki [36] in 1972. This treatment begins by introducing, for p0 > 2p, the n + 1 integers ν0 , ν1 , · · · , νn from the following continued fraction decomposition of p0 /p, p0 = ν0 + 1 + p ν1 +

1 1 ν2 +···+ νn1 +2

,

(2.3)

where νn ≥ 0 and all other νj ≥ 1. For the case p0 < 2p we replace p by p0 − p. We say that this is a n + 1 zone decomposition and that there are νj types of quasi particles in zone j. From these integers we define (where µ is an integer, 0 ≤ µ ≤ n + 1) Pµ−1 for 1 ≤ µ ≤ n + 1 . j=0 νj (2.4) tµ = −1 for µ = 0 We refer to tn+1 as the number of types of quasi particles in the system. When an index j satisfies (2.5) tµ + 1 ≤ j ≤ tµ+1 + δn,µ , we say that the index j is in the µth zone and that 1 + tµ and t1+µ are the boundaries of this zone. Note that by definition zone 0 (n) has ν0 + 1 (νn + 1) allowed values of j while all other zones have νµ allowed values of j. We will sometimes refer to j = tn+1 + 1 and j = 0 as “virtual” positions. We will explicitly consider below the case p0 > 2p. The case p0 < 2p will be treated separately in Sect. 12. According to Takahashi and Suzuki [36], there are also νi types of quasi particles in zones i = 0, . . . , n − 1 for the XXZ chain. In zone n there are, however, νn + 2 types of quasiparticles, and in addition there is an extra zone n + 1 with one quasi particle in the XXZ chain. It is the omission of the three quasi particles of zone n and n + 1 which truncates the XXZ chain to the model M (p, p0 ). From the νj we define the set of integers yµ recursively as y−1 = 0, y0 = 1, y1 = ν0 + 1, yµ+1 = yµ−1 + (νµ + 2δµ,n )yµ , (1 ≤ µ ≤ n), (2.6) and further set lj = yµ−1 + (j − 1 − tµ )yµ for 1 + tµ ≤ j ≤ tµ+1 + δn,µ .

(2.7)

(µ) |0 ≤ µ ≤ n, j is in µth zone } We then define what we call the Takahashi length {l1+j

330


(µ) l1+j

=

j+1 yµ−1 + (j − tµ )yµ

for µ = 0 and 0 ≤ j ≤ t1 for 1 ≤ µ ≤ n and 1 + tµ ≤ j ≤ t1+µ + δn,µ ,

(2.8)

(µ) which differs from l1+j only at the boundaries j = tµ . Note that l1+j is monotonic in j th while l1+j is not. To indicate that j lies in the µ zone, i.e. 1 + tµ ≤ j ≤ t1+µ , we will in the following always write jµ instead of j. We also define a second set of integers zµ for as

z−1 = 0, z0 = 1, z1 = ν1 + 2δ1,n , zµ+1 = zµ−1 + (νµ+1 + 2δµ+1,n )zµ , (1 ≤ µ ≤ n − 1) (2.9) and zµ−2 + (jµ − tµ )zµ−1 for 1 ≤ µ ≤ n and 1 + tµ ≤ jµ ≤ t1+µ + δn,µ (µ) = l˜1+j µ 0 for µ = 0. (2.10) (µ) We refer to l˜1+j as a truncated Takahashi length. It is clear that z is obtained from µ µ the same set of recursion relations as the yµ except that zone zero is removed in the partial fraction decomposition of p0 /p. The removal of this zone zero is equivalent to 0 considering a new XXZ chain with an anisotropy 10 = − cos π{ pp }, where {x} denotes the fractional part of x. We note that the l(µ) (l˜(µ) ) are the dimensions of the unitary 1+jµ

1+jµ

p

p0

representations of the quantum group su(2)q± with q+ = e p0 (q− = eiπ{ p } ). The final result we need from [36] is the specialization of their equation (1.10) to the case of the 0th Fourier component. Then, using the notation where the integers nk (mk ) with 1 ≤ k ≤ tn+1 are the number of particle (hole) excitations of type k we find what we call (m, n) system (Eq. (2.18) of [26]) iπ

1 1 (mk−1 + mk+1 ) + u¯ k for 1 ≤ k ≤ tn+1 −1 and k 6= ti , i = 1, · · · , n, 2 2 1 1 nti + mti = (mti −1 + mti − mti +1 ) + u¯ ti , for i = 1, · · · , n, 2 2 1 1 ntn+1 + mtn+1 = (mtn+1 −1 + mtn+1 δνn ,0 ) + u¯ tn+1 , 2 2 (2.11) where by definition m0 = L, all u¯ k are integers and all mk are nonnegative integers. Let us emphasize here that whereas mk is always nonnegative, nk may at times take on negative values. We denote (2.11) symbolically as n k + mk =

n = Mm +

1 L ¯ e¯ 1 + u, 2 2

where we define the tn+1 -dimensional vectors e¯ k by δj,k 1 ≤ k ≤ tn+1 (¯ek )j = 0 k = 0, 1 + tn+1 . Additionally, we shall require the 1 + tn+1 -dimensional vectors ek defined as n (ek )j = δj,k 1 ≤ k ≤ 1 + tn+1 0 k = 0. Solving (2.12) for m yields

(2.12)

(2.13)

(2.14)


m = M−1 (n −

331

L 1 ¯ e¯ 1 − u). 2 2

(2.15)

The asymmetric matrix −2M is almost of block diagonal form with each block except the last being the tadpole Cartan matrix Aνi , and the last block being a regular Aνn Cartan matrix (unless νn = 0). Note that in the simplest case p = 1 the matrix −2M becomes the Cartan matrix Ap0 −3 . For these reasons it is natural to think of −2M as a new mathematical construct: a Cartan matrix of fractional size. Algebraic structures associated with −2M will be investigated elsewhere. From the system of equations (2.11) one may deduce the following partition problem for L by multiplying equation i in (2.11) by li and summing all equations up, tn+1 X

(ni −

i=1

u¯ i m(tn+1 ) L )li + l(1+tn+1 ) = , 2 2 2

(2.16)

where lj is given by (2.7) and l(1+tn+1 ) = p0 − 2yn . This partition problem can be employed to carry out an analysis along the lines of [24]. We will also need it later in the proof of the initial conditions and to determine allowed variable changes in the proof of the recurrences. Finally we note that it is useful to consider the cases p0 > 2p and p < p0 < 2p separately because the case p0 > 2p may be obtained from the case p0 < 2p by a “duality map” [26]. We concentrate first on the case p0 > 2p where ν0 ≥ 1. In this case the ν0 types of particles in the 0th zone are treated differently than the remaining particles. At times we will find it necessary to use nk , 1 ≤ k ≤ ν0 as the independent variables in the 0th zone and mk as the independent variables in all other zones. Thus we define the vector of independent variables as ˜ = {n1 , n2 , · · · , nν0 , mν0 +1 , mν0 +2 , · · · , mtn+1 } m

(2.17)

and the vector of dependent variables as n˜ = {m1 , m2 , · · · , mν0 , nν0 +1 , · · · , ntn+1 }.

(2.18)

From (2.11) we find that ¯ 1,ν0 + ˜ + LE n˜ = −Bm

L B 1 1 e¯ ν +1 + u¯ + + e¯ ν0 +1 (u¯ +T · V) + u¯ − 2 0 2 2 2

(2.19)

where the nonzero elements of the matrix B are given in terms of the ν0 × ν0 matrix, 1

1 2 2 .. .

1  CT−1 =  1  .. . 1 2

1 2 3 .. .

... ... ... .. .

1 2 3 .. .

3

...

ν0

    

(2.20)

(where − 21 CT is the matrix of the first ν0 rows and columns of the matrix M of (2.12)) as

332


Bi,j = 2(CT−1 )i,j for 1 ≤ i, j ≤ ν0 , Bν0 +1,j = Bj,ν0 +1 = j for 1 ≤ j ≤ ν0 , n ν0 1X δj,ν0 +1 + (1 − δj,ti ) for ν0 + 1 ≤ j ≤ tn+1 − 1, 2 2

t

Bj,j =

i=2

(2.21)

Btn+1 ,tn+1 = 1, 1 X δj,ti for j > ν0 , Bj,j+1 = − + 2 n

i=2

Bj+1,j

1 = − for j > ν0 . 2

The vector u¯ is decomposed as u¯ = u¯ + + u¯ −

(2.22)

with ν0 X

u¯ + =

u¯ i e¯ i ,

(2.23)

i=1 tn+1 X

u¯ − =

u¯ i e¯ i ,

(2.24)

i=ν0 +1

V=

ν0 X

i¯ei ,

(2.25)

i=1

and ¯ a,b = E

b X

e¯ i .

(2.26)

i=a

The splitting of (2.22) can be done for any vector. We will, for example, also use this ˜ in the following. splitting for the vector m

3. Summary of Results Now that we have summarized the needed results of [36] we may complete the specification of the bosonic and fermionic sums which appear in our identities. We first complete the bosonic specification in Subsect. 3.1 and give the definitions needed for the fermionic sums in Subsect. 3.2. The final results will be outlined in subsection C, but the detailed identities will be presented in Sect. 10. We conclude in Subsect. 3.3 with a discussion of the special case q = 1. 3.1. Bosonic polynomials. In order to complete the specification of the bosonic polynomials Br,s (L, b) of (1.12) we represent both b and s as series in Takahashi lengths. Thus we write


b=

β X i=1

(µi ) l1+j µ

333

(3.1)

i

with 0 ≤ µ1 < µ2 < · · · < µβ ≤ n and 1 + tµi ≤ jµi ≤ t1+µi + δn,µi with the further restriction that if jµi = t1+µi then µi+1 ≥ µi + 2.

(3.2)

This decomposition is unique. We often say that a b of this form lies in zone µβ . Hence if b is in zone µ then (µ) (µ+1) ≤ b ≤ l2+t − 1. l2+t µ µ+1

(3.3)

A similar decomposition is made for s. From the decomposition of b we specify r(b) as follows: ( Pβ r(b) =

˜(µi ) i=1 l1+jµi Pβ (µi ) ˜ i=1 l1+jµi

+ δµ1 ,0

if 1 ≤ b ≤ p0 − ν0 − 1, if p0 − ν0 ≤ b ≤ p0 − 1,

(3.4)

(µ) (µ) where l˜1+j as defined by (2.10). This generalizes the case considered in [26] of b = l1+j µ µ being a single Takahashi length with µ ≥ 1 where we had (µ) (µ) r(l1+j ) = l˜1+j . µ µ

(3.5)

One may prove that X (µ ) p l˜1+ji µ − aµ1 , c = i p0 β

bb

(3.6)

i=1

where bxc is the greatest integer contained in x and n aµ 1 =

1 0

if µ1 ≥ 2 and µ1 even, otherwise.

(3.7)

Using (3.6) the map (3.4) may be alternatively expressed as r(b) = where

bb pp0 c + θ(µ1 even) bb pp0 c n θ(A) =

1 0

for 1 ≤ b ≤ p0 − ν0 − 1 , for p0 − ν0 ≤ b ≤ p0 − 1 if A is true, if A is false.

(3.8)

(3.9)

When r is expressed using relation (3.8) we refer to the expression as the Forrester-Baxter form for r after the closely related formula of [33]. The proof of (3.6) is straightforward and will be omitted. The following elementary properties of the b → r map are clear from (3.4) and (2.10): 1. 2.

r(b + 1) = r(b) or r(b) + 1, if r(b + 1) = r(b) + 1 then r(b + 2) = r(b + 1)

(3.10) (3.11)

334


3. if b is as in (3.1), (3.2) with µ1 ≥ 1 then and r(b + x)−r(x) =

4.

r(b − 1) = r(b) = r(b + 1) − 1 = r(b + 2) − 1 ( Pβ

˜(µi ) i=1 l1+jµi (n) l˜2+t −θ(yn 1+n

(3.12)

(n) for b < p0 − yn = l2+t , 1 ≤ x ≤ yµ1 1+n

− y1 < x)

(n) for b = l2+t , 1 ≤ x ≤ yn −1 1+n 0

r(1) = r(2) = 1, r(p − 1) = p − 1.

,

(3.13) (3.14)

Whereas the bosonic polynomials Br,s (L, b; q) depend on the Takahashi decomposition of b only through the map r(b), the corresponding fermionic polynomials depend sensitively on the details of the Takahashi decomposition of b and s. In [26] we treated the simplest case where both b and s consist of one single Takahashi length. In this paper we will still consider s to be of the form (µs ) (µs ) (or p0 − l1+j ) 1 + tµs ≤ js ≤ tµs +1 s = l1+j s s

(3.15)

but b is often left arbitrary (µs and js are from now on reserved to specify s as in (3.15) and are not to be confused with µi or jµi of the Takahashi decomposition (3.1)). However, the complexity of the final fermionic polynomials will depend on the details of the decomposition of b. 3.2. Fermionic polynomials. Our fermionic sums Fr,s (L, b; q) will be constructed out of two elementary fermionic objects fs (L, u; q) and f˜s (L, u˜ ; q) where the vectors u and u˜ depend on b. The objects fs (L, u; q) and f˜s (L, u˜ ; q) are polynomials (in q 1/4 ) and differ from the polynomials of [26] in that the value at q = 0 is not normalized to 1. This choice of normalization is made for later convenience and we trust that no confusion will result from referring to fs (L, u; q) and f˜s (L, u˜ ; q) as polynomials in the sequel. In order to define fs (L, u; q) and f˜s (L, u˜ ; q) we need to discuss two generalizations of the q-binomials in (1.9) which allow n to be negative for n + m < 0, but will automatically vanish if m < 0.

Definition of q-binomials. We use here the definitions of [37] (1) n+1 (0) (q )m m+n m+n for m ≥ 0, n integer, (q)m = = m q n q 0 otherwise, where (a; q)n = (a)n =

n−1 Y

(1 − aq j ), 1 ≤ n; (a)0 = 1.

(3.16)

(3.17)

j=0

The reason for distinguishing between the two q-binomials with superscript zero or one is that the first one is more convenient if the variables nj are taken as independent in the (m, n)-system whereas the latter one is more convenient if the mj are taken as independent. We remark that when m and n are nonnegative we have the symmetry

(0,1)

m+n n

= q

(0,1)

m+n m

. q

(3.18)


335

We also note that we have the special values for L positive (0) (0) −L −L = 1, = 0, 0 q −L q (1) (1) −L −L = 0, = 1. 0 q −L q

(3.19)

Both functions satisfy the two recursion relations without restrictions on L or m (0,1) (0,1) (0,1) L−1 L L−m L − 1 = +q , (3.20) m q m−1 q m q (0,1) (0,1) (0,1) L L−1 m L−1 =q + . m q m q m−1 q

(3.21)

This extension of the q-binomial coefficients is needed for the fermionic function Fr(b),s (L, b; q) for general values of b and s. In [26] we used the definition (1.9) and thus obtained more limited results. Definition of fs (L, u; q). Let us now define fs (L, u; q) with u ∈ Z 1+t1+n , where L is a nonnegative integer. There are actually several equivalent forms, differing only by which set of mk and nk in the (m, n)–system (2.11) is taken to be independent. In the following ˜ n and m as independent variables, three equivalent forms for fs (L, u; q) are given with m, ˜ m and n can be determined by Eqs. (2.19), respectively. The corresponding variables n, (2.15) and (2.12), respectively. X

fs (L, u; q) =

×

tY n+1

˜

¯ 1,ν0 + ˜ + LE ((Itn+1 − B)m

+ 21 Bu¯ + + 21 e¯ ν0 +1 (u¯ +T · V) + 21 u¯ − )j m ˜j

L ¯ 2 eν0 +1

X

T

θ(mtn+1 ≡ u1+tn+1 (mod 2))q Q(n,m)+A

n∈Ztn+1

=

˜ m

˜ − ∈2Ztn+1 −ν0 +w− (u1+t ¯) m ,u n+1 m ˜ + ∈Zν0

j=1

=

T

˜ q Q(m)+A

X ¯ m∈2Ztn+1 +w(u1+tn+1 ,u)

T

q Q(n,m)+A

˜ m

tY n+1 j=1

˜ m

tY n+1 j=1

mj + nj nj

((Itn+1 + M)m + mj

u¯ 2

+

(θ(j>ν0 )) q

(0) q

(1) L ¯ 2 e1 )j . q

(3.22) ¯ a,b and V have been defined in Sects. 1, 2 and Z is the set of integers Here B, M, Itn+1 , E ¯ {. . . , −2, −1, 0, 1, 2, . . .}. The vectors m, n satisfy the system (2.11) with u, ¯ + u¯ = u(s)

tn+1 X i=1

where for s as in (3.15) and 1 ≤ k ≤ t1+n ,

e¯ i ui ,

(3.23)

336


¯ (u(s)) k =

δk,js − δk,js

Pn

for 1 + tµs ≤ js ≤ tµs +1 and µs ≤ n − 1, (3.24) for tn < js , µs = n.

i=µs +1 δk,ti

¯ Note that the vectors u¯ and u(s) in (3.23) have dimension tn+1 while u has dimension 1 + tn+1 . ˜ m) ˜ and ˜ and Q(n, m), the linear term AT m Let us now define the quadratic forms Q( ¯ the parity restriction vector w(u1+tn+1 , u): ˜ We define the quadratic form Q( ˜ m) ˜ as Definition of the quadratic forms Q and Q. 1 T ˜ m) ˜ = m ˜ Bm, ˜ Q( 2

(3.25)

where B is defined by (2.21). An equivalent form makes use of (2.17)–(2.19) to write 1 T ¯ 1,ν0 − L e¯ ν0 +1 − B u¯ + − 1 e¯ ν +1 (u¯ +T · V) − u¯ − ). ˜ (ñ − LE Q(n, m) = − m 2 2 2 2 0 2

(3.26)

˜ We write The linear term AT m. A = A(b) + A(s) ,

(3.27)

where the tn+1 -dimensional vector A(b) is obtained from u as 1 (b) Ak = − 2 uk for k in an even, nonzero zone, 0 otherwise,

(3.28)

¯ as and the tn+1 -dimensional vector A(s) is obtained from u(s) 1 ¯ − 2 u(s) ¯ k − 21 VT · u(s)δ k,ν0 +1 for k in an odd zone, = A(s) 1 T k ¯ + − 2 e¯ k Bu(s) for k in an even zone.

(3.29)

¯ The summation variables m ˜ j (j = The parity restrictions and the vector w(u1+tn+1 , u) ν0 + 1, . . . , t1+n ) in the first line of (3.22) and mj (j = 1, . . . , t1+n ) in the third sum of (3.22) are subject to even/odd restrictions which are determined from u and s by the requirement that the entries of all q-binomials in (3.22) are integers as long as u ∈ Z1+tn+1 . To formulate this analytically we define wk(j) for 1 ≤ k ≤ tn+1 , tµ0 + δµ0 ,0 + 1 ≤ j ≤ tµ0 +1 + δµ0 ,n , for some 0 ≤ µ0 ≤ n as wk(j)

 0 = j−k  w(j) + w(j) 1+tµ+1 k+1

and then define ¯ = w(u1+tn+1 , u)

tn+1 X k=1

k≥j tµ0 ≤ k ≤ j − 1 tµ ≤ k < tµ+1 , for some µ < µ0 

e¯ k u1+t1+n wk(1+tn+1 ) +

tn+1 X j=1

(3.30)

(3.31)

 wk(j) u¯ j  .

(3.32)


337

Then the variables mj in (3.22) satisfy ¯ (mod 2Zt1+n ). m ≡ w(u1+t1+n , u)

(3.33)

Note that mtn+1 ≡ u1+tn+1 ≡ P (mod 2), where parity P ∈ {0, 1}. Clearly, the 1 + t1+n component of the vector u determines the parity of the mt1+n variable in the fundamental fermionic polynomials (3.22). The limit L → ∞. In order to obtain character identities from the polynomial identities we need to consider the limit L → ∞. Only the first expression for fs (L, b; q) in (3.22) is suitable for this limit since Q(n, m) which appears in the other two expressions depends on L. Hence in the limit L → ∞ the expression for the polynomial (3.22) reduces to X T ˜ ˜ ˜ m q Q(m)+A fs (u; q) = lim fs (L, u; q) = L→∞



˜− m



ν ˜ + ∈Z 0 m ≥0 t −ν 0 +w(u ∈2Z n+1

¯) 1+tn+1 ,u

(1) νY tn+1 0 +1 ˜ + 21 u¯ − )j 1  Y ((Itn+1 − B)m  × , (q)m˜ j m ˜j q j=1

(3.34)

j=2+ν0

which depends on the vector u¯ + only through the linear term A. Definition of f˜s (L, u˜ ; q). The fermionic polynomial f˜s (L, u˜ ; q) is defined for vectors of the form u˜ = eν0 −j0 −1 − eν0 + u10 , where 0 ≤ j0 ≤ ν0 is in the zone 0 and u10 ∈ Z 1+t1+n is any vector with no components in the zero zone, i.e. (u10 )i = 0 for 1 ≤ i ≤ ν0 . We define f˜s (L, eν0 −j0 −1 − eν0 + u10 ; q)  L+j0  q − 2 [fs (L + 1, eν0 −j0 − eν0 + u10 ; q)   −fs (L, e1+ν0 −j0 − eν0 + u10 ; q)] for 1 ≤ j0 ≤ ν0 , = L L  (q 2 − q − 2 )fs (L − 1, u10 ; q)   L +q − 2 fs (L, eν0 −1 − eν0 + u10 , q) for j0 = 0.

(3.35)

When j0 = ν0 , the left-hand side of this definition loses its meaning because we have no e−1 . However, for conformity, we introduce the notation f˜s (L, e−1 − eν0 + u10 ; q) to mean the first line of the right-hand side of (3.35) with j0 = ν0 . We will show in Sect. 6 that for j0 = ν0 this definition reduces to f˜s (L, e−1 − eν0 + u10 ; q) = 0,

(3.36)

while for j0 = ν0 − 1, f˜s (L, e0 − eν0 + u10 ; q) = q =q

L−(ν0 −1) 2

fs (L − 1, e1 − eν0 + u10 ; q)

L−(ν0 −1) 2

fs (L, −eν0 + u10 ; q).

(3.37)

In the limit L → ∞ we have lim f˜s (L, u˜ ; q) = 0.

L→∞

(3.38)

338


The identities. The polynomial identities proven in this paper are all of the form Fr(b),s (L, b; q) = q Nr(b),s Br(b),s (L, b; q),

(3.39)

where r(b) as given in (3.4) and Fr(b),s (L, b; q) is of the form Fr(b),s (L, b; q) =

X

L

q cu + 2 gu fs (L, u, q) +

X

q c˜u˜ f˜s (L, u˜ , q).

(3.40)

˜ U˜ (b) u∈

u∈U (b)

Here U (b) and U˜ (b) are sets of vectors determined by b. The sets U (b), U˜ (b) and the exponents cu , c˜u˜ , gu ≥ 0 and Nr(b),s will be explicitly computed. We note however that there are very special values for b for which the representation of Fr(b),s (L, b; q) in (3.40) is not correct. This phenomenon will be discussed in Sect. 8. The results depend sensitively on the details of the expansion of b (3.1). For the cases of 1,2 and 3 zones the identities for all values of b are given in Sect. 7. However for the general case of n + 1 zones with n ≥ 3 there are many special cases to consider. The process of obtaining a complete set of results is tedious and in this paper we present explicit results only for the following special cases of b (and r): (µ) (µ) , r = l˜1+j + δµ,0 1 : b = l1+j µ µ (µ) (µ) (µ) 2a : l1+j − ν0 + 1 ≤ b ≤ l1+j − 1, r = l˜1+j µ µ µ

2b :

1 ≤ µ, tµ + 1 ≤ jµ ≤ t1+µ − 1 + δµ,n + 1 ≤ b ≤ l(µ) + ν0 + 1, r = 1 + l˜(µ)

(µ) l1+j µ

1+jµ

1+jµ

1 ≤ µ, tµ + 1 ≤ jµ ≤ t1+µ − 1 + δµ,n 3: b=

β X

(µ) l1+j , r =1+ µ

µ=0

β X

(µ) l˜1+j , 1≤β≤n µ

µ=1

0 ≤ j0 ≤ ν 0 − 1 1 + tµ ≤ jµ ≤ tµ+1 − 3 1 ≤ µ ≤ β − 2 1 + tβ−1 ≤ jβ−1 ≤ tβ − 2 1 + tβ ≤ jβ ≤ tβ+1 − 1 + δβ,n 4: b=

β X µ=α

(µ) l1+j , r= µ

β X

(µ) l˜1+j , 1 ≤ α, α + 1 ≤ β ≤ n µ

µ=α

1 + tµ ≤ jµ ≤ tµ+1 − 3 α ≤ µ ≤ β − 2 1 + tβ−1 ≤ jβ−1 ≤ tβ − 2 1 + tβ ≤ jβ ≤ tβ+1 − 1 + δβ,n , where α = 1 and α ≥ 2 are treated separately. The results are given in Sect. 10 as follows:

(3.41)


339

Polynomial Identities. 1: 2a : 2b : 3: 4:

10.3 10.4 10.5 10.10 10.15.

(3.42)

Character identities. 1 and 2a : 2b : 3: 4, α = 1 : 4, α ≥ 2 :

10.20 10.21 10.22 10.23 10.24.

(3.43)

These results hold for νi 6= 1, i = 1, · · · , n − 1 and in addition (10.3) and (10.20) are valid even if some or all νi = 1 or νn = 0 (or both) with minor modifications of overall factors. Further results are to be found in Sect. 11 and 12 and (C.19) of Appendix C. (µs ) (µs ) (p0 − l1+j ). The The identities of Sect. 10 (11) are valid for p0 > 2p and for s = l1+j s s identities of Sect. 12 are valid for p0 < 2p. 3.3. The special case q = 1. We close this section with a brief discussion of the special case q = 1. The details of this case will be presented elsewhere, but it is useful to sketch the results here in order to give a characterization of the vectors u and u˜ that appear in (3.40) which is complementary to the constructive procedure developed in Sects. 7 and 8. When q = 1 the bosonic function (1.12) simplifies because the dependence on r vanishes and the fermionic form (3.40) simplifies because f˜s (L, u˜ ; q = 1) = fs (L, u˜ ; q = 1).

(3.44)

For these reasons many of the distinctions between the special cases noted in the previous section disappear and we find the single identity valid for all b, X fs (L, u; q = 1), (3.45) Br(b),s (L, b; q = 1) = u∈W (b)

where the set of vectors W (b) is obtained from the Takahashi decomposition of b (3.1) as follows: If in (3.1) β = 1 so that (µ) , (3.46) b = l1+j µ then we define as in [26] W (b) having only the single element u = e jµ −

n X

eti , 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n , µ = 0, 1, · · · , n.

(3.47)

i=µ+1

This vector was called an r string in [26]. For the general case we write the Takahashi decomposition of an arbitrary b as

340


b=

(µ ) l1+jβµ β

+ x where x =

β−1 X i=1

(µi ) l1+j , µ

(3.48)

i

where β is defined from (3.1). If x = 0 we define a vector v(0) as: v(0) = ejµβ .

(3.49)

If x 6= 0 and it is not true that µβ = n and jn = 1 + t1+n , then we define two vectors as for tµβ + 1 ≤ jµβ ≤ −1 + t1+µβ , e1+jµβ v(0) = e + (1 − δ )e for jµβ = t1+µβ , 1+t1+µβ µβ ,n t1+µβ (3.50) v(1) = ejµβ . Furthermore if x 6= 0 and we do not have µβ = n and jn = 1 + t1+n we define the two numbers (3.51) b(0) = x and b(1) = yµβ − x. In the exceptional case where x 6= 0, µβ = n and jn = 1 + t1+n we define only one vector and one number v(1) = e1+t1+n , (3.52) b(1) = yn − x. We then expand both b(0) and b(1) (and in the exceptional case only b(1) ) again in a Takahashi series and repeat the process as many times as needed until x = 0 in which case the process terminates. This recursion leads to a branched chain of vectors v(i1 ) , v(i1 ,i2 ) , · · · , v(i1 ,i2 ,···,if ) , where 1 ≤ f ≤ µβ + 1; i1 , · · · , if = 0, 1. Notice however that f might vary from branch to branch. Let us define µf to be the lowest µ such that (i ,...,if ) there exist an i such that tµ < i ≤ t1+µ + δµ,n and vi 1 6= 0. Then the set W (b) consists of all vectors v(i1 ) + v(i1 ,i2 ) + · · · + v(i1 ,i2 ,···,if ) −

n X

et k ,

(3.53)

k=µf +1

where all vectors v are determined by the above described recursive procedure. The explicit solution of this recursion relation involves the recognition that there are many separate cases of the Takahashi decomposition of b which lead to sets W (b) which may differ even in the number of vectors in the set. Certain of these special cases correspond to the cases distinguished in the previous section. The complete solution of this recursive definition will be given elsewhere where we will use it to give explicit Rogers-Schur-Ramanujan identities for general values of b. We note in the cases consid(0) is present in the Takahashi decomposition ered in the previous subsection that when l1+j 0 then there are vectors which have components in zone zero of the form u˜ = eν0 −j0 −a − eν0 + u10 , a = 1, 2,

(3.54)

where the vector u10 is defined above (3.35). These are the vectors u˜ ∈ U˜ (b) of (3.40). Note that the set U˜ (b) is empty if in the Takahashi decomposition of b (3.1) we have µ1 ≥ 1. The remaining vectors are the vectors u ∈ U (b) of (3.40).


341

4. The Bosonic Recursion Relations In this section we derive recursion relations for the bosonic polynomials Br,s (L, b; q) defined by (1.12) and the b → r map (3.4). Here and in the remainder of the paper we will often suppress the argument q of all functions unless explicitly needed. Moreover we find it convenient to remove an L independent power of q by defining 1 B˜ r(b),s (L, b) = q 2 (φr(b),s −φr(s),s ) Br(b),s (L, b),

(4.1)

where φx,y is defined as φx+1,y − φx,y = 1 − y +

ξ X

(ηi ) l1+j η

i=1

i

(4.2)

when x has the decomposition in terms of truncated Takahashi lengths x=

ξ X i=1

(ηi ) l˜1+j , η i

(4.3)

with 1 ≤ η1 < η2 < · · · < ηξ ≤ n and 1 + tηi ≤ jηi ≤ t1+ηi + δn,ηi . We note that because φx,y appears only as a difference in (4.1) that boundary conditions on φx,y are not needed. This change in normalization allows us to prove the following recursion relations which contain no explicit dependence on s. For 2 ≤ b ≤ p0 − ν0 − 1,  ˜ Br(b),s (L − 1, b + 1) + B˜ r(b),s (L − 1, b − 1)    +(q L−1 − 1)B˜ if µ1 = 0, j0 ≥ 1 r(b),s (L − 2, b) , B˜ r(b),s (L, b) = L−1 ˜ ˜ 2 B  q (L − 1, b + 1) + B (L−1, b −1) if µ1 ≥ 1 r(b)+1,s r(b),s   L B˜ r(b),s (L−1, b + 1) + q 2 B˜ r(b)−1,s (L − 1, b −1) if µ1 = 0, j0 = 0 (4.4) where µ1 and j0 are obtained from the Takahashi decomposition of b (3.1). For the remaining cases we have B˜ 1,s (L, 1) = B˜ 1,s (L − 1, 2)

(4.5)

and

  B˜ p−1,s (L−1, b + 1) + B˜ p−1,s (L−1, b −1) B˜ p−1,s (L, b) = +(q L−1 − 1)B˜ p−1,s (L − 2, b)  ˜ Bp−1,s (L − 1, p0 − 2)

if p0 − ν0 ≤ b ≤ p0 − 2 if b = p0 − 1. (4.6) The recursion relations (4.4)–(4.6) have a unique solution once appropriate boundary conditions are given. One set of boundary conditions which will specify B˜ r(b),s (L, b) as the unique solution are the values obtained from Br(b),s (0, b) = δs,b , 1 ≤ b ≤ p0 − 1.

(4.7)

However, it will prove useful to recognize that this is not the only way in which boundary conditions may be imposed on (4.4)–(4.6) to give (1.12) as the unique solution. One alternative is the condition which is readily obtained from the term j = 0 in (1.12),

342


Br(b),s (L, b) =

0 1

for L = 0, 1, 2, · · · , |s − b| − 1 . for L = |s − b|

(4.8)

To prove (4.4)–(4.6) for the b → r map of Sect. 3 we first consider the case where b and r are unrelated and recall the elementary recursion relations for q–binomial coefficients (3.20) and (3.21). If we use (3.20) in the definition (1.12) for Br,s (L, b) we find Br,s (L, b) = Br,s (L − 1, b − 1) + q

L+b−s 2

Br+1,s (L − 1, b + 1),

(4.9)

Br−1,s (L − 1, b − 1) + Br,s (L − 1, b + 1).

(4.10)

and if we use (3.21) we find Br,s (L, b) = q

L−b+s 2

Furthermore if we write (4.9) as q−

L+b−s 2

[Br,s (L, b) − Br,s (L − 1, b − 1)] = Br+1,s (L − 1, b + 1)

(4.11)

[Br,s (L − 1, b + 1) − Br,s (L − 2, b)] = Br+1,s (L − 2, b + 2)

(4.12)

and q−

L+b−s 2

and subtract (4.12) from (4.11) we may use (4.10) on the right-hand side to find q−

L+b−s 2

[Br,s (L, b) − Br,s (L − 1, b − 1) − Br,s (L − 1, b + 1) + Br,s (L − 2, b)] =q

L+s−b −1 2

Br,s (L − 2, b)

,

(4.13) and thus we have Br,s (L, b) = Br,s (L − 1, b − 1) + Br,s (L − 1, b + 1) + (q L−1 − 1)Br,s (L − 2, b). (4.14) If we now relate r to b using the map of Sect. 3 and use the Definition (4.1) then Eqs. (4.9), (4.10) and (4.14) are the three equations of (4.4) and the first equation of (4.6). We also note from (1.12) that ! ∞ X L L j 2 pp0 −jsp j 2 pp0 +jsp −q q B0,s (L, 0) = L+s L−s 0 0 2 − jp q 2 − jp q j=−∞ ! ∞ X L L j 2 pp0 −jsp j 2 pp0 +jsp = q = 0, −q L+s L+s 0 0 2 − jp q 2 + jp q j=−∞ (4.15) where to get the second line we first use (3.18) and then let j → −j. Combining this with (4.10) at r = 1, b = 1 we obtain (4.5). Finally we note that in an analogous fashion we may prove (4.16) Bp,s (L, p0 ) = 0, and thus the last equation of (4.6) follows.


343

5. Recursive Properties for Fundamental Fermionic Polynomials fs (L, u) Our goal is to construct fermionic objects from the fundamental fermionic polynomials (3.22) which will satisfy the bosonic recursion relations (4.4)-(4.6) obeyed by B˜ r(b),s (L, b). To do this we will use the following recursive properties for the fundamental fermionic polynomials fs (L, u), where here and in the remainder of the paper we restrict our attention to νi ≥ 2 for i = 1, · · · , n − 1. The analysis for νi = 1 is analogous to that for νi ≥ 2, but since the recursion relations given below should be slightly modified, this case will not be treated here for reasons of space. Let us first introduce the following notation for tµ + 1 ≤ jµ ≤ tµ+1 + δµ,n : u0 (jµ ) = ejµ + u0 − et1+µ (1 − δµ,n ), u±1 (jµ ) = ejµ ±1 + u0 − et1+µ (1 − δµ,n )

,

(5.1)

Pν 0 0 ui = 0 for µ = 0 and for µ ≥ 1 where u0 is an arbitrary vector only restricted by i=1 0 the components must satisfy uj = 0 for j ≤ jµ + 1 − δjµ ,t1+n . These conditions are used in the proof of the recursive properties. Further define (t) = Ea,b

b X

eti .

(5.2)

i=a

Then if j0 is in the zone µ = 0, where 0 ≤ j0 ≤ ν0 = t1 we find (for L ≥ 1)  fs (L − 1, u1 (0)), j0 = 0     fs (L − 1, u1 (j0 )) + fs (L − 1, u−1 (j0 )) fs (L, u0 (j0 )) = +(q L−1 − 1)fs (L − 2, u0 (j0 )), 1 ≤ j0 ≤ ν0 − 1   − 1, u1 (ν0 ) + eν0 ) + fs (L − 1, u−1 (ν0 ))   fs (LL−1 +(q − 1)fs (L − 2, u0 (ν0 )), j0 = ν0 = t1 .

(5.3)

For jµ in the zone 1 ≤ µ ≤ n, where 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n we have four separate recursive properties (for L ≥ 1): 1. for jµ = 1 + tµ , fs (L, u0 (1 + tµ )) = q + q θ(µ≥2)(

ν0 +1 3 L−1 2 − 4 + 4 θ(µ

ν0 −θ(µ even) L−1 ) 2 − 4

+ θ(µ ≥ 2)q + 2θ(µ ≥ 3)

q

(t) fs (L − 1, u1 (1 + tµ ) − E1,µ )

(t) fs (L − 1, u−1 (1 + tµ ) − E1,µ )

ν0 −θ(µ odd) L−1 2 − 4

µ−1 X

odd)

(t) fs (L − 1, e−1+tµ + u0 (1 + tµ ) − E1,µ )

ν0 −θ(i odd) L−1 2 − 4

(t) fs (L − 1, −E1,i + e−1+ti + u0 (1 + tµ ))

i=2

+ [1 + θ(µ ≥ 2)]fs (L − 1, e−1+ν0 − eν0 + u0 (1 + tµ )) + (q L−1 − 1)fs (L − 2, u0 (1 + tµ )); (5.4) 2. for 2 + tµ ≤ jµ ≤ −1 + tµ+1 + δµ,n ,

344

A. Berkovich, B. M. McCoy, A. Schilling ν0 +1 3 L−1 2 − 4 + 4 θ(µ

fs (L, u0 (jµ )) = q +q

ν0 −θ(µ even) L−1 2 − 4

µ X

+ 2θ(µ ≥ 2)

odd)

(t) fs (L − 1, u1 (jµ ) − E1,µ )

(t) fs (L − 1, u−1 (jµ ) − E1,µ ) ν0 −θ(i odd) L−1 2 − 4

q

(t) fs (L − 1, u0 (jµ ) − E1,i + e−1+ti )

(5.5)

i=2

+ 2fs (L − 1, eν0 −1 − eν0 + u0 (jµ )) + (q L−1 − 1)fs (L − 2, u0 (jµ )); 3. for 1 ≤ µ ≤ n − 1, and jµ = tµ+1 , fs (L, u0 (t1+µ )) = q +q

ν0 −θ(µ odd) L−1 2 − 4

ν0 −θ(µ even) L−1 2 − 4

+ 2θ(µ ≥ 2)

µ X

(t) fs (L − 1, −E1,µ + et1+µ + u1 (tµ+1 ))

(t) fs (L − 1, −E1,µ + u−1 (tµ+1 ))

q

ν0 −θ(i odd) L−1 2 − 4

(t) fs (L − 1, −E1,i + e−1+ti + u0 (t1+µ ))

i=2

+ 2fs (L − 1, e−1+ν0 − eν0 + u0 (t1+µ )) + (q L−1 − 1)fs (L − 2, u0 (t1+µ )); (5.6) 4. for µ = n and jn = 1 + tn+1 , fs (L, e1+t1+n ) = q

ν0 −θ(n even) L−1 2 − 4

+ 2θ(n ≥ 2)

n X

q

(t) fs (L − 1, et1+n − E1,n )

ν0 −θ(i odd) L−1 2 − 4

(t) fs (L − 1, −E1,i + e−1+ti + e1+t1+n ) ,

(5.7)

i=2

+ 2fs (L − 1, eν0 −1 − eν0 + e1+t1+n ) + (q L−1 − 1)fs (L − 2, e1+t1+n ) where we remind the reader that the 1 + t1+n component of the vector u determines the parity of the variable mt1+n in the fundamental fermionic polynomial (3.22). We prove these recursive properties in Appendix A. 6. Properties of f˜s (L, u˜ ) In this section we will prove the Recursive properties of f˜s (L, u˜ ). The function f˜s (L, u0 (ν0 − j0 − 1)), defined by (3.35) and (5.1) with u0 → u10 satisfies the recursive properties for 1 ≤ j0 ≤ ν0 − 1 f˜s (L, u0 (ν0 − j0 − 1)) = f˜s (L − 1, u−1 (ν0 − j0 − 1)) + f˜s (L − 1, u1 (ν0 − j0 − 1)) + (q L−1 − 1)f˜s (L − 2, u0 (ν0 − j0 − 1)). (6.1) To prove (6.1) we adopt the simplified notation fj0 (L) = fs (L, u0 (j0 )). Then for 1 ≤ j0 ≤ ν0 − 1 the relation (5.3) reads fj0 (L) = f1+j0 (L − 1) + f−1+j0 (L − 1) + (q L−1 − 1)fj0 (L − 2) which we rewrite as

(6.2)


345

fj0 (L) − f1+j0 (L − 1) = f−1+j0 (L − 1) − fj0 (L − 2) + q L−1 fj0 (L − 2).

(6.3)

Then if we replace j0 by ν0 − j0 , L by L + 1 and set I(L, j0 ) = fν0 −j0 (L + 1) − f1+ν0 −j0 (L),

(6.4)

I(L, j0 ) = I(L − 1, j0 + 1) + q L fν0 −j0 (L − 1).

(6.5)

we have We now eliminate fν0 −j0 (L − 1) by first writing (6.5) as [I(L, j0 ) − I(L − 1, 1 + j0 )]q −L = fν0 −j0 (L − 1),

(6.6)

and then by setting L → L − 1 and j0 → j0 − 1 to get the companion equation [I(L − 1, j0 − 1) − I(L − 2, j0 )]q −L+1 = f1+ν0 −j0 (L − 2). Subtracting (6.7) from (6.6) and multiplying by q [I(L, j0 ) − I(L − 1, 1 + j0 )]q −

L−j0 2

(6.7)

we obtain

L+j0 2

− [I(L − 1, −1 + j0 ) − I(L − 2, j0 )]q −

L−2+j0 2

=q

L−j0 2

I(L − 2, j0 ).

(6.8)

Recalling the definition (3.35) we see that f˜s (L, u0 (ν0 − j0 − 1)) = q −

L+j0 2

I(L, j0 ),

(6.9)

and using this along with (6.8) we obtain (6.1). Note from (5.3) that f0 (L) = f1 (L − 1) which implies that (6.10) f˜s (L, e−1 − eν0 + u10 ) = 0 = I(L, ν0 ), and using (6.6) we have f˜s (L, e0 − eν0 + u10 ) = q

L−(ν0 −1) 2

fs (L − 1, e1 − eν0 + u10 ) = q

L−(ν0 −1) 2

fs (L, −eν0 + u10 ). (6.11)

We also need the

Limiting Relation. lim f˜s (L, u˜ ) = 0.

L→∞

(6.12)

To prove (6.12) we use (6.6) and (6.10) to obtain the system of ν0 − j0 equations I(L, j0 ) − I(L − 1, j0 + 1) = q L fν0 −j0 (L − 1), I(L − 1, j0 + 1) − I(L − 2, j0 + 2) = q L−1 fν0 −j0 −1 (L − 2), I(L − 2, j0 + 2) − I(L − 3, j0 + 3) = q L−2 fν0 −j0 −2 (L − 3), ··· I(L − (ν0 − j0 − 1), ν0 − 1) − 0 = q L−(ν0 −j0 −1) f1 (L − (ν0 − j0 )), which, if added together give

(6.13)

346


I(L, j0 ) =

ν0 −j 0 −1 X

q L−i fν0 −j0 −1 (L − 1 − i).

(6.14)

i=0

Thus multiplying by q −

L+j0 2

we find

L f˜s (L, u0 (ν0 − j0 − 1)) = q 2

ν0 −j 0 −1 X

j0

q − 2 −i fν0 −j0 −i (L − 1 − i),

(6.15)

i=0

and hence since limL→∞ fj0 (L) is finite we see from (6.15) that the limiting relation (6.12) holds for u = u0 (ν0 − j0 − 1), 1 ≤ j0 ≤ ν0 − 1. The case j0 = 0 requires a separate treatment. First we note that L f˜s (L, u0 (ν0 − 1)) = f˜s (L, u0 (ν0 − 2)) + q 2 fs (L − 1, u0 (ν0 )),

(6.16)

which follows from (3.35). In this equation we let L → ∞ and using (6.15) with j0 = 1 we obtain (6.12) for u˜ = u0 (ν0 − 1). 7. Construction of Fr(b),s (L, b) We now turn to the details of the construction of the fermionic representations Fr(b),s (L, b), L ≥ 0 of the bosonic polynomials B˜ r(b),s (L, b). Our method is to construct ˜ which by construction fermionic functions Fr(b),s (L, b) in terms of fs (L, u) and f˜s (L, u) satisfy the bosonic recursion relations (4.4) and (4.6). We begin by choosing as a starting value for b = 1, (t) ). F1,s (L, 1) = fs (L, −E1,n

(7.1)

The boundary bosonic recursion relation (4.5) requires that (t) ) F1,s (L − 1, 2) = fs (L, −E1,n

(7.2)

from which if we let L → L+1 and use the first fermionic recursive properties in (5.3) we (t) find with u0 = −E2,n , (t) F1,s (L, 2) = fs (L, e1 − E1,n ).

(7.3)

We now continue this procedure in a recursive fashion. We construct Fr(b),s (L, b) for all b by defining Fr(b),s (L, b) in terms of Fr(b−1),s (L, b − 1) and Fr(b−2),s (L, b − 2) through the bosonic recursion relations (4.4) for the b → r map defined in (3.4) and then simplifying the expressions by using the fermionic recursive properties of Sects. 5 and 6. The b → r map is important since it prescribes which bosonic recursion relation is being used to construct Fr(b),s (L, b) (i.e. whether one uses the depth-2 recurrence or one of the depth-1 recurrences). This process is continued until we reach b = p0 − 1 for which the last identity of (4.6) must hold. This recursive construction can be carried out for any starting function F1,s (L, 1) but the last equation of (4.6) will only hold if F1,s (L, 1) has been properly chosen. The recursive process used to generate fermionic polynomials will be referred to as an evolution. The map from two initial polynomials F1,s (L, 1), F1,s (L, 2) to polynomials Fr(b),s (L, b), Fr(b+1),s (L, b + 1) will be called a flow of length b.


347

After Fr(b),s (L, b) has been constructed for all p0 − 1 values of b to satisfy the bosonic recursion relations (4.4) –(4.6) we will complete the analysis by studying the behavior for small values of L to show that with a suitable normalization q Nr,s the initial conditions (4.7) or (4.8) are satisfied. For 1 ≤ b ≤ ν0 + 1 (i.e. b in zone 0) this general recursive construction can be explicitly carried out for an arbitrary n + 1 zone problem. In this case the Takahashi decomposition (3.1) of b consists of the single term (0) = 1 + j 0 , 0 ≤ j0 ≤ ν0 . b = l1+j 0

(7.4)

As long as 1 ≤ j0 ≤ ν0 , F1,s (L, b) satisfies the first equation of (4.4) because r = 1 does not change. Comparing this recursion relation with the fermionic recursive property (5.3) and using the values for F1,s (L, 1) and F1,s (L, 2) of (7.1) and (7.3) we conclude that for an n + 1 zone problem with b in zone 0. (t) (0) ), b = l1+j = 1 + j0 and 0 ≤ j0 ≤ ν0 . F1,s (L, b) = fs (L, ej0 − E1,n 0

(7.5)

(1) we have entered into zone one. However we still have r = 1 When b = 2 + ν0 = l2+ν 0 and an identical computation gives (t) ). F1,s (L, 2 + ν0 ) = fs (L, e1+ν0 − E2,n

(7.6)

To proceed further into zone 1 we must cross a boundary where r changes from 1 to 2. Here we use the second bosonic recursion relation in (4.4) for b = 2 + ν0 which has the Takahashi expansion (3.1) with β = 1 and µ1 = 1, j1 = 1 + ν0 , (1) b = 2 + ν0 = l2+ν 0

(7.7)

and find q

L−1 2

F2,s (L − 1, ν0 + 3) = F1,s (L, ν0 + 2) − F1,s (L − 1, ν0 + 1)

(7.8)

which, after using (7.6) and (7.5) becomes q

L−1 2

(t) (t) F2,s (L − 1, ν0 + 3) = fs (L, e1+ν0 − E2,n ) − fs (L − 1, −E2,n ).

(7.9)

(t) ) by an expression in terms of fermionic To reduce this we replace fs (L, e1+ν0 − E1,n polynomials with arguments L − 1 and L − 2 using relation (5.4) with µ = 1. In this case several of the terms in the general expression (5.4) vanish and we have (t) )=q fs (L, e1+ν0 − E2,n

ν0 −2 L−1 2 − 4

(t) (t) fs (L − 1, e2+ν0 − E1,n ) + fs (L − 1, −E2,n )

(t) (t) + fs (L − 1, e−1+ν0 + e1+ν0 − E1,n ) + (q L−1 − 1)fs (L − 2, e1+ν0 − E2,n ). (7.10) Using this in (7.9) and setting L → L + 1 we find

F2,s (L, ν0 + 3) = q −

ν0 −2 4

(t) fs (L, e2+ν0 − E1,n )

(t) (t) ) + (q 2 − q − 2 )fs (L − 1, e1+ν0 − E2,n ) + q − 2 fs (L, e−1+ν0 + e1+ν0 − E1,n L

=q

L

ν −2 − 04

L

(t) (t) fs (L, e2+ν0 − E1,n ) + f˜s (L, e−1+ν0 + e1+ν0 − E1,n ),

(7.11)

348


where in the last line we have used the case j0 = 0 in Definition (3.35). In a similar fashion we consider (1) b = 3 + ν0 = l2+ν + l1(0) . 0

(7.12)

We use the third recursion relation in (4.4) written as L

F2,s (L − 1, ν0 + 4) = F2,s (L, ν0 + 3) − q 2 F1,s (L − 1, ν0 + 2)

(7.13)

which, after using (7.6) and (7.11) becomes F2,s (L − 1, ν0 + 4) = q −

ν0 −2 4

(t) fs (L, e2+ν0 − E1,n )

(t) (t) ) − fs (L − 1, e1+ν0 − E2,n )]. + q − 2 [fs (L, e−1+ν0 + e1+ν0 − E1,n L

(7.14)

We now reduce this by using the first equation in (5.3), the first line in the definition (3.35) and setting L → L + 1 to obtain the result F2,s (L, ν0 + 4) = q −

ν0 −2 4

(t) (t) fs (L, e1 + e2+ν0 − E1,n ) + f˜s (L, eν0 −2 + e1+ν0 − E1,n ). (7.15)

Let us review what has been done. We started with F1,s (L, 1) and constructed all the values of F1,s (L, b) with 1 ≤ b ≤ 2 + ν0 to satisfy the first bosonic recursion relation in (4.4) where r does not change. We refer to these recursion relations where r does not change as “moving b on the plateau”. In this process we did not create any new terms in the linear combination. We then constructed F2,s (L, 3 + ν0 ) by using the second recursion relation in (4.4) in which the term with b + 1 has r + 1. In this step we created the new term f˜. We refer to this new term as a reflected term (much as there is a reflected wave at a boundary in an optics problem). We then created F2,s (L, 4 + ν0 ) by using the third recursion relation in (4.4) where b − 1 has r − 1. This process did not create any additional terms. We refer to this process of using the two equations in (4.4) where r changes by one as “transiting an r boundary”. In most cases when b moves on the plateau we apply (5.3) to the fs – terms and (6.1) to the f˜s –terms in (3.40). Note that the fermionic recurrences we employ may still vary from term to term in (3.40). Again in most cases while transiting the r– boundary we use (5.4)–(5.7). However, there are important exceptional cases (related to the so–called dissynchronization effect discussed in Sect. 8 and Appendix B) where this rule breaks down and, as a result, we are forced to apply (5.3) to some terms in (3.40) and (5.4)– (5.7) to others (for examples of this phenomena see (B.2), (B.8) and (B.13)). With this overview in mind we can continue the construction process as far as we like. To be explicit we present the results of the construction for b in zones 1 and 2 in a problem with four or more zones.


349

b in zone 1. (1) = (j1 − t1 )(ν0 + 1) + 1, t1 + 1 ≤ j1 ≤ t2 , 1 : b = l1+j 1 (1) r = l˜1+j = j1 − ν 0 , 1 (t) Fr,s (L, b) = q c(j1 ) fs (L, ej1 − E2,n ).

(1) (0) (1) 2 : b = l1+j + l1+j = l1+j + 1 + j0 , 0 ≤ j0 ≤ t1 − 1, t1 + 1 ≤ j1 ≤ t2 − 1, 1 0 1 (1) r = l˜1+j + 1 = j1 − ν0 + 1, 1 ν0−2 4

Fr,s (L, b) = q c(j1 )−

(t) (t) fs (L, ej0 +ej1+1 −E1,n ) + q c(j1 ) f˜s (L, eν0−j0−1 +ej1 −E1,n ).

(1) (0) 3 : b = l1+t + l1+j = ν1 (ν0 + 1) + 2 + j0 , 0 ≤ j0 ≤ t1 − 1, 2 0 (1) r = l˜1+t + 1 = ν1 + 2, 2

Fr,s (L, b) = q c(t2 )−

ν0 −1 4

(t) fs (L, ej0 − et1 + e1+t2 − E3,n )

(t) + q c(t2 ) f˜s (L, eν0 −j0 −1 + et2 − E1,n ,)

(7.16) where 1 c(j1 ) = − (ν0 − 2)(j1 − ν0 − 1) for t1 + 1 ≤ j1 ≤ t2 . 4

(7.17)

b in zone 2. Here we distinguish separate cases depending on whether 1 ≤ j1 ≤ t2 − 2 or j1 = t2 − 1. The restriction (3.2) says that j1 = t2 does not occur. We also do not consider the cases b > y3 . (2) 1 : b = l1+j = y1 + (j2 − t2 )y2 , 1 + t2 ≤ j2 ≤ t3 , 2 (2) r = l˜1+j = 1 + (j2 − t2 )ν1 , 2 (t) Fr,s (L, b) = q c(j2 ) fs (L, ej2 − E3,n ).

(2) (1) (0) (2) 2 : b = l1+j + l1+j + l1+j = l1+j + (j1 − t1 )y1 + 2 + j0 , 2 1 0 2

0 ≤ j0 ≤ t1 − 1, t1 + 1 ≤ j1 ≤ t2 − 2, t2 + 1 ≤ j2 ≤ t3 − 1, r = l˜(2) + l˜(1) + 1 = l˜(2) + (j1 − t1 ) + 1, 1+j2

1+j1

Fr,s (L, b) = q

1+j2

c(j2 )+c(j1 )−

+ q c(j2 )+c(j1 )− + q c(j2 )+c(j1 )− + q c(j2 )+c(j1 )−

ν0 −2 4 ν0 2

+ 41

ν0 +1 4

ν0 2

+ 21

(t) fs (L, e1+j0 + et2 −(j1 −t1 )−1 + ej2 − E1,n )

(t) f˜s (L, eν0 −j0 −2 + et2 −(j1 −t1 ) + ej2 − E1,n ) (t) fs (L, ej0 + ej1 +1 + ej2 +1 − E1,n )

(t) f˜s (L, eν0 −j0 −1 + ej1 + ej2 +1 − E1,n ).

350


(2) (0) (2) 3 : b = l1+j + l1+j = l1+j + 1 + j0 , 0 ≤ j0 ≤ t1 , t2 + 1 ≤ j2 ≤ t3 − 1, 2 0 2 (2) r = l˜1+j + 1, 2 ν0

(t) ) Fr,s (L, b) = q c(j2 )− 4 fs (L, ej0 + et2 −1 + ej2 − E1,n (t) + q c(j2 ) f˜s (L, eν0 −j0 −1 − et1 + ej2 − E3,n )

+ q c(j2 )−

ν0 +1 4

(t) fs (L, ej0 + ej2 +1 − E1,n ).

(2) (1) (2) +l1+j = l1+j +(j1 − t1 )y1 + 1, t1 +1 ≤ j1 ≤ t2 − 2, t2 + 1 ≤ j2 ≤ t3 − 1, 4 : b = l1+j 2 1 2 (2) (1) (2) r = l˜1+j + l˜1+j = l˜1+j + (j1 − t1 ), 2 1 2 ν0

(t) ) Fr,s (L, b) = q c(j2 )+c(j1 )− 4 fs (L, et1 −1 + et2 −(j1 −t1 ) + ej2 − E1,n

+ q c(j2 )+c(j1 )− + q c(j2 )+c(j1 )−

ν0 2

+L 2

ν0 +1 4

(t) fs (L, et2 −(j1 −t1 )−1 + ej2 − E1,n )

(t) fs (L, ej1 + ej2 +1 − E2,n ).

(7.18) (2) (2) + lt(1) = l1+j + (ν1 − 1)y1 + 1, t2 + 1 ≤ j2 ≤ t3 − 1, 5 : b = l1+j 2 2 2 (2) (2) r = l˜1+j + l˜t(1) = l˜1+j + ν1 − 1, 2 2 2 ν0

(t) ) Fr,s (L, b) = q c(j2 )+c(t2 −1)− 4 fs (L, ej2 − E2,n

+ q c(j2 )+c(t2 −1)−

ν0 +1 4

(t) fs (L, e−1+t2 + e1+j2 − E2,n ).

(2) (0) (2) 6 : l1+j + lt(1) + l1+j = l1+j + (ν1 − 1)y1 + 2 + j0 , 2 2 0 2

0 ≤ j0 ≤ t1 − 1, t2 + 1 ≤ j2 ≤ t3 − 1, r = l˜(2) + l˜t(1) + 1 = l˜(2) + ν1 , 1+j2

1+j2

2

Fr,s (L, b) = q

c(j2 )+c(t2 −1)−

+ q c(j2 )+c(t2 −1)−

ν0 2

+ q c(j2 )+c(t2 −1)−

ν0 +1 4

+ 41

ν0 4

(t) f˜s (L, eν0 −j0 −1 + ej2 − E1,n )

(t) fs (L, ej0 + et2 + ej2 +1 − E1,n )

(t) f˜s (L, eν0 −j0 −1 + et2 −1 + ej2 +1 − E1,n ),

(7.19) where

1 3 + ν1 (ν0 − 2) − (j2 − t2 ) for t2 + 1 ≤ j2 ≤ t3 . (7.20) 2 4 Thus far we have used our constructive procedure to generate all polynomials Fr(b),s (L, b) for b in zones 0, 1 and 2 where the total number of zones is 4 or greater. However, to complete the process we must carry out the construction for b in the final zone and show that the closing relation in (4.6) is satisfied. The construction for (n) has two features not present in any other zone. The first is that the map b → r b > l1+t 1+n of (3.4) has changed and the second is that the parity restriction on mt1+n , which was even before, is now sometimes allowed to be odd. We recall that this parity is specified in our notation by the parity of the 1 + t1+n component of the vector u. Thus we compute the following results for the final zone. c(j2 ) =


351

b in zone 1 in a two zone problem. 1: The first equation of (7.16) now may be extended to all 1 + t1 ≤ j1 ≤ 1 + t2 with (t) replaced by zero and c(t2 + 1) given by (7.17). −E2,n (t) replaced 2: The second equation of (7.16) now holds for all 1+t1 ≤ j1 ≤ t2 with −E2,n by zero. 3: Equation three of (7.16) is omitted and replaced by (1) (0) + l1+j = (ν1 + 1)(ν0 + 1) + 2 + j0 , 0 ≤ j0 ≤ t1 − 1, b = l2+t 2 0 (1) r = l˜2+t = ν1 + 1 = p − 1, 2

Fr,s (L, b) = q

c(t2 +1)

(7.21)

fs (L, et1 −j0 −1 − et1 + e1+t2 ),

where c(t2 + 1) is given by (7.17) instead of (7.20).

b in zone 2 in a three zone problem. 1: Equation 1 in (7.18) is now valid for 1 + t2 ≤ j2 ≤ t3 + 1 and Eqs. 2–6 in (7.18) and (7.19) are now valid for t2 + 1 ≤ j2 ≤ t3 + 1 with the convention that wherever e2+t3 ˜ the term is omitted and c(t3 + 1) appears in the argument of some fs (L, u) or f˜s (L, u) is given by (7.20). 2: We have the following additional closing equation: (2) (0) b = l2+t + lt(1) + l1+j = p0 − ν0 + j0 − 1, 2 3 0

0 ≤ j0 ≤ t1 , (2) r = l˜2+t + l˜t(1) = ν1 (ν2 + 2) = p − 1, 2 3

(7.22)

1

Fr,s (L, b) = q c(t3 +1)+c(t2 )− 2 fs (L, eν0 −j0 − et1 − et2 + e1+t3 ), where c(t3 + 1) is given by (7.20). For the problem with 2 and 3 zones we have now constructed a complete set of fermionic polynomials for all 1 ≤ b ≤ p0 − 1 which satisfy the bosonic recursion relations (4.4), (4.5) and the first equation in (4.6) by construction and the second equation in (4.6) by use of the first equation of (5.3). In order to complete the proof of the bose/fermi identities it remains to show that the fermionic polynomials satisfy the boundary conditions for Br(b),s (L, b) at L = 0 (4.7) and to compute the normalization constant in (1.11). This is easily done and thus we obtain the final result that for the (µs ) , cases of 2 and 3 zones with s = l1+j s Fr(b),s (L, b) = q 2 (φr(b),s −φr(s),s )+c(js ) Br(b),s (L, b) 1

where Fr(b),s (L, b) is given by the formulae of this section.

(7.23)

352


8. The Inductive Analysis of Evolution In the previous section we constructed all fermionic polynomials Fr(b),s (L, b), 1 ≤ b ≤ p0 − 1 in the problem with 1,2 and 3 zones. The procedure we used is completely general, but becomes somewhat tedious to execute when the number of zones increases beyond three. This is because the results are sensitive to the details of the Takahashi decomposition (3.1). On the other hand, there are certain cases of (3.1) such as b = (µ) , 1+tµ ≤ jµ ≤ t1+µ +δµ,n , 0 ≤ µ ≤ n, where the form of Fr(b),s (L, b) remains very l1+j µ simple for any µ. The question arises if one can treat certain classes of decompositions (3.1) without having explicit formulas for all fermionic polynomials Fr(b),s (L, b), 1 ≤ b ≤ p0 − 1. In this section we shall provide a positive answer to this question by proving inductively a set of explicit formulas for certain flows of length yµ − 1. This inductive analysis is possible because as can be seen from (3.13) the construction of (µ) , 1 + tµ ≤ jµ ≤ t1+µ − 1 + δµ,n , µ ≥ 2 from the pair Fr(b),s (L, b), b = l2+j µ (µ) involves exactly the same steps {Fr(b0 )+1,s (L, b0 + 1), Fr(b0 )+1,s (L, b0 + 2)}, b0 = l1+j µ as that of Fzµ−1 ,s (L, yµ ) from the pair {F1,s (L, 1), F1,s (L, 2)}. Furthermore, recalling that y1+µ = yµ−1 + νµ yµ , it is natural to decompose the flow of length yµ+1 − 1 into flows of smaller length and to take this decomposition as a starting point of our inductive analysis of evolution. In this direction, we first discuss the notation for the flows in terms of the b → r map, whose properties are summarized in (3.10)–(3.14).

Notation. The flow −→x of length x + 1 denotes the sequence of steps corresponding to the b → r map as defined in (3.4) (or equivalently (3.8)) between b = 1 and b = 2 + x. r

(8.1) 1 1

2+x

b

Piece of b → r map between 1 and 2+x (schematic)

According to (3.10) and (3.11) this sequence is made up of three steps: a)

b)

c)

(8.2) Different kinds of steps in the b → r map

(where we note that steps like

(8.3) Two kinds of steps which are not allowed


353

are not allowed). The b → r map specifies exactly which bosonic recursion relation is being used to define the (in this case fermionic) polynomials. Case a) in (8.2) denotes that r is the same for all three objects Fr,s (L, b − 1), Fr,s (L, b) and Fr,s (L, b + 1) involved and hence one uses the first recursion relation (4.4). Case b) indicates the use of the second and case c) the use of the last recursion relation in (4.4). Let us further denote by =⇒2 the flow according to the steps as defined by the following graph: (8.4) 2

piece of the b → r map of length 3 defining

The b → r map as defined by (3.4) (or (3.8)) has an unambiguous initial point (r1 , b1 ) = (1, 1). By −→x we mean only the sequence of x steps found on the b → r map as specified in (8.1). The notation −→x does not fix the initial point (r1 , b1 ). The initial point (r1 , b1 ) can be placed anywhere . Having agreed on that, we can now piece together flow 1 and flow 2 such that the last segment of the first flow is identical to the first segment of the second flow by identifying the two final points of the first flow as the two initial points of the second flow. For example adjoining flow c of (8.2) to flow b of (8.2) gives flow (8.4) and adjoining a of (8.2) to c of (8.2) gives (8.5) We then denote by −→x(1) −→x(2) the flow given by piecing together the flow −→x(1) and −→x(2) . Note that in general it is not true that −→x(1) +x(2) = −→x(1) −→x(2) . With these conventions we now show that there is a decomposition of −→yµ+1 −2 in terms of −→yµ−1 −2 , −→yµ −2 and =⇒2 given by the following graph: yµ+1-2

= yµ-2

2 yµ-1-2

2

yµ-2

2

yµ-2

2

yµ-2

(8.6)

A decomposition of the flow of length yµ+1-1, µ>1 with νµ>1

In other words we need to show that piecing together −→yµ −2 , −→yµ−1 −2 and =⇒2 as shown in (8.6) amounts to −→yµ+1 −2 . This can be easily done by recalling (3.12) with b = yµ and b = yµ−1 + kyµ for 1 ≤ k ≤ νµ − 1, 2 ≤ µ and proving two additional results: r(b + yµ ) − r(b) = r(yµ ) = zµ−1 , for 1 ≤ b ≤ yµ−1 (8.7) and (µ) r(b + yµ−1 + kyµ ) − r(b) = r(yµ−1 + kyµ ) = l˜1+t for 1 ≤ b ≤ yµ . µ +k

(8.8)

To check (8.7) we use the Takahashi decomposition of b (3.1) with µβ ≤ µ − 2. Pβ (µi ) (µ−1) + l1+t is a valid decomposition in Takahashi If b 6= yµ−1 then b + yµ = i=1 l1+j µi µ lengths and according to (3.4) we have r(yµ + b) − r(b) = l˜(µ−1) = zµ−1 . Similarly for 1+tµ

354


(µ) (µ−2) b = yµ−1 we have r(yµ−1 + yµ ) − r(yµ−1 ) = l˜2+t − l˜1+t = zµ−1 . Equation (8.8) may µ µ−1 be checked in a similar fashion. Notice however that the order of the arrows in the decomposition as shown in the previous figure is crucial. If we moved for example −→yµ−1 −2 =⇒2 to the one before last position as shown in the next figure the decomposition does not agree with all the steps as defined by −→yµ+1 −2 . yµ+1-2

= yµ-2

2

yµ-2

yµ-2

2

yµ-2

2

2 yµ-1-2

2 yµ-2

(8.9)

A wrong decomposition of the flow of length yµ+1-1, µ>1

This can be easily seen from (3.12) with b = yµ−1 + (νµ − 1)yµ and b = kyµ , 1 ≤ k ≤ νµ − 1, 2 ≤ µ and the following lemma: Lemma 1.1. a. For 1 ≤ k ≤ νµ − 2, µ ≥ 2 we have r(b + kyµ ) − r(b) = kzµ−1 for all 1 ≤ b ≤ yµ except for b = bµ,−1 =

Pµ−1 i=0

(8.10)

yi for which

r(bµ,−1 + kyµ ) − r(bµ,−1 ) = kzµ−1 + (−1)µ .

(8.11)

b. for 1 ≤ b ≤ yµ−1 we have r(b + (νµ − 1)yµ ) − r(b) = (νµ − 1)zµ−1 .

(8.12)

r(b + lt(µ) ) − r(b) = l˜t(µ) . 1+µ 1+µ

(8.13)

c. for 1 ≤ b ≤ yµ

We shall also require the companion of (8.10) and (8.11) Lemma 1.2. For µ ≥ 3 we have r(b + yµ−1 ) − r(b) = zµ−2 for all 1 ≤ b ≤ yµ except for b = bµ,1 =

Pµ−2 i=0

(8.14)

yi for which

r(bµ,1 + yµ−1 ) − r(bµ,1 ) = zµ−2 + (−1)µ−1 .

(8.15)

These results can be checked in the same fashion as Eqs. (8.7) and (8.8). The decomposition (8.9) fails precisely because there is a special value b = bµ,−1 for which (8.10) is not valid. However, one can find a slightly modified version of (8.9) which does indeed hold. For this we need to define two further types of arrows.


355

y −2

µ Definition. We denote by −→−1 the flow according to the steps defined by the b → r map (3.4) between 1 + yµ and 2yµ .

r

(8.16) 1 1

2yµ

1+yµ

b

b → r map between 1+yµ and 2yµ (schematic) y −2

We denote by −→1 µ the flow according to the steps defined by the b → r map (3.4) between 1 + yµ−1 and yµ−1 + yµ . r

(8.17) 1 1

yµ-1+yµ

1+yµ-1

b

b → r map between 1+yµ-1 and yµ-1+yµ (schematic) We further identify yµ −2

yµ −2

−→ = −→ 0

(8.18)

Slightly generalizing the above discussion we recapitulate. Let Oi (L, bi , q) for i = 1, 2 be polynomials (in q 1/4 ) depending on L where L is a nonnegative integer. Let Oi (L, bi , q) for i = 3, 4 be polynomials depending on L obtained recursively from O1 (L, b1 , q), O2 (L, b2 , q) by the flow −→xt of length 1 + x. We denote this as x

{O1 (L, b1 , q) , O2 (L, b2 , q)} −→ {O3 (L, b3 , q), O4 (L, b4 , q)} , t

(8.19)

where the symbol x above the arrow denotes that O4 (L, b4 , q) follows from O1 (L, b1 , q) and O2 (L, b2 , q) after x steps. Parameters bi , i = 1, 2, 3, 4 with b2 = b1 + 1, b3 = b1 + x, b4 = b1 + x + 1 and q associated with Oi will often be suppressed. The symbol t below the arrow denotes which sequences of the recursion relations in (4.4) are

356


being used. The sequence t can be thought of pictorially as a continuous graph made up of horizontal and diagonal segments where by horizontal segment we mean only the segment between (r, b) and (r, b + 1) and by diagonal segment we mean the segment between (r, b) and (r + 1, b + 1). The only restriction on the sequence t is that a diagonal segment must be preceded and followed by horizontal segments unless the diagonal segment is the first or last segment. As in the case of the notation −→x , the notation −→xt does not fix the initial point. Each pair of adjacent segments, called a step, in the flow −→xt represents one of the three recursion relations in (4.4) in a manner exactly analogous to the discussion following (8.2). More precisely, 1

{O1 (L, b1 , q), O2 (L, b1 + 1, q)} −→{O3 (L, b1 + 1, q), O4 (L, b1 + 2, q)}, t

(8.20)

where O3 (L, b, q) = O2 (L, b, q)

(8.21)

and

 O2 (L + 1, b1 + 1, q) − O1 (L, b1 , q) + (1 − q L )O2 (L − 1, b1 + 1, q)     if t is a of (8.2)   −L/2 [O2 (L + 1, b1 + 1, q) − O1 (L, b1 , q)] O4 (L, b1 +2, q) = q if t is b of (8.2)    (L+1)/2  O1 (L, b1 , q)   O2 (L + 1, b1 + 1, q) − q if t is c of (8.2). (8.22) Note that for L = 0 the last term in the top line of (8.22) vanishes. Therefore one does not need to know the polynomials O1 (L, b1 , q), O2 (L, b1 + 1, q) for L < 0 to determine O4 (L, b1 + 2, q) for L ≥ 0. To compare three different flows (8.16)–(8.18), let us set the initial points of the yµ −2 y −2 , −→0 µ to be (r1 , b1 ) = (1, 1). Then according to (8.10), (8.11) and flows −→±1 y −2 (8.14) ,(8.15) the pieces of the b → r map used to define −→aµ , a = ±1 differ from y −2 the one used to define −→0 µ at exactly one point. We refer to this phenomena as the Pµ−1−δ dissynchronization effect. Recalling the definition for a = ±1 that bµ,a = i=0 a,1 yi , y −2 y −2 µ ≥ 2 + δa,1 we find that the difference between −→aµ , a = ±1 and −→0 µ can be illustrated by the following two figures. Here the solid line denotes the piece of the y −2 y −2 b → r map for −→aµ , a = ±1 and the dashed line that for −→0 µ .

bµ,a-1

bµ,a

bµ,a+1

Dissynchronization effect for µ-δa,1 even with a=-1,+1

bµ,a-1

bµ,a

bµ,a+1

(8.23)

Dissynchronization effect for µ-δa,1 odd with a=-1,+1

More precisely, all elementary segments in these two flows are identical except in the y −2 interval (bµ,a − 1, bµ,a + 1). For µ − δa,1 even (respect. odd) the flow −→aµ , a = ±1 restricted to the interval [bµ,a − 1, bµ,a + 1] is given by c of (8.2) (b of (8.2)). The flow, y −2 −→0 µ , restricted to the same interval is given by b of (8.2) (c of (8.2)).


357 y

We now may prove the following two decompositions of −→aµ+1 −1, 0, 1

−2

, where a =

yµ+1-2 a

= yµ-2

2 yµ-1-2

-δa,1

2

2

yµ-2 δa,-1

0

2

yµ-2 0

yµ-2 0

(8.24)

The decomposition of the flow of length yµ+1-1 for νµ>1 in the cases µ >1, a=0,1 and µ >2, a=-1 used to prove Proposition 1 and

yµ+1-2 a

= yµ-2

2

-δa,1

yµ-2 -δa,0-δa,1

2

yµ-2 -1

2

yµ-2 -1

2 yµ-1-2 0

2 yµ-2 0

(8.25)

The decomposition of the flow of length yµ+1-1, µ>1 with νµ>2 used to prove Proposition 2 y −2

In (8.24) the arrows =⇒2 −→0 µ appear νµ − 2 times and in (8.25) the arrows yµ −2 =⇒2 −→−1 appear νµ − 3 times. The decomposition (8.24) for a = 0 is (8.6). To prove (8.24) for a = ±1 we use (8.24) with a = 0 and (8.23) with a = ±1, µ replaced by µ + 1 to convert the top graph of (8.24) with a = 0 into the top graph of (8.24) with a = ±1, by altering just the single step centered at bµ+1,a , a = ±1 as prescribed by (8.23). It is easy to see from (8.23) that this procedure converts the first (fifth) arrow of the bottom yµ−2 y (−→1 µ−2 ) for a = 1(−1) and has no effect on other graph (8.24) with a = 0 into −→−1 arrows of the bottom graph (8.24). This completes the proof of (8.24). The decomposition (8.25) is the correct modification of (8.9). When a = 0 it follows from (8.10)-(8.13) and (8.23) with a = −1. To prove (8.25) for a = 1(−1) we again replace one single step of the top graph of (8.25) with a = 0. This converts the first yµ −2 y −2 (−→0 µ ) and thus completes the proof. (third) arrow of the bottom graph into −→−1 Let us finally define some useful sets of terms Definition. For 1 ≤ µ ≤ n, P−1 (L, µ, uµ0 ), P0 (L, µ, uµ0 ) and P1 (L, µ, uµ0 ) are defined as follows: P−1 (L, µ, uµ0 ) =

µ X

q2− L

ν0 −θ(i odd) 4

(t) fs (L, −E1,i + e−1+ti + uµ0 ) + fs (L, e−1+t1 − et1 + uµ0 ),

i=2

=

µ X i=2

q

ν0 −1−θ(i even) 4

(t) f˜s (L, −E1,i + e−1+ti + uµ0 ) + fs (L, e−1+t1 − et1 + uµ0 )

(8.26)

358


P0 (L, µ, uµ0 ) =

µ X

q−

ν0 −θ(i odd) 4

(t) fs (L, −E1,i + e−1+ti + uµ0 ) + f˜s (L, eν0 −1 − eν0 + uµ0 )

,

(8.27)

i=2

and P1 (L, µ, uµ0 ) =

µ X

q−

ν0 −θ(i odd) 4

(t) fs (L, e1 − E1,i + e−1+ti + uµ0 ) + f˜s (L, eν0 −2 − eν0 + uµ0 )

, (8.28)

i=2

where uµ0 is any 1 + tn+1 -dimensional vector with non-zero entries only in zone µ or higher, i.e. (uµ0 )i = 0 for i ≤ tµ . (We consider here the case that νi ≥ 2, 1 ≤ i ≤ n − 1.) Equipped with the above notations and explanations we are now in the position to formulate two propositions which will be important in the sequel. Proposition 1. Let a = −1, 0, 1 and P−1 (L, µ, uµ0 ) be defined as in (8.26). Then we have for a = 0, µ ≥ 1 and a = −1, µ ≥ 2 and a = 1, µ ≥ 3, o n (t) (t) + uµ0 ), fs (L, e1 − E1,µ + uµ0 ) fs (L, −E1,µ (8.29) . µ 1 yµ −2 × −→ q c(tµ )+ 2 a(−1) P−1 (L, µ, uµ0 ), fs (L, uµ0 ) a

Proposition 2. Let a = −1, 0, 1 and P0 (L, µ, uµ0 ) and P1 (L, µ, uµ0 ) be defined as in (8.27) and (8.28). Then we have for a = 0, µ ≥ 1 and a = −1, µ ≥ 2 and a = 1, µ ≥ 3, P0 (L, µ, uµ0 ), P1 (L, µ, uµ0 ) n L−ν0 +θ(µ odd) o yµ −2 (t) +c(tµ )+ 21 a(−1)µ 2 fs (L, −E1,µ + uµ0 ), 0 . × −→ q (8.30) a o n µ 1 1 (t) = q c(tµ )− 2 θ(µ even)+ 2 a(−1) f˜s (L, −E1,µ + uµ0 ), 0 Here c(tµ ) is defined recursively by c(tµ+1 ) = c(tµ−1 ) + νµ c(tµ ) −

ν0 − θ(µ even) ν0 + 1 3 + (νµ − 1)(− + θ(µ odd)), (8.31) 4 4 4

where c(t0 ) =

ν0 , 4

c(t1 ) = 0.

(8.32)

We will also need c(j0 ) = 0 for 1 ≤ j0 ≤ t1 ,

1 ν0 + 1 3 µ + θ(µ odd) + c(tµ ) + c(tµ−1 ) c(jµ ) = (−) + (jµ − tµ ) − 2 4 4 for tµ + 1 ≤ jµ ≤ tµ+1 + 2δµ,n and 1 ≤ µ ≤ n.

(8.33)

Propositions 1 and 2 are very important because they enable us to prove many identities without explicitly constructing all fermionic polynomials in a one by one fashion. From them we will prove Theorem 1 (8.50) and as a result obtain the Rogers–Schur– (µ) . The reason for the introduction Ramanujan identities for M (p, p0 ) models at b = l1+j µ


359

of the several sets of terms P0 and P±1 is that (as can be seen quite explicitly in (10.5) (µ) with j0 = 0, 1) the polynomials Fr(b+a),s (L, b + a) for b = l1+j , a = 1, 2, 1 ≤ µ ≤ µ n, 1 + tµ ≤ jµ ≤ tµ+1 − 1 + δµ,n can be written as {Fr(b+1),s (L, b + 1), Fr(b+2),s (L, b + 2)} = q c(jµ )− +q

c(jµ )

ν0 +1 3 4 + 4 θ(µ

odd)

(t) (t) {fs (L, e1+jµ − E1,n ), fs (L, e1 + e1+jµ − E1,n )} ,

{P0 (L, µ, ejµ −

(t) Eµ+1,n ), P1 (L, µ, ejµ

−

(8.34)

(t) Eµ+1,n )}

and then we see from (8.29) and (8.30) that the first pair on the right-hand side of (8.34) may be studied independently from the second pair under the flow −→b−2 , b ≤ yµ (but we note this independence is not true when b > yµ ). We note too that the proofs given below of the two propositions are also quite independent. In particular, for (8.29) we use decomposition (8.24) while for (8.30) we use (8.25). The final tool we need for our inductive proof are the following lemmas which we use to treat the evolution along =⇒2 in the decompositions (8.24) and (8.25): µ

Lemma 2.1. Define g(µ, n, jµ ) = − ν04+1 + 43 θ(µ odd) + (−1) 4 θ(n > µ)δjµ ,tµ+1 , 1 ≤ (t) + pe ˜ 1+t1+n ) = 0 µ ≤ n, 1 + δµ,1 + tµ ≤ jµ ≤ tµ+1 + δµ,n , and fs (L, ea + e2+tn+1 − E1,n for a = 0, 1 and p˜ ∈ Z. Then we have P−1 (L, µ − δ1+tµ ,jµ , ejµ − θ(n > µ)et1+µ + u00 (jµ )) +q2− L

ν0 −θ(µ even) 4

(t) fs (L, e−1+jµ − θ(n > µ)et1+µ − E1,µ + u00 (jµ )), − θ(n > µ)et1+µ + u00 (jµ ))

fs (L, ejµ n (t) fs (L, e1+jµ + θ(n > µ)δt1+µ ,jµ etµ+1 − E1,µ+θ(n>µ) =⇒q + u00 (jµ )), , (8.35) o (t) fs (L, e1 + e1+jµ + δt1+µ ,jµ θ(n > µ)etµ+1 − E1,µ+θ(n>µ) + u00 (jµ )) + P0 (L, µ, ejµ − θ(n > µ)et1+µ + u00 (jµ )), P1 (L, µ, ejµ − θ(n > µ)et1+µ + u00 (jµ )) 2

g(µ,n,jµ )

where u00 (jµ ) =

0 0 uµ+1 − δjµ ,t1+µ (uµ+1 )1+t1+n e1+t1+µ pe ˜ 1+t1+n , p˜ ∈ Z

if µ < n if µ = n.

(8.36)

Lemma 2.2. For 1 + tµ ≤ jµ ≤ t1+µ − 1 − δµ,1 , 1 ≤ µ ≤ n − 1 we have 0 {P−1 (L, µ, et1+µ −(jµ −tµ ) − et1+µ + uµ+1 )+

q2− L

ν0 +1 3 4 + 4 θ(µ

odd)

(t) 0 fs (L, et1+µ −(jµ −tµ )+1 − E1,µ+1 + uµ+1 ),

0 fs (L, et1+µ −(jµ −tµ ) − et1+µ + uµ+1 )} 2

=⇒ q −


(t) 0 {fs (L, et1+µ −(jµ −tµ )−1 − E1,µ+1 + uµ+1 ),

(t) 0 fs (L, e1 + et1+µ −(jµ −tµ )−1 − E1,µ+1 + uµ+1 )} 0 + {P0 (L, µ − δjµ ,t1+µ −1 , et1+µ −(jµ −tµ ) − et1+µ + uµ+1 ), 0 )}. P1 (L, µ − δjµ ,t1+µ −1 , et1+µ −(jµ −tµ ) − et1+µ + uµ+1

(8.37)

360


These lemmas follow immediately from the recursive properties (5.3)-(5.7) of fs and the definition (3.35) of f˜s . Proof of Propositions 1 and 2. We prove Propositions 1 and 2 by induction on µ. For y −2 the proof of Proposition 1 we use the decomposition of −→aµ+1 as given in (8.24) and for the proof of Proposition 2 the decomposition (8.25). Since in the decompositions yµ −2 y −2 yµ−1 −2 and −→0 µ−1 but never −→±1 Propo(8.24) and (8.25) one uses flows −→±1,0 sitions 1 and 2 for µ + 1 and all a will follow if Propositions 1 and 2 are true for a = 0 and µ, µ − 1 and a = −1, 1 and µ. One may check that it is sufficient to prove the following initial conditions for Propositions 1 and 2: a) Propositions 1 and 2 for a = 0 and µ = 1, 2, b) Propositions 1 and 2 for a = −1 and µ = 2, c) Proposition 1 for a = −1 and µ = 3. With these initial conditions Propositions 1 and 2 with a = 0 follow for all µ ≥ 1, Propositions 1 and 2 with a = −1 follow for all µ ≥ 2 and Propositions 1 and 2 with a = 1 follow for all µ ≥ 3. Proof of the initial conditions. Here we prove point a) only. Points b) and c) are treated in Appendix B. We first consider Proposition 1 for µ = 1, 2 and a = 0. Taking into account (7.5) with j0 = 0, 1, ν0 − 1, ν0 and case 2 of (7.16) with j0 = −1 + t1 , j1 = −1 + t2 along with case 1 of (7.16) with j1 = t2 and recalling definition of P−1 (8.26) we see that (t) (t) (t) (t) − E1+i,n ), fs (L, e1 − E1,i − E1+i,n ) {fs (L, −E1,i yi −2 c(ti )

−→ q 0

(t) (t) {P−1 (L, i, −E1+i,n ), fs (L, −E1+i,n )}

(8.38)

(t) by ui0 in (8.38) we obtain Proposition 1 for with i = 1, 2. Now if we replace −E1+i,n µ = 1, 2. Note that the above replacement is legitimate because the recursive properties of Sects. 5 and 6 allow us to repeat the constructions of Sect. 7 for any vector ui0 . The proof of Proposition 2 for µ = 1, 2 and a = 0 is only slightly more involved. Making use of case 2 of (7.16) with j0 = 0, 1, ν0 −1, case 1 of (7.16) with j1 → j1 +1 on one hand and case 3 of (7.18) with j0 = 0, 1, 1 + t2 ≤ j2 ≤ t3 − 2, case 6 of (7.19) with j0 = −1 + t1 and case 1 of (7.18) with j2 → j2 + 1 on the other hand we obtain upon (i) < yn , recalling the definitions (8.26)-(8.28) and property (3.13) with b = l1+j i

n

o (t) (t) (t) (t) fs (L, −E1,i + e1+ji − E1+i,n ), fs (L, e1 − E1,i + e1+ji − E1+i,n ) n o (t) (t) + P0 (L, i, eji − E1+i,n ), P1 (L, i, eji − E1+i,n ) n o yi −2 c(ti )− ν0 +1 + 3 δi,1 (t) (t) 4 4 P−1 (L, i, e1+ji − E1+i,n −→ q ), fs (L, e1+ji − E1+i,n ) 0 o n 1 (t) ), 0 + q c(ti )− 2 δi,2 f˜s (L, eji − E1,n (8.39) (t) to derive from with i = 1, 2. Next we use Proposition 1 with µ = i, ui0 = e1+ji − E1+i,n (8.39), q−

ν0 +1 3 4 + 4 δ1,i


361

o

n

(t) (t) ), P1 (L, i, eji − E1+i,n ) P0 (L, i, eji − E1+i,n o n 1 yi −2 (t) × −→ q c(ti )− 2 δi,2 f˜s (L, eji − E1,n ), 0 .

(8.40)

0

Thus, replacing eji −

(t) E1+i,n

by ui0 in (8.40) we obtain Proposition 2 with µ = 1, 2.

We conclude this subsection with the following comment. In deriving (8.40) from (8.39) we succeeded in separating the evolution of a {P0 , P1 } pair from that of a {fs , fs } (2) +1 ≤ pair. This separation can be made in the formulas (7.18) and (7.19) as long as l1+j 2 (2) b ≤ l2+j2 , 1 + t2 ≤ j2 ≤ t3 − 1. All descendents of the {fs , fs } pair will have e1+j2 in their arguments and all descendants of the pair {P0 , P1 } will have ej2 in their arguments instead. Using this identification principle we easily obtain for ν1 ≥ 3, ν0 ≥ 2, {P0 (L, 2, u20 ), P1 (L, 2, u20 )} y1 +1

−→ q − + q−

ν0 −1 2

(t) (t) {fs (L, e1 + e−2+t2 − E1,2 + u20 ), fs (L, e2 + e−2+t2 − E1,2 + u20 )}

ν0 −2 4

(t) (t) {f˜s (L, e−2+t1 + e−1+t2 − E1,2 + u20 ), f˜s (L, e−3+t1 + e−1+t2 − E1,2 + u20 )} (8.41) by comparing case 2 of (7.18) with j0 = 0, 1; j1 = 1 + t1 and case 3 of (7.18)with (t) by u20 . This result will be used in Appendix B to prove j0 = 0, 1 and replacing ej2 − E3,n Proposition 2 with a = −1, µ = 2.

Proof of Proposition 1 for µ − 1, µ → µ + 1. Let us now show that Proposition 1 with y −2 a = −1, 0, 1 follows inductively for µ + 1. We will decompose −→aµ+1 according to (8.24) which allows us to use Proposition 1 and 2 for µ and µ − 1 for which they are true by assumption. Let us start by applying proposition 1 with µ, a0 = −δa,1 and uµ0 = n o (t) (t) 0 0 0 uµ+1 − etµ+1 to fs (L, −E1,µ+1 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) and subsequently using Lemma 2.1 with µ → µ − 1, jµ−1 = tµ to obtain n o (t) (t) 0 0 fs (L, −E1,µ+1 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) yµ −2 c(t )+ 1 δ (−1)µ+1 0 0 −→ q µ 2 a,1 ), fs (L, −etµ+1 + uµ+1 ) P−1 (L, µ, −etµ+1 + uµ+1 −δa,1 ν0 −θ(µ even) n µ+1 1 2 (8.42) (t) 0 4 q− fs (L, −E1,µ−1 =⇒ q c(tµ )+ 2 δa,1 (−1) + e1+tµ − etµ+1 + uµ+1 ), o (t) 0 fs (L, e1 − E1,µ−1 + e1+tµ − etµ+1 + uµ+1 ) 0 0 + P0 (L, µ − 1, −etµ+1 + uµ+1 ), P1 (L, µ − 1, −etµ+1 + uµ+1 ) . where we have noticed from (8.26) that P−1 (L, µ, uµ0 ) = P−1 (L, µ − 1, etµ − etµ + uµ0 ) +q2− L

ν0 −θ(µ−1 even) 4

(t) fs (L, e−1+tµ − etµ − E1,µ−1 + uµ0 ).

(8.43)

We point out that the appearance of µ−1 instead of µ in the right-hand side of (8.42) after yµ steps explains the use of decomposition (8.24) instead of (8.25) for the proof of Proposition 1. We now apply Proposition 1 with µ → µ−1, a0 = 0 to the first pair and Proposition 2 with µ → µ − 1 a0 = 0 to the second pair in the right-hand side of (8.42) which yields

362


o (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 yµ−1 +yµ −2 c(1+t )+ 1 δ (−1)µ+1 0 µ 2 a,1 −→ q ) P−1 (L, µ − 1, e1+tµ − etµ+1 + uµ+1 n

(8.24)

+q 2 − L


(8.44) o

(t) 0 0 fs (L, −E1,µ−1 − etµ+1 + uµ+1 ), fs (L, e1+tµ − etµ+1 + uµ+1 ) .

even) + c(tµ ) + c(tµ−1 ) which follows from (8.33). Here we have used c(1 + tµ ) = − ν0 −θ(µ 4 The symbol (8.24) under the arrow in (8.44) means that we have evolved the initial state according to the first yµ−1 + yµ − 2 steps of the decomposition given by (8.24). Next applying Lemma 2.1 with µ and jµ = 1 + tµ we obtain o n (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 ν0 +1 3 n yµ−1 +yµ c(1+t )+ 1 δ (−1)µ+1 (t) 0 µ 2 a,1 q − 4 + 4 θ(µ odd) fs (L, −E1,µ −→ q + e2+tµ − etµ+1 + uµ+1 ), (8.24) o (t) 0 fs (L, e1 − E1,µ + e2+tµ − etµ+1 + uµ+1 ) 0 0 + P0 (L, µ, e1+tµ − etµ+1 + uµ+1 ), P1 (L, µ, e1+tµ − etµ+1 + uµ+1 ) . (8.45) Now we may use Propositions 1 and 2 again with µ and a0 = δa,−1 which yields o n (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 yµ−1 +2yµ −2 c(2+t )+ 1 a(−1)µ+1 0 µ 2 −→ q ) P−1 (L, µ, e2+tµ − etµ+1 + uµ+1 (8.24) o ν0 −θ(µ even) L (t) 0 0 4 +q 2 − fs (L, −E1,µ + e1+tµ − etµ+1 + uµ+1 ), fs (L, e2+tµ − etµ+1 + uµ+1 ) , (8.46) where we used c(2 + tµ ) = c(1 + tµ ) − ν04+1 + 43 θ(µ odd) + c(tµ ) which again follows from µ+1 1 0 ) (8.33). Notice that the final entry of (8.44) q c(1+tµ )+ 2 (−1) δa,1 fs (L, e1+tµ − etµ+1 + uµ+1 0 and the final entry of (8.46) q c(2+tµ )+ 2 a(−1) fs (L, e2+tµ − etµ+1 + uµ+1 ) only differ in that e1+tµ has become e2+tµ and the phase factor has changed. Applying now repeatedly y −2 =⇒2 and −→0 µ according to the decomposition (8.24) and using Lemma 2.1 and Propositions 1 and 2 for µ we obtain o n (t) (t) 0 0 + uµ+1 ), fs (L, e1 − E1,µ+1 + uµ+1 ) fs (L, −E1,µ+1 yµ−1 +(jµ −tµ )yµ −2 c(j )+ 1 a(−1)µ+1 0 −→ q µ 2 ) P−1 (L, µ, ejµ − etµ+1 + uµ+1 (8.24) o ν0 −θ(µ even) L (t) 0 0 2 +q 2 − fs (L, −E1,µ + e−1+jµ − etµ+1 + uµ+1 ), fs (L, ejµ − etµ+1 + uµ+1 ) . (8.47) The arrow in (8.47) denotes the flow after the first yµ−1 + (jµ − tµ )yµ − 2 steps (tµ + 1 < jµ ≤ tµ+1 ) according to the decomposition (8.24). Finally setting jµ = tµ+1 in (8.47) and using the easily verifiable identity 1

µ+1

L − ν0 + θ(µ odd) L ν0 − θ(µ even) + c(tµ ) − c(t1+µ ) = − (8.48) 2 2 4 along with (8.43) with µ → µ + 1 we obtain Proposition 1 for µ + 1. This concludes the proof of Proposition 1. c(−1 + t1+µ ) +


363

Proof of Proposition 2 for µ − 1, µ → µ + 1. Let us show that Proposition 2 holds for µ + 1 and a = −1, 0, 1 inductively. First we assume νµ > 2. Recalling the definition of uµ0 we see from (8.27) and (8.28) that 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 n ν0 −θ(µ even) (t) 0 4 fs (L, −E1,µ+1 + e−1+tµ+1 + uµ+1 ), = q− o (t) 0 0 0 fs (L, e1 − E1,µ+1 + e−1+tµ+1 + uµ+1 ) + P0 (L, µ, uµ+1 ), P1 (L, µ, uµ+1 ) .

(8.49)

We evolve these polynomials according to the decomposition (8.25). Thus we first evolve the first pair on the rhs of (8.49) using Proposition 1 and the second pair using Proposi0 followed by Lemma 2.2 with tion 2 for µ, a0 = −δa,1 and uµ0 = e−1+tµ+1 − etµ+1 + uµ+1 jµ = 1 + tµ to obtain 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 yµ −2 c(t )− ν0 −θ(µ even) + 1 δ (−1)µ+1 0 4 2 a,1 −→ q µ ) P−1 (L, µ, e−1+tµ+1 − etµ+1 + uµ+1

−δa,1

+q 2 − L

ν0 +1 3 4 + 4 θ(µ

ν0 −θ(µ even)

2

odd)

o

(t) 0 0 fs (L, −E1,µ + uµ+1 ), fs (L, e−1+tµ+1 − etµ+1 + uµ+1 ) µ+1

1

+ 2 δa,1 (−1) 4 =⇒q c(tµ )− ν0 −θ(µ even) n (t) 0 4 × q− fs (L, −E1,µ + e−2+tµ+1 − etµ+1 + uµ+1 ),

o

(t) 0 fs (L, e1 − E1,µ + e−2+tµ+1 − etµ+1 + uµ+1 )

0 0 + P0 (L, µ, e−1+tµ+1 − etµ+1 + uµ+1 ), P1 (L, µ, e−1+tµ+1 − etµ+1 + uµ+1 ) . (8.50) In the next step we apply Proposition 1 and Proposition 2 with µ and a0 = −δa,0 − δa,1 which yields 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 µ+1 1 2yµ −2 c(2+t )−c(t 0 µ µ−1 )+ 2 a(−1) −→ q ) P−1 (L, µ, e−2+tµ+1 − etµ+1 + uµ+1

(8.25)

+q

ν0 +1 3 L 2 − 4 + 4 θ(µ

odd)

(t) fs (L, −E1,µ

fs (L, e−2+tµ+1

+ e−1+tµ+1 − etµ+1 + 0 − etµ+1 + uµ+1 ) ,

(8.51)

0 uµ+1 ),

2y −2

µ where −→(8.25) denotes the flow according to the first 2yµ −2 steps of the decomposition

even) a,1 (8.25) and we used the identity 2c(tµ ) − ν0 −θ(µ + (−1)µ+1 = c(2 + tµ ) − 2 2 1 µ+1 c(tµ−1 ) + 2 a(−1) which follows from (8.33). Applying now further jµ − tµ − 2 (tµ + 3 ≤ jµ ≤ tµ+1 − 1) times the combination yµ −2 =⇒2 −→−1 using Lemma 2.2 and Proposition 1 and 2 with µ and a0 = −1 we obtain δa,0+2δ

364


0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 µ+1 1 (jµ −tµ )yµ −2 c(j )−c(t 0 µ−1 )+ 2 a(−1) −→ q µ ) P−1 (L, µ, etµ+1 −(jµ −tµ ) − etµ+1 + uµ+1

(8.25)

+q2−

ν0 +1 3 4 + 4 θ(µ

(t) 0 fs (L, −E1,µ + etµ+1 −(jµ −tµ )+1 − etµ+1 + uµ+1 ), 0 fs (L, etµ+1 −(jµ −tµ ) − etµ+1 + uµ+1 ) . (8.52) Setting jµ = tµ+1 − 1 in the last formula gives 0 0 P0 (L, µ + 1, uµ+1 ), P1 (L, µ + 1, uµ+1 ) (νµ −1)yµ −2 v 0 −→ q P−1 (L, µ, e1+tµ − etµ+1 + uµ+1 ) (8.25) (8.53) ν0 +1 3 L (t) 0 + q 2 − 4 + 4 θ(µ odd) fs (L, −E1,µ + e2+tµ − etµ+1 + uµ+1 ), 0 ) , fs (L, e1+tµ − etµ+1 + uµ+1 L

odd)

where v = c(−1 + tµ+1 ) − c(tµ−1 ) + 21 a(−1)µ+1 . Doing the next step we enter zone µ − 1. Using Lemma 2.2 with jµ = tµ+1 − 1 we get 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 n (νµ −1)yµ v− ν0 −θ(µ even) (t) 0 4 −→ q − etµ+1 + uµ+1 ), fs (L, −E1,µ−1 (8.25) o (t) 0 fs (L, e1 − E1,µ−1 − etµ+1 + uµ+1 ) 0 0 + q v P0 (L, µ − 1, e1+tµ − etµ+1 + uµ+1 ), P1 (L, µ − 1, e1+tµ − etµ+1 + uµ+1 ) . (8.54) The appearance of µ − 1 instead of µ on the right-hand side of (8.54) which occurred after (νµ − 1)yµ steps explains the choice of decomposition (8.25) instead of (8.24) in the proof of Proposition 2. Now using Proposition 1 and Proposition 2 with µ → µ − 1 and a0 = 0 we find 0 0 P0 (L, µ + 1, uµ+1 ), P1 (L, µ + 1, uµ+1 ) ν0 −θ(µ even) (νµ −1)yµ +yµ−1 −2 v+c(t 0 µ−1 )− 4 −→ q ) P−1 (L, µ − 1, −etµ+1 + uµ+1 (8.25) o ν0 −θ(µ even) L (t) 0 0 4 +q 2 − fs (L, −E1,µ−1 + e1+tµ − etµ+1 + uµ+1 ), fs (L, −etµ+1 + uµ+1 ) . (8.55) To complete the proof, we evolve (8.55) according to =⇒2 and use the fermionic recursive property (5.6), the first equation of (5.3) and definition (3.35) to obtain 0 0 ), P1 (L, µ + 1, uµ+1 ) P0 (L, µ + 1, uµ+1 (νµ −1)yµ +yµ−1 v− ν0 −θ(µ even) +c(t 0 0 µ−1 ) 4 −→ q ), P1 (L, µ, −etµ+1 + uµ+1 ) , P0 (L, µ, −etµ+1 + uµ+1 (8.25)

and then apply Proposition 2 with µ and a0 = 0 to arrive at 0 0 P0 (L, µ + 1, uµ+1 ), P1 (L, µ + 1, uµ+1 ) n o ν0 −θ(µ even) L−ν0 +θ(µ odd) yµ+1 −2 (t) +c(tµ−1 )+ +c(tµ ) 0 4 2 −→ q v− fs (L, −E1,µ+1 + uµ+1 ), 0 , (8.25)

(8.56)

(8.57)


365

where we used (νµ − 1)yµ + yµ−1 + yµ − 2 = yµ+1 − 2. Finally from (8.33) we derive ν0 − θ(µ even) L − ν0 + θ(µ odd) + c(tµ−1 ) + + c(tµ ) 4 2 L − ν0 + θ(µ + 1 odd) 1 = + c(tµ+1 ) + a(−1)µ+1 . 2 2 v−

(8.58)

Combining (8.57) and (8.58) we derive Proposition 2 for µ + 1. This concludes the proof of proposition 2 for νµ > 2. When νµ = 2 a separate treatment is needed because the decomposition (8.25) does not hold. However, in this case the decomposition (8.24) with νµ = 2 can be used and with this the proof of Proposition 2 for νµ = 2 follows. Recalling lemma 2.1, the equations (8.44), (8.47) and the definition of Takahashi length (2.8) we organize the results proven in this section as Theorem 1. For 1 ≤ µ ≤ n, 1 + δµ,1 + tµ ≤ jµ ≤ tµ+1 + δµ,n , we have n o (t) (t) fs (L, −E1,n + P e1+t1+n ), fs (L, e1 − E1,n + P e1+t1+n ) n µ (t) −→ q c(jµ ) P−1 (L, µ − δ1+tµ ,jµ , ejµ − Eµ+1,n + P e1+t1+n )

(µ) l1+j −2

o (t) (t) fs (L, e−1+jµ − E1,n + P e1+t1+n ), fs (L, ejµ − Eµ+1,n + P e1+t1+n ) n 2 (t) =⇒ q c(jµ ) q g(µ,n,jµ ) fs (L, e1+jµ + δt1+µ ,jµ θ(n > µ)etµ+1 − E1,n + P e1+t1+n ), o (t) fs (L, e1 + e1+jµ + δt1+µ ,jµ θ(n > µ)etµ+1 − E1,n + P e1+t1+n ) n o (t) (t) P0 (L, µ, ejµ − Eµ+1,n + P e1+t1+n ), P1 (L, µ, ejµ − Eµ+1,n + P e1+t1+n ) , (8.59) (t) = 0, P = 0, 1 and g(µ, n, jµ ) is defined just above (8.35). where En+1,n +q 2 − L


Before we move on, we observe that the r(b) graph of Sect. 3 for yµ −y1 −1 ≤ b ≤ yµ is (up to a shift) for µ ≥ 2 (8.60) yµ-y1-1 yµ-y1

yµ

Using the fermionic recursive properties of Sect. 5 for fs and (6.1) for f˜s as well as definition (3.35) one can easily show that o y n o n 1 (t) (t) (t) + uµ0 ), fs (L, −E2,µ + uµ0 ) −→ f˜s (L, −E1,µ + uµ0 ), 0 (8.61) fs (L, e1+t1 − E2,µ (8.60)

Notice that if we use (8.62) yµ-y1-1

instead, we obtain

yµ-1

yµ

366


o

n

(t) (t) + uµ0 ), fs (L, −E2,µ + uµ0 ) fs (L, e1+t1 − E2,µ o n y1 −1 (t) (t) −→ fs (L, e1 − E1,µ + uµ0 ), fs (L, −E1,µ + uµ0 ) (8.62) n o 1 (t) −→ fs (L, −E1,µ + uµ0 ), 0 .

(8.63)

(8.2) b

Comparing (8.61) with Proposition 2 with a = 0 we infer that for µ ≥ 2, P0 (L, µ, uµ0 ), P1 (L, µ, uµ0 ) n o yµ −y1 −2 c(t )− 1 θ(µ even) (t) (t) fs (L, e1+t1 − E2,µ −→ q µ 2 + uµ0 ), fs (L, −E2,µ + uµ0 ) .

(8.64)

We are now ready to discuss the evolution along the final stretch of the r(b) map

r

p-1

(8.65)

1 1

p’-2y1+1 p’-1-y1

p’-1

b

Last stretch of the b→ r map (n) ≤ b ≤ p0 − y1 − 1 is the same Notice that the piece of the b → r map with 1 + l2+t n+1 (up to a shift) as the map b → r of Sect. 3 with 1 ≤ b ≤ yn − y1 − 1. The piece of the b → r map restricted to the interval [p0 − y1 − 1, p0 − 1] is graph (8.62) with µ = n and the last segment removed. Recalling Theorem 1 (with µ = n and jn = 1 + tn+1 ) we have o n (t) (t) + P e1+t1+n ), fs (L, e1 − E1,n + P e1+t1+n ) fs (L, −E1,n (8.66) (n) 0 l2+t

=p −yn

−→

n+1

q c(1+tn+1 ) {P0 (L, n, (1 − P )e1+tn+1 ), P1 (L, n, (1 − P )e1+tn+1 )} .

Applying (8.64) and (8.63) with µ = n, un0 = (1 − P )e1+tn+1 to the right-hand side of (8.66), we arrive at Theorem 2. o n (t) (t) + P e1+t1+n ), fs (L, e1 − E1,n + P e1+t1+n ) fs (L, −E1,n p0 −3 c(tn+1 +1)+c(tn )− 1 θ(n even) 2

−→q

n

o

(8.67)

(t) (t) × fs (L, e1 − E1,n + (1 − P )e1+t1+n ), fs (L, −E1,n + (1 − P )e1+tn+1 ) .

If we now identify the fermionic polynomials generated in the evolution (8.67) with (o) (L, b) (for P = 1), then we have Fr(b),s (L, b) (for P = 0) and with Fr(b),s


367

(t) F1,s (L, 1) = fs (L, −E1,n ), (t) F1,s (L, 2) = fs (L, e1 − E1,n ),

···

(8.68) 0

and

Fp−1,s (L, p − 2) = q

c(1+t1+n )+c(tn )− 21 θ(n even)

fs (L, e1 −

Fp−1,s (L, p0 − 1) = q

c(1+t1+n )+c(tn )− 21 θ(n

(t) fs (L, −E1,n + e1+t1+n )

even)

(t) E1,n

+ e1+t1+n ),

(o) (t) (L, 1) = fs (L, −E1,n + e1+t1+n ), F1,s (o) (t) F1,s (L, 2) = fs (L, e1 − E1,n + e1+t1+n ),

···

(8.69)

(o) Fp−1,s (L, p0

− 2) = q

(o) Fp−1,s (L, p0 − 1) = q

c(1+t1+n )+c(tn )− 21 θ(n even)

fs (L, e1 −

c(1+t1+n )+c(tn )− 21 θ(n

(t) fs (L, −E1,n ).

even)

(t) E1,n ),

Finally we use the first equation in (5.3) to verify that Fp−1,s (L, p0 −2), Fp−1,s (L, p0 − (o) (o) (L, p0 − 2), Fp−1,s (L, p0 − 1)) satisfy the closing equation in (4.6). Thus we 1) (Fp−1,s have shown that the constructive procedure defined in Sects. 7 and 8 with the initial values for b = 1 and b = 2 specified by the first two equations in (8.68) or in (8.69) gives rise to fermionic polynomials which satisfy all bosonic recursion relations (4.4)–(4.6) for 1 ≤ b ≤ p0 − 1. 9. Normalization Constants and Boundary Conditions (µs ) From the conclusion of the previous section it follows that when s = l1+j , 1 + tµs ≤ s js ≤ t1+µs , L + b + s ≡ 0(mod2) the identity 0 pX −1

Fr(b),s (L, b) =

ks,s0 B˜ r(b),s0 (L, b)

(9.1)

s0 =1 s0 ≡s(mod2)

will hold for L > 0, provided constants ks,s0 can be chosen so that (9.1) holds for L = 0. Using (4.1), (4.7) it is trivial to verify that for ks,s0 = Fr(s0 ),s (0, s0 )

(9.2)

with s0 ≡ s(mod2), 1 ≤ s0 ≤ p0 − 1 Eq. (9.1) indeed holds for L = 0. However, (9.2) is of very little use because Fr(b),s (L, b) have not been explicitly constructed for all b ∈ [1, p0 − 1]. Fortunately, it turns out that the constants ks,s0 can be determined from (9.1) with L = 0, 1, · · · , p0 − 1 and b = 1, s, p0 − 1. In this direction we first calculate the threshold values of L, i.e. the lowest values of L such that Fr(b),s (L, b)6=0 for b = 1, s, p0 − 1. We know that Fr(b),s (L, b) with b = 1, p0 − 1, s is given by X

Fr(b),s (L, b) = q N

n∈Zt1+n mt

1+n

≡P ( mod 2)

T

q Q(n,m)+A

˜ m

tY n+1 j=1

nj + mj nj

(0) .

(9.3)

368


˜ is defined in (2.17), Here m ( 0 N = c(js ) c(tn+1 + 1) + c(tn ) − 21 θ(n even)

and P =

0 1

if b = 1 if b = s if b = p0 − 1

(9.4)

if b = 1, s if b = p0 − 1.

(9.5)

n and m are related by (2.11) with 0 ¯ (t) ¯ (t) + e¯ js − E u¯ = −E 1,n 1+µs ,n for b = 1, p − 1,

¯ (t) u¯ = 2¯ejs − 2E 1+µs ,n for b = s where ¯ (t) = E a,b

b X

,

(9.6)

e¯ ti .

(9.7)

i=a

Even though n, m ∈ Zt1+n , a bit of analysis shows that effectively For b = 1, s : ni ≥ δi,js − δi,tj , tj ≥ js mi ≥ 0

(9.8) 0

For b = p − 1 : ni ≥ 0, mi > 0. Moreover, if ni takes on negative values then mi = 0. Therefore, the threshold configurations for the three cases are ¯ (t) n = e¯ js − θ(µs < n)E 1+µs ,n , mtn+1 ≡ 0 (mod 2) for b = 1,

n = 0, mtn+1 ≡ 1 (mod 2) for b = p0 − 1, ¯ (t) n = e¯ js − θ(µs < n)E 1+µs ,n , mtn+1 ≡ 0 (mod 2) for b = s,

(9.9)

with the corresponding thresholds from (2.16) Ltr = Ltr =

µs X i=1 n X i=1 0

(yi − yi−1 ) + ljs = s − 1 for b = 1, (yi − yi−1 ) +

n X

(yi − yi−1 ) − ljs + l1+tn+1

(9.10)

i=µs +1 0

=p − 1 − s for b = p − 1, Ltr =0 for b = s, with lj defined by (2.7) and µs by (3.15). Finally the threshold for Br(b),s (L, b) is Ltr = |s − b|.

(9.11)

Hence for b = 1, L = 0, 1, . . . , s − 2 and L + 1 + s ≡ 0( mod 2) we have from (9.1)

Rogers–Schur–Ramanujan Identities for Minimal Models of CFT 0 pX −1

0=

ks,s0 B˜ 1,s0 (L, 1).

369

(9.12)

s0 =1 s0 ≡s( mod 2)

Clearly, (9.12) and (9.11) imply that ks,s0 = 0 for 1 ≤ s0 ≤ s − 1, s0 ≡ s( mod 2).

(9.13)

Analogously, if we evaluate (9.1) at b = p0 − 1, L = 0, 1, 2, . . . , p0 − 2 − s and L + p0 − 1 + s ≡ 0( mod 2) we obtain

0=

0 pX −1

ks,s0 B˜ p−1,s0 (L, p0 − 1).

(9.14)

s0 =1 s0 ≡s( mod 2)

Equations (9.11) and (9.14) imply that ks,s0 = 0 for 1 + s ≤ s0 , s0 ≡ s( mod 2).

(9.15)

Combining (9.13) and (9.15) yields ks,s0 = δs0 ,s q a(js ) .

(9.16)

To determine a(js ) we evaluate (9.1) at b = s, L = 0 to get Fr(s),s (L = 0, b = s) = q a(js ) B˜ r(s),s (L = 0, b = s),

(9.17)

or since Fr(s),s (L = 0, b = s) = q c(js ) and B˜ r(s),s (L = 0, b = s) = 1 we have a(js ) = c(js ).

(9.18)

Fr(b),s (L, b) = q c(js ) B˜ r(b),s (L, b),

(9.19)

This finally leads to

where L + b + s ≡ 0( mod 2). Analogously, we can show that Fr(b),s (L, b) = 0 for L + b + s ≡ 1( mod 2).

(9.20)

This concludes the proof of the boundary conditions which leads to the equality of the fermionic and the bosonic form of M (p, p0 ) polynomial character identities.

370


10. Rogers–Schur–Ramanujan Type Identities for M (p, p0 ) Minimal Models We now may give the Rogers–Schur–Ramanujan type identities for the the four cases of (3.41). From (9.19) we see that all identities are of the form Fr(b),s (L, b) = q 2 (φr(b),s −φr(s),s )+c(js ) Br(b),s (L, b). 1

(10.1)

We thus state the results by giving the appropriate functions Fr(b),s (L, b). (µ) 10.1. Rogers–Schur–Ramanujan identities for b = l1+j a pure Takahashi lengths. µ (µ) a pure Takahashi length follow The Rogers–Schur–Ramanujan identities for b = l1+j µ immediately from Theorem 1 of Sect. 8 and the proof of the boundary conditions of Sect. 9. Since (t) ), F1,s (L, 1) = fs (L, −E1,n (10.2) (t) F1,s (L, 2) = fs (L, e1 − E1,n ),

we infer from (8.59) that (µ) (t) ) = q c(jµ ) fs (L, ejµ − Eµ+1,n ), Fr(b),s (L, l1+j µ

(10.3)

(µ) where c(jµ ) was defined in (8.33) and r(b) = δµ,0 + l˜1+j . Thus we have explicitly proven µ the result announced in [26]. We would like to stress that (10.3) holds even in the cases where some or all νi = 1 or νn = 0 (or both) with appropriate modification of the exponent c(j µ ).

10.2. The vicinity of the Takahashi length. Using the fermionic recursive properties of Sect. 5 and 6 and the Theorem 1 (of Sect. 8), it is easy to extend the Rogers–Schur– Ramanujan identities of the previous subsection to case 2 of (3.41). (µ) (µ) − ν0 + 1 ≤ b ≤ l1+j − 1, µ ≥ 1. Case 2a: l1+j µ µ ν0 −1−θ(µ odd) (µ) (t) c(jµ ) 4 f˜s (L, ej0 −1 − E1,n q − j ) = q + e−1+jµ ) Fr(b),s (L, l1+j 0 µ + θ(µ ≥ 2)

µ X

q

ν0 −1−θ(i even) 4

(t) (t) f˜s (L, ej0 −1 − E1,i + e−1+ti + ejµ − Eµ+1,n )

i=2

(t) ) +fs (L, eν0 −j0 − et1 + ejµ − Eµ+1,n

(10.4)

1 ≤ j0 ≤ ν0 − 1.

(µ) (µ) + 1 ≤ b ≤ l1+j + ν0 + 1, µ ≥ 1. Case 2b: l1+j µ µ

ν0 +1 3 (µ) (t) c(jµ ) + 1 + j ) = q + e1+jµ ) q − 4 + 4 θ(µ odd) fs (L, ej0 − E1,n Fr(b),s (L, l1+j 0 µ + θ(µ ≥ 2)

µ X

q−

ν0 −θ(i odd) 4

(t) (t) fs (L, ej0 − E1,i + e−1+ti + ejµ − Eµ+1,n )

(10.5)

i=2

(t) ) , 0 ≤ j0 ≤ ν 0 , +f˜s (L, eν0 −j0 −1 − et1 + ejµ − Eµ+1,n where recalling (3.36) we note that case 2a agrees with (10.3) if j0 = 0. r in this section is a function of b as given in (3.41).


371

10.3. Further cases. The Propositions 1 and 2 of Sect. 8 are very powerful and can be used to to study all cases 1 ≤ b ≤ p0 − 1. As an illustration we present the results for the fermionic forms of cases 3 and 4 of (3.41). The details of the derivation are in Appendix C. To state the results we need several definitions (with α and β defined by the Takahashi decomposition of case 4 in (3.41)). Definitions.

j = |jα , jα+1 , . . . , jβ >, i = |iα , iα+1 , . . . , iβ−1 , 0 >,

(10.6)

with ik = 0, 1 for α ≤ k ≤ β − 1. When iα = 0 we define ai and bj from i = | 0, · · · , 0, 1, · · · , 1, 0, · · · , 0, · · · , 1, · · · , 1, 0, · · · , 0 >, | {z } | {z } | {z } | {z } | {z } a1

a2

b2

(10.7)

al

bl

where 1 ≤ ai , 1 ≤ i ≤ l and 1 ≤ bj , 2 ≤ j ≤ l. If l = 1, then i = | 0, · · · , 0 >. Note | {z } a1 Pl that α − 1 + a1 + j=2 (aj + bj ) = β. Similarly when iα = 1 we write i = | 1, · · · , 1, 0, · · · , 0, 1, · · · , 1, 0, · · · , 0, · · · , 1, · · · , 1, 0, · · · , 0 >, | {z } | {z } | {z } | {z } | {z } | {z } a1

b1

b2

a2

bl

(10.8)

al

where 1 ≤ aj , bj with 1 ≤ j ≤ l. From i we further define if iµ = 0 jµ + 1 Riµ (jµ + 1) = tµ+1 − (jµ − tµ ) − 1 for iµ = 1.

(10.9)

Results. With these definitions we may have the following results for the fermionic polynomials of cases 3 and 4 of (3.41). Case 3 of (3.41). Here α = 0 and we have  X  q rf(3,1) (i) fs (L, ej0 +i1 − et1 + u(3) (i, j)) Fr(b),s (L, b) = q c(3) (j)  i1 ,...,iβ−1 =0,1 i0 =0

X

+



(10.10)

 q rf(3,2) (i) f˜s (L, eν0 −j0 −i1 −1 − et1 + u(3) (i, j)) ,

i1 ,...,iβ−1 =0,1 i0 =1

where c(3) (j) =

β X µ=1

u(3) (i, j) =

β−1 X µ=1

for i0 = 0 we set

(c(jµ ) −

ν0 + 1 3 + θ(µ odd)), 4 4

(t) eRiµ (jµ +1)+|iµ+1 −iµ |−|iµ −iµ−1 | + e1+jβ −i−1+β − E2,n ,

(10.11)

(10.12)

372

A. Berkovich, B. M. McCoy, A. Schilling l Pj−1 1 1X rf(3,1) (i) = δa1 ,1 + (−1)a1 + k=2 (ak +bk ) θ(bj even), 4 2

(10.13)

j=2

and for i0 = 1 Pj−1 δb ,1 1 X ν0 + (−1)b1 − 1 + (−1) k=2 (ak +bk ) θ(bj even), 4 4 2 l

rf(3,2) (i) =

(10.14)

j=2

where we have used the convention that

Pb i=a

= 0 if b < a.

Case 4 of (3.41). Here α ≥ 1 and we have  X  q rf(4,1) (i) fs (L, u(4,1) (i, j)) Fr(b),s (L, b) = q c(4) (j)  iα+1 ,...,iβ−1 =0,1 iα =0

X

+

q rf(4,2) (i)+δ1,α ( 2 − L

ν0 4

)

fs (L, u(4,2) (i, j))

iα+1 ,...,iβ−1 =0,1 iα =1

+q

rf(4,2) (i)−

where c(4) (j) = c(jα ) +

β X

(−1)α +δ1,α 4

fs (L, u(4,3) (i, j))

(c(jµ ) −

µ=α+1

,

ν0 + 1 3 + θ(µ odd)), 4 4

(10.16)

u(4,1) (i, j) = ejα +iα+1 + u(4) (i, j), u(4,2) (i, j) = −etα + etα+1 −(jα −tα )−1−iα+1 + u(4) (i, j), u(4,3) (i, j) = etα −1 − etα + etα+1 −(jα −tα )−iα+1 + u(4) (i, j), u(4) (i, j) =

β−1 X

(10.15)

(10.17)

(t) eRiµ (jµ +1)+|iµ+1 −iµ |−|iµ −iµ−1 | + ejβ +1−iβ−1 − Eα+1,n

µ=α+1

for iα = 0 rf(4,1) (i) =

l Pj−1 X 1 (−1)α (−1)a1 + k=2 (ak +bk ) θ(bj even) 2

(10.18)

j=2

and for iα = 1



rf(4,2) (i) = (−1)α 

b1

(−1) 4

+

l 1X

2

Pj−1 (−1)

k=1

 (ak +bk )

θ(bj even) .

(10.19)

j=2

10.4. Character identities. It remains to take the limit L → ∞ to produce character identities from the polynomial identities. These character identities are somewhat simpler because f˜s (L, u˜ ) vanishes as L → ∞. We remark that in this limit the explicit dependence on b vanishes and only the dependence on r remains. All polynomial identities which have different values of b but the same value of r give the same character identity. Thus we obtain the following fermionic forms for the bosonic forms of the


373

characters q 2 (φr(j),s −φr(s),s )+c(js ) Br(j),s (q). In all formulas of this section r is a function of j as given in (3.41). 1

Cases 1 and 2a. (t) ). Fr(j),s (q) = q c(jµ ) fs (ejµ − Eµ+1,n

(10.20)

Case 2b. ν0 +1 3 (t) + e1+jµ ) Fr(j),s (q) = q c(jµ ) q − 4 + 4 θ(µ odd) fs (−E2,n +θ(µ ≥ 2)

µ X

q

ν −θ(i odd) − 0 4

!

(t) (t) fs (−E2,i + e−1+ti + ejµ − Eµ+1,n ) .

(10.21)

i=2

Case 3. X

Fr(j),s (q) = q c(3) (j)

q rf(3,1) (i) fs (u(3) (i, j)).

(10.22)

i1 ,...,iβ−1 =0,1 i0 =0

Case 4 with α = 1.  X

 Fr(j),s (q) = q c(4) (j) 

q rf(4,1) (i) fs (u(4,1) (i, j))

i2 ,...,iβ−1 =0,1 i1 =0

+

X



(10.23)

 q rf(4,2) (i) fs (u(4,3) (i, j)) .

i2 ,...,iβ−1 =0,1 i1 =1

Case 4 with α ≥ 2.  X

 Fr(j),s (q) = q c(4) (j) 

q rf(4,1) (i) fs (u(4,1) (i, j))

iα+1 ,...,iβ−1 =0,1 iα =0

+

X

(10.24)

q rf(4,2) (i) fs (u(4,2) (i, j))

iα+1 ,...,iβ−1 =0,1 iα =1

+q rf(4,2) (i)−

(−1)α 4

fs (u(4,3) (i, j))

.

374


11. Reversed Parity Identities For all the identities presented thus far we have started with the term with b = 1, where the fermionic polynomial (3.22) has mt1+n restricted to even values. Indeed mt1+n has (n) = p0 − 2yn . even parity for all the results presented in this paper as long as b ≤ l1+t 1+n (e) These fermionic sums are what were called even sums Fr(b),s (L, b) in [26]. According to Theorems 1 and 2 of Sect. 8 it is equally possible to start with the fermionic polynomial which appears in the first equation in (8.69) with mt1+n odd and the entire construction of this paper may be carried out exactly as before. The only difference is that the parity of mt1+n will be reversed in all the formulas presented earlier, which amounts to the replacement u → u + e1+tn+1 . The step which must be different is the analysis of the boundary conditions and the computation of the normalization constant. This is done (o) (L, b) (8.69) we below and calling the reversed parity fermionic polynomials Fr(b),s find from each of the identities proven above the corresponding reversed parity identity (o) ¯ (L, b) = q a(s,r) Br(b),s¯ (L, b), Fr(b),s

(11.1)

(µs ) where s¯ = p0 − s with s = l1+j and a(s, ¯ r) is given by the two equivalent expressions s

1 a(s, ¯ r) = c(js ) + c(1 + tn+1 ) + c(tn ) − θ(n even) 2 1 + (φr,s¯ − φr(s),s + φ1,s − φp−1,s¯ ) 2 1 1 = c(js ) + (φr,s¯ − φr(s),s ) + (φp−1,s − φp−1,s¯ + φ1,s − φ1,s¯ ). 2 4

(11.2)

(µ) (11.1) gives the result announced in [26]. We note that when b = l1+j µ

Computation of the normalization constant. From Sect. 9 it suffices to determine the constant a(s) ¯ in the identity (o) ¯ ˜ Br(b),s¯ (L, b), (L, b) = q a(s) Fr(b),s

(11.3)

since from the definition of B˜ r(b),s (L, b) (4.1) it follows that 1 a(s, ¯ r) = a(s) ¯ + (φr,s¯ − φr(s), ¯ s¯ ). 2

(11.4)

In (11.3) we first set b = p0 − 1, r = p − 1 to obtain (o) ¯ 2 (φp−1,s¯ −φr(s), ¯ s¯ ) Fp−1,s (L, p0 − 1) = q a(s)+ Bp−1,s¯ (L, p0 − 1). 1

(11.5)

On the other hand from (7.1) and the last equation in (8.69), (o) (L, p0 − 1) = q c(1+t1+n )+c(tn )− 2 θ(n even) F1,s (L, 1) Fp−1,s 1

= q c(1+t1+n )+c(tn )− 2 θ(n even)+c(js )+ 2 (φ1,s −φr(s),s ) B1,s (L, 1), 1

1

(11.6)


375

where in the last line we used (10.1). Comparing (11.4) and (11.6) and using the symmetry of Br(b),s (L, b) (1.13) with r(b) = b = 1, B1,s (L, 1) = Bp−1,s¯ (L, p0 − 1),

(11.7)

we find the first expression for a(s) ¯ 1 a(s) ¯ = c(1 + t1+tn ) + c(tn ) − θ(n even) + c(js ) 2 1 + (φ1,s − φr(s),s + φr(s), ¯ s¯ − φp−1,s¯ ). 2

(11.8)

A second expression for a(s) ¯ is obtained by setting b = 1, r = 1 in (11.3) (o) ¯ 2 (φ1,s¯ −φr(s), ¯ s¯ ) (L, 1) = q a(s)+ B1,s¯ (L, 1). F1,s 1

(11.9)

However we also find from the second line of (8.68), the first line of (8.69) and (10.1) (o) (L, 1) = q −c(1+t1+n )−c(tn )+ 2 θ(n even) Fp−1,s (L, p0 − 1) F1,s 1

= q −c(1+t1+n )−c(tn )+ 2 θ(n even)+c(js )+ 2 (φp−1,s −φr(s),s ) Bp−1,s (L, p0 − 1). (11.10) Comparing (11.9) and (11.10) and using the symmetry (1.13) with s = b = 1 and s → p0 − s B1,s¯ (L, 1) = Bp−1,s (L, p0 − 1), (11.11) 1

1

we obtain

1 a(s) ¯ = c(js ) − c(1 + t1+n ) − c(tn ) + θ(n even) 2 1 + (φp−1,s − φr(s),s + φr(s), ¯ s¯ − φ1,s¯ ). 2 Equations (11.8) and (11.12) imply the following identity 1 1 c(1 + t1+n ) + c(tn ) − θ(n even) = (φp−1,s + φp−1,s¯ − φ1,s − φ1,s¯ ), 2 4

(11.12)

(11.13)

and using this identity we obtain (11.2) from (11.8) and (11.4). 12. The Dual Case p < p0 < 2p In the XXZ chain (2.1) we see from (2.2) that the regime p < p0 < 2p corresponds to 1 > 0 and since the XXZ chain has the symmetry HXXZ (1) = −HXXZ (−1) the (m, n) systems for ±1 are the same. Thus we consider the relation between M (p, p0 ) and M (p0 − p, p0 ) which is obtained by the transformation q → q −1 in the finite L polynomials Fr(b),s (L, b; q) and Br(b),s (L, b; q). To implement this transformation it is mandatory that the L → ∞ limit be taken only in the final step. (p,p0 ) (L, b; q) given by (1.12) where we have made the dependence Consider first Br(b),s on p and p0 explicit. Then if we note that from the definitions (3.16)

376


we find

(0,1)

n+m m

= q −mn

q −1

0

(p,p ) Br(b),s (L, b, q −1 ) = q −

L2 2

q

(0,1) n+m , m q

(b−s)2 4

0

(12.1)

0

(p −p,p ) Bb−r(b),s (L, b; q).

(12.2)

Similarly we let q → q −1 in fs (L, u; q) and f˜s (L, u˜ ; q) given by (3.22) and (3.35) to obtain P 2 fs (L, u; q −1 ) = q

− L4 − L 2

ν0

j=1

uj

fs(d) (L, u; q),

(12.3)

2

L f˜s (L, u˜ ; q −1 ) = q − 4 f˜s(d) (L, u˜ ; q),

where

X

fs(d) (L, u; q) =

q− 2 m 1

T

Mm+A0 T m+C 0

¯ m∈2Zt1+n +w(u1+tn+1 ,u)

×

tY n+1 j=1

((Itn+1 + M)m + mj

u¯ 2

+

(1) L ¯ 2 e1 )j ,

(12.4)

q

M was defined in (2.11), (2.12), and the tn+1 -dimensional vector A0 is 0

0

A0 = A (b) + A (s)

with 0

Ak(b) 0

Ak(s)

= =

− 21 uk 0

for k in an odd zone , for k in an even zone

− 21 u¯ k (s) 0

(12.5)

(12.6)

for k in an even zone . for k in an odd zone

and

1 T ¯ + + y¯ T B¯y . u¯ (s)+ Bu(s) 8 u¯ is given by (3.23) and (3.24), and C0 = −

y¯ =

ν0 X

e¯ i ui ,

(12.7)

(12.8)

i=1

f˜s(d) (L, eν0 −j0 −1 − eν0 + u10 ; q)  0 j0 = ν0  L+j0  2L+1  − (d) 0  q 2 [q 4 fs (L + 1, eν0 −j0 − eν0 + u1 ; q)  −fs(d) (L, e1+ν0 −j0 − eν0 + u10 ; q)] 1 ≤ j0 ≤ ν0 − 1, =  L 0  2 f (d) (L, e  q − e + u , q) ν0 −1 ν0 s  1  1 −(q L − 1)q − 4 fs(d) (L − 1, u10 ; q) for j0 = 0

(12.9)

and the vector u10 ∈ Z 1+t1+n was defined as any 1 + t1+n -dimensional vector Pν0 with no uj = 0 components in the zero zone. We note that for the vectors u in (3.40) that j=1 unless (12.10) u = −eν0 + u10


377

in which case the sum is −1. Thus since the L2 dependent factors in the transforms (12.2) and (12.3) are the same we obtain from all the polynomial identities of M (p, p0 ) of the form (10.1) polynomial identities for M (p0 − p, p0 ) of the form (d) (L, b; q) = q − 2 (φr(b),s −φr(s),s )− 2 c(js )+ Fr(b),s 1

1

(b−s)2 4

0

0

(p −p,p ) Bb−r(b),s (L, b; q),

(12.11)

(d) where Fr(b),s (L, b; q) for M (p0 −p, p0 ) is obtained from the corresponding Fr(b),s (L, b; q) of Sect. 10 as (d) (L, b; q) = Fr(b),s

X

X

q −cu fs(d) (L, u; q) +

q −c˜u˜ f˜s(d) (L, u˜ ; q)

(12.12)

˜ U˜ (b) u∈

u∈U (b)

where the exponents cu , c˜u˜ and the sets U (b), U˜ (b) as defined in (3.40) are explicitly given in Sect. 10. Pν0 uj in fs (L, u; q −1 ) in (12.3) is needed We note in particular that the term − L2 j=1 in case 4 of (3.41) with α = 1 in order to cancel the explicit factor of q − 2 in (10.15). In the limit L → ∞ we find from (12.4), L

lim fs(d) (L, u; q) = fs(d) (u; q)

(12.13)

L→∞

with fs(d) (u; q) =

X

q− 2 m 1

T

Mm+A0 T m+C 0

¯ m∈2Zt1+n +w(u1+t1+n ,u)

(1) tn+1 1 Y ((Itn+1 + M)m + u2¯ )j × , (q)m1 mj q

(12.14)

j=2

and from (12.9) lim f˜s(d) (L, eν0 −j0 −1 − eν0 + u10 ; q) =

L→∞

j0

q 2 − 4 fs(d) (eν0 −j0 − eν0 + u10 ; q) 0 1

j0 = 6 ν0 j0 = ν 0 . (12.15)

Thus we find from (12.11), (12.12) the corresponding character identities (d) Fr(b),s (b; q) = q − 2 (φr(b),s −φr(s),s )− 2 c(js )+ 1

1

(b−s)2 4

0

0

(p −p,p ) Bb−r(b),s (b; q)

(12.16)

and (d) (b; q) = Fr(b),s

X u∈U (b)

q −cu fs(d) (u; q) +

X

j0

q −c˜u˜ + 2 − 4 fs(d) (u˜ 0 ; q), 1

(12.17)

˜ U˜ (b) u∈

where u˜ 0 is obtained from (3.54) by replacing j0 by j0 − 1. We note that in contrast to the case p0 > 2p the terms involving u˜ contribute in the L → ∞ limit.

378


13. Discussion The techniques developed in this paper are of great generality and may be extended to derive many further results which extend those presented in the previous sections. We will thus conclude this paper by discussing several extensions and applications which will be elaborated elsewhere. 13.1. Extension of the fundamental fermionic polynomials (3.22). There are two features in the definition of the polynomials fs (L, u; q) which may strike one as perhaps arbitrary and unmotivated; namely the fact that the vector A of (3.27)- (3.29) is of a different ¯ form depending on whether the components of u(b) and u(s) are in odd or even zones and the choice of the branch (0) or (1) of the q-binomial coefficients and it may be asked whether other choices are possible. For the choice of branch of the q-binomials it is clear from appendix A that the fermionic recursive properties of Sect. 5 will be equally valid if the interchange of branches (0) ↔ (1) is made everywhere. The only place the branch is explicitly used is in Sect. 9 where the polynomials Fr(b),s (L, b) are identified with a linear combination of the bosonic polynomials B˜ r(b),s (L, b). A change of branch will lead to different linear combinations. In a similar fashion different linear terms are possible which will also lead to different linear combinations. An analogous phenomena was studied in some detail for the model SM (2, 4ν) in [38]. 13.2. Extension to arbitrary s. Throughout this paper we have for simplicity confined our attention to values of s given by a pure Takahashi length. However, an examination of the proof of Appendix A shows that in fact the fundamental fermionic recursion ¯ relations are valid for all vectors u(s). With this generalization all values of s can be treated in a manner parallel to the way general values of b were treated above and the ¯ final result is a double sum over both sets of vectors u(b) and u(s). This generalization will be presented in full elsewhere. 13.3. Extension of m,n system. The (m, n) system (2.11) of this paper has the feature that the term L/2 appears only in the first component of the equation. However it is also useful to consider L/2 in other (possibly several) positions. Each choice will lead to further identities. Certain features of this extension have been seen in [22, 29, 33, 40]. As long as L/2 remains in the zero zone we are in the regime of weak anisotropy in the sense (1) (1) of [40] and we will obtain identities for coset models (A(1) 1 )N × (A1 )M /(A1 )N +M with N integer and M fractional. However if L/2 is in a higher zone we are in the region of strong anisotropy and identities for cosets with both fractional levels can be obtained. 13.4. Bailey pairs. We conclude by demonstrating that all polynomial identities of this paper may be restated in terms of new Bailey pairs and thus, by means of Bailey’s lemma [20], [41], [42], we may use the results of this paper to produce bose/fermi character identities for models other than M (p, p0 ). Definition of Bailey pair. A pair of sequences (αj , βj ) is said to form a Bailey pair relative to a if βn =

n X j=0

αj . (q)n−j (aq)n+j

(13.1)


379

All of our polynomial identities may be cast into this form. This is easily seen, following [25], by writing the bosonic polynomial (1.12) (where we suppress the dependence of r on b) with L = 2n + b − s (when L + b − s even) as Br,s (2n + b − s, b; q) = (q

b−s+1

)2n

∞ X j=−∞

0

0

q j(jpp +rp −sp) (q)n−jp0 (q b−s+1 )n+jp0 0

q (jp+r)(jp +s) − (q)n−(jp0 +s) (q b−s+1 )n+(jp0 +s)

!

(13.2) .

Thus comparing the identity (1.14) with (13.1) we obtain the following Bailey pair relative to a = q b−s : βn = q − 2 (φr,s −φr(s),s )−c(js ) Fr,s (2n + b − s, b, q)/(aq)2n ,  0 0  q j(jpp +rp −sp) for n = jp0 , (j ≥ 0)   j(jpp0 −rp0 +sp) for n = jp0 + s − b, (j ≥ 1) . αn = q (jp+r)(jp0 +s)  for n = jp0 + s, (j ≥ 0) −q   (jp−r)(jp0 −s) for n = jp0 − b, (j ≥ 1) −q 1

(13.3)

If two of the restrictions in (13.3) are the same the formula should be read as the sum of both. The utility of the Bailey pair follows from the lemma of Bailey [41] Lemma. Given a Bailey pair and a pair (γn , δn ) satisfying γn =

∞ X j=n

then

∞ X n=0

δj , (q)j−n (aq)j+n

αn γn =

∞ X

β n δn .

(13.4)

(13.5)

n=0

We note in particular two sets of (γn , δn ) pairs. One is the original pair of Bailey [41] (ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n , (aq/ρ1 )n (aq/ρ2 )n (q)N −n (aq)N +n (ρ1 )n (ρ2 )n (aqρ1 ρ2 )N −n (aq/ρ1 ρ2 )n , δn = (q)N −n (aq/ρ1 )N (aq/ρ2 )N

γn =

(13.6)

which has been used to produce characters of the N = 1 and N = 2 supersymmetric models [43] and a new pair of [44] which gives characters of the coset models (A(1) 1 )N × (1) ) /(A ) , where M may be fractional. (A(1) 1 M 1 N +M The presentation of the detailed consequences of this observation are too lengthy for inclusion in this paper and will be published separately [45].

380


Appendix A. Proof of Fundamental Fermionic Recursive Properties In this appendix we prove the fundamental fermionic recursive properties (5.3) for 0 ≤ j0 < ν0 and (5.5) in detail. The proof of the remaining cases can be carried out in a similar fashion and will be left to the reader. We begin by introducing some shorthand notations. We use the vectors u0 (jµ ) and u±1 (jµ ) as defined in (5.1). Furthermore we need the operator which projects onto the components of a vector in an odd zone n πi =

for t1+2l + 1 ≤ i ≤ t2+2l , otherwise

1 0

(A.1)

and π˜ = 1 − π,

(A.2)

which projects onto components of even zones. We also need the vector Oa,b =

0P b i=a

if b < a πi e¯ i

.

(A.3)

We also make the following definition which generalizes the fermionic polynomial (3.22): q

δ(n,m)

¯ A ¯ = B

X

q

δ(n,m)+8s (n,m,u,L)

tY n+1

n,mt1+n ≡P ( mod 2)

j=1

(0) nj + mj + A¯ j , nj + B¯ j q

(A.4)

where P = 0, 1 with P ≡ u1+t1+n (mod 2), and nj , mj are related by (2.11) for given L ¯ and u, tn+1 X ¯ (u0 (jµ ))i e¯ i + u(s), 1 + tµ ≤ jµ ≤ t1+µ (A.5) u¯ = i=1

and 8s (n, m, u, L) = Q(n, m) + Lf (n, m),

(A.6)

where the quadratic term Q is defined in (3.26) and the linear term Lf is defined through A in (3.27)-(3.29). We note in particular 0 = fs (L, u0 (jµ )). 0

(A.7)

Finally we define ¯ L = set of solutions to (m, n) − system (2.11) with L, u. ¯ {n, m, u}

(A.8)

It will be necessary to make variable changes in n, m in (A.4). Hence we will need some identities relating different objects. For example it is easy to check that the set of ¯ L is equal to the set of solutions {n, m, u¯ + e¯ 1 }L−1 . One may also solutions {n, m, u} show that 8s (n, m, u, L) = 8s (n, m, u + e1 , L − 1). Thus fs (L, u) = fs (L − 1, u + e1 ),

(A.9)


381

which proves (5.3) for j0 = 0. One may also show that ¯ L−2 , ¯ L − {¯e1 , 0, 0} = {n, m, u} {n, m, u} m1 + 8s (n, m, u, L) = (L − 1) + 8s (n − e¯ 1 , m, u, L − 2), P ν0

where u is any vector such that

i=1

q m1

−¯e1 −¯e1

(A.10)

ui = 0. From this follows

= q L−1 fs (L − 2, u0 (jµ )).

(A.11)

Similarly one can prove the identities:

¯ 1,ν0 − e¯ ν0 +1 −2E −¯eν0 +1

1:

¯ 1,ν0 +1 −¯eν0 − E −¯eν0 +1

2:

¯ 1,j0 −E 0

3:

= fs (L − 2, u0 (jµ )),

= fs (L − 1, eν0 −1 − eν0 + u0 (jµ )),

= fs (L − 1, u1 (j0 )),

4 : q mj0 −mj0 −1 +1

5 : q mj0 −mj0 −1 +1

¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 −E e¯ j0 −1 − e¯ j0

¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 −2E e¯ j0 −1 − e¯ j0

¯ 1,jµ −E 0

ν0 +1 3 L−1 2 − 4 + 4 θ(µ

6: q

OT 1,jµ n

7: q

OT 1,jµ −1 n+(mjµ −mjµ −1 +1)θ(µ even)

=q

=q

ν0 −θ(µ even) L−1 2 − 4

T

ν0 −θ(k odd) L−1 2 − 4

= fs (L − 1, u−1 (j0 )), j0 > 1,

odd)

= fs (L − 2, u0 (j0 )), j0 > 1,

(t) fs (L − 1, u1 (jµ ) − E1,µ ), 0 < µ ≤ n,

¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E e¯ jµ −1 − e¯ jµ

(t) fs (L − 1, u−1 (jµ ) − E1,µ ), 0 < µ ≤ n,

8 : q O1,tk n+(ntk +m1+tk )θ(k even) =q

¯ 1,1+tk −¯etk − E −¯e1+tk

(t) fs (L − 1, u0 (jµ ) + e−1+tk − E1,k ), 2 ≤ k ≤ µ ≤ n.

(A.12) The derivation of (A.12) requires the verification the following intermediate equations:

382


¯ 1,ν0 , 0} = {n, m, u} ¯ L − {¯eν0 +1 , 2E ¯ L−2 , 10 : {n, m, u} ¯ 1,ν0 , u, L − 2), 8s (n, m, u, L) = 8s (n − e¯ ν0 +1 , m − 2E ¯ 1,ν0 , 0} = {n, m, e¯ ν0 −1 − e¯ ν0 + u} ¯ L − {¯eν0 +1 , e¯ ν0 + E ¯ L−1 , 20 : {n, m, u} ¯ 8s (n, m, u, L) = 8s (n − e¯ 1+ν0 , m − e¯ ν0 − E1,ν0 , u + eν0 −1 − eν0 , L − 1), ¯ 1,j0 , 0} = {n, m, u¯ + e¯ j0 +1 − e¯ j0 }L−1 , ¯ L − {0, E 30 : {n, m, u} ¯ 1,j0 , u + ej0 +1 − ej0 , L − 1), 8s (n, m, u, L) = 8s (n, m − E ¯ 1,j0 −1 , 0} = {n, m, u¯ + e¯ j0 −1 − e¯ j0 }L−1 ¯ L + {¯ej0 −1 − e¯ j0 , −E 40 : {n, m, u} (mj0 − mj0 −1 + 1) + 8s (n, m, u, L) ¯ 1,j0 −1 , u + ej0 −1 − ej0 , L − 1), = 8s (n + e¯ j0 −1 − e¯ j0 , m − E ¯ 1,j0 −1 , 0} = {n, m, u} ¯ L + {¯ej0 −1 − e¯ j0 , −2E ¯ L−2 , j0 > 1 50 : {n, m, u} 8s (n, m, u0 (j0 ), L) + 1 + mj0 − mj0 −1 ¯ 1,j0 −1 , u0 (j0 ), L − 2), = 8s (n + e¯ j0 −1 − e¯ j0 , m − 2E ¯ 1,jµ , 0} = {n, m, u¯ + e¯ jµ +1 − e¯ jµ − E ¯ (t) }L−1 ¯ L − {0, E 60 : {n, m, u} 1,µ 8s (n, m, u0 (jµ ), L) + OT1,jµ n =

L − 1 ν0 + 1 3 ¯ 1,jµ , u1 (jµ ) − E(t) , L − 1), µ 6= 0, − + θ(µ odd) + 8s (n, m − E 1,µ 2 4 4

¯ 1,jµ −1 , 0} = {n, m, u¯ + e¯ jµ −1 − e¯ jµ − E ¯ (t) }L−1 ¯ L + {¯ejµ −1 − e¯ jµ , −E 70 : {n, m, u} 1,µ 8s (n, m, u0 (jµ ), L) + OT1,jµ −1 n + θ(µ even)(1 + mjµ − mjµ −1 ) =

L−1 2

ν0 − θ(µ even) ¯ 1,jµ −1 , u−1 (jµ ) − E(t) , L − 1), + 8s (n + e¯ jµ −1 − e¯ jµ , m − E 1,µ 4 (t) 0 ¯ 1,tk , 0} = {n, m, u¯ + e¯ −1+tk − E ¯ }L−1 ¯ L − {¯e1+tk , e¯ tk + E 8 : {n, m, u} 1,k −

8s (n, m, u0 (jµ ), L) + OT1,tk n + (ntk + m1+tk )θ(k even) =

L−1 2

ν0 − θ(k odd) ¯ 1,tk , u0 (jµ )+e−1+tk −E(t) , L − 1), + 8s (n − e¯ 1+tk , m − e¯ tk − E 1,k 4 (A.13) ¯ (t) are given by (2.26), (5.2) and (9.7). ¯ a,b , E(t) , E where E a,b a,b After these preliminaries we are in the position to prove the fermionic recurrences (5.3) for 0 ≤ j0 < ν0 and (5.5). Our method is the technique of telescopic expansion introduced in [22] and further developed in [21] and [28]. With this technique we may expand fs (L, u0 (jµ )), −

fs (L, u0 (jµ )) =

X jµ ¯ ¯ 1,l T −E 0 OT n −E1,jµ + , = q 1,jµ q O1,l n+π˜ l ml 0 −¯el 0

(A.14)

l=1

where (3.20) has been used in odd zones and (3.21) has been used in even zones. We shall further need the telescopic expansions


q

θ(k odd)nt1+k −1+t X1+k

¯ 1,1+t1+k −¯et1+k − E −¯e1+t1+k

383

=

¯ l,1+t1+k ¯ 1,t1+k − E −E q −¯el − e¯ 1+t1+k l=dk ¯ dk ,1+t1+k ¯ 1,t1+k − E T −E , + q Odk ,t1+k n −¯e1+t1+k

and

OT l+1,t

1+k

n+(ml −1)θ(k even)

(A.15)

¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E = e¯ jµ −1 − e¯ jµ jµ −1 X OT ¯ ¯ ejµ −1 )+π˜ l ml −E1,jµ − El,jµ −2 l+1,jµ −1 (n+¯ q −¯el + e¯ jµ −1 − e¯ jµ l=dk ¯ ¯ θ(µ even)+OT ejµ −1 ) −E1,jµ −2 − E1+tµ ,jµ 1+tµ ,jµ −1 (n+¯ , +q e¯ jµ −1 − e¯ jµ

q θ(µ even)

(A.16)

with dk = 1 + tk + δk,0 , where again (3.20) has been used in odd zones and (3.21) has been used in even zones. Using case 6 in (A.12) we may rewrite (A.14) as fs (L, u0 (jµ )) = q

ν0 +1 3 L−1 2 − 4 + 4 θ(µ

odd)

(t) fs (L − 1, u1 (jµ ) − E1,µ )+

µ X

Ik ,

(A.17)

k=0

where Ik =

gk X

T

q O1,l n+π˜ l ml

l=dk

¯ 1,l −E , 0≤k≤µ −¯el

(A.18)

t1+k for k < µ and may now process the terms Ik (0 ≤ k ≤ µ − 1) as for k = µ jµ follows. In the lth term (l = 6 dk ) in (A.18) we perform the change of variables and gk =

n → n + e¯ l − e¯ l−1 − e¯ 1+t1+k , ¯ l,t1+k m → m − 2E

(A.19)

to obtain

T

Ik = q O1,tk n+mdk θ(k even) +

−1+t X1+k l=dk

¯ 1,dk −E −¯edk

q (O1,t1+k +O1+l,t1+k )n+m1+t1+k θ(k odd)+(ml −1)θ(k even) T

T

¯ 1,1+t1+k ¯ l,t1+k − E −E . −¯el − e¯ 1+t1+k (A.20)

For k = 0, µ > 0 we use (A.11) and (A.12) 1,2 and obtain

384


I0 = q

m1

−¯e1 −¯e1

¯ 1,1+ν0 ¯ 1,ν0 − e¯ 1+ν0 −¯eν0 − E −2E + − −¯eν0 +1 −¯e1+ν0

(A.21)

= fs (L − 1, eν0 −1 − eν0 + u0 (jµ )) + (q L−1 − 1)fs (L − 2, u0 (jµ )). Applying the telescopic expansion (A.15) to the last sum in (A.20), we obtain for k 6= 0,

¯ 1,dk −E −¯edk ¯ 1,1+t1+k T −¯et1+k − E + q O1,t1+k n+m1+t1+k θ(k odd) q nt1+k θ(k odd) −¯e1+t1+k ¯ ¯ T −E1,t1+k − Edk ,1+t1+k . −q Odk ,t1+k n −¯e1+t1+k T

Ik = q O1,tk n+mdk θ(k even)

(A.22)

For k 6= 0, we perform one more change of variables n → n + e¯ 1+t1+k − e¯ tk − e¯ 1+tk , ¯ 1+tk ,t1+k m → m + 2E

(A.23)

in the last term of (A.22) to arrive at ¯ 1,1+t1+k T −¯et1+k − E Ik = q O1,t1+k n+(m1+t1+k +nt1+k )θ(k odd) −¯e1+t1+k ¯ T −E1,1+tk + q O1,tk n+m1+tk θ(k even) −¯e1+tk ¯ 1,1+tk −¯etk − E OT 1,tk n+m1+tk θ(k even)+(mtk −1)θ(k odd) −q −¯etk − e¯ 1+tk k+1 X T ¯ 1,1+ta eta − E O1,ta n+(nta +m1+ta )θ(a even) −¯ q , = −¯eta − e¯ 1+ta

(A.24)

a=k

where to get the last line (3.20) or (3.21) has been used for k even or odd, respectively. Thus recalling (A.12) 2,8 we have for 1 ≤ k ≤ µ − 1, Ik =

k+1 X

q (1−δa,1 )(

ν0 −θ(a odd) L−1 ) 2 − 4

(t) fs (L − 1, eta −1 − E1,a + u0 (jµ )).

(A.25)

a=k

It remains to process Iµ . To this end we perform the change of variables n → n − e¯ l−1 + e¯ l + e¯ jµ −1 − e¯ jµ ¯ l,jµ −1 m → m − 2E in the lth term (l 6= dµ ) appearing in the sum (A.18) with k = µ. This yields

(A.26)


Iµ = q

OT 1,jµ −1 n+mdk θ(µ even)

jµ −1

+

X

q

¯ 1,dµ −E −¯edµ

385

T OT 1,jµ −1 n+Ol+1,jµ −1 (n+ejµ −1 )+(ml +mjµ −mjµ −1 )θ(µ even)

l=dµ

¯ l,jµ −2 ¯ 1,jµ − E −E , −¯el + e¯ jµ −1 − e¯ jµ

(A.27) where we used the empty sum convention. Using the telescopic expansion (A.16) we find Iµ = q +q

OT 1,jµ −1 n+mdk θ(µ even)

¯ 1,dµ −E −¯edµ

¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E e¯ jµ −1 − e¯ jµ ¯ ¯ OT ejµ −1 ) −E1,jµ −2 − E1+tµ ,jµ 1+tµ ,jµ −1 (n+¯ . −q e¯ jµ −1 − e¯ jµ

OT 1,jµ −1 n+(mjµ −mjµ −1 +1)θ(µ even)

(A.28)

For µ = 0 (A.28) yields

−¯e1 I0 = q + θ(j0 > 1)q mj0 −mj0 −1 +1 −¯e1 ¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 ¯ 1,j0 −1 + e¯ j0 −1 − e¯ j0 −2E −E − × e¯ j0 −1 − e¯ j0 e¯ j0 −1 − e¯ j0 m1

(A.29)

= fs (L − 1, u−1 (j0 )) + (q L−1 − 1)fs (L − 2, u0 (j0 )). where we used (A.9), (A.11), (A.12) 4,5. For µ > 0 we need to change variables again n → n − e¯ tµ − e¯ 1+tµ − e¯ jµ −1 + e¯ jµ , ¯ 1+tµ ,jµ −1 m → m + 2E

(A.30)

in the last term of the right-hand side of (A.28) to get ¯ 1,1+tµ −E Iµ = q −¯e1+tµ ¯ ¯ jµ −1 − e¯ jµ OT n+(mjµ +mjµ −1 +1)θ(µ even) −E1,jµ −1 + e + q 1,jµ −1 e¯ jµ −1 − e¯ jµ ¯ 1,1+tµ −¯etµ − E OT 1,jµ −1 n+m1+tµ θ(µ even)+(mtµ −1)θ(µ odd) −q −¯etµ − e¯ 1+tµ ¯ 1,jµ −1 + e¯ jµ −1 − e¯ jµ −E OT 1,jµ −1 n+(mjµ −mjµ −1 +1)θ(µ even) =q e¯ jµ −1 − e¯ jµ ¯ T etµ − E1,1+tµ O n+(m1+tµ +ntµ )θ(µ even) −¯ + q 1,jµ −1 , −¯e1+tµ OT 1,jµ −1 n+m1+tµ θ(µ even)

(A.31)

386


where we used the elementary binomial recurrences (3.20) and (3.21). Recalling Eqs. 7, 8 of (A.12) we finally obtain for µ > 0, Iµ = q +q

ν0 −θ(µ odd) L−1 2 − 4

(t) fs (L − 1, u0 (jµ ) + etµ −1 − E1,µ )

ν0 −θ(µ even) L−1 2 − 4

(t) fs (L − 1, u−1 (jµ ) − E1,µ ).

(A.32)

The desired results (5.3) (for 2 ≤ j0 ≤ ν0 − 1) and (5.5) follow by combining formulas (A.17), (A.21), (A.25), (A.29) and (A.32).

Appendix B. Proof of the Initial Conditions for Propositions 1 and 2 In this appendix we prove the initial conditions for Propositions 1 and 2 as discussed in µ a Sect. 8. These proofs explain the origin of the factors q 2 (−1) in Propositions 1 and 2 and also demonstrate that at the dissynchronization point the form (3.40) does not hold. Proof of Proposition 1 with a = −1 and µ = 2. To prove Proposition 1 with a = −1, µ = 2 −2 : 2 we will use the following decomposition of the flow −→y−1 y2-2 -1

= y1-2 0

1

2

(8.2)b (8.5)

y1-2 0

2

y1-2 0

2

y1-2

(B.1)

0

The decomposition of the flow of length y2-1 used to prove Proposition 1 and 2 with µ=2, a=-1 whose proof is elementary and will be omitted. We recall that u20 is any 1 + tn+1 –dimensional vector with (u20 )j = 0 for j ≤ t2 . Then first using Proposition 1 with µ = 1, a = 0 and then using (5.3) with j0 = ν0 we obtain {fs (L, −et1 − et2 + u20 ), fs (L, e1 − et1 − et2 + u20 )} y1 −2

−→ {fs (L, e−1+t1 − et1 − et2 + u20 ), fs (L, −et2 + u20 )} 0 L 1 −→ fs (L, −et2 + u20 ), q − 2 fs (L, e1+t1 − et2 + u20 )

(B.2)

(8.2)b

L 1 + q 2 − L fs (L − 1, −et2 + u20 ) . q2

It may be seen that the second polynomial in the last pair of (B.2) for which b is at the dissynchronization value b2,−1 is not of the form (3.40). We may further evolve using flow −→2(8.5) to find


L u20 ), q − 2 fs (L, e1+t1

u20 )

387

L 2

1

u20 )

− et2 + + (q − L )fs (L − 1, −et2 + fs (L, −et2 + q2 ν0 −2 1 2 −→ q − 2 q − 4 {fs (L, −et1 + e2+t1 − et2 + u20 ), fs (L, e1 − et1 + e2+t1 − et2 + u20 )} (8.5) + {P0 (L, 1, e1+t1 − et2 + u20 ), P1 (L, 1, e1+t1 − et2 + u20 ), } . (B.3) where we used (5.4) with j1 = 1 + t1 and the first equation in (5.3) as well as the definitions (3.35), (8.27), (8.28). Now applying Propositions 1 and 2 with a = 0 and µ = 1 for −→0y1 −2 and lemma 2.1 for =⇒2 according to (B.1), we finally obtain upon recalling the definition (8.26), n

o y −2 1 2 (t) (t) + u20 ), fs (L, −E1,2 − et2 + u20 ) −→ q − 2 q c(t2 ) {P1 (L, 2, u20 ), fs (L, u20 )} . fs (L, −E1,2 −1

(B.4) − 21 is Therefore Proposition 1 for a = −1 and µ = 2 is proven. We note that the factor q µ a the origin of the factor q 2 (−1) in Proposition 1 with a = −1. Proof of proposition 2 with a = −1 and µ = 2. For the proof of Proposition 2 with a = −1, µ = 2 we shall again use the decomposition (B.1). However, in the present case it is convenient to rewrite the piece −→1(8.2)b −→2(8.5) in the bottom graph of (B.1) using the identity 1

2

2

1

−→ −→ = =⇒ −→ .

(8.2)b (8.5)

(8.2)a

(B.5)

To simplify our analysis, let us further assume that ν1 ≥ 3, ν0 ≥ 2. We start by applying Propositions 1 and 2 with a = 0 and µ = 1, {P0 (L, 2, u20 ), P1 (L, 2, u20 )} n o ν0 (t) (t) =q − 4 fs (L, e−1+t2 − E1,2 + u20 ), fs (L, e1 + e−1+t2 − E1,2 + u20 ) + {P0 (L, 1, u20 ), P1 (L, 1, u20 )} n ν0 −2 L y1 −2 − ν0 −→ q 4 P−1 (L, 1, e−1+t2 − et2 + u20 ) + q 2 − 4 fs (L, −et1 + u20 ) ,

(B.6)

0

fs (L, e−1+t2 − et2 + u20 )} . Evolving this once more according to =⇒2 and using Lemma 2.2 with µ = 1, j1 = 1 + t1 we find from (B.6), y1

{P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→ o n ν0 (t) (t) + u20 ), fs (L, e1 + e−2+t2 − E1,2 + u20 ) q − 2 fs (L, e−2+t2 − E1,2 +q

ν − 40

(B.7)

{P0 (L, 1, e−1+t2 − et2 + u20 ), P1 (L, 1, e−1+t2 − et2 + u20 )} .

Evolving now according to −→1(8.2)a and using the fermionic recurrence (5.3) for fs and (6.1)for f˜s as well as definitions (8.27), (8.28), we derive for ν0 ≥ 2, ν1 ≥ 3,

388

A. Berkovich, B. M. McCoy, A. Schilling y1 +1

{P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→ (B.1) n o ν (t) (t) − 20 fs (L, e1 + e−2+t2 − E1,2 q + u20 ), fs (L, e2 + e−2+t2 − E1,2 + u20 ) o n ν0 (t) (t) + q − 4 f˜s (L, e−2+t1 + e−1+t2 − E1,2 + u20 ), f˜s (L, e−3+t1 + e−1+t2 − E1,2 + u20 ) , (B.8) 2 −2 where the arrow in (B.8) represents the flow −→y−1 restricted to the first 1 + y1 steps. We now note that the polynomials in the rhs of (B.8) and in the rhs of (8.41) differ 2 −2 only by the overall factor of q −1/2 . Furthermore, according to (8.23) the flow −→y−1 and the flow −→y0 2 −2 have the same steps starting with the step y1 +2. These observations along with Proposition 2 for a = 0, µ = 2 clearly imply that y2 −2

{P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→ q − 2 −1

1

n

q c(t2 )+

L−ν0 2

(t) fs (L, −E1,2 + u20 ), 0

o (B.9)

which concludes the proof of Proposition 2 with a = −1, µ = 2, ν0 ≥ 2, ν1 ≥ 3. The proof of the remaining cases with ν1 = 2 and ν1 ≥ 3, ν0 = 1 can be carried out in a similar fashion and is left to the reader. Finally, we note that the factor q −1/2 in (B.9) is (−)µ the origin of the factor q a 2 in Proposition 2 with a = −1, µ. Proof of Proposition 1 with a = −1 and µ = 3. To prove Proposition 1 with a = −1, µ = 3 3 −2 we shall use a decomposition of −→y−1 obtained from (8.24) with µ = 2, a = −1 by 2 replacing the second arrow =⇒ from the left by the composite arrow −→1(8.2)a −→1(8.2)b . This replacement is necessary because (8.24) is not valid as it stands for µ = 2. In what y3 −2 follows the notation −→b−2 −1 , y2 < b ≤ y3 is understood as the flow −→−1 restricted to the first b − 2 steps. For convenience we consider ν0 ≥ 2 and we start by applying Proposition 1 with µ = 2, a = 0 and subsequently using (8.43) with µ = 2, u20 = −et3 + u30 and lemma 2.1 with µ = 1, j1 = t2 to obtain n

o y 2 (t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) −→ fs (L, −E1,3 −1

q

+

ν0 −1 4

{fs (L, −et1 + e1+t2 − et3 + u30 ), fs (L, e1 q c(t2 ) {P0 (L, 1, −et3 + u30 ), P1 (L, 1, −et3 + u30 )} .

c(t2 )−

− et1 + e1+t2 − et3 + u30 )}

(B.10)

Application of Propositions 1 and 2 with a = 0 and µ = 1 to the rhs of (B.10) yields n

o y +y −2 2 1 (t) (t) fs (L, −E1,3 + u30 ), fs (L, e1 − E1,3 + u30 ) −→ −1 n ν0 −1 ν0 −1 L q c(t2 )− 4 fs (L, e−1+t1 − et1 + e1+t2 − et3 + u30 ) + q 2 − 4 fs (L, −et1 − et3 + u30 ), fs (L, e1+t2 − et3 + u30 )} . (B.11) Next evolving the rhs of (B.11) according to −→1(8.2)a and making use of (5.4) with j2 = 1 + t2 we derive


389

o (t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) fs (L, −E1,3 n ν0 +1 L y2 +y1 −1 c(t2 )− ν0 −1 (t) 4 fs (L, e1+t2 − et3 + u30 ), q 2 − 4 fs (L, e2+t2 − E1,3 −→ q + u30 ) n

−1

o ν0 L (t) (t) + u30 ) + fs (L, e−1+t1 + et2 + e1+t2 − E1,3 + u30 ) . +q 2 − 4 fs (L, e−1+t2 + e1+t2 − E1,3 (B.12) To proceed further we observe that the first two steps of −→1(8.2)b −→y1 2 −2 are the same as =⇒2 . The straightforward use of properties (5.3), (6.1) and definition (3.35) then gives n

oy

(t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) fs (L, −E1,3

2 +y1 +1

−→ q 2 {z1 (L, u30 ), z2 (L, u30 )} 1

−1

(B.13)

where za (L, u30 ) = q c(t2 )−

ν0 −1 4

q−

ν0 +1 4

(t) fs (L, ea + e2+t2 − E1,3 + u30 )+

ν0 (t) (t) q − 4 fs (L, ea + e−1+t2 + e1+t2 − E1,3 + u30 ) + f˜s (L, et1 −1−a + et2 + e1+t2 − E1,3 + u30 ) (B.14) with a = 1, 2. On the other hand, as can be seen from case 3 of (7.18) with j2 = (t) t2 + 1, j0 = 1, 2 and −E4,n replaced by u30 , n

oy

(t) (t) + u30 ), fs (L, e1 − E1,3 + u30 ) fs (L, −E1,3

2 +y1 +1

−→ {z1 (L, u30 ), z2 (L, u30 )} .

(B.15)

Comparing (B.13) and (B.15) we see that the expressions in the rhs of these expressions y3 −2 differ only by a factor of q 1/2 . According to (8.23) with µ = 3, a = −1 the flow −→−1 and the flow −→0y3 −2 have identical steps starting with the step y2 + y1 + 2. This fact together with (B.13), (B.15) and Proposition 1 with µ = 3, a = 0 imply that n

o y −2 1 3 (t) (t) fs (L, −E1,3 + u30 ), fs (L, e1 − E1,3 + u30 ) −→ q c(t3 )+ 2 {P−1 (L, 3, u30 ), fs (L, u30 )} , −1

(B.16) which concludes the proof of Proposition 1 with a = −1, µ = 3, ν0 ≥ 2. The proof of the special case ν0 = 1 follows along similar lines and will be omitted.

Appendix C. Proof of 10.10 and 10.15 We sketch the proof of (10.10) and (10.15) in several stages. First we use the results of Sect. 8 to prove (C.12) and (C.13). From Theorem 1 of Sect. 8 and (C.13) we then prove (10.15) for the case β = α + 1 and after that we use both (C.12) and (C.13) to prove the general case of (10.15) with 1 ≤ α ≤ β − 2 by induction. We conclude by proving (10.10). It is useful to introduce the notation j˜µ = jµ − tµ , 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n . In 0 addition, we define the flow −→ab −2 with a = 0, −1 and yµ < b0 ≤ yµ+1 as part of the y −2 flow −→aµ+1 restricted to the first b0 − 2 steps. We also note the equality of flows

390

A. Berkovich, B. M. McCoy, A. Schilling j˜ µ yµ −2 2

j˜ µ yµ

−→ = −→ =⇒, 2 ≤ j˜µ ≤ νµ + δµ,n , 2 − |a| ≤ µ ≤ n a

a

and yµ

yµ −2 2

−→ = −→ =⇒, 2 − |a| ≤ µ ≤ n a

(C.1)

(C.2)

0

with a = 0, −1. These follow from (3.12) for a = 0 if we note the Takahashi decomposition ( (µ) (µ−1) lj µ + l tµ for 2 ≤ j˜µ ≤ νµ + δµ,n , 1 ≤ µ ≤ n (C.3) j˜µ yµ = (µ−1) l1+tµ for j˜µ = 1, 1 ≤ µ ≤ n. The identical argument works for a = −1 if we replace jµ by 1 + jµ . Next we use (C.1) and (C.2) along with (8.50)-(8.52) and Lemma 2.2 of Sect. 8 to obtain j˜ µ yµ

0 0 ), P1 (L, µ + 1, uµ+1 )} −→ {P0 (L, µ + 1, uµ+1 a

q

c(jµ )−c(tµ−1 )−

ν0 −θ(µ even) 1 + 2 a(−1)µ+1 4

µ+1

1

+q c(jµ )−c(tµ−1 )+ 2 a(−1)

(t) {fs (L, −E1,µ+1 + et1+µ −j˜ µ −1 + (t) fs (L, e1 − E1,µ+1 + et1+µ −j˜ µ −1

0 uµ+1 ), 0 + uµ+1 )}

(C.4)

0 {P0 (L, µ, et1+µ −j˜ µ − et1+µ + uµ+1 ), 0 )} P1 (L, µ, et1+µ −j˜ µ − et1+µ + uµ+1

with 1 + |a| ≤ j˜µ ≤ νµ − 2, 2 ≤ µ ≤ n − 1, a = 0, −1. Furthermore, from (8.23) and (8.25) we derive (µ) l1+j −2

j˜ µ yµ yµ−1 −2

−→ = −→ −→ µ

a

a

0

for 2 ≤ µ ≤ n, 1 ≤ j˜µ ≤ νµ − 1 + 2δµ,n , a = 0, −1. (C.5)

Then combining (C.5) with (C.4), Proposition 1 from Sect. 8 and (8.50) with a = 0 and µ replaced by µ − 1 gives (µ) l1+j −2

0 0 ),P1 (L, µ + 1, uµ+1 )} −→ {P0 (L, µ + 1, uµ+1 µ

a

q c(jµ )−

ν0 −θ(µ even) 1 + 2 a(−1)µ+1 4

(t) 0 0 {Y (L, µ, jµ , uµ+1 ), fs (L, −Eµ,µ+1 + et1+µ −j˜ µ −1 + uµ+1 )

+ q−

(−1)µ 4

(t) 0 fs (L, e−1+tµ − Eµ,µ+1 + et1+µ −j˜ µ + uµ+1 )} (C.6) with 1 + |a| ≤ j˜µ ≤ νµ − 2, 2 ≤ µ ≤ n − 1, a = 0, −1 and (t) 0 0 Y (L, µ, jµ , uµ+1 ) =P−1 (L, µ − 1, −Eµ,µ+1 + et1+µ −j˜ µ −1 + uµ+1 )

+q

(−1)µ 4

+q2− L

(t) 0 P−1 (L, µ − 1, e−1+tµ − Eµ,µ+1 + et1+µ −j˜ µ + uµ+1 ) (C.7)


(t) 0 fs (L, −E1,µ−1 + et1+µ −j˜ µ + uµ+1 ).

The analogues of (C.6) and (C.7) for µ = 1 are


391

(1) l1+j −2

1 {P0 (L, 2, u20 ), P1 (L, 2, u20 )} −→

a

q c(j1 )−

ν0 4

+ a2

(C.8)

ν0

(t) {Y (L, 1, j1 , u20 ), q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 + u20 )+ L

(t) fs (L, e−1+t1 + et2 −j˜ 1 − E1,2 + u20 )}

and

Y (L, 1, j1 , u20 ) = fs (L, et2 −j˜ 1 − et2 + u20 )

(C.9)

with 1+|a| ≤ j˜1 ≤ ν1 −2, a = 0, −1. To derive (C.8)-(C.9) for a = 0 we compare case 3 of (7.18) with j0 = 0, 1 with case 2 of (7.18) with j0 = ν0 −1, and j1 replaced by j1 −1 and (t) by u20 . By comparing case 4 of (7.18) to find the desired result after replacing ej2 − E3,n (B.7)-(B.8) and (C.8) with j1 = 2, a = 0 and recalling that according to (8.23) the 2 −2 flows −→y0 2 −2 and −→y−1 have the same steps, starting with the step 1 + y0 + y1 we easily verify (C.8) for a = −1, 2 ≤ j˜1 ≤ ν1 − 2. (µ−1) l2+j −2

Next we apply the flow −→−1 µ−1 to the right-hand side (rhs) of (C.4) with a = 0. Taking into account (8.47) with a = −1, µ replaced by µ−1 and jµ replaced by jµ−1 +1 and (C.6) with a = −1, µ replaced by µ − 1, µ ≥ 3 or (C.8) with a = −1 with j1 replaced by 1 + j1 for µ = 2 we derive (µ−1) l2+j −2

ν0 −θ(µ odd)

1

µ+1

+ 2 (−1) 4 rhs of (C.4) −→ q c(jµ )+c(1+jµ−1 )− −1 0 ), {Y (L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ − et1+µ + uµ+1 µ−1

q δµ,2 ( 2 − L

+q +q

ν0 4

)

×

(t) 0 fs (L, −Eµ−1,µ+1 + etµ −j˜ µ−1 −2 + et1+µ −j˜ µ + uµ+1 )

(−1)µ −δµ,2 4

fs (L, e−1+tµ−1 −

(t) Eµ−1,µ+1

+ etµ −j˜ µ−1 −1 + et1+µ −j˜ µ +

(C.10) 0 uµ+1 )}

(−1)µ 4

0 {X(L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ −1 − et1+µ + uµ+1 ), (t) 0 + e1+jµ−1 + et1+µ −j˜ µ −1 + uµ+1 )} fs (L, −Eµ,µ+1

with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 3, 1 + tµ ≤ jµ ≤ tµ+1 − 2, 2 ≤ µ ≤ n − 1 and 0 0 ) = θ(jµ > y1 ) P−1 (L, µ − δ1+tµ ,jµ , ejµ − et1+µ + uµ+1 )+ X(L, µ, jµ , uµ+1 q2− L


(t) 0 fs (L, e−1+jµ − E1,1+µ + u1+µ )

(C.11)

+ δµ,1 δj1 ,y1 fs (L, −et2 + u20 ) for 1 ≤ µ ≤ n. To proceed further we note the following equality of flows: j˜ µ yµ

−→

(µ−1) l2+j −2

−→ µ−1

−1

=

(µ) (µ−1) l1+j +l1+j −2 µ

−→

µ−1

=

(µ−1) (µ) −2 l1+j −2 2 l1+j µ µ−1

−→ =⇒

−→

(C.12)

with µ ≥ 2, 1 ≤ j˜µ ≤ νµ − 2 + 2δµ,n , 1 ≤ j˜µ−1 ≤ νµ−1 − 1. The first equation in (C.12) follows from (8.25) with a = 0. The second equation in (C.12) is just properties

392


(µ) (3.12)–(3.13) of the map b → r with b = l1+j . Equation (C.12) together with (C.6) with µ a = 0 and (C.10) imply the flow (t) 0 0 {Y (L, µ, jµ , uµ+1 ), fs (L, −Eµ,µ+1 + et1+µ −j˜ µ −1 + uµ+1 )

+ q−

(−1)µ 4

(µ−1) l1+j −2

2

(t) 0 fs (L, e−1+tµ − Eµ,µ+1 + et1+µ −j˜ µ + uµ+1 )} ν0 +1

3

=⇒ −→ q c(jµ−1 )− 4 + 4 θ(µ odd) × 0 {Y (L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ − et1+µ + uµ+1 ), µ−1

q δµ,2 ( 2 − L

+q +q

(−1)µ 4

ν0 4

−

)

(C.13)

(t) 0 fs (L, −Eµ−1,µ+1 + etµ −j˜ µ−1 −2 + et1+µ −j˜ µ + u1+µ )}

δµ,2 4

(t) 0 fs (L, e−1+tµ−1 − Eµ−1,µ+1 + etµ −j˜ µ−1 −1 + et1+µ −j˜ µ + uµ+1 )}

(−1)µ 4

0 {X(L, µ − 1, 1 + jµ−1 , et1+µ −j˜ µ −1 − et1+µ + uµ+1 ), (t) 0 + e1+j˜ µ−1 + et1+µ −j˜ µ −1 + uµ+1 )} fs (L, −Eµ,µ+1

with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 3, 1 + tµ ≤ jµ ≤ tµ+1 − 2, 2 ≤ µ ≤ n − 1. Next we use the recursive properties of Sect. 5 along with (8.47) and (C.6) with a = 0, µ replaced by µ − 1 and (C.8) with a = 0 for µ = 2 to obtain 0 ), fs (L, ejµ {X(L, µ, jµ , uµ+1

− et1+µ +

{X(L, µ − 1, jµ−1 , e1+jµ − et1+µ +

+q

(−1)µ 4

2 0 u1+µ )} =⇒

(µ−1) l1+j −2

−→ µ−1

(t) 0 uµ+1 ), fs (L, −Eµ,µ+1

q c(jµ−1 )−

ν0 +1 3 4 + 4 θ(µ

+ ejµ−1 + e1+jµ +

odd)

×

0 u1+µ )}

0 {Y (L, µ − 1, jµ−1 , ejµ − et1+µ + uµ+1 ),

q δµ,2 ( 2 − L

+q

(−1)µ 4

ν0 4

−

)

(t) 0 fs (L, −Eµ−1,µ+1 + etµ −j˜ µ−1 −1 + ejµ + uµ+1 )

δµ,2 4

(t) 0 fs (L, e−1+tµ−1 − Eµ−1,µ+1 + etµ −j˜ µ−1 + ejµ + uµ+1 )}

(C.14) with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 2, 1 + tµ ≤ jµ ≤ t1+µ − 1 + δµ,n , 2 ≤ µ ≤ n. Theorem (t) 0 specified to be −Eµ+2,n along with 1 (of Sect. 8) with P = 0 and (C.14) with uµ+1 (µ) (µ−1) l1+j +l1+j −2 µ

−→

µ−1

(µ−1) (µ) −2 l1+j −2 2 l1+j µ µ−1

= −→ =⇒

−→

(C.15)

imply (µ) (µ−1) l1+j +l1+j −2 µ

ν0 +1

3

(t) (t) ), fs (L, e1 − E1,n )} −→ q c(jµ )+c(jµ−1 )− 4 + 4 θ(µ odd) × {fs (L, −E1,n (t) (t) ), fs (L, ejµ−1 + e1+jµ − Eµ,n )} {X(L, µ − 1, jµ−1 , e1+jµ − E1+µ,n

+q

(−1)µ 4

µ−1

(C.16)

(t) {Y (L, µ − 1, jµ−1 , ejµ − E1+µ,n ),

q δµ,2 ( 2 − L

+q

(−1)µ 4

ν0 4

−

)

(t) fs (L, etµ −j˜ µ−1 −1 + ejµ − Eµ−1,n )

δµ,2 4

(t) fs (L, e−1+tµ−1 + etµ −j˜ µ−1 + ejµ − Eµ−1,n )}


393

with 1 + tµ−1 ≤ jµ−1 ≤ tµ − 2, 1 + tµ ≤ jµ ≤ tµ+1 + δµ,n − 1, 2 ≤ µ ≤ n. The second terms of the right-hand side of (C.16) agrees with the right-hand side of (10.15) when β = α + 1 = µ which thus completes the proof of this special case. To complete the proof of (10.15) in the general case 1 ≤ α ≤ β − 2 we use the equality of flows (α) b(α+1,β)−2 2 l1+jα −2

−→

where b(α, β) =

β X

b(α,β)−2

=⇒ −→ = −→ ,

(C.17)

(µ) l1+j ; 1 ≤ α, 1 + α ≤ β ≤ n, µ

µ=α

1 + tµ ≤ jµ ≤ tµ+1 − 3, α ≤ µ ≤ β − 2, 1 + tβ−1 ≤ jβ−1 ≤ tβ − 2, 1 + tβ ≤ jβ ≤ tβ+1 − 1 + δβ,n .

(C.18)

Equation (C.17) is just properties (3.12)-(3.13) of the b → r map with b = b(α + 1, β). Then by starting with (C.16) and by use of (C.13) and (C.14) we may prove by induction (α + 1 → α) that b(α,β)−2

(t) (t) {fs (L, −E1,n ), fs (L, e1 − E1,n )} −→ X c(4) (j)+rf(4,1) (i) q {X(L, α, jα + iα+1 , u(4) (i, j) + et1+α ), fs (L, u(4,1) (i, j))} iα+1 ,···,iβ−1 =0,1 iα =0

X

+

q c(4) (j)+rf(4,2) (i) {Y (L, α, jα + iα+1 , u(4) (i, j) + et1 +α ),

iα+1 ,···,iβ−1 =0,1 iα =1

q δ1,α ( 2 − L

ν0 4

)

fs (L, u(4,2) (i, j)) + q −

(−1)α +δ1,α 4

fs (L, u(4,3) (i, j))},

(C.19) where c(4) (j), rf(4,1) (i), rf(4,2) (i), u(4) (i, j), u(4,1) (i, j), u(4,2) (i, j). and u(4,3) (i, j) are defined in Sect. 10.3. The second terms on the right-hand side of (C.19)agree with the right-hand side of (10.15) which thus completes the proof. It remains to prove (10.10). To do this we first compare cases 1 and 2 of (7.16) with (t) E3,n replaced by u20 to infer that (0) 2 l1+j −2

0 {X(L, 1, j1 , u20 ), fs (L, ej1 − et2 + u20 )} =⇒ −→

{q − q−

ν0 −2 4

ν0 −2 4

(t) (t) fs (L, ej0 −1 + ej1 +1 − E1,2 + u20 ) + f˜s (L, eν0 −j0 + ej1 − E1,2 + u20 ),

(C.20)

(t) (t) fs (L, ej0 + ej1 +1 − E1,2 + u20 ) + f˜s (L, eν0 −j0 −1 + ej1 − E1,2 + u20 )}

with 1 ≤ j0 ≤ ν0 − 1, 1 + t1 ≤ j1 ≤ t2 − 1. Furthermore we have {X(L, 1, j1 , u20 ), fs (L, ej1 − et2 + u20 )} −→ 1

(8.2) b

{fs (L, ej1 −et2 + u20 ), q −

ν0 −2 4

(t) (t) fs (L, ej1 +1 − E1,2 + u20 ) + f˜s (L, eν0 −1 + ej1 − E1,2 + u20 )} (C.21)

394


with 1 + t1 ≤ j1 ≤ t2 − 1. Analogously by comparing cases 2 and 4 of (7.18) with (t) replaced by u20 we find ej2 − E3,n ν0

(t) {Y (L, 1, j1 , u20 ), q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 + u20 ) L

(0) 2 l1+j −2

0 (t) + u20 )} =⇒ −→ + fs (L, e−1+t1 + et2 −j˜ 1 − E1,2

{q − q−

ν0 −2 4

1 (t) fs (L, ej0 + et2 −j˜ 1 −1 − E1,2 + u20 ) + q 2 f˜s (L, eν0 −j0 −1 + et2 −j˜ 1 + u20 ),

ν0 −2 4

1 (t) (t) fs (L, ej0 +1 + et2 −j˜ 1 −1 − E1,2 + u20 ) + q 2 f˜s (L, eν0 −j0 −2 + et2 −j˜ 1 − E1,2 + u20 )} (C.22) with 1 ≤ j0 ≤ ν0 − 1, 1 + t1 ≤ j1 ≤ t2 − 2 and ν0

(t) + u20 ) {Y (L, 1, j1 , u20 ), q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 L

(t) + fs (L, e−1+t1 + et2 −j˜ 1 − E1,2 + u20 )} −→ 1

(8.2) b

ν0

(t) (t) {q 2 − 4 fs (L, et2 −j˜ 1 −1 − E1,2 + u20 ) + fs (L, e−1+t1 + et2 −j˜ 1 − E1,2 + u20 ), L

q−

ν0 −2 4

1 (t) (t) fs (L, e1 + et2 −j˜ 1 −1 − E1,2 + u20 ) + q 2 f˜s (L, eν0 −2 + et2 −j˜ 1 − E1,2 + u20 )} (C.23) with 1 + t1 ≤ j1 ≤ t2 − 2. The result (10.10) follows from combining (C.19) with α = 1 with (C.20)–(C.23).

Acknowledgement. The authors are grateful to G.E. Andrews for his interest and encouragement, to K. Voss and S.O. Warnaar for discussions and careful reading of the manuscript, and to T. Miwa for many helpful comments. One of us (AB) is pleased to acknowledge the hospitality of the ITP of SUNY Stony Brook where part of this work was done. This work is supported in part by the NSF under DMR9404747.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Rogers, L.J.: Proc. Lond. Math. Soc. 25, 318 (1894) Schur, I., Preuss, S.-B.: Akad. Wiss. Phys.–Math. Kl, 302 (1917) Rogers, L.J. and Ramanujan, S.: Proc. Camb. Phil. Soc. 19, 214 (1919) MacMahon, P.A.: Combinatory Analysis. Vol. 2, Cambridge: Cambridge University Press, 1916 Andrews, G.E.: The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol 2, G.-C. Rota ed. Reading, MA: Addison Wesley 1976 Slater, L.J.: Proc. Lond. Math. Soc. (2) 54, 147 (1951–52) Baxter, R.J.: J. Stat. Phys. 26, 427 (1981) Andrews, G.E., Baxter, R.J. and Forrester, P. J.: J. Stat. Phys. 35, 193 (1984) Date, E., Jimbo, M., Miwa, T. and Okado, M.: Phys. Rev. B35, 2105 (1987); Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Nucl. Phys B 290, 231 (1987) and Adv. Stud. in Pure Math. 16, 17 (1988) Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B.: J. Stat. Phys. 34, 763 (1984) and Nucl. Phys. B 241, 333 (1984) Feigin, B. and Fuchs, D.B.: Funct. Anal. Appl. 17, 241 (1983) Feigin, B. and Fuchs, D.B.: Lecture Notes in Math. 1060, Springer-Verlag 1984 ed. L.D. Faddeev and A.A. Malcev) 230 Lepowsky, J. and Primc, M.: Structure of the standard modules for the affine Lie algebra A(1) 1 , Contemporary Mathematics, Vol. 46, Providence, RI: AMS, 1985


14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

395

Kedem, R. and McCoy, B.M.: J. Stat. Phys. 71, 865 (1993) Dasmahapatra, S., Kedem, R., McCoy, B.M. and Melzer, E.: J. Stat. Phys. 74, 239 (1994) Kedem, R., Klassen, T.R., McCoy, B.M. and Melzer, E.: Phys. Letts. B 304, 263 (1993) Kedem, R., Klassen, T.R., McCoy, B.M. and Melzer, E.: Phys. Letts. B 307, 68 (1993) Dasmahapatra, S., Kedem, R., Klassen, T.R., McCoy, B.M. and Melzer, E.: Int. J. Mod. Phys. B 7, 3617 (1993) Rocha-Caridi, A.: In: Vertex Operators in Mathematics and Physics, ed. J. Lepowsky, S. Mandelstam and I.M. Singer, Berlin: Springer, 1985 Andrews, G.E.: Proc. Nat. Sci. USA 71, 4082 (1974) Andrews, G.E.: Pac. J. Math. 114, 267 (1984) Berkovich, A., McCoy, B.M. and Orrick, W.: J. Stat. Phys. 83, 795 (1996) Berkovich, A.: Nucl. Phys. B 431, 315 (1994) Foda, O. and Warnaar, S.O.: Lett. Math. Phys. 36, 145 (1996) Warnaar, S.O.: J. Stat. Phys. 82, 657 (1996) Foda, O. and Quano, Y-H.: Int. J. Mod. Phys. A 12, 1651 (1997) Berkovich, A. and McCoy, B.M.Lett. Math. Phys. 37 49 (1996) Melzer, E.: Int. J. Mod. Phys. A 9, 1115 (1994) Schilling, A.: Nucl. Phys. B 459, 393 (1996) and Nucl. Phys. B 467, 247 (1996) Andrews, G.E.: Scripta Math. 28, 297 (1970) Foda, O. and Quano, Y.-H.: Int. J. Mod. Phys. A 10, 2291 (1995) Kirillov, A.N.: Prog. Theor. Phys. Suppl. 118, 61 (1995) Warnaar, S.O.: Commun. Math. Phys. 184, 203 (1997) Forrester, P.J. and Baxter, R.J.: J. Stat. Phys. 38, 435 (1985) Andrews, G.E., Baxter, R.J., Bressoud, D.M., Burge, W.H., Forrester, P.J. and Viennot, G.: Europ. J. Combinatorics 8, 341 (1987) Warnaar, S.O., Pearce, P.A., Seaton, K.A. and Nienhuis, B.: J. Stat. Phys. 74, 469 (1994) Takahashi, M. and Suzuki, M.: Prog. of Theo. Phys. 48, 2187 (1972) Gaspar, G. and Rahman, M.: Basic Hypergeometric Series. Cambridge: Cambridge Univ. Press, 1990, Appendix I Berkovich, A. and McCoy, B.M.: Int. J. of Math. and Comp. Modeling (in press), hepth/9508110 Baver, E. and Gepner, D.: Phys. Lett. B 372, 231 (1996) Berkovich, A., Gomez, C. and Sierra, G.: Nucl. Phys. B 415, 681 (1994) Bailey, W.N.: Proc. Lond. Math. Soc. (2) 50, 1 (1949) Agarwal, A.K., Andrews, G.E. and Bressoud, D.M.: J. Indian Math. Soc. 51, 57 (1987) Berkovich, A., McCoy, B.M. and Schilling, A.: Physica A 228, 33 (1996) Schilling, A. and Warnaar, S.O.: Int. J. Mod. Phys. B 11, 189 (1997) and A Higher–Level Bailey Lemma: Proof and Application, To appear in The Ramanujan Journal (q-alg/9607014) Berkovich, A., McCoy, B.M., Schilling, A. and Warnaar, S.O.: Bailey flows and Bose-Fermi identities (1) (1) for the conformal coset models (A(1) 1 )N × (A1 )N 0 /(A1 )N +N 0 ,. Nucl. Phys. B 499, 621 (1997)

Communicated by T. Miwa

Commun. Math. Phys. 191, 397 – 407 (1998)

Communications in


Box Dimensions and Topological Pressure for some Expanding Maps? Huyi Hu Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. E-mail: [email protected] Received: 6 November 1996 / Accepted: 16 May 1997

Abstract: We consider an expanding map defined on an open region of the plane and study the box dimensions of its invariant sets. Under the condition that the map leaves invariant a “strong unstable foliation” F, we prove that the box dimension of an invariant set is given by δF + δT , where δT is its dimension transverse to F and δF is the root of a certain function involving topological pressure.

0. Introduction Self-similar sets, or fractals, can often be realized as invariant sets of expanding maps. When the map is conformal, i.e. when its derivative expands by the same amount in all directions, Bowen’s formula gives a relation between fractal dimension and topological pressure. More precisely, Bowen’s formula says that the Hausdorff or box dimension of a set 3 is the unique number δ such that the topological pressure P (f |3 , δφ) = 0, where φ(x) = − log |Df (x)|. (See [B1], also see [R].) In this paper we generalize Bowen’s result to a class of expanding maps that are not conformal. Let f be a C 2 map from an open set U ⊂ R2 to R2 and let 3 ⊂ U be a compact invariant set of f on which f is expanding. Suppose that f leaves invariant a foliation F of U along which it expands more strongly than in complementary directions. We project 3 onto a curve that is transversal to the foliation to obtain a set γ. We will show that the box dimension of γ exists and call it the transverse box dimension of 3, denoted by δT . The Lipschitzness of F guarantees that δT is independent of the choice of | det Df (x)| . the transversal curve. Define φF (x) = − log |Df (x)|F and φT (x) = − log |Df (x)|F It is easy to see that there exists a unique real number δF such that the topological ? This work was done while I was in the University of Maryland and was supported by NSF under grants DMS-8802593 and DMS-9116391, and by DOE (Office of Scientific Computing.)

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pressure P (f |3 , δF φF + δT φT ) = 0. We will show that the box dimension of 3 is given by δF + δT . We mention an earlier result of Bedford ([Be], also see [BU]) in which he studied a class of self-similar sets that are graphs of continuous functions. Bedford proved that the box dimension of such a graph is equal to 1 + t, where t is the root of a pressure function similar to ours. (The setting in [Be] is actually quite different. See the Remark in Sect. 1 for more detail.) In recent years there have been a number of results on the Hausdorff and box dimensions of invariant sets of “nonconformal” maps of the plane. (See, e.g. [AJ,F,GL,KP,Mc, PU,PW].) In most (though certainly not all, see e.g. [PW]) of these results, the maps are assumed to be piecewise affine; sometimes they are assumed to have fixed rates of expansion in horizontal and vertical directions. Our setting can be viewed as slightly more general than these. Our result, however, is only for box dimension, which is less subtle than Hausdorff dimension. It would be interesting to know if a similar formula for Hausdorff dimension holds.

1. Assumptions and Statement of Results Let U ⊂ R2 be an open set and let f : U → R2 be a C 2 map. A closed subset 3 ⊂ U is called invariant if f 3 = 3. A closed invariant set 3 is called locally maximal if there exists an open neighborhood ⊂ U of 3 such that any invariant set 30 , 3 ⊂ 30 ⊂ , coincides with 3. It is easy to see that if 3 is locally maximal, then f −1 3 ∩ = 3. The map f is called expanding on 3 if there exist constants κ > 1 and C > 0 such that for all x ∈ 3, |Df n (x)v| ≥ Cκn |v| v ∈ R2 . Without loss of generality, we may assume C = 1. In this paper we always assume that 3 is a locally maximal f -invariant set and f is expanding on 3. It can be proved under this setting that ∀β > 0, ∃ > 0 such that for any -pseudo orbit in 3, there exists a point in 3 β-shadowing it. Therefore 3 has Markov partitions with diameter less than . That is, there are finite number of subsets 31 , · · · , 3S with S [ 3i such that int 3i ∩ int 3j = ∅ if i 6= j, and each f 3i is a union of sets 3j . 3= i=1

We refer the reader to [B2] for more details. We also assume that f satisfies the following. Assumption A. (i) There exists a continuous family of cones C(x) such that Dfx C(x) ⊂ int C(f x) ∀x ∈ ∩ f −1 . (ii) There exists a line bundle E, E(x) ⊂ int C(x), such that Dfx E(x) = E(f x) ∀x ∈ ∩ f −1 . In this assumption, (i) implies that f has different expanding rates (see Lemma 4.1 ∞ \ [ for a proof), while (ii) implies that E(x) = Dfyn C(y) is the strong expanding n=1 y∈f −n x

direction (see the Remark after Lemma 4.1) and the direction is independent of preimages of x. Therefore, the strong unstable foliation F is well defined, which is a family of C 2 curves tangent to E. We can use F to state an equivalent assumption.

Box Dimensions for Expanding Maps

399

Assumption A’. There exists a foliation F of C 2 on such that (i) ∃C ≥ 1, λ < 1 such that ∀n ≥ 0, | det Df n (x)| ≤ Cλn |Df n (x)|F |2 (ii) f F (x) ⊃ F(f x) ∀x ∈

T

∀x ∈

n \

f −i ;

i=0

f −1 .

We say that F is Lipschitz, if for any smooth curve 0 ⊂ transversal to F -leaves, the map π : x → 0 ∩ F(x) is Lipschitz whenever it is defined. It is easy to see that Df induces a contract map F which sends the set of all line bundle L = {L(x) ∈ C(x) : x ∈ } to itself, and E is the unique fixed point of F . Moreover, by the same argument as in [HP] or [HPS], we can get that {E(x)} is C 1 and therefore, the strong unstable foliation F is C 1 . However, we only need that the foliation F is Lipschitz. We state it as the following. Fact. Under above assumptions, F is Lipschitz. Let i ⊂ be an open set containing a small neighborhood of 3i for each 3i . We may assume that the diameter of 3i is small enough such that restricted to i f is injective. Since E(x) is Lipschitz, we may also assume that ∀x, y ∈ i , the angle between E(x) and E(y) is small. So up to a coordinate system change, restricted to each i we can regard F (x) as a foliation given by parallel lines. For each i , take a C 1 curve 0i in the convex hull of i transversal to F-leaves. Let π : i → 0i be a continuous map defined by sliding along the F -leaves, i.e. for x ∈ i , π(x) = 0i ∩ F(x). The Lipschitzness means that π is Lipschitz. We assume that 0i is taken in such a way S S [ [ that π(x) is defined for all x ∈ i . Put γi = π3i , πi = π|i , 0 = 0i and γ = γi . i=1

i=1

The upper and lower box dimension of a bounded set A in a metric space are defined by dimB (A) = lim sup β→0

log N (A, β) − log β

and

dimB (A) = lim inf β→0

log N (A, β) − log β

respectively, where N (A, β) denotes the minimal number of balls of radius β that cover A. If dimB (A) = dimB (A), then we call the common value the box dimension of A and denote it by dimB (A). Since the projection π is Lipschitz, it is easy to see that both dimB (γ) and dimB (γ) are independent of the choice of 0. We will prove in Sect. 3 that dimB (γ) = dimB (γ). Therefore it makes sense to write δT = dimB (γ) and call it the transverse box dimension of 3, Denote |Df (x)|F = |Df (x)|E(x) | and |Df (x)|T = |det Df (x)| |Df (x)|F , where the subscripts F and T refer to “Foliation” and “Transversion” respectively. Put φF (x) = − log |Df (x)|F

and

φT (x) = − log |Df (x)|T .

Both are Lipschitz functions. Let P (f, φ) denote the topological pressure of f for the function φ (see e.g. [W] for a definition). P (f |3 , tφF + δT φT ) decreases as t is increased and goes to ±∞ as t goes to ∓∞. Now we state our result.

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Theorem. Let f be a C 2 map from an open set U ⊂ R2 to R2 and let 3 ⊂ U be a locally maximal compact invariant set of f on which f is expanding and topologically transitive. Suppose Assumption A or A’ is satisfied. Then the box dimension dimB (3) of 3 is equal to δF + δT , where δT is the transverse box dimension of 3, and δF = t is the unique real number such that the topological pressure P (f |3 , tφF + δT φT ) = 0. Remark. We mention some differences between Bedford’s setting [Be] and ours. In [Be] the sets are assumed to be graphs of continuous functions and the maps are assumed to preserve the weak unstable foliation rather than the strong unstable foliation. Bedford’s assumption is natural in the setting of fractal graphs. For general invariant sets, however, it seems more natural to work from the strong unstable directions than from the weak ones. See e.g. [LY]. We introduce some more notations. Suppose f 3i ⊃ 3j for some i and j. We denote by 3ij the component of the preimage of 3j contained in 3i , i.e. 3ij = 3i ∩ f −1 3j . Generally, if f 3ij ⊃ 3ij+1 for j = 0, · · · , k − 1, then we write 3i0 i1 ···ik =

k \

f −l 3ij ,

j=0

i0 i1 ···ik =

k \

f −l ij .

j=0

Since f is expanding on 3, we may regard that restricted to i0 i1 ···ik , f k : i0 i1 ···ik → ik is a diffeomorphism. Also we write 0i0 i1 ···ik = f −k 0ik ∩ i0 i1 ···ik ,

γi0 i1 ···ik = f −k γik ∩ i0 i1 ···ik .

Note that we have 3i0 i1 ···ik ⊂ 3i0 and i0 i1 ···ik ⊂ i0 , but in general 0i0 i1 ···ik 6⊂ 0i0 and γi0 i1 ···ik 6⊂ γi0 . Instead, the relations are π0i0 i1 ···ik ⊂ 0i0 and πγi0 i1 ···ik ⊂ γi0 . A word i0 i1 · · · ik , k ≤ ∞, is called admissible if f 3ij ⊃ 3ij+1 for j = 0, · · · , k − 1. In our proof we will use “≈” for the meaning “up to a constant factor”, where the constant depends only on f and 3. 2. Examples Example 2.1. Let R be a unit square in R2 . Take real numbers b > a > 1. Put n rectangles R1 , · · · , Rn of width a−1 and height b−1 in R, aligned with the axis of R. We require that these n small rectangles are disjoint and that their projections onto a horizontal line either coincide or are disjoint. Therefore these small rectangles form p columns, p ≤ n. Define a C 2 map f on R2 such that f maps each of Ri affinely onto R. Write fi = f |Ri : Ri → R, i = 1, · · · , n. The n inverses fi−1 determine an f -invariant set n S 3 ⊂ Ri in a usual way (see [H]). It is easy to see that 3 is a locally maximal invariant i=1

set of f , and f is expanding and topologically mixing on 3. Moreover, restricted to each small rectangle, f preserves vertical lines and expands faster along the vertical direction than the horizontal direction.


401

log p . Note that ∀x ∈ 3, log a φT (x) = − log a and φF (x) = − log b. Since both functions are constants, we have (see [W] Sect. 9.2) Clearly, the transverse box dimension of 3 is δT =

P (f |3 , tφF + δT φT ) = htop (f |3 ) − t · log b −

log p · log a = log n − t log b − log p. log a

Put P (f |3 , tφF + δT φT ) = 0 and then solve for t. We get t = result, dimB (3) =

log n − log p log p + . log b log a

log n − log p . So by our log b

log 2 . log 2 + = log 5 log 3.8 2 4 T S f −k Ri .) 0.431 + 0.519 = 0.950. (In the figure the black rectangles form the set In Fig. 1, a = 3.8, b = 5, n = 4 and p = 2. We have dimB (3) =

k=0

Figure 1

i=1

Figure 2

Example 2.2. We perturb the position of the small rectangles in Example 1. The perturbation is not necessarily small, as long as the rectangles stay in the unit square and disjoint. Let π be a projection from the unit square into a horizontal line, and let πi = π|Ri . Then γ = π3 is the invariant set for the contracting maps πi f −1 π −1 , i = 1, · · · , n. By Falconer’s result ([F], Theorem 5.4 and 5.3), for almost every such perturbation, the log n }. It is equal to 1 if transverse box dimension δT = dimB (γ) is equal to min{1, log a n ≥ a. Let 3 be such a set. Then we have 0 = P (f |3 , tφF + δT φT ) = htop (f |3 ) − t · log b − 1 · log a = log n − t log b − log a. Therefore t =

log n − log a log n − log a and the box dimension of 3 is + 1. log b log b

In Fig. 2 we take a = 3.8, b = 5 and n = 4. The box dimension of a set obtained in log 4 − log 3.8 . + 1 = 0.032 + 1 = 1.032. the above way is log 5

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Remark. By the theorem stated in last section, we can see that the box dimensions of the invariant sets in the above examples do not change if we move the small rectangles along the vertical direction, as long as these rectangles are separated from each other.

3. Transverse Box Dimension Lemma 3.1 (Distortion estimates). There exists a constant J > 1 such that for any admissible word i0 i1 · · · ik , J −1 ≤

|Df k (z)|F ≤J |Df k (y)|F

and

J −1 ≤

|Df k (z)|T ≤ J, |Df k (y)|T

provided both y, z ∈ i0 i1 ···ik . Proof. Note that both |Df (x)|F and |det Df (x)| are Lipschitz, and so is |Df (x)|T . Also, f is uniformly expanding. With these facts, the arguments are standard. (For the arguments, see e.g. [M] pp. 173–174, [G] p. 71, or [MS] p. 353.) Lemma 3.2. There exists a constant K > 1 such that for any bounded subset A ⊂ R2 and for any constants β > 0, a > 1, a2 KN (A, aβ) ≥ N (A, β) ≥ N (A, aβ). Proof. This is clear.

Lemma 3.3. There exists a constant L > 1 such that for any i = 1, · · · , S, LN (γi , β) ≥ N (γ, β). Proof. Since f is topologically transitive on 3, we can find k > 0 such that ∀i = 1, · · · , S, f k i ⊃ . Put i0 = i. For each 1 ≤ j ≤ S, we fix one admissible word i0 i1 · · · ik with ik = j. Let γi0 i1 ···ik denote the preimage of γj under f k |i0 i1 ···ik . Note S [ that the projection πi : γi0 i1 ···ik → γi is an at most S to 1 map. We have ik =1

SN (γi , β) ≥

S X

N (γi0 i1 ···ik , β).

ik =1

By Lemma 3.2, N (γi0 i1 ···ik , β) ≥ N (γj , β|Df k |) ≥ (K|Df k | )−1 N (γj , β), 2

k

k

where |Df | = max|Df (x)|. Since N (γ, β) = x∈

S X

N (γj , β), the result follows.

j=1

Lemma 3.4. There exists b > 0 such that for any β > 0 small, for any m, n > 0, N (γ, β m+n ) ≥ bN (γ, β m )N (γ, β n ).


403

Proof. Since N (γ, β n ) denote the minimal number of β n -balls covering γ, we can find yj ∈ γ, j ≥ 1, · · · , 21 N (γ, β n ), such that each B(yj , 21 β n ) is contained in some i and B(yj 0 , 21 β n ) ∩ B(yj 00 , 21 β n ) = ∅ if j 0 6= j 00 . ˜ j , 1 β n ) = π −1 B(yj , 1 β n ), Suppose B(yj , 21 β n ) ∈ i0 for some i0 . Denote B(y i0 2 2 where πi : i → γi is the projection defined in Sect. 1. We can find an admissible word i0 i1 · · · ik such that ˜ j , 1 βn) i0 ···ik ⊂ B(y 2

but

˜ j , 1 β n ). i0 ···ik−1 6⊂ B(y 2

Recall that f k : i0 ···ik → ik is a diffeomorphism. By the distortion estimates, it is easy to check that for any α < 21 β n , z ∈ γi0 ···ik , B f k z, αJ −1 |Df k (x)| ¯ T ∩γik ⊂ f k B(z, α)∩γi0 ···ik ⊂ B f k z, αJ|Df k (x)| ¯ T ∩ γ ik , where x¯ = x¯ i0 ···ik ∈ i0 ···ik . That is, under the map f k , the image of an α-ball in γi0 ···ik is a subset of γik with diameter between 2αJ −1 |Df k (x)| ¯ T and 2αJ|Df k (x)| ¯ T. Therefore, N (γi0 ···ik , α) ≈ N (γik , α|Df k (x)| ¯ T ). n ¯ −1 By the distortion estimates, we have |Df k (x)| T ≈ diam(πi0 i0 ···ik ) ≈ β . Hence,

N (γi0 ···ik , α) ≈ N (γik , αβ −n ). Note that πi0 : γi0 ···ik → B(yj , 21 β n ) ∩ γi0 is an injective Lipschitz map. We have 1 N B(yj , β n ) ∩ γ, α ≥ N πi0 γi0 ···ik , α ≈ N (γi0 ···ik , α). 2 So there exist a constant c > 0 independent of the choice β, n, and yj such that 1 N B(yj , β n ) ∩ γ, α ≥ cN γik , αβ −n . 2 By Lemma 3.3 and this inequality, n 1 2 N (γ,β )

N (γ, α) ≥

X j=1

1 1 N B(yj , β n ) ∩ γ, α ≥ N (γ, β n ) · cL−1 N γ, αβ −n 2 2

1 ≥ cL−1 N (3, β n )N γ, αβ −n . 2 In particular, if we take α = β m+n , then the result follows by putting b =

1 −1 cL . 2

Proposition 3.5. dimB (γ) = dimB (γ). Proof. By Lemma 3.4, the sequence {log bN (γ, β n )} is superadditive. So the limit log bN (γ, β n ) n→∞ − log β n lim

exists. This implies the result of the proposition.

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H. Hu

4. Proof of the Theorem Lemma 4.1. Under Assumption A, there exist C ≥ 1 and λ < 1 such that ∀n ≥ 0, | det Df n (x)| |Df n (x)|T = ≤ Cλn |Df n (x)|F |Df n (x)|2F

∀x ∈

n \

f −i .

i=0

Proof. For each x ∈ , take a coordinate system such that the cone C(x) is the union of the first and the third quadrants. Since C(x) is continuous, we may require that the change of the coordinate is continuous as x varies. a(x) b(x) Let denote the matrix of the map Df (x) under the coordinate systems c(x) d(x) and let 1(x) = |a(x)d(x) − b(x)c(x)|. The fact that Dfx C(x) ⊂ int C(f x) implies that for each x, a(x), b(x), c(x) and d(x) have the same sign. By continuity, we have b0 = min{|b(x)| : x ∈ ∩ f −1 } > 0 and c0 = min{|c(x)| : x ∈ ∩ f −1 } > 0. Also write 10 = max{1(x) : x ∈ ∩ f −1 }. 0 0 vf i x vx = Dfxi vx . We have ∈ E(x) and denote vf i x = Take a vector vx = vx00 vf00i x 2 2 vf0 x vf00x = a(x)d(x) + b(x)c(x) vx0 vx00 + a(x)c(x)vx0 + b(x)d(x)vx00 > 1(x) 1 +

2b0 c0 0 00 v x vx . 10

Therefore, vf0 n x vf00n x > vx0 vx00

n−1 Y

1(f i x) 1 +

i=0

2b0 c0 . 10

Since vx is away from the boundary of C(x), |vx |2 ≈ vx0 vx00 ∀x ∈ . Also | det Df n (x)| ≈ n−1 Y 1(f i x). So the above inequality gives i=0

|Df n (x)vx |2 > C −1 | det Df n (x)| 1 +

2b0 c0 n |vx |2 10

(1)

for some C > 1 independent of x and n. By the definition of | · |F and | · |T , we get | det Df n (x)| 2b0 c0 −n |Df n (x)|T = 0, we can find a constant C1 > 1 such that ∀β > 0, C1−1

|Df k (x

i0 ···ik )|F

δT −α

|Df k (xi0 ···ik )|T

≤ N (γ, β|Df k (xi0 ···ik )|T ) ≤ C1

For a function φ, denote Sk φ(x) = We have |Df k (x)| δT F

|Df k (x)|T

k−1 X

|Df k (x

i0 ···ik )|F

|Df k (xi0 ···ik )|T

δT +α

.

(5)

φ(f i x). Recall the definition of φT and φF .

i=o

n o = exp δT Sk φT (x) − δT Sk φF (x) n o n o = exp δT Sk φT (x) + δF Sk φF (x) exp −δT − δF Sk φF (x) .

Let µ be the Gibbs state of the function δT φT +δF φF . The fact that P (f, δT φT +δF φF ) = 0 implies that n o exp δT Sk φT (xi0 ···ik ) + δF Sk φF (xi0 ···ik ) ≈ µ3i0 ···ik for any xi0 ···ik ∈ 3i0 ···ik (see e.g. [B2]). Also, by (2) o n exp −δT − δF Sk φF (xi0 ···ik ) = |Df k (xi0 ···ik )|δFT +δF ≈ β −(δT +δF ) ,

406

H. Hu

and by (4), |Df k (x

i0 ···ik )|F

|Df k (x

α

i0 ···ik )|T

≈ β|Df k (xi0 ···ik )|T

−α

< β −α .

So the second inequality of (5) can be written as N (γ, β|Df k (xi0 ···ik )|T ) ≤ C2 µ3i0 ···ik β −(δT +δF +α) for some C2X > 1. Note µ3i0 ···ik = 1. Therefore by (3) we obtain that i0 ···ik ∈C

N (3, β) ≤ C

X

N (γ, β|Df k (xi0 ···ik )|T ) ≤ CC2 β −(δT +δF +α) .

i0 ···ik ∈C

Consequently, dimB (3) = lim sup β→0

log N (3, β) ≤ δT + δF + α. − log β

Similarly, by using the first inequality of (5) we can get dimB (3) = lim inf β→0

log N (3, β) ≥ δT + δF − α. − log β

Since these two inequalities are true for any α > 0, we know that dimB (3) = δT + δF . Acknowledgement. It is my pleasure to thank Professor Lai-Sang Young for her valuable help in forming the assumptions. I thank Professor M. Boyle and Professor J.Yorke for stimulating conversations. I would also thank the referee for carefully reading the manuscript and for suggesting various improvements.

References [AJ] [B1]

Alexander, J.C., Yorke, J.A.: Fat baker’s transformation. Ergodic Theory Dyn. Syst. 4, 1–23 (1984) ´ Bowen, R.: Hausdorff dimension of quisi-circles. Inst. Hautes. Etudes Sci. Publ. Math. 50, 11–25 (1979) [B2] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes Math. 470, New York: Springer-Verlag, 1975 [Be] Bedford, T.: The box dimension of self-affine graphs and repellers. Nonlinearity 2, 53–71 (1989) [BU] Bedford, T., Urbański, M.: The box and Hausdorff dimension of self-affine sets. Ergodic Theory Dyn. Syst. 10, 627–644 (1990) [F] Falconer, K.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103, 339–350 (1988) [G] Godbillon, C.: Dynamical Systems on Surfaces. Berlin–Heidelberg–New York: Springer-Verlag, 1983 [GL] Gatzouras, D., Lalley, S.: Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41, 533–568 (1992) [H] Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981) [HP] Hirsch, M., Puch, C.: Stable manifolds and hyperbolic sets. In: Proc. Symp. in Pure Math. Vol 14, Providence RI: AMS, 1970, pp. 133–164 [HPS] Hirsch, M., Puch, C., Shub, M.: Invariant manifolds. Lect. Notes Math. 470, Berlin–Heidelberg–New York: Springer-Verlag, 1977


[KP] [LY] [M]

407

Kenyon, R., Peres, Y.: Hausdorff dimensions of affine-invariant sets. Preprint. Ledrappier, F., Young, L-S.: The metric entropy of diffeomorphisms. Ann. Math. 122, 509–574 (1985) Mañe´ , R.: Ergodic Theory and Differentiable Dynamics. Berlin–Heidelberg–New York: SpringerVerlag, 1987 [Mc] McMullen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96, 1–9 (1984) [MS] De Melo, W., Van Strien, S.: One-Dimensional Dynamics. Berlin–Heidelberg–New York: SpringerVerlag, 1993 [PU] Przytycki, F., Urbanski, M.: On the Hausdorff dimension of some fractal sets. Studia Math. 93, 155–186 (1989) [PW] Pesin, Y., Weiss. H.: On the dimension of a general class of geometrically defined deterministic and random cantor-like sets. Preprint [R] Ruelle, D.: Bowen’s formula for the Hausdorff dimension of self-similar sets. In: Scaling and Selfsimilarity in Physics, Progress in Physics. Vol. 7, Boston: Birkhäuser, 1983, pp. 351–358 [W] Walters, P.: An Introduction to Ergodic Theory. New York–Heidelberg–Berlin: Springer-Verlag, 1981 Communicated by Ya. G. Sinai

Commun. Math. Phys. 191, 409 – 466 (1998)

Communications in


Dyonic Sectors and Intertwiner Connections in 2+1-Dimensional Lattice ZN -Higgs Models João C. A. Barata1 , Florian Nill2 1 Instituto de F´ısica da Universidade de S˜ ao Paulo, P. O. Box 66318, São Paulo 05315 970, SP, Brasil. E-mail: [email protected] 2 Institut f¨ ur Theoretische Physik der Freien Universität Berlin, Arnimallee 14, Berlin 14195, Germany. E-mail: [email protected]

Received: 13 December 1996 / Accepted: 19 May 1997

Abstract: We construct dyonic states ωρ in 2+1-dimensional lattice ZN -Higgs models, i.e. states which are both, electrically and magnetically charged. These states are parametrized by ρ = (ε, µ), where ε and µ are ZN -valued electric and magnetic charge distributions, respectively, living on the spatial lattice Z2 . The associated Hilbert spaces Hρ carry charged representations πρ of the observable algebra A, the global transfer matrix t and a unitary implementation of the group Z2 of spatial lattice ! translations. We X X ε(x), µ(p) ∈ ZN × ZN these prove that for coinciding total charges qρ = x

p

representations are dynamically equivalent and we construct a local intertwiner connection U (0) : Hρ → Hρ0 , where 0 : ρ → ρ0 is a path in the space of charge distributions Dq = {ρ : qρ = q}. The holonomy of this connection is given by ZN -valued phases. This will be the starting point for a construction of scattering states with anyon statistics in a subsequent paper. Contents 1 2 2.1 2.2 2.3 2.4 3 3.1 3.2 3.3 3.4 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 The Basic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 The local algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Local transfer matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Ground States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 External charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 The Construction of Dyonic Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Dyonic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Global transfer matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Global charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 The dyonic self energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 State Bundles and Intertwiner Connections . . . . . . . . . . . . . . . . . . . . . . . 437

410

J. C. A. Barata, F. Nill

4.1 4.2 4.3 4.4 A A.1 A.2 B B.1 B.2 B.3 B.4 B.5 B.6 B.7

The local intertwiner algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 The intertwiner connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 The representation of translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 A Brief Sketch of the Polymer and Cluster Expansions . . . . . . . . . . . . . 447 Expansions for the vacuum sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Expansions for the dyonic sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 The Remaining Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Proof of Proposition 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Proof of Propositions 3.1.2 and 3.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Proof of Theorem 3.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Proof of Proposition 3.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Completing the proof of Proposition 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . 460 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Proof of Proposition 4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

1. Introduction In this paper we continue our project initiated in [1] of a constructive analysis of states with anyonic statistics in a 2+1 dimensional lattice gauge theory. We investigate a general class of models with the discrete Abelian gauge group ZN for arbitrary N ∈ N, N ≥ 2, and with discrete Higgs fields. The vacuum expectations in this theory are represented by classical expectations of an (euclidean) statistical mechanics model given by the thermodynamic limit of finite volume expectations of the form R dαdϕ Bcl (α, ϕ) e−S3 R , (1.1) hBcl i3 := dαdϕ e−S3 with a generalized Wilson action: X X Sg (dα(p)) + Sh (dϕ(b) − α(b)) . S3 (ϕ, α) := p

(1.2)

b

Here ϕ and α are ZN -valued Higgs and gauge fields, respectively, on a euclidean spacetime lattice Z3 . Hence ϕ lives on sites, α lives on bonds, d denotes the lattice exterior derivative and the above sums go over all elementary positively oriented bonds b and plaquettes p in a finite space-time volume 3 of our lattice. Bcl is some classical observable, i.e. a gauge invariant function of the gauge field α and of the Higgs field ϕ with finite support. Here, gauge invariance means invariance under the simultaneous transformations ϕ → ϕ + λ and α → α + dλ for arbitrary λ : Z3 7→ ZN , with finite support. Above, the integrations over α and ϕ are actually finite sums, since these variables are discrete (to be precise, the discrete Haar measure on ZN is employed). In order to have charge conjugation symmetry and reflection positivity (i.e. a positive transfer matrix) the actions Sg and Sh will be chosen as even functions on ZN ≡ {0, . . . , N − 1} taking their minimal value at 0 ∈ ZN . Thus we have a general Fourier expansion N −1 1 X 2πmn , (1.3) βg/h (m) cos Sg/h (n) = − √ N N m=0

Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models

411

n ∈ ZN , where we call βg (m) and βh (m), m ∈ ZN , the gauge and Higgs coupling constants, respectively1 . We also require βg/h (m) = βg/h (N − m), i.e. βg/h and −Sg/h are in fact Fourier transforms of each other. This model describes a ZN -Higgs model where the radial degree of freedom of the Higgs field is frozen, i.e. |φ| = 1 and φ(x) = 2πi e N ϕ(x) . We will be interested in the so-called “free charge phase” of this model, which, roughly speaking, is obtained whenever, for all 0 6= n ∈ ZN , Sg (n) − Sg (0) ≥ cg 0, Sh (n) − Sh (0) ≤

c−1 h

1,

(1.4) (1.5)

for large enough positive constants cg and ch . In this region convergent polymer and cluster expansions are available and have been analyzed in detail in [1], see also Appendix A for a short review. The first analysis of the structure of the charged states in this phase had been performed for the case of the group Z2 by Fredenhagen and Marcu in [2]. In that work it had been shown that electrically charged states exist in d + 1 dimensions, d ≥ 2 in the “free charge phase” of the model. The ideas employed by the authors involved a wide combination of methods from Algebraic Quantum Field Theory and Classical Statistical Mechanics. Later on the existence of electrically charged particles in the same model had been shown in [3] and the existence of multi-particle scattering states of these particles had been proven in [4], combining methods and results of [2] and of [5]. In [1] we extended some of these results to the ZN -Higgs model mentioned above and showed, after previous results of Gaebler on the Z2 case [6], the existence of magnetically charged states in 2 + 1 dimensions. In [1] we also proved the existence of electrically and of magnetically charged particles in this model. Our intention here is to show the existence of dyonic states ωρ in the “free charge phase” of our ZN -Higgs model, i.e. states carrying simultaneously electric and magnetic charges, ρ = (ε, µ). This had been performed in the Z2 case in [6]. We construct the associated charged representations of the observable algebra A as the GNS-triples (πρ , Hρ , ρ ) obtained from ωρ . These representations ! fall into equivalence classes laX X beled by the total charges qρ = ε(x), µ(p) ∈ ZN × ZN . We show that for x

p

each choice of q ∈ ZN × ZN the state bundle [ (ρ, Hρ ) Bq := ρ∈Dq

over the discrete base space Dq := {ρ : qρ = q} is equipped with a non-flat connection, i.e. a collection of unitary parallel transporters Uρ0 , ρ : Hρ → Hρ0 depending on paths 0 from ρ to ρ0 in Dq such that πρ0 = Ad Uρ0 ,ρ ◦ πρ . The holonomy of this connection is given by ZN -valued phases which appear as winding numbers between “electric” Wilson loops and “magnetic” vortex loops in the euclidean functional integral picture. In an upcoming paper [7] this construction will be the starting point for an analysis of the anyonic statistics of scattering states in these models. At this point we should also mention the previous work of J. Fröhlich and P. A. Marchetti on the construction of dyonic and anyonic states in the framework of euclidean 1 The values of β (0) and β (0) only determine additive constants to the action and will be fixed by g h convenient normalization conditions.

412


lattice field theories ([8, 9] and [10]). In their approach the analogue of our Hilbert spaces Hρ are obtained by Osterwalder-Schrader reconstruction methods which makes it difficult to discuss the representation theoretic background of the observable algebra. In difference with their approach we work with the Hamiltonian description. In particular, we construct our states as functionals on the quasi-local observable algebra generated by the time-zero fields. As usual, in this approach the euclidean description reappears in the form of local transfer matrices TV , whose ground state expectation values are given by functional integrals of the type (1.1). Still, we interpret and motivate our constructions mostly algebraically and consider the functional integral techniques only as a technical tool. We now describe the plan of this paper in some more detail. In Sect. 2 we introduce our basic setting. We follow the standard canonical quantization prescription of gauge theories in A0 = 0 gauge to define the local algebras generated by the time-zero fields. We then define an “euclidean dynamics” in terms of local transfer matrices whose ground states give rise to finite volume euclidean functional integral expectations with actions (1.2)-(1.3). We also review the notion of external charges (related to a violation of Gauss’ law) in order to distinguish them from the dynamical charges (“superselection sectors”) that we are interested in this work. Section 3 is devoted to the construction of dyonic sectors. We start with generalizing the Fredenhagen-Marcu prescription to obtain dyonic states ωρ on our observable algebra A. We then construct the associated GNS-representations (πρ , Hρ , ρ ) of A and implement the euclidean dynamics by a global transfer matrix on Hρ . We show ˆ ⊃ A (i.e. the ∗-algebra generated that as representations of the “dynamic closure” A by A and the global transfer matrix) these representations are irreducible and pairwise equivalent provided their total charges qρ ∈ ZN ×ZN coincide. (We also conjecture that they are dynamically inequivalent, if their total charges disagree.) Finally we show that the infimum of the energy spectrum in the dyonic sectors is uniquely fixed by requiring charge conjugation symmetry and cluster properties of correlation functions for infinite space-like separation. M Hρ by defining electric In Sect. 4 we construct an intertwiner algebra on Hq = ρ∈Dq

and magnetic “charge transporters” Eq (b), Mq (b) ∈ B(Hq ) living on bonds b in Z2 and fulfilling local Weyl commutation relations. In terms of these intertwiners we obtain a unitary connection U (0) : Hρ → Hρ0 intertwining πρ and πρ0 for any path 0 : ρ → ρ0 in Dq . The holonomy of this connection is given by ZN -valued phases. We conclude by applying our connection to construct a unitary implementation of the translation group in the dyonic sectors. We remark at this point that we do not touch the question of the existence of dyonic particles in these models, i.e. of particles in the dyonic sectors carrying simultaneously electric and magnetic charges. To study the existence of such particles requires adaptation of the known Bethe-Salpeter kernels methods for situations involving charged particles in lattice models. This will be performed elsewhere. Clearly, if these particles exist they should be expected to show anyonic statistics among themselves. A good part of the methods and results used here has been extracted from [1] and we will often refer to this paper when necessary. In particular, we will not repeat the proof of the convergence of the polymer expansion we are going to use since this point has been discussed in detail in [1], see however Appendix A for a short review. In fact, although polymer expansions are the main technical tool of this work (as well as of all other works on ZN -Higgs models cited above), our aim here is to formulate theorems


413

and present results in a way which can be followed without any detailed knowledge on cluster expansions. Following this strategy, we abandon all statistical mechanics aspects of our proofs to Appendix B and reserve the main body of this work to exploit algebraic and quantum field theoretical concepts. Remarks on the notation.. Due to a different focus our notation will differ in many points from that of [1]. We will change our notation according to our needs of emphasis and clarity. The symbol indicates “end of proof”. Products of operators run from the left n Y Aa means A1 · · · An . For an invertible operator B, Ad B denotes the to the right, i.e. a=1

automorphism B · B −1 . If A ⊂ B(H) is an algebra acting on a Hilbert space H then we denote by A0 the commutant of A, i.e. the set of all operators of B(H) which commute with all elements of A. Here B(H) is the algebra of all bounded operators acting on H. 2. The Basic Setting We will always consider the lattice Zd , d = 2, 3 as a chain complex and denote by (Zd )p the elementary positively oriented p-cells in Zd . We also use the standard terminology “sites”, “bonds” and “plaquettes” for 0-, 1- and 2-cells, respectively. By a (finite) volume V ⊂ Zd we mean the closed chain sub-complex generated by a (finite) union of elementary d-cells in (Zd )d . We denote by Vp the set of elementary oriented p-cells in V where, by definition, a p-cell is contained in Vp ⊂ (Zd )p if and only if it lies in the boundary of some (p+1)-cell contained in Vp+1 . We denote by Cp (V ) ≡ ZVp the set of p-chains in V and by C p (V ) ≡ C p (V, ZN ) the set of ZN -valued cochains with support in V (i.e. group homomorphisms α : Cp (V ) → ZN ). As usual we identify C p (V ) with the group of ZN -valued functions on (Zd )p with support in Vp . Hence, for V ⊂ W we have the natural inclusion C p (V ) ⊂ C p (W ). We also denote C p := C p (Zd ) and define p ⊂ C p as the set of p-cochains with finite support. Often we will identify an elemenCloc tary p-cell c ∈ (Zd )p with its characteristic p-cochain (i.e. taking the value 1 ∈ ZN on c and 0 ∈ ZN else). |V | Considered as a finite Abelian group C p (V ) ∼ = ZN p is self-dual for all finite V , the pairing C p × C p → U (1) being given by the homomorphism   X 2πi α(c)β(c) . (2.1) (α, β) 7→ eihα, βi := exp  N c∈Vp

We denote by d : C p → C p+1 and d∗ : C p → C p−1 the exterior derivative and its adjoint, such that ∗ eihα, dβi = eihd α, βi p p−1 for all α ∈ Cloc and all β ∈ Cloc . In the main body of this paper we will be working with “time-zero” fields, i.e. cochains defined on the spatial lattice Z2 . The translation to the euclidean functional integral formalism will bring us to a space-time lattice Z3 . To unload the notation we will use the same symbols for both pictures as long as the meaning becomes obvious from the context. Hence, in both pictures ϕ ∈ C 0 will denote the Higgs field, α ∈ C 1 will denote the gauge field and a gauge transformation consists of a mapping (ϕ, α) 7→ (ϕ+λ, α+dλ) 0 . with λ ∈ Cloc

414


2.1. The local algebras. As is well known, a d + 1-dimensional lattice system described by (1.1,1.2) can typically also be described as a quantum spin system, using the transfer matrix formalism. By this we mean an operator algebra living on a d-dimensional spatial lattice, together with discrete “euclidean time” translations given by e−tH , t ∈ Z, where T ≡ e−H is the transfer matrix. In this formulation expectations like (1.1) represent the vacuum or ground state of the “euclidean” dynamics defined by the transfer matrix. We have described in detail the quantum spin system of our model in [1] (see also [2]). It corresponds to the Weyl form of the usual canonical quantization prescription in temporal (α0 = 0) gauge. Let us recall here its main ingredients. On the spatial lattice Z2 we introduce the local algebra of time-zero Higgs and gauge fields in the following way. To each x ∈ (Z2 )0 we associate the unitary ZN -fields PH (x) and QH (x) and to each b ∈ (Z2 )1 we associate the unitary ZN -fields PG (b) and QG (b) (the subscripts G and H stand for “gauge” and “Higgs”, respectively) satisfying the relations: (2.2) PH (x)N = QH (x)N = PG (b)N = QG (b)N = 1l, and the ZN -Weyl algebra relations PG (α)QG (β) = e−ihα, βi QG (β)PG (α),

(2.3)

PH (γ)QH (δ) = e−ihγ, δi QH (δ)PH (γ),

(2.4)

1 ∈ Cloc γ(x)

where Y α, β PH (x)

0 Cloc

and γ, δ ∈ play the rôle of test functions, i.e. PH (γ) := , etc. Operators localized at different sites and bonds commute and the

x∈(Z2 )0

G-operators commute with the H-operators. We denote [δQH ](α) := QH (d∗ α), [δ ∗ PG ](β) := PG (dβ), etc.,2 where d is the exterior derivative on cochains and d∗ is its adjoint. We will realize these operators by attaching to each lattice point x a Hilbert space Hx and to each lattice bond b a Hilbert space Hb , where Hx ∼ = Hb ∼ = CN . The operators QH (x), PH (x), QG (b) and PG (b) are given on Hx , and Hb , respectively, as matrices with matrix elements: PH (x)a, b = PG (b)a, b = δa, b+1(mod N ) and QH (x)a, b = QG (b)a, b = δa, b e

2πi N a

(2.5)

,

(2.6)

for a, b ∈ {0, . . . , N − 1}. The operators QH and QG have to be interpreted as the ZN versions of the Higgs field 2πi 2πi and gauge field, respectively: QH (x) = e N ϕ(x) , QG (x) = e N α(b) , with ϕ and α taking values in ZN . The operators PH and PG are their respective canonically conjugated “exponentiated momenta”, i.e. shift operators by one ZN -unit. Hence these operators indeed provide the Weyl form of the canonical quantization in α0 = 0 gauge, see also Eq. (2.11) below. We denote by Floc the ∗-algebra generated by these operators. Denoting by F(V ) the by QH (x), PH (x), QG (b) and PG (b) for x ∈ O V0 , b ∈ V C∗ -sub-algebra generated [ 1, V ⊂ O F(V ). The algebra F(V ) acts on HV := Hx Hb . Z2 finite, one has Floc = |V | 0, large enough. For this a polymer and cluster expansion has been used. The above region of analyticity (for real couplings) is contained in the “free charge region” of the phase diagram. This phase is characterized by the absence of screening and confinement. All results of our present work, specifically those concerned with the existence and the properties of the charged states are valid for gc and hc sufficiently small. 2.3. Ground States. In this subsection we recall the important definition of a ground state and discuss some of its basic features. We start with introducing two useful concepts. First, the adjoint γ ∗ of an automorphism γ of a unital ∗-algebra C is defined through ∗ γ (A) := (γ(A∗ ))∗ , A ∈ C. We have γ ∗∗ = γ and γ is a ∗-automorphism iff γ = γ ∗ . For the composition of automorphisms one has (α ◦ β)∗ = α∗ ◦ β ∗ and consequently α∗ −1 = α−1 ∗ . For an invertible element A ∈ C one also has (Ad A)∗ = Ad A∗ −1 . Finally, if ω is a γ-invariant state on C then trivially it is also γ ∗ -invariant. Second, we say that a state ω on a ∗-algebra C has the cluster property for the automorphism γ if, for all A, B ∈ C, one has lim ω(Aγ n (B)) = ω(A)ω(B).

n→∞

We now come to a central definition which has first been introduced in [2].

(2.25)


419

Definition 2.3.1. Let γ be a (not necessarily ∗-preserving) automorphism of a unital ∗-algebra C. A state ω on C is called a “ground state” with respect to γ and C if it is γ-invariant and if 0 ≤ ω(A∗ γ(A)) ≤ ω(A∗ A),

∀A ∈ C.

Actually γ-invariance follows from (2.26) (see, e.g. [1]).

(2.26)

We will motivate this abstract definition below when we discuss the ground state of the finite volume transfer matrix. The following lemma will be very useful for proving that certain states are ground states with respect to our euclidean dynamics α (or suitable modifications of α to be introduced in Sect. 3.1). This lemma was already implicitly used in [2]. Lemma 2.3.2. Let γ be an automorphism on a ∗-algebra C satisfying γ ∗ = γ −1 and let ω be a γ-invariant state on C which has the cluster property for γ (actually one just needs that, for each A ∈ C, the sequence ω(A∗ γ a (A)), a ∈ N, is bounded). Then, for all A ∈ C, (2.27) |ω(A∗ γ(A))| ≤ ω(A∗ A). The proof is easy. See [1]. We will now exhibit translation invariant ground states of the automorphism α. First, let us explain in this context the heuristic motivation of Definition 2.3.1. If V ∈ HV is the Frobenius eigenvector of the finite volume transfer matrix TV with eigenvalue kTV kHV then, in face of the positivity of TV , the inequalities 0 ≤ V , A∗ TV ATV−1 V ≤ V , A∗ AV (2.28) obviously hold for any A ∈ F(V ). This motivates to consider the relation (2.26) with γ = α as a characterization of a state replacing the vector states V for infinite volumes. One should notice here that the usual characterization of a ground state of a quantum spin system as a state ω for which lim ω(A∗ [HV , A]) ≥ 0 for all local A is inadequate for transfer matrix systems, due V ↑Z2

to the highly non-local character of the Hamilton operator HV ≡ − ln TV (at least in more than two space-time dimensions). Following [2], a translation invariant ground state for α with respect to the algebra F can be obtained by TrHV TVn BTVn EV0 , (2.29) ω0 (B) = lim lim V ↑Z2 n→∞ TrHV TV2n EV0 for B ∈ Floc , where EV0 is an in principle arbitrary operator with positive matrix elements. Before the limits V ↑ Z2 and n → ∞ are taken the expression in the righthand side of (2.29) is identical to the classical expectation (1.1) in a volume V × {−n, . . . , n} ⊂ Z3 of a suitable classical observable Bcl associated with B. A convenient choice for EV0 giving free boundary conditions in time-direction is (see [2]) EV0 := eAV /2 FV0 eAV /2 where FV0 :=

X (ϕ, α), (ϕ0 , α0 )

|ϕ, αihϕ0 , α0 |

(2.30) (2.31)

420


Here the sum in (2.31) goes over all classical time-zero configurations with support in V , i.e. 0-cochains ϕ, ϕ0 ∈ C 0 (V ) and 1-cochains α, α0 ∈ C 1 (V ) . The existence of the thermodynamic limit in (2.29) can be established using Griffiths’ inequalities or the cluster expansions. The cluster expansions also provide a way to prove the translation and rotation invariance of the limit state ω0 . Another important fact derived from the cluster expansion is that the restriction of ω0 to A has the cluster property for the automorphism α. One of the most useful aspects of Definition 2.3.1 of a ground state is the possibility to define infinite volume transfer matrices. Indeed, if ω0 is a ground state of α with respect to F and (π0 (F), 0 , H0 ) is the GNS-triple associated with ω0 , we define, following [2], the infinite volume transfer matrix T0 as the element of B(H0 ) given on the dense set π0 (F)0 by T0 π0 (A)0 := π0 (α(A))0 ,

(2.32)

A ∈ F. One checks that this is a well-defined positive operator with 0 ≤ T0 ≤ 1. If moreover ω0 satisfies the cluster property (2.25), then 0 ∈ H0 is the unique (up to a phase) eigenvector of T0 with eigenvalue 1. Hence we call π0 the vacuum representation of (F, α). Since ω0 is translation invariant we can also define a unitary representation of the translation group in the vacuum sector through U0 (x)π0 (A)0 := π0 (τx (A))0 ,

(2.33)

A ∈ F, x ∈ Z2 . The momenta in the vacuum sector are therefore defined by U0 (x) = exp(iP · x), with sp P ∈ [−π, π]2 and the vacuum 0 is also the unique (up to a phase) translation invariant vector in H0 . Next we remark that ω0 is also charge conjugation invariant. This can be seen from (2.29) by using iC (TV ) = TV , iC (EV0 ) = EV0 and the fact that iC F(V ) is unitarily implemented on HV . Hence, charge conjugation is implemented as a symmetry of the vacuum sector by the unitary operator C0 ∈ B(H0 ) given on π0 (F)0 by C0 π0 (A)0 := π0 (iC (A))0

(2.34)

When constructing charged states ωρ in Sect. 3 this will no longer hold, i.e. there we will have ωρ ◦ iC = ω−ρ . 2.4. External charges . Let π be a representation of the field algebra F on some separable Hilbert space Hπ and for q ∈ C 0 let Hπq ⊂ Hπ be the subspace of vectors ψ ∈ Hπ satisfying 2πi (2.35) π(G(x))ψ = e N q(x) ψ for all x ∈ Z2 . According to common terminology we call Hπq the subspace with external electric charge q, since the operators π(G(x)) implement the gauge transformations on Hπ . Clearly, since A commutes with G we have π(A) Hπq ⊂ Hπq .

(2.36)

Moreover, using QH (x)G(x) = e2πi/N G(x)QH (x) for all x ∈ Z2 we have π(QH (q 0 ))Hπq = Hπq−q 0 for all q 0 ∈ Cloc .

0

(2.37)

Dyonic Sectors and Intertwiner Connections in 2+1-D Lattice ZN -Higgs Models 0

421

0

The representations of A on Hπq and on Hπq−q , π q and π q−q , respectively, cannot 2πi be unitarily equivalent for q 0 6= 0 with finite support since, by (2.35), G(x) − e N q(x) 1l ∈ 0 Ker π q and, hence Ker π q 6= Ker π q−q for q 0 6= 0. However, since QH (q)AQH (q)∗ = A one has for any ψq ∈ Hπq and A ∈ A π q (A)ψq = π(A)ψq = π(QH (q 0 ))∗ π ◦ Ad QH (q 0 )(A) π(QH (q 0 ))ψq 0 = π(QH (q 0 ))∗ π q−q ◦ Ad QH (q 0 )(A) π(QH (q 0 )ψq , (2.38) 0

thus showing that π q and π q−q ◦ Ad QH (q 0 ) are unitarily equivalent.4 Now Hπq might be zero for general representations π and general external charge distributions q (e.g. for the vacuum representation π0 and q with infinite support). In this work we are only interested in representations π containing a non-trivial subspace Hπ0 6= 0 of zero external charge (and therefore, by (2.37), also Hπq 6= 0 for all external charges with finite support). Moreover, Hπ0 will always be cyclic under the action of π(Floc ) and therefore we will always have5 Hπ =

M

Hπq

(2.39)

0 q∈Cloc

In fact, such a decomposition is always obtained for GNS-representations (πω , ω , Hω ) 0 or, equivalently, associated with states ω on F, provided ω ∈ Hωq for some q ∈ Cloc ω(F G(x)) = e2πi q(x)/N ω(F )

(2.40)

for all F ∈ F and all x ∈ (Z2 )0 . In this case we call ω an eigenstate (with external 0 ) of the gauge group algebra G. Note that ω is an eigenstate of G charge q ∈ Cloc with external charge q if and only if ω ◦ Ad QH (q 0 ) is an eigenstate with external charge 0 q + q 0 . Correspondingly, πω (QH (q 0 )∗ ) ω ∈ Hωq+q will also be a cyclic vector for πω (F). Hence, without loss, we may restrict ourselves to eigenstates ω of G with zero external charge. In fact, we will only be studying the restrictions of representations π(A) to Hπ0 as the “physical” subspace of Hπ , i.e. the subspace on which “Gauss’ Law”, π(G(x)) = 1l, holds. We emphasize that this notion of external (or “background”) charge is not to be confused with the concept of (dynamically) charged states and the associated charged representations of A. By this we mean representations of A with zero external charge, which are inequivalent to the vacuum representation (at least when extended to a suitable ˆ ⊃ A), e.g. by the appearance of different mass spectra or the “dynamic closure” A absence of a time and space translation invariant “vacuum” vector. By analogy with the terminology of quantum field theory we call an equivalence class of such charged representations a superselection sector. They are the main interest of our work. Hence, from now on by charged states we will always mean dynamically charged states in this later sense. 4 Note, however, that this equivalence does not respect the dynamics, since Ad Q (q) does not commute H with α. 0 5 Use that F loc = ⊕q Aloc QH (q) is a grading labeled by q ∈ Cloc , i.e. the irreducible representations of G in Floc .

422


3. The Construction of Dyonic Sectors In [1] with the help of the cluster expansions (see Appendix A) we were able to show the existence of electrically and of magnetically charged sectors in the “free charges” phase of the ZN -Higgs model. We also proved the existence of massive charged 1-particle states in these sectors. This section is devoted to the construction of states and the associated sectors which are at the same time electrically and magnetically charged. Following the common use we call them dyonic states. Hereby we generalize and improve ideas from [6], where such states have been first constructed for the Z2 -Higgs model. In [7] we will continue this analysis by constructing scattering states in the dyonic sectors and identifying these to constitute an “anyonic Fock space” over the above mentioned 1-particle states. We start in Sect. 3.1 with reformulating the Fredenhagen-Marcu construction for charged states ωρ by defining them as the thermodynamic limit of ground states of modified local transfer matrices TV (ρ). For ρ = (ε, µ) these modified transfer matrices correspond to modified Hamiltonians, where the kinetic term in the Higgs fields is replaced by 1 1 (πH , πH ) → (πH + ε, πH + ε) 2 2 and the magnetic self energy is replaced by 1 1 (dα, dα) → (dα + µ, dα + µ). 2 2 In the functional integral these states are represented by euclidean expectations in the background of infinitely long vertical Wilson and vortex lines sitting above sites x ∈ (Z2 )0 and plaquettes p ∈ (Z2 )2 , respectively, of the time-zero plane Z2 . Their ZN values are given by values of the electric charges ε(x) and the magnetic charges µ(p), respectively. In Sect. 3.2 we construct the charged representations πρ of A associated with the states ωρ and the global transfer matrices implementing the euclidean dynamics in πρ . In Sect. 3.3 we prove that πρ and πρ0 are dynamically equivalent whenever their total charges coincide, qρ = qρ0 . We also conjecture that otherwise they are dynamically inequivalent and give some criteria for a proof. In Sect. 3.4 we show that the infimum of the energy spectrum in the dyonic sectors may be uniquely normalized by imposing charge conjugation symmetry and the requirement of decaying interaction energies between two charge distributions in the limit of infinite spatial separation. We recall once more that, without mentioning explicitly, all results of this section are valid in the free charge phase, i.e. for gc and hc (defined in (2.23), (2.24)) sufficiently small. 3.1. Dyonic states. We first recall the idea behind the Fredenhagen-Marcu (FM) string operator and its use in the construction of electrically charged states [2]. Starting from the usual method to create localized electric dipole states by applying a Mandelstam string operator to the vacuum x, y := φ(x)φ(y)∗ e

i

Ry x

Ai dz i

,

(3.1)


423

Fredenhagen and Marcu [2] proposed a modification so as to keep the energy of the dipole state x, y finite as y → ∞. Using our lattice notation the FM-proposal reads6 M ∗ n F x,y := lim cn QH (x)QH (y) T0 QG (sxy ). n→∞

(3.2)

Here sx,y is an arbitrary path connecting x to y in our spatial lattice, T0 = e−H is the global transfer matrix in the vacuum sector and cn > 0 is a normalization constant M FM to get kF x,y k = 1. Note that x,y still lies in the vacuum sector and has zero external charge. In a second step one may then send one of the charges to infinity7 to obtain dynamically charged states as expectation values on Aloc , M FM (3.3) A ∈ Aloc . ωx (A) := lim F x,y , Ax,y , y→∞

Note that as a limit of eigenstates of G with zero external charge, ωx is also an eigenstate of G with zero external charge. Using duality transformations, an analogous procedure for the construction of magnetically charged states has been given in [1]. In order to be able to discuss dyonic states within a common formalism, we now pick up an observation of [1] to reformulate the above construction as follows. First we recall from [2] that in our range of couplings we have M ∗ F x,y = QH (x)QH (y) ψx, y ,

(3.4)

δ −δ

where ψx,y ∈ H0x y is the unique (up to a phase) ground state vector of the restricted δ −δ δ −δ global transfer matrix T0 H0x y , where H0x y ⊂ H0 is the subspace of an external electric charge-anticharge pair sitting at x and y, respectively. In fact, by looking at (3.2) we have (3.5) ψx, y = s − lim cn T0n QG (sx, y ), n→∞

and using our cluster expansion it is easy to check that the limit exists independently of the chosen string sx,y connecting x to y. To avoid the use of external charges (for which we do not have a magnetic analogue) M we now equivalently reformulate this by saying that F x,y is the unique (up to a phase) ground state vector of the modified transfer matrix T0 (δx − δy ) defined by T0 (δx − δy ) := QH (x)QH (y)∗ T0 QH (y)QH (x)∗ H00 ,

(3.6)

where H00 ⊂ H0 is the subspace without external charges. Since this modified transfer matrix generates a modified dynamics given by α0 = Ad (Q(x)Q(y)∗ ) ◦ α ◦ Ad (Q(y)Q(x)∗ ), M FM 0 we conclude that the state A 7→ (F x,y , A x,y ) is a ground state of α when restricted to the observable algebra, A ∈ A. A similar statement holds for magnetic dipole states and magnetically modified transfer matrices [1], see also below. To treat electric and magnetic charges simultaneously we now generalize this con0 2 and DM ≡ Cloc denote the set of ZN -valued struction as follows. Let DE ≡ Cloc 0-cochains (≡ electric charge distributions) and 2-cochains (≡ magnetic charge distributions), respectively, with finite support in our spatial lattice Z2 , and let D = DE ×DM . 6 By a convenient abuse of notation we often drop the symbol π when referring to the vacuum 0 representation. 7 Actually, in [2] the two limits, (3.2) and (3.3), were performed simultaneously.

424


For ρ = (ε, µ) ∈ D, supp ρ ⊂ V0 × V2 , we generalize (3.6) and define the modified local transfer matrices in a finite volume V ⊂ Z2 as the element of A(V ) given by TV (ρ) := Z(µ)1/2 QH (ε)TV QH (ε)∗ Z(µ)1/2 , where we have used the notations

Y

QH (ε) :=

QH (x)ε(x)

(3.7)

(3.8)

x∈(Z2 )0

Y

and Z(µ) :=

Z (µ(p)) (p).

(3.9)

p∈(Z2 )2

Here the gauge invariant operator Z (n) (p), n ∈ {0, . . . , N − 1}, is defined by (see also [1])   −1  1 N  X 2πi jn − 1 (δQG (p))j . βg (j) exp (3.10) Z (n) (p) = exp √  N  N j=0

On our local Hilbert spaces HV it acts by Z (n) (p) |ϕ, αi =

e−Sg (dα(p)+n) |ϕ, αi . e−Sg (dα(p))

(3.11)

Hence, this operator can be interpreted as the operator creating a vortex with magnetic charge n at the plaquette p. The definition (3.7) is also motivated by the fact that under duality transformations we roughly have TV (ε, 0) ↔ TV ∗ (0, ε∗ ), see [1] for the precise statement. Also note that iC (TV (ρ)) = TV (−ρ) and that TV (ρ) is still gauge invariant, i.e. TV (ρ) ∈ Aloc . In a formal continuum notation, these modified transfer matrices correspond to modified Hamiltonians, where the kinetic term in the Higgs fields is replaced by 1 1 (πH , πH ) → (πH + ε, πH + ε) 2 2

(3.12)

and the magnetic self energy is replaced by 1 1 (dα, dα) → (dα + µ, dα + µ). 2 2

(3.13)

Together with these modified transfer matrices we also have a modified euclidean dynamics αρ given by the automorphism of F, αρ (A) := lim TV (ρ)ATV (ρ)−1 , V ↑Z2

A ∈ F,

(3.14)

such that α0 ≡ α. As in the case of α0 , for each A ∈ Floc the limit above is already reached at finite V . Moreover αρ (A) = A and iC ◦ αρ = α−ρ ◦ iC .

(3.15)

We emphasize that we introduce these modifications not as a substitute of our original “true” dynamics, but for technical reasons only. From (3.7) we conclude


αρ = Ad Kρ ◦ α0 ◦ Ad Kρ∗ = Ad Lρ ◦ α0 , where

∈ Floc

Kρ := Z(µ)1/2 QH () and

Lρ := Kρ α0 (Kρ∗ ) = TV (ρ)TV (0)−1

425

(3.16) (3.17)

∈ Aloc ,

(3.18)

the last equality holding for V large enough. Note also that for ρ = (µ, ) and ρ0 = (µ0 , 0 ) such that supp µ ∩ supp µ0 = ∅ one has Kρ+ρ0 = Kρ Kρ0 .

(3.19)

If the distance between supp ρ and supp ρ0 is large enough one also has Lρ+ρ0 = Lρ Lρ0 .

(3.20)

Next, we define the total charge, qρ ∈ ZN × ZN , of a distribution ρ = (, µ) ∈ D as ! X X ε(x), µ(p) (3.21) qρ = e , mµ := x

and put

Dq := {ρ ∈ D |

p

qρ = q}.

(3.22)

Then, for all ρ = (ε, µ) ∈ D0 , (i.e. with vanishing total charge), there exist 1-cochains, sε and sµ , with finite support, such that d∗ sε = −ε

and

dsµ = −µ.

(3.23)

This allows to generalize the FM-construction and define, for ρ = (ε, µ) ∈ D0 , the family of states ωρ on A as ground states of the modified dynamics αρ by the following method. As already observed in [2] and [1], ground states for the modified transfer matrices can be obtained by taking the thermodynamic limit of the state defined by the formula n ωV, ρ (A) = lim ωV, ρ (A),

(3.24)

n→∞

where n ωV, ρ (A)

T rHV TV (ρ)n ATV (ρ)n EVρ = , T rHV TV (ρ)2n EVρ

A ∈ F(V ),

(3.25)

and where EVρ is some suitably chosen matrix to adjust the boundary conditions. There are two possibilities we will discuss. The first one is EVρ = 1l, thus getting periodic boundary conditions in euclidean time direction for the classical expectations associated n with ωV, ρ (A). In order to understand what happens algebraically one checks that, for µ = 0, the desired ground state simply becomes ω0 ◦ Ad QH (), which is a state with external electric charge. Hence, the resulting representation of A would be equivalent to the vacuum representation. A similar statement holds for µ 6= 0. Since this is not the kind of state we are interested in let us look at the second case. There we choose EVρ in such a way that we get free boundary conditions in the n euclidean time direction for the classical expectations associated with ωV, ρ (A), together with horizontal electric and magnetic strings “conducting” the charges among the points

426


of the support of and µ, respectively, and located in the highest and in the lowest timeslices of Vn := V × {−n, . . . , n}. Hence we choose 1-cochains s and sµ obeying (3.23) and put h h i i∗ (3.26) EVρ := Z(µ)−1/2 M (sµ )QGH (sε ) EV0 Z(µ)−1/2 M (sµ )QGH (sε ) . Here EV0 has been defined in (2.30), QGH (b) are the gauge invariant link operators (2.10) and (3.27) M (sµ ) := e−AV /2 e−BV PG (sµ )eBV eAV /2 ≡ e−AV /2 PG (sµ )eAV /2 . S Note that M (sµ ) ∈ A(V ) for all V containing supp µ supp sµ and that M (sµ ) is actually independent of V provided the distance between supp sµ and the boundary ∂V is ≥ 2. The motivation for this definition comes from its effect in the euclidean functional integral, where QGH (sε ) produces a superposition of electric Mandelstam strings connecting the charges described by ε along the support of sε . Similarly, the operator Z(µ)−1/2 M (sµ ) creates magnetic Mandelstam strings joining the plaquettes of supp µ via the support of sµ . In the functional integral this magnetic string will appear in the form of shifted (or frustrated) vertical plaquettes placed between the first two and the last two time-slices of the space-time volume Vn . This can be seen by looking at the product TV (ρ)Z(µ)−1/2 M (sµ ) appearing in (3.25) due to the definition (3.26). There, the factor Z(µ)−1/2 e−AV /2 e−BV gets canceled by a corresponding factor in TV (ρ) and the matrix elements of PG (sµ )eBV (coming next according to (3.27)) are given by hϕ0 , α0 | PG (sµ )eBV |ϕ, αi # " X X 0 0 Sg (α(b) − α (b) + sµ (b)) − Sh (ϕ(x) − ϕ (x)) . = exp −

(3.28)

x∈V0

b∈V1

After transforming the functional integral expression for (3.25) to unitary gauge the shift sµ (b) in (3.28) appears on the vertical plaquette spanned by the horizontal bond b and the time-like bond htn−1 , tn i (and similarly, but with opposite sign at ht−n , t−n+1 i). The above construction of charged states can be understood as analogous to the construction of Fredenhagen and Marcu, except that, at finite volume, the horizontal Mandelstam strings are already located at the highest (respectively lowest) time-slices. Given the definition (3.26), the limit (3.24) is actually independent of the choice of the horizontal strings sε and sµ satisfying (3.23) and we can take the thermodynamic limit to obtain (3.29) ωρ (A) := lim ωV, ρ (A). V ↑Z2

EV0 implies G(x)EVρ

= EVρ for all x ∈ V and therefore ωρ provides Note that G(x)EV0 = an eigenstate of G with zero external charge. Also note that the boundary conditions (3.26)-(3.27) now imply ωρ ◦ iC = ω−ρ . (3.30) Rewriting (3.29) in terms of euclidean expectation values in the unitary gauge we get an expectation of a classical function Acl, ρ associated with A in the presence of both, “electric” Wilson loops LE (sε , n) living on bonds and “magnetic” vortex loops LM (sµ , n) living on plaquettes, i.e.


ωρ (A) = hAcl, ρ iρ := lim lim ZL−1E , LM V

n

Z

da Acl, ρ (a)e−SG (da+LM )−SH (a) e

427 2πi N (a, LE )

,

(3.31) where a denotes the euclidean lattice ZN -gauge field a := dϕ − α (unitary gauge) and ZLE , LM is the normalization such that ωρ (1l) = 1. Here the vertical parts of the loops LE and LM run from euclidean time t = −n to t = +n and are spatially located at the supports of ε and µ, respectively. The horizontal parts of LE and LM are given by the 1-cochains ±sε/µ (3.23), shifted to the euclidean time slice ±n, respectively. To be more precise, LE (sε , n) ∈ C 1 (Vn ) is the unique 1-cochain on the euclidean space-time lattice Z3 satisfying d∗ LE (sε , n) = 0 together with the condition that its horizontal part is nonzero only on the time slices ±n, where it coincides with ±sε . Similarly, LM (sµ , n) ∈ C 2 (Vn ) is the unique 2-cochain on Z3 satisfying dLM (sµ , n) = 0 plus the condition that the horizontal part of its dual 1-cochain is non-vanishing only on the time slices ±(n − 1/2), where it coincides with the dual of ±sµ . We will refer to such expectations by the following symbolic picture: . ............................................................................. ... C . .. C .. .. C ... .. . .. C .. .. VnC . . .. C ... .. C .. Acl .. . C . . . C .................................................................................. C euclid. . ωρ (A) = lim n, V .. C . ............................................................................ .. ... C .. .. C . . . .. . .. .. C .. .. Vn C .. C .. ... C ... .. C ... . . C ................................................................................ C euclid.

(3.32)

The dotted box indicates the space-time volume Vn . The shaded region in the numerator indicates the support of the classical function Acl associated with an observable A. The boldface vertical lines indicate the vertical part of the loops LM , i.e. the stacks of horizontal plaquettes whose projection onto the time-zero plane is given by µ. They are the euclidean realization of magnetic vortices located at µ. The dashed vertical lines indicate the vertical part of the loops LE . Their projection onto the time-zero plane is given by ε. They are the euclidean realization of electric charges located at ε. For finite n all vertical lines actually close to loops via the strings ±sε/µ running in (or just inside of) the horizontal boundary of Vn . In the limit n → ∞ the choice of these horizontal parts becomes irrelevant. In fact, applying our cluster expansion techniques [1] we have Proposition 3.1.1. For all ρ ∈ D0 the limit (3.29) provides a well defined state ωρ on F which is independent of the choice of 1-chains (sε , sµ ) satisfying (3.23). Moreover, ωρ A provides a ground state with respect to the modified dynamics αρ fulfilling the cluster property.

428


Generalizing Eq. (3.2) we will show in Theorem 3.3.2 below, that for all ρ ∈ D0 , i.e. with vanishing total charge, the states ωρ are actually induced by vectors in the vacuum sector H00 . To get charged states ωρ for ρ ∈ Dq , q 6= 0, we now have to move a counter-charge to infinity. To this end let ρ · a denote the translate of ρ by a ∈ Z2 . Then ρ − ρ0 · a ∈ D0 for all ρ, ρ0 ∈ Dq . Proposition 3.1.2. For A ∈ Floc , 0 6= q ∈ ZN × ZN arbitrary and ρ, ρ0 ∈ Dq let ωρ (A) := lim ωρ−ρ0 ·a (A).

(3.33)

a→∞

Then the limit exists independently of the chosen sequence a → ∞ and it is independent of ρ0 . Moreover ωρ A provides a ground state with respect to the modified dynamics αρ fulfilling the cluster property. Clearly, (3.33) implies that now (3.30) also holds for qρ 6= 0. Next, we note that the symmetry group S of our spatial square lattice (i.e. consisting of translations by a ∈ Z2 and rotations by 21 nπ) acts naturally from the right on Dq by (ρ · g)(x) := ρ(g · x) and we obviously have ωV, ρ ◦ τg = ωg−1 V, ρ·g for all g ∈ S. Hence we get the Corollary 3.1.3. For all g ∈ S and all ρ ∈ D we have ωρ·g = ωρ ◦ τg , where τg denotes the natural action of S on F. We now give an interpretation of these states as charged states in the following sense. For V ⊂ Z2 , finite, define the charge measuring operators 8E (V ) :=

Y

δ ∗ PG (x)

and

x∈V

8M (V2 ) :=

Y

δQG (p).

(3.34)

p∈V2

R TheseR operators are lattice analogues of the continuum expressions exp −i ∇E and exp i dA, respectively, i.e. they measure the total electric charge inside V and the total magnetic flux through V , respectively. As in Theorem 6.2 of [1] we have Proposition 3.1.4. If V ⊂ Z2 is e.g. a square centered at the origin, one has, under the conditions of Proposition 3.1.2: ωρ (8E (V )) 2πie , = exp N V ↑Z2 ω0 (8E (V )) ωρ (8M (V2 )) 2πimµ lim , = exp N V ↑Z2 ω0 (8M (V2 )) lim

(3.35) (3.36)

where e and mµ were defined in (3.21). We omit the proof here, since it is analogous to the equivalent one found in [1]. Finally, we check that our charges have Abelian composition rules. Since, as opposed to the DHR-theory of superselection sectors [12], the states ωρ are not given in terms of localized endomorphisms, we take the following statement as a substitute for this terminology.


429

Proposition 3.1.5. Let ρ, ρ0 ∈ D and for each a ∈ Z2 let ωρ and ωρ+ρ0 ·a be given by Propositions 3.1.1 or 3.1.2, respectively. Then lim ωρ+ρ0 ·a = ωρ .

a→∞

Proposition 3.1.5 is actually a special case of the following more general factorization property, which will be important in the construction of scattering states in [7]. Theorem 3.1.6. Let ρ1 , ρ2 ∈ D and for a ∈ Z2 put ρ(a) := ρ1 + ρ2 · a. Then, for all A, A0 , B and B 0 ∈ Aloc and all n ∈ N0 , n (A0 τa−1 (B 0 )) = lim ωρ(a) τa−1 (B)Aαρ(a) |a|→∞ ωρ1 Aαρn1 (A0 ) ωρ2 Bαρn2 (B 0 ) . (3.37)

A sketch of the proofs of Propositions 3.1.1, 3.1.2 and Theorem 3.1.6 is given in Appendices B.1–B.3. 3.2. Global transfer matrices. Given the family of dyonic states ωρ on A we denote by (πρ , Hρ , ρ ) the associated GNS representation πρ of A on the Hilbert space Hρ with cyclic vector ρ ∈ Hρ . (From now on we will no longer consider external charges, and therefore we simplify the notation by putting Hρ ≡ Hρ0 .) On Hρ we define a modified global transfer matrix Tρ by putting for A ∈ A, Tρ πρ (A)ρ = e−Eρ πρ (αρ (A))ρ ,

(3.38)

where Eρ ∈ R is an additive normalization constant, which will be determined later to appropriately adjust the self energies of the charge distributions ρ relative to each other. Theorem 3.2.1. For all ρ ∈ D Eq. (3.38) uniquely defines a positive bounded operator on Hρ , 0 ≤ Tρ ≤ e−Eρ , (3.39) which is invertible on πρ (A)ρ and implements the modified dynamics, i.e. Ad Tρ ◦ πρ = πρ ◦ αρ . Moreover, C ρ is the unique eigenspace of Tρ with maximal eigenvalue e−Eρ ≡ kTρ k. As a warning we recall that the charged states ωρ are ground states only with respect to the modified euclidean dynamics αρ and only when restricted to the observable algebra A ⊂ F. Hence, although πρ and Tρ could easily be extended to F and πρ (F)ρ , respectively, ρ would not necessarily be a ground state of Tρ in this enlarged Hilbert space containing external charges. However, since we will never look at external charges we will not be bothered by such a possibility. Proof of Theorem 3.2.1. First Tρ is well defined on πρ (A)ρ , since πρ (A)ρ = 0 implies ωρ (BA) = 0 for all B ∈ A and therefore e2Eρ kTρ πρ (A)k2 = ωρ αρ (A)∗ αρ (A) = ωρ αρ−1 (αρ (A)∗ )A = 0, by αρ -invariance of ωρ . Since αρ is invertible, Tρ is invertible on πρ (A)ρ . Moreover, by the ground state property we have

430


πρ (A)ρ , Tρ πρ (A)ρ = e−Eρ ωρ (A∗ αρ (A)) ≤ e−Eρ ωρ (A∗ A),

(3.40)

which implies 0 ≤ Tρ ≤ e−Eρ . The identity Ad Tρ ◦ πρ = πρ ◦ αρ is obvious. Finally, let Pρ ∈ B(Hρ ) be the projection onto the eigenspace of Tρ with eigenvalue e−Eρ and ◦

P ρ = Pρ − |ρ ihρ |. Then, for all A, B ∈ A, lim ωρ (A∗ αρn (B)) = lim enEρ (πρ (A)ρ , Tρn πρ (B)ρ ) n

n

= (πρ (A)ρ , Pρ πρ (B)ρ ) ◦

= ωρ (A∗ )ωρ (B) + (πρ (A)ρ , P ρ πρ (B)ρ ). ◦

Hence the cluster property implies P ρ = 0.

(3.41)

Given the modified global transfer matrices Tρ we now use relation (3.16) to define on Hρ the family of transfer matrices Tρ (ρ0 ) := πρ (Lρ0 L−1 ρ )Tρ ,

(3.42)

where Lρ has been given in (3.18). Hence, Tρ ≡ Tρ (ρ) and we have Corollary 3.2.2. For all ρ, ρ0 ∈ D the operators Tρ (ρ0 ) are positive and bounded on Hρ and (3.43) Ad Tρ (ρ0 ) ◦ πρ = πρ ◦ αρ0 . Proof of Corollary 3.2.2. For the purpose of this proof we consider temporarily the enlarged GNS-triple (πρ (F), ρ , Hρ ), where Hρ now is the closure of πρ (F)ρ and contains vector states with external electric charges. Equation (3.43) follows immediately from (3.16). To see that Tρ (ρ0 ) is positive we use Lρ = Kρ α0 (Kρ∗ ), for Kρ defined in (3.17), to conclude Tρ (ρ0 ) = πρ (Kρ0 )Tρ (0)πρ (Kρ∗0 ) = πρ (Kρ0 Kρ−1 )Tρ (ρ)πρ (Kρ∗ −1 Kρ∗0 ), which is positive since Tρ (ρ) ≡ Tρ is positive.

(3.44)

In view of this result we call Tρ (0) the unmodified (“true”) global transfer matrix, since it implements the original dynamics α ≡ α0 . We also have the following important Corollary 3.2.3. Let Aρ ⊂ B(Hρ ) be the ∗-algebra generated by πρ (A) and Tρ (0). Then the commutant of Aρ is trivial: A0ρ = C 1l. Proof of Corollary 3.2.3. Since Tρ (ρ) ∈ Aρ we have [B, Tρ (ρ)] = 0 for all B ∈ A0ρ which implies Bρ = λρ for some λ ∈ C by the uniqueness of the ground state vector ρ of Tρ (ρ). Hence Bπρ (A)ρ = πρ (A)Bρ = λπρ (A)ρ for all A ∈ A and therefore B = λ1l. In view of Corollary 3.2.3 we may consider πρ as an irreducible representation of ˆ Algebraically A ˆ is defined to be the crossed ˆ ⊃ A such that Aρ = πρ (A). an extension A product of A with the semi-group N0 acting by n 7→ αn ∈ Aut A. Described in terms of ˆ is the ∗-algebra generated by A and a new selfadjoint element generators and relations A t ≡ t(0) with commutation relation tA = α(A)t for all A ∈ A. Note that the construction


431

(2.32) shows that any GNS-representation πω of A from an α-invariant state ω naturally ˆ Moreover, Tω = πω (t) is positive provided the first extends to a representation of A. inequality in (2.26) holds. Using (3.18) we may also define ˆ t(ρ) := Lρ t ∈ A.

(3.45)

Then t(ρ) = t(ρ)∗ (since α(L∗ρ ) = Lρ )) and t(ρ) A = αρ (A) t(ρ) for all A ∈ A. By (3.42) ˆ by defining for fixed ρ and all this implies that also πρ extends to a representation of A ρ0 πρ (t(ρ0 )) := cρ Tρ (ρ0 ), (3.46) where cρ > 0 may be chosen arbitrarily. 3.3. Global charges. In this section we show that two charged representations of A, πρ and πρ0 , are dynamically equivalent if their total charges coincide, qρ1 = qρ2 . Here we take as an appropriate notion of equivalence the following Definition 3.3.1. Two representations, π and π 0 , of A are called dynamically equivalent, ˆ and if there exists a unitary intertwiner if they both extend to representations of A U : Hπ → Hπ0 such that U π(A) = π 0 (A) U for all A ∈ A and U π(t) = c π 0 (t) U for some constant c > 0. The use of the constant c is to allow for the possibility of different zero-point normalizations of the energy. Clearly, by rescaling the global transfer matrices one can always achieve c = 1. To prove that for qρ = qρ0 the representations πρ and πρ0 are dynamically equivalent in this sense we first show Proposition 3.3.2. Let q ∈ ZN × ZN be fixed and let ρ, ρ0 ∈ Dq . Then there exists a unique (up to a phase) unit vector 8ρ,ρ0 ∈ Hρ such that Tρ (ρ0 ) 8ρ,ρ0 = kTρ (ρ0 )k 8ρ,ρ0 . Moreover, 8ρ,ρ0 ∈ Hρ is cyclic for πρ (A) and it induces the state ωρ0 , i.e. (8ρ,ρ0 , πρ (A)8ρ,ρ0 ) = ωρ0 (A),

∀A ∈ A.

(3.47)

Proof. We adapt the proof of Theorem 6.4 of [2] to our setting. Hence we pick 1-cochains 1 such that (d∗ è , d`m ) = ρ − ρ0 ∈ D0 and define `ε , `m ∈ Cloc 8nρ,ρ0 :=

Tρ (ρ0 )n ρ,ρ0 , kTρ (ρ0 )n ρ,ρ0 k

(3.48)

where ρ,ρ0 := πρ (Aρ,ρ0 (è , `m )) ρ , 0 −1/2

Aρ,ρ0 (è , `m ) := Z(µ )

(3.49) 1/2

M (`m )QGH (è )Z(µ)

,

(3.50)

and where µ, µ0 denote the magnetic components in ρ, ρ0 . Using the definitions (3.243.29) we check in Appendix B.5 that the sequence of states (8nρ,ρ0 , πρ (A)8nρ,ρ0 ) converges to ωρ0 (A) for all A ∈ Aloc . We also show, that 8nρ,ρ0 is actually a Cauchy sequence in Hρ and therefore 8ρ,ρ0 := lim 8nρ,ρ0 ∈ Hρ (3.51) n

exists. By construction it is an eigenvector of Tρ (ρ0 ) with eigenvalue

432


λρ0 = lim n

kTρ (ρ0 )n+1 ρ,ρ0 k = lim(8nρ,ρ0 , Tρ (ρ0 )2 8nρ,ρ0 )1/2 . n kTρ (ρ0 )n ρ,ρ0 k

(3.52)

Let now Hρ,ρ0 := πρ (A)8ρ,ρ0 ⊂ Hρ . Then Tρ (ρ0 )Hρ,ρ0 ⊂ Hρ,ρ0 and therefore ˆ ρ,ρ0 ⊂ Hρ,ρ0 by (3.46). Hence Hρ,ρ0 = Hρ by Corollary 3.2.3. By the clusπρ (A)H ter property of ωρ0 with respect to αρ0 we conclude λρ0 = kTρ (ρ0 )k and similarly as in (3.41) the associated eigenspace must be 1-dimensional. For later purposes we emphasize that Proposition 3.3.2 in particular implies that the choice of (è , `m ) in (3.49) only influences the phase of the limit vector 8ρ,ρ0 in (3.51). Using Proposition 3.3.2 we now get the equivalence of charged representations whenever their total charge coincides. Theorem 3.3.3. Let ρ, ρ0 ∈ D and qρ = qρ0 . Then πρ and πρ0 are dynamically equivalent and kTρ (ρ00 )k kTρ (0)k = ∀ρ00 ∈ D. (3.53) kTρ0 (0)k kTρ0 (ρ00 )k Proof. Let U : Hρ0 → Hρ be given on πρ0 (A)ρ0 by U πρ0 (A)ρ0 := πρ (A)8ρ,ρ0 .

(3.54)

Then U extends to a unitary intertwining πρ0 and πρ . Moreover, since Tρ0 (ρ0 ) and Tρ (ρ0 ) implement αρ0 on Hρ0 and Hρ , respectively, one immediately concludes U Tρ0 (ρ0 )kTρ0 (ρ0 )k−1 = Tρ (ρ0 )kTρ (ρ0 )k−1 U.

(3.55)

Using (3.42) this implies U Tρ0 (ρ00 )kTρ0 (ρ0 )k−1 = Tρ (ρ00 )kTρ (ρ0 )k−1 U

∀ρ00 ∈ D

(3.56)

and therefore πρ and πρ0 are dynamically equivalent in the sense of Definition 3.3.1. Finally, (3.56) immediately implies (3.53). We now recall that fixing kTρ (ρ)k ≡ e−Eρ for a given ρ ∈ D amounts to fixing kTρ (ρ00 )k for any other ρ00 ∈ D. Hence, Theorem 3.3.3 shows that the energy normalization Eρ , ρ ∈ Dq , in (3.38) may be fixed up to a constant depending only on the total charge q by requiring kTρ (ρ00 )k = kTρ0 (ρ00 )k for all ρ, ρ0 ∈ Dq and some (and hence all) ρ00 ∈ D. In particular we now take ρ00 = 0 and require kTρ (0)k = e−(q) ,

∀ρ ∈ Dq

(3.57)

for some - as yet undetermined - function (q) ∈ R describing the “minimal self-energy” in the sector with total charge q ∈ ZN ×ZN . In the next subsection we will appropriately fix (q) such that 2(q) ≡ (q) + (−q) gives the minimal energy needed to create a pair of charge-anticharge configurations of total charge ±q from the vacuum and separate them apart to infinite distance. We close this subsection by remarking that we expect conversely (at least for a generic choice of couplings in the free charge phase) that the representations πρ and πρ0 are dynamically inequivalent if qρ 6= qρ0 . Inequivalence to the vacuum sector for qρ 6= 0 can presumably be shown by analogue methods as in [2], i.e. by proving the absence of a translation invariant vector in Hρ (see Sect. 4.3 for the implementations of translations


433

in the charged sectors). More generally, one could try to prove that kTρ (ρ0 )k is not an eigenvalue of Tρ (ρ0 ) if qρ 6= qρ0 . To this end one would have to show that qρ 6= qρ0 implies s − lim n

Tρ (ρ0 )n πρ (A)ρ = 0 kTρ (ρ0 )kn

(3.58)

for all A ∈ Aloc . As in Proposition 5.5 of [2] a sufficient condition for this would be (πρ (A)ρ , Tρ (ρ0 )n πρ (A)ρ )2 = 0. n→∞ (πρ (A)ρ , Tρ (ρ0 )2n πρ (A)ρ ) lim

(3.59)

Using (3.42) this would amount to finding the asymptotics of the vertical string expectation value ! n−1 Y ∗ k −1 n αρ (Lρ0 Lρ )αρ (A) . (3.60) ωρ A k=0

Similarly as in the Bricmont-Fröhlich criterion for the existence of free charges [13] Eq. (3.59) would follow provided (3.60) would decay like n−α e−n·const for some α > 0 (in [13] α = d/2). Since we have not tried to prove this we only formulate the Conjecture 3.3.4. If qρ 6= qρ0 then kTρ (ρ0 )k is not an eigenvalue of Tρ (ρ0 ). Together with Proposition 3.3.2 and Theorem 3.3.3 the Conjecture 3.3.4 would imply that πρ and πρ0 are dynamically equivalent if and only if their total charges coincide. 3.4. The dyonic self energies. In this subsection we determine the “dyonic selfenergies" (q) introduced in (3.57). In order to get charge conjugation invariant self energies it will be useful to start with implementing the charge conjugation as an intertwiner Cρ : Hρ → H−ρ by putting for A ∈ A and ρ ∈ D, Cρ πρ (A)ρ := π−ρ (iC (A))−ρ .

(3.61)

Lemma 3.4.1. Cρ extends to a well defined unitary Hρ → H−ρ such that i) Ad Cρ ◦ πρ = π−ρ ◦ iC . ii) Cρ∗ = C−ρ . iii) Cρ Tρ (ρ0 )eEρ = T−ρ (−ρ0 )eE−ρ Cρ . Proof. By Eq. (3.30) Cρ is well defined, unitary and obeys i). Part ii) follows from i2C = id. Finally, in the case ρ0 = ρ part iii) follows from Eq. (3.15). By (3.42) the case ρ0 6= ρ follows from iC (Lρ ) = L−ρ By charge conjugation symmetry we would of course like to have Eρ = E−ρ . We now fix the self-energies (q) in (3.57) of the sectors q ∈ ZN × ZN by first putting the vacuum energy to zero, (0) = 0, i.e. kTρ (0)k = 1,

∀ρ ∈ D0 .

(3.62)

This fixes e−Eρ ≡ kTρ (ρ)k = kπρ (Lρ )Tρ (0)k for all ρ ∈ D0 and in particular implies Eρ = E−ρ , ∀ρ ∈ D0 . In fact, we have Lemma 3.4.2. For fixed q ∈ ZN × ZN we have (q) = (−q) if and only if Eρ = E−ρ , ∀ρ ∈ Dq .

434


Proof. Putting ρ0 = 0 in Lemma 3.4.1 iii) we see that kTρ (0)k = kT−ρ (0)k, ∀ρ ∈ Dq , is equivalent to Eρ = E−ρ , ∀ρ ∈ Dq , since Cρ is unitary. We now show that under the conditions (0) = 0 and (q) = (−q) the self-energies of the charged sectors are completely determined if we adjust them such that 2(q) ≡ (q) + (−q) becomes equal to the minimal energy needed to create a pair of chargeanticharge configurations of total charge ±q from the vacuum and separate them apart to infinity. Physically this means that for infinite separation we can consistently normalize the interaction energy between two charges to zero. Mathematically this is expressed by a factorization formula similar to Theorem 3.1.6 for matrix elements of the global transfer matrix . As it will turn out, this is further equivalent to an additivity property of the modified ground state energies, i.e. Eρ+ρ0 ·a → Eρ + Eρ0 as |a| → ∞. Theorem 3.4.3. There exists precisely one assignment of self-energies ZN ×ZN 3 q → (q) ∈ R such that kTρ (0)k = e−(q) , ∀ρ ∈ Dq and such that the following conditions hold i) (0) = 0, ii) (q) = (−q), iii) The factorization formula (3.37) extends to transfer matrices, i.e. πρ(a) (Aτ−a (B))ρ(a) , Tρ(a) (0)n πρ(a) (A0 τ−a (B 0 ))ρ(a) lim |a|→∞ = πρ1 (A)ρ1 , Tρ1 (0)n πρ1 (A0 )ρ1 πρ2 (B)ρ2 , Tρ2 (0)n πρ2 (B 0 )ρ2 , (3.63) where we used the notation of Theorem 3.1.6. To prove Theorem 3.4.3 and in particular the cluster property (3.63) we have to control norm ratios of (modified) transfer matrices, for which we have to invoke our polymer expansions. First we have Lemma 3.4.4. Let ρ, ρ0 ∈ Dq for some q ∈ ZN × ZN and choose 1-cochains è , `m ∈ 1 such that (d∗ è , d`m ) = ρ − ρ0 . Let Aρ,ρ0 ≡ Aρ,ρ0 (è , `m ) ∈ Aloc be defined as in Cloc (3.50) and put # "ν−1 Y (ν) ∗ k −1 αρ (Lρ0 Lρ ) αρν (Aρ,ρ0 ) . (3.64) Xρ,ρ0 := Aρ,ρ0 k=0

Then we have (2ν+1)

ωρ (Xρ,ρ0 ) kTρ (ρ0 )k . = lim (2ν) ν→∞ kTρ (ρ)k ωρ (Xρ,ρ 0)

(3.65)

Proof. By Proposition 3.3.2 and Eqs. (3.48)–(3.51) we have for all q ∈ ZN × ZN and all ρ, ρ0 ∈ Dq , (ρ,ρ0 , Tρ (ρ0 )2ν+1 ρ,ρ0 ) , ν→∞ (ρ,ρ0 , Tρ (ρ0 )2ν ρ,ρ0 )

kTρ (ρ0 )k = (8ρ,ρ0 , Tρ (ρ0 )8ρ,ρ0 ) = lim where

ρ,ρ0 := πρ (Aρ,ρ0 (è , `m )) ρ has been given in (3.49). The lemma follows by using (3.42) to rewrite

(3.66)


0 ν

Tρ (ρ ) =

"ν−1 Y

435

# αρk (Lρ0 L−1 ρ )

Tρ (ρ)ν ,

k=0

and recalling Tρ (ρ)ρ = kTρ (ρ)kρ by Theorem 3.2.1.

Note that the norm ratio (3.65) is obviously independent of the choice of the scale eEρ ≡ kTρ (ρ)k. We also recall from Proposition 3.3.2 that the choice of (è , `m ) only influences the phase of 8ρ,ρ0 and hence the limit ν → ∞ in (3.65) is in fact independent of this choice. When expressing the r.h.s. of (3.65) in terms of euclidean path integrals we get a ratio of expectations of the form (3.31), where the classical function corresponding (ν) to Xρ,ρ 0 is itself a superposition of Wilson loops and vortex loops. These are built by the same recipes as before, i.e. with horizontal parts given by ±(è , `m ) in the time slices t = ν and t = 0, respectively, and with vertical parts joining the endpoints of these strings. Hence we get the ratio of euclidean path integral expectations of two such loop configurations with time-like extension ν + 1 and ν, respectively, in the limit ν → ∞. Here these expectations have to be taken in the background determined by ρ, i.e. in the presence of the vertical parts of LE and LM described in (3.31), where there the limit n → ∞ and V → ∞ has to be taken first.8 Next we have to control certain factorization properties of the above norm ratios in the limit of the charge distributions being separated to infinity. Proposition 3.4.5. Let ρ1 , ρ2 ∈ D and for a ∈ Z2 put ρ1 (a) = ρ1 − ρ1 · a, ρ2 (a) = ρ2 − ρ2 · a. Then i) The limit kTρ1 (a) (ρ2 (a))k cρ1 , ρ2 := lim >0 (3.67) |a|→∞ kTρ1 (a) (ρ1 (a))k exists independently of the sequence a → ∞ and satisfies cρ1 ·g, ρ2 ·g = cρ1 , ρ2 , ∀g ∈ S. Moreover, for b ∈ Z2 we have the factorization property lim cρ1 +ρ01 ·b, ρ2 +ρ02 ·b = cρ1 , ρ2 cρ01 , ρ02 .

|b|→∞

ii) If qρ1 = qρ2 then c ρ1 , ρ 2 =

kTρ1 (ρ2 )k2 . kTρ1 (ρ1 )k2

(3.68)

(3.69)

Proof. Part i) is proven in detail in Appendix B. Here we just remark that it can roughly be understood from the perimeter law of Wilson and vortex loop expectations, which guarantees that ratios of loop expectations converge for infinitely large loops if the difference of their perimeters stays finite (in the above case this difference is given by two lattice units, due to the difference by one unit in time extension, see (3.65)). To prove ii) we pick in (3.66) the choice (3.70) ρ1 (a), ρ2 (a) = πρ1 (a) Aρ1 (a), ρ2 (a) (è (a), `m (a)) ρ1 (a) , where we take (è (a), `m (a)) = (è − è · a, `m − `m · a) This is why in our polymer expansion only the vertical parts of LE and LM matter, since the horizontal parts run in the time-like boundary of Vn and since the contributions from polymers reaching from the support of a classical function Acl to this boundary decay to zero for n → ∞. 8

436


for some fixed configuration (è , `m ) satisfying (d∗ è , d`m ) = ρ1 − ρ2 . With this choice it is not difficult to check that for |a| large enough, Aρ1 (a), ρ2 (a) (è (a), `m (a)) = Aρ1 , ρ2 (è , `m )τ−a (Aρ1 , ρ2 (è , `m )).

(3.71)

Similarly, for fixed ν and |a| large enough, = Xρ(ν) τ (Xρ(ν) ). Xρ(ν) 1 , ρ2 −a 1 , ρ2 1 (a), ρ2 (a)

(3.72)

When proving part i) in Appendix B.4 we show that plugging (3.72) into (3.65) the limit ν → ∞ is uniform in a and hence we may take the limit |a| → ∞ first. However, this takes us into the setting of the factorization formula (3.37) implying (ν) (ν) 2 lim ωρ(a) (Xρ(a), ρ0 (a) ) = ωρ1 (Xρ1 , ρ2 ) ,

|a|→∞

which proves part ii).

(3.73)

Proof of Theorem 3.4.3. The idea of the proof consists of translating the physically motivated conditions i)-iii) into conditions on the family of constants Eρ , ρ ∈ D. We have already remarked in (3.62) that i) may be obtained by suitably fixing Eρ for all ρ ∈ D0 . In Lemma 3.4.2 we have noticed that ii) is equivalent to Eρ = E−ρ for all ρ. We now show that iii) holds if and only if for all ρ1 , ρ2 ∈ D, lim

|a|→∞

To this end we use

Eρ1 +ρ2 ·a = Eρ1 + Eρ2 .

n Tρ (ρ)n = πρ (Lρ )Tρ (0) = πρ (Yρ(n) )Tρ (0)n ,

where Yρ(n) :=

n−1 Y

α0k (Lρ ).

(3.74)

(3.75)

(3.76)

k=0

Furthermore, for |a| large enough and ρ(a) = ρ1 + ρ2 · a we have Lρ(a) = Lρ1 τ−a (Lρ2 ) = τ−a (Lρ2 )Lρ1 .

(3.77)

Using that α0 commutes with τ−a we get for large enough |a| and A, A0 , B and B 0 ∈ Aloc , πρ(a) (Aτ−a (B))ρ(a) , Tρ(a) (0)n πρ(a) (A0 τ−a (B 0 ))ρ(a) n = ωρ(a) τ−a (B ∗ [Yρ(n) ]−1 )A∗ [Yρ(n) ]−1 αρ(a) (A0 τ−a (B 0 )) e−nEρ (a) . 2 1 Here we have chosen |a| large enough (depending on n) such that A∗ commutes with −1 τ−a Yρ(n) . On the other hand, we get for i = 1, 2 and all A, A0 ∈ Aloc , 2 πρi (A)ρi , Tρi (0)n πρi (A0 )ρi = ωρi A∗ [Yρ(n) ]−1 αρni (A0 ) e−nEρi . (3.78) i Thus, by Theorem 3.1.6 the factorization formula (3.63) is equivalent to (3.74). Hence, there can be at most one solution of the conditions i)–iii) above. Indeed, Eqs. (3.74) together with Eρ = E−ρ imply Eρ =

1 lim Eρ−ρa 2 |a|→∞

(3.79)


437

for all ρ ∈ D. Since ρ − ρ · a ∈ D0 and since condition i) completely fixes Eρ = E−ρ for all ρ ∈ D0 , Eq. (3.79) fixes Eρ = E−ρ uniquely for all ρ ∈ D. We are left to prove that given Eρ = E−ρ for all ρ ∈ D0 via condition i) the limit (3.79) indeed exists for all ρ ∈ D and satisfies (3.74). However this is an immediate consequence of Proposition 3.4.5. Indeed, putting ρ2 = 0 in (3.67) and using kTρ1 (a) (0)k = 1 by condition i) we get for all ρ ∈ D, lim eEρ−ρ·a = cρ, 0 ,

|a|→∞

(3.80)

and hence (3.79) implies 1 ln cρ, 0 . (3.81) 2 Note that for qρ = 0 this is consistent with (3.69). The cluster property (3.74) now is an immediate consequence of (3.68). This concludes the proof of Theorem 3.4.3. Eρ =

kT (0)k

We remark that we have not worked out an analytic expression for the ratio kTρρ(ρ)k in the case qρ 6= 0, which is why we do not have further analytic knowledge of the dyonic self-energies (q). In particular we have not tried to confirm the natural “stability conjecture” (q) > 0 for all q 6= 0. In the purely electric (or purely magnetic) sectors the existence of 1-particle states [1] implies that e−(qρ ) ≡ kTρ (0)k lies in the spectrum of Tρ (0), i.e. (qρ ) is precisely the 1-particle self-energy at zero momentum. 4. State Bundles and Intertwiner Connections In this section we consider an appropriate analogue of what in the DHR-theory of super selection sectors would be called the state bundle. In our setting, by this we mean the ˆ obtained from collection of all GNS triples (Hρ , πρ , ρ ) of cyclic representation of A the family of states ωρ , ρ ∈ D. Modulo Conjecture 3.3.4, these representations fall into equivalence classes labelled by their total charges qρ . Hence we obtain, for each value of the global charge q ∈ ZN × ZN , a Hilbert-bundle Bq over the discrete base space Dq , which is simply given as the disjoint union [ ˙ Hρ . (4.1) Bq := ρ∈Dq

ˆ The fibers Hρ of this bundle are all naturally isomorphic as A-modules by Theorem ˆ acts irreducibly on Hρ by Corollary 3.2.3, these isomorphisms are 3.3.3. Since πρ (A) all uniquely determined up to a phase. When trying to fix this phase ambiguity one is naturally led to the problem of constructing an intertwiner connection on Bq , i.e. a family of intertwiners U (0) : Hρ → Hρ0 satisfying πρ0 = Ad U (0) ◦ πρ and depending on paths 0 in Dq from ρ to ρ0 . Here, by a path in Dq we mean a finite sequence (ρ0 , ρ1 , . . . , ρn ) of charge distributions ρi ∈ Dq , such that ρi and ρi+1 are “nearest neighbours” in Dq in a suitable sense to be explained below. In order to be able to formulate a concept of locality for these intertwiners we will also consider the Hilbert direct sum M Hρ (4.2) Hq := ρ∈Dq

438


ˆ be represented by on which we let A ∈ A

M

Πq (A) :=

πρ (A) .

(4.3)

ρ∈Dq

The above mentioned connection will then be given in terms of an intertwiner ZN ˆ Locality in this framework Weyl algebra Wq acting on Hq and commuting with Πq (A). is formulated by the statement that Wq is generated by elementary “electric string operators” Eq (b) and elementary “magnetic string operators” Mq (b) satisfying for all oriented bonds b, b0 ∈ (Z2 )1 , Eq (b)N = Mq (b)N = 1l,

(4.4)

and the local ZN -Weyl commutation relations Eq (b)Eq (b0 )

= Eq (b0 )Eq (b),

Mq (b)Mq (b0 ) = Mq (b0 )Mq (b), Eq (b)Mq (b0 ) = e

2πi N δb, b0

(4.5)

Mq (b0 )Eq (b).

These operators will map each fiber Hρ , ρ ∈ Dq , isomorphically onto a “neighbouring one”, i.e. (4.6) Eq (b)Hρ = Hρ+(d∗ δb , 0) , Mq (b)Hρ = Hρ+(0, dδb ) ,

(4.7)

1 denotes the 1-cochain taking value one on b and zero otherwise. Hence, where δb ∈ Cloc one might think of Eq (b) (Mq (b)) as creating an electric (magnetic) charge-anticharge pair sitting in the boundary (coboundary) of the bond b. Defining charge distributions in Dq to be nearest neighbours if they differ by either such an elementary electric or magnetic dipole, the connection in Bq along a path (ρ0 , ρ1 , . . . , ρn ) in Dq is now given in the obvious way as an associated product of Eq (b)’s and Mq (b0 )’s, mapping each fiber Hρi isomorphically onto its successor Hρi+1 according to (4.6)-(4.7). We remark that in a DHR-framework one would expect these intertwiners to be given on the dense subspace πρ (A)ρ ⊂ Hρ in terms of unitary localized charge transporters Sel (b) and Smag (b) ∈ Aloc by

Eq (b)πρ (A)ρ = πρ+(d∗ δb , 0) (A Sel (b))ρ+(d∗ δb , 0) , Mq (b)πρ (A)ρ = πρ+(0, dδb ) (A Smag (b))ρ+(0, dδb ) ,

(4.8) (4.9)

which would be consistent and well defined if ωρ+(d∗ δb , 0) = ωρ ◦ Ad Sel (b), ωρ+(0, dδb ) = ωρ ◦ Ad Smag (b).

(4.10) (4.11)

Intuitively one could think of Sel as a kind of “electric” Mandelstam string operator as in (3.1) and of Smag (b) as its dual “magnetic” analogue. Unfortunately, due to the non-local energy regularization in (3.2) (and more generally, in (3.25)), the existence of such localized charge transporters Sel and Smag in Aloc satisfying (4.10)-(4.11) is very questionable. We also remark that, as opposed to the DHR-framework, in our lattice model the states ωρ are not given in the form ωρ = ω0 ◦ γρ for some localized automorphism γρ on


439

A. This is why we do not have a field algebra extension of A carrying a global ZN × ZN symmetry, i.e. we are not able to define charged fields intertwining πρ ◦ γρ0 and πρ+ρ0 . It is therefore astonishing and, in our opinion, asks for further conceptual explanation that in this model one is nevertheless able to construct an intertwiner algebra with local commutation relations as given in (4.5). We will come back to this question in [7], where the existence of the local intertwiner algebra (4.4)-(4.5) will be the basic input for the construction of Haag-Ruelle scattering states in the charged sectors. 4.1. The local intertwiner algebra. We now proceed to the construction of the intertwiner algebra Wq obeying (4.4)-(4.5). For ρ, ρ0 ∈ Dq , let Pρ (ρ0 ) : Hρ → Hρ be the one-dimensional projection onto the ground state of Tρ (ρ0 ). Using the notation of (3.49)-(3.50) we then define for arbitrary 1 , 1-cochains l ∈ Cloc Pρ (ρ − (d∗ l, 0))πρ (Aρ, ρ−(d∗ l, 0) (l, 0))ρ , kPρ (ρ − (d∗ l, 0))πρ (Aρ, ρ−(d∗ l, 0) (l, 0))ρ k Pρ (ρ − (0, dl))πρ (Aρ, ρ−(0, dl) (0, l))ρ φmag , (l) = ρ kPρ (ρ − (0, dl))πρ (Aρ, ρ−(0, dl) (0, l))ρ k φel ρ (l) =

(4.12) (4.13)

which are well defined unit vectors in Hρ , by Proposition 3.3.2. We may then define ˆ by putting on the dense ˆ q (l) on Hq commuting with Πq (A) unitary operators Eˆq (l), M set πρ (A)ρ , A ∈ A, ρ ∈ Dq , Eˆq (l)πρ (A)ρ := πρ+(d∗ l, 0) (A)φel ρ+(d∗ l, 0) (l), mag ˆ Mq (l)πρ (A)ρ := πρ+(0, dl) (A)φρ+(0, dl) (l),

(4.14) (4.15)

which are well defined as in (3.54). It turns out that these intertwiners do not yet obey local commutation relations, however the violation of locality can be described by a kind of “coboundary” equation. Theorem 4.1.1. There exists an assignment of phases zρel (l), zρmag (l) ∈ U (1) such that 1 and all ρ ∈ Dq , for all l1 and l2 ∈ Cloc i) ii) iii)

el el zρ+(d ∗ l , 0) (l1 ) zρ (l2 ) 2 . Eˆq (l1 + l2 )ρ , Eˆq (l1 )Eˆq (l2 )ρ = zρel (l1 + l2 )

(4.16)

z mag dl2 ) (l1 ) zρmag (l2 ) ˆ q (l1 + l2 )ρ , M ˆ q (l1 )M ˆ q (l2 )ρ = ρ+(0, mag M . zρ (l1 + l2 )

(4.17)

z el dl1 ) (l1 ) zρmag (l1 ) ˆ q (l1 )ρ = eihl1 , l2 i ρ+(0, ˆ q (l1 )Eˆq (l2 )ρ , Eˆq (l2 )M (4.18) . M mag el zρ+(d ∗ l , 0) (l1 ) zρ (l2 ) 2

iv) If d∗ l = 0 then zρel (l) = 1. If dl = 0 then zρmag (l) = 1. el/mag

v) zρel/mag (l) = zρ·g

(l · g), ∀g ∈ S.

(4.19) (4.20) (4.21)

440


The proof of this theorem and the precise definition of the phases zρel (l) and zρmag (l) will be given in Appendix B.6. 1 , We now use these phases to define on Hq the unitaries Zqel (l) and Zqmag (l), l ∈ Cloc by Zqel (l) Hρ := zρel (l)1lHρ , Zqmag (l) Hρ := zρmag (l)1lHρ .

(4.22) (4.23)

ˆ and we may define new Then Zqel (l) and Zqmag (l) clearly also commute with Πq (A) intertwiners Eq (l) := Eˆq (l)Zqel (l)−1 , ˆ q (l)Zqmag (l)−1 , Mq (l) := M

(4.24) (4.25)

which now obey local commutation relations. 1 the operators Eq (l) and Mq (l) are unitaries in B(Hq ) Theorem 4.1.2. For all l ∈ Cloc ˆ They satisfy the Weyl algebra relations commuting with Πq (A).

Eq (l1 )Eq (l2 ) = Eq (l1 + l2 ), Mq (l1 )Mq (l2 ) = Mq (l1 + l2 ), Eq (l1 )Mq (l2 ) = eihl1 , l2 i Mq (l2 )Eq (l1 ),

(4.26) (4.27) (4.28)

and restricted to each subspace Hρ ⊂ Hq , ρ ∈ Dq , we have Eq (l)Hρ = Hρ+(d∗ l, 0) , Mq (l)Hρ = Hρ+(0, dl) .

(4.29) (4.30)

Proof. It remains to prove the identities (4.26)–(4.28), for which it is enough to check them when applied to ρ ∈ Hρ , ∀ρ ∈ Dq . First we get Eq (l1 + l2 )ρ = zρel (l1 + l2 )−1 φel ρ+(d∗ l1 +d∗ l2 , 0) ,

(4.31)

which is a unit vector in the image of the one dimensional projection Pρ+(d∗ l1 +d∗ l2 , 0) (ρ). On the other hand we have Eq (l1 )Eq (l2 )ρ = zρel (l2 )−1 Eq (l1 )φel ρ+(d∗ l2 , 0) (l2 ).

(4.32)

ˆ and πρ0 +(d∗ l1 , 0) (A) ˆ we have Since Eq (l1 ) Hρ0 intertwines πρ0 (A) Eq (l1 )Tρ0 (ρ) Hρ0 = Tρ0 +(d∗ l1 , 0) (ρ)Eq (l1 ) Hρ0 , implying

(4.33)

(4.34) Eq (l1 )Pρ0 (ρ) Hρ0 = Pρ0 +(d∗ l1 , 0) (ρ)Eq (l1 ) Hρ0 . Hence (4.32) implies (4.35) Eq (l1 )Eq (l2 )ρ ∈ Pρ00 (ρ)Hρ , 00 ∗ ∗ where ρ = ρ + (d l1 + d l2 , 0). Comparing (4.31) and (4.35) we conclude that Eq (l1 )Eq (l2 )ρ and Eq (l1 + l2 )ρ only differ by a phase. This phase must be one since, by (4.16) and (4.24)–(4.25), (4.36) Eq (l1 + l2 )ρ , Eq (l1 )Eq (l2 )ρ = 1. Equations (4.27)-(4.28) are proven similarly.


441

Putting Eq (b) ≡ Eq (δb ) and Mq (b) ≡ Mq (δb ), Theorem 4.1.2 provides the local intertwiner algebra Wq announced in (4.4)–(4.5). 4.2. The intertwiner connection. We now turn to the bundle theoretic point of view, where we consider the above intertwiners as a connection on Bq . As already explained, we call a pair ρ, ρ0 ∈ Dq nearest neighbours if they differ by an elementary electric or magnetic dipole, i.e. if there exists a bond b ∈ (Z2 )1 such that ρ0 − ρ = (±d∗ δb , 0) or ρ0 − ρ = (0, ±d δb ). In the first case we put Uρ0 , ρ = Eq (b)±1 :

H ρ → Hρ 0 ,

(4.37)

and in the second case we put Uρ0 , ρ = Mq (b)±1 :

H ρ → Hρ 0 .

(4.38)

If 0 = (ρ0 , . . . , ρn ) is a path in Dq , i.e. a finite sequence of nearest neighbour pairs, then we put U(0) := Uρn ρn−1 · · · Uρ1 ρ0 :

Hρ 0 → Hρ n

(4.39)

as the associated “parallel transport”. If 0 is a closed path, i.e. ρn = ρ0 , then U (0) must ˆ be a phase by the irreducibility of πρ0 (A). In order to determine these “holonomy phases” one may use the commutation relation (4.28) which allows to restrict ourselves to purely electric or purely magnetic loops 0 (i.e. where U(0) only consists of a product of Eq (b)’s or Mq (b)’s, respectively). Using (4.26)–(4.27) such loop operators U(0) are always of the form U (0) = Eq (le ) Hρ or U (0) = Mq (lm ) Hρ , where the loop condition on 0 implies d∗ le = 0 or dlm = 0, respectively. Note that this is consistent with (4.29)–(4.30), i.e. these loop operators must map Hρ onto itself. To compute the holonomy phases we now use the fact that d∗ le = 0 and dlm = 0 2 0 and k ∈ Cloc such that implies that there exist uniquely determined cochains s ∈ Cloc d ∗ s = le ,

dk = lm .

(4.40)

We also recall that in these cases Theorem 4.1.1 iv) implies Eq (le ) = Eˆq (le ) and Mq (lm ) = ˆ q (lm ). Hence, being a phase when restricted to Hρ , it is enough to apply these operators M to ρ where, by the definitions (4.22)–(4.23) and (4.24)–(4.25), they yield φel ρ (le ) and (lm ), respectively. φmag ρ Thus, we get our holonomy phases from 2 0 Proposition 4.2.3. Let s ∈ Cloc and k ∈ Cloc such that d∗ s = le and dk = lm . Then, for ρ = (, µ) we have ihµ, si ρ , φel ρ (le ) = e

(lm ) φmag ρ

=e

−ih, ki

ρ .

(4.41) (4.42)

442


By the above arguments Proposition 4.2.3 implies Eq (le ) Hρ = eihµ, si , Mq (lm ) Hρ = eih, ki .

(4.43)

If we think of s being the characteristic function of a surface encircled by an electric loop le , then the holonomy phase (4.41) is just the magnetic flux through this surface. Interchanging electric and magnetic (and passing to the dual lattice), Eq. (4.42) may be stated analogously. This state of affairs will be the origin of anyon statistics of scattering states in [7]. Proposition 4.2.3 will be proven in Appendix B.7. 4.3. The representation of translations. Using our local intertwiner algebra Wq we are now in the position to define on each fiber Hρ a unitary representation Dρ (a), a ∈ Z2 , of the group of the lattice translation, such that Ad Dρ (a) ◦ πρ = πρ ◦ τa and such that Dρ (a) commutes with Tρ (0), ∀a ∈ Z2 . Moreover, for fixed q, these representations are all equivalent, i.e. U(0)Dρ (a) = Dρ0 (a)U(0) for all paths 0 : ρ → ρ0 . First we need a lift of the natural action of translations on Dq to bundle automorphisms on Bq . For the sake of generality let us formulate this by including also the lattice rotations. Lemma 4.3.4. For g ∈ S let Vρ (g) : Hρ → Hρ·g be given by Vρ (g)πρ (A)ρ := πρ·g (τg−1 (A))ρ·g .

(4.44)

Then Vρ (g) is a well defined unitary intertwining πρ ◦ τg with πρ·g 9 , i.e. Vρ (g)πρ (τg (A)) = πρ·g (A)Vρ (g), Vρ (g)Tρ (ρ0 ) = Tρ·g (ρ0 · g)Vρ (g),

(4.45) (4.46)

for all A ∈ A. Proof. Vρ (g) is well defined and unitary since ωρ ◦ τg = ωρ·g . Equations (4.45)-(4.46) follow from αρ ◦τρ·g = αρ·g and Eρ·g = Eρ (by (3.81) and S-invariance of cρ, ρ0 ). Note that the definition (4.44) implies the obvious identity Vρ·g (h)Vρ (g) = Vρ (gh).

(4.47)

for all g, h ∈ S, which means that the family of fiber isomorphisms Vρ (g), ρ ∈ Dq , may indeed be viewed as a lift of the natural right action of S on Dq to an action by unitary bundle automorphism on Bq . Equivalently, we now consider M Vq (g) := Vρ (g) (4.48) ρ∈Dq

as a unitary representation of S on Hq , satisfying Ad Vq (g) ◦ Πq = Πq ◦ τg−1 .

(4.49)

Moreover, we have 9 Note that the ∗-automorphism τ ∈ Aut A may be extended to a ∗-automorphism on A ˆ by putting g τg (t) = t.


443

Lemma 4.3.5. Vq (g)Eq (l) = Eq (l · g)Vq (g), Vq (g)Mq (l) = Mq (l · g)Vq (g).

(4.50) (4.51)

Proof. By Theorem 4.1.1 v) we have Vq (g)Zqel/mag (l) = Zqel/mag (l · g)Vq (g).

(4.52)

ˆ q . However this Hence it is enough to prove the claim with Eq , Mq replaced by Eˆq , M is a straight forward consequence of the definitions and the fact that by (4.46), Vρ (g)Pρ (ρ0 ) = Pρ·g (ρ0 · g)Vρ (g),

(4.53)

and, therefore, el/mag

Vρ (g)φel/mag (l) = φρ·g ρ

(l · g).

(4.54)

We remark that Lemma 4.3.5 implies that the connection U (0) is S-invariant, i.e. Vρ0 (g)U(0) = U(0 · g)Vρ (g)

(4.55)

for all paths 0 : ρ → ρ0 and all g ∈ S. In order to arrive at an implementation of the translation group Z2 ⊂ S mapping each fiber Hρ onto itself we now have to compose Vρ (a)∗ , a ∈ Z2 , with a parallel transporter Uρ (a) : Hρ → Hρ·a along a suitable path 0 : ρ → ρ · a. To this end it is enough to consider the cases a = ei , i = 1, 2, where e1 = (1, 0) and e2 = (0, 1) denote the generators of Z2 . 0 there A minute’s thought shows that for any electric charge distribution ∈ Cloc 1 exists a unique 1-cochain è (, i) ∈ Cloc with support only on bonds in direction i, such that the translated image · ei of by one unit in direction i satisfies · ei = + d∗ è (, i).

(4.56)

2 there exists a unique 1-cochain Similarly, for a magnetic charge distribution µ ∈ Cloc `m (µ, i) with support only on the bonds perpendicular to the direction i, such that

µ · ei = µ + d`m (µ, i).

(4.57)

For ρ = (, µ) ∈ Dq we now define Uρ (ei ) : Hρ → Hρ·ei by putting Uρ (ei ) := Eq (è (, i))Mq (`m (µ, i)) Hρ = Mq (`m (µ, i))Eq (è (, i)) Hρ ,

(4.58)

where the second equality follows since, by construction, `m (µ, i) and è (, i) always have disjoint support. With this construction we now define the unitaries Dρ (ei ) : Hρ → Hρ , i = 1, 2, by Dρ (ei ) := Vρ (ei )∗ Uρ (ei ).

(4.59)

444


Theorem 4.3.6. The unitaries Dρ (ei ), i = 1, 2, generate a representation of Z2 impleˆ i.e. menting the translation automorphism on A, Dρ (e1 )Dρ (e2 ) = Dρ (e2 )Dρ (e1 ), Dρ (ei )πρ (A) = πρ (τei (A))Dρ (ei ),

(4.60) (4.61)

ˆ for all A ∈ A. Proof. Equation (4.61) immediately follows from (4.58) and (4.45)-(4.46). To prove (4.60) we first use Lemma 4.3.5 to conclude that Dρ (e2 )Dρ (e1 ) = Vρ (e2 )∗ Vρ·e2 (e1 )∗ Uρ·e1 (e2 )Uρ (e1 ), Dρ (e1 )Dρ (e2 ) = Vρ (e1 )∗ Vρ·e1 (e2 )∗ Uρ·e2 (e1 )Uρ (e2 ).

(4.62) (4.63)

Since (4.47) implies Vρ·e2 (e1 )Vρ (e2 ) = Vρ·e1 (e2 )Vρ (e1 ) = Vρ (e1 + e2 ),

(4.64)

we are left to check Uρ (e2 )−1 Uρ·e2 (e1 )−1 Uρ·e1 (e2 )Uρ (e1 ) = 1lHρ .

(4.65)

Using the definition (4.58) and the Weyl algebra relations (4.26)-(4.28), Eq. (4.65) is equivalent to (4.66) Mq (Lm (µ)) Eq (Le ()) Hρ = (u1 u2 u3 )−1 , where Lm (µ) = `m (µ, 1) + `m (µ · e1 , 2) − `m (µ · e2 , 1) − `m (µ, 2), Le () = è (, 1) + è ( · e1 , 2) − è ( · e2 , 1) − è (, 2),

(4.67) (4.68)

and where ui ∈ U (1) are the phases obtained by commuting in (4.65) all factors of Eq ’s to the right of Mq ’s, i.e. u1 (, µ) = exp −ihè (, 2), `m (µ, 1) − `m (µ · e2 , 1)i , (4.69) u2 (, µ) = exp −ihè ( · e2 , 1), `m (µ · e1 , 2)i , (4.70) u3 (, µ) = exp −ihè ( · e1 , 2), `m (µ, 1)i . (4.71) To verify (4.66) we decompose and µ into a sum over “monopoles”, i.e. cochains supported on a single site or a single plaquette, respectively. Using the obvious fact that è and `m are ZN -module maps, i.e. è (n1 + 2 , i) = nè (1 , i) + è (2 , i), `m (nµ1 + µ2 , i) = n`m (µ1 , i) + `m (µ2 , i),

(4.72) (4.73)

0 2 and µ1, 2 ∈ Cloc , we conclude that (4.66) holds if and only for all n ∈ ZN , 1, 2 ∈ Cloc if it holds for all pairs of monopole distributions ρ = (, µ) = (δx , δp ), x ∈ (Z2 )0 , p ∈ (Z2 )2 . In fact, we have Y Y ui (δx , δp )(x)µ(p) , ui (, µ) = x∈(Z2 )0 p∈(Z2 )2

and similarly


(, µ) , Mq (`m (µ))Eq (è ())(, µ) Y

Y

445

=

(δx , δp ) , Mq (`m (δp ))Eq (è (δx ))(δx , δp )

(x)µ(p)

.

(4.74)

x∈(Z2 )0 p∈(Z2 )2

Now, for = δx one easily verifies è (δx , j) = −δhx−ej , xi ,

(4.75)

implying è (δx · ei , j) = −δhx−ei −ej , x−ei i . For µ = δp and p = hy, y + e1 , y + e1 + e2 , y + e2 i we get `m (δp , 1) = δhy, y+e2 i , `m (δp , 2) = −δhy, y+e1 i . Plugging this into (4.69)-(4.71) we get 2πi δx, y+e2 − δx, y , u1 = exp N 2πi u2 = exp − δx, y+e2 , N 2πi u3 = exp − δx, y+e1 +e2 , N implying (u1 u2 u3 )−1 = exp

2πi δx, y + δx, y+e1 +e2 . N

(4.76) (4.77)

(4.78) (4.79) (4.80)

(4.81)

Next we look at (4.67)–(4.68) to compute Lm (δp ) = δhy, y+e2 i − δhy−e1 , yi − δhy−e2 , yi + δhy, y+e1 i = −dδy , (4.82) Le (δx ) = −δhx−e1 , xi − δhx−e1 −e2 , x−e1 i + δhx−e1 −e2 , x−e2 i + δhx−e2 , xi = −d∗ δq , (4.83) where q is the oriented plaquette q = hx − e1 − e2 , x − e2 , x, x − e1 i. This implies 2πi Mq (Lm (δp ))Eq (Le (δx ))ρ = exp δx−e1 −e2 , y + δx, y ρ . (4.84) N Comparing (4.81) with (4.84) we have proven (4.66) and therefore Theorem 4.3.6.

We remark that our definition (4.59) is consistent with the translation invariance of the vacuum 0 , since ρ = 0 implies U0 (ei ) = 1l and V0 (ei )0 = 0 . Next we show that our intertwiner connection also intertwines the representations Dρ of the translation group Z2 in Hρ , ∀ρ ∈ Dq . This is formulated most economically by putting M Dq (ei ) := Dρ (ei ), (4.85) ρ∈Dq

implying ˆ for all A ∈ A.

Dq (ei )Πq (A) = Πq (τei (A))Dq (ei ),

(4.86)

446


Proposition 4.3.7. For i = 1, 2 the intertwiner algebra Wq commutes with Dq (ei ). Proof. Putting Uq (ei ) := Vq (ei )Dq (ei ) and using Lemma 4.3.5 we have to show Uq (ei )Eq (`0 ) = Eq (`0 · ei )Uq (ei ), Uq (ei )Mq (`0 ) = Mq (`0 · ei )Uq (ei ),

(4.87) (4.88)

1 . Since Uq (ei ) provides an intertwiner mapping Hρ → Hρ·ei it is again for all `0 ∈ Cloc enough to check these identities on ρ , ∀ρ ∈ Dq . Using (4.58), (4.59) and commuting Mq (`m (µ, i)) to the left, Eq. (4.87) is equivalent to 0

Eq (è ( + d∗ `0 , i))Eq (`0 )(, µ) = eih` ·ei , `m (µ, i)i Eq (`0 · ei )Eq (è (, i))(, µ) . (4.89) To prove (4.89) we compute L(`0 , i) := è ( + d∗ `0 , i) + `0 − `0 · ei − è (, i) = è (d∗ `0 , i) + `0 − `0 · ei ,

(4.90)

1 yielding d∗ L(`0 , i) = 0. Let S(`0 , i) ∈ Cloc be such that d∗ S(`0 , i) = L(`0 , i). Then, by (4.43), Eq. (4.89) is equivalent to 0

0

eih` ·ei , `m (µ, i)i = eihS(` , i),

µi

.

(4.91)

Similarly as in the proof of Theorem 4.3.6 it is enough to check (4.91) for all `0 = δb , b ∈ (Z2 )1 , and all µ = δp , p ∈ (Z2 )2 . Hence, let b = hx, x + ej i and p = hy, y + e1 , y + e1 + e2 , y + e2 i. Then d∗ δb = δx+ej − δx and (4.75) gives L(δhx, x+ej i , i) = è (δx+ej , i) − è (δx , i) + δhx, x+ej i − δhx−ei , x+ej −ei i = δhx, x+ej i − δhx+ej −ei , x+ej i + δhx−ei , xi − δhx−ei , x+ej −ei i(4.92) . Thus, if j = i then L(δhx, x+ej i , i) = 0 implying (4.91), since in this case `m (µ, i) is perpendicular to the direction j. We are left to check the case j 6= i, which gives L(δhx, x+ej i , i) = (−1)j d∗ δq−ei , where q = hx, x + e1 , x + e1 + e2 , x + e2 i. Thus we get for i 6= j, 2πi (−1)j δx−ei , y . (4.93) eihS(δhx, x+ej i ), δp i = exp N On the other hand, Eqs. (4.76)–(4.77) give for i 6= j, `m (δp , i) = (−1)j δhy, y+ej i ,

(4.94)

and hence (4.93)–(4.94) imply (4.91) in the case i 6= j. Thus we have proven (4.87). Equation (4.88) is proven by similar methods. 4.4. Conclusions. In this work we have investigated the dyonic sector structure of 2 + 1dimensional lattice ZN -Higgs models described by the Euclidean action (1.2) in a range of couplings (2.23)-(2.24) corresponding to the “free charge phase” of the Euclidean statistical mechanics model (1.1). We have worked in the Hamiltonian picture by formulating the model in terms of its observable algebra A generated by the time-zero fields. A Euclidean (modified) dynamics αρ = lim Ad TV (ρ) has been defined on A in terms of local (modified) transfer V


447

matrices TV (ρ), where ρ = (ε, µ) is a superposition of electric and magnetic ZN -charge distributions with finite support on the spatial lattice Z2 . Dyonic states ωρ have been constructed as ground states of the modified dynamics αρ on A. The associated charged representations (πρ , Hρ , ρ ) of A extend to irreducible representations of the “dynamic closure” Aˆ ⊃ A, where Aˆ = hA, ti is the abstract ∗-algebra generated by A and a global positive transfer matrix t implementing the “true” (i.e. unmodified) dynamics α0 . πρ and πρ0 are equivalent as representations of ˆ provided their total charges coincide, qρ = qρ0 . We have conjectured that the total A, charges qρ ∈ ZN × ZN indeed label the sectors of the model, i.e. πρ 6∼ πρ0 if qρ 6= qρ0 . The infimum of the energy spectrum (q) of each sector has been shown to be uniquely fixed by the conditions (0) = 0 and (q) = (−q) and the requirement of decaying interaction energies for infinite spatial separation. of the state bundle Bq = [In Sect. 4 we have analyzed structural algebraic aspectsM (ρ, Hρ ), Dq := {ρ | qρ = q}, by constructing on Hq = Hρ a local intertwiner ρ∈Dq

algebra Wq commuting with

M

ρ∈Dq

ˆ The generators of Wq are given by electric πρ (A).

ρ∈Dq

and magnetic “charge transporters”, Eq (b) and Mq (b), localized on bonds b in Z2 and fulfilling local Weyl commutation relations (Theorem 4.1.2). In terms of these charge transporters we have obtained a unitary connection U (0) : Hρ → Hρ0 intertwining πρ and πρ0 for any path 0 : ρ → ρ0 in Dq . The holonomy of this connection is given by ZN -valued phases related to the electric and magnetic charges enclosed by 0 (see (4.43)). Finally, the connection 0 7→ U (0) has been used to construct on each Hρ a unitary ˆ representation of the group of spatial lattice translations acting covariantly on πρ (A) and being intertwined by U (0). We remark that the existence of such representations in the charged sectors of our model is by no means an obvious feature. In [7] the holonomy phases of our connection 0 will be the main ingredient for establishing the anyonic nature of multiparticle scattering states of electrically and magnetically charged particles whose existence has been shown in [1]. Appendix A. A Brief Sketch of the Polymer and Cluster Expansions A.1. Expansions for the vacuum sector. In this Appendix we present the basics of the polymer and cluster expansion developed in [1]. We intend to present here only the most relevant facts to make the main ideas in the proofs of this Appendix understandable. For more details see [1]. Definition A.1.1. For a 1-cochains E with d∗ E = 0 and a 2-cochain D with dD = 0, both with finite support, we define the “winding number of E around D” as 2πi D hu , Ei . (A.1) [D : E] := exp N One easily sees that this definition does not depend of the choice of the particular configuration uD .

448


Let us now prepare the definition of our polymers and their activities. Define the sets n P = P ∈ (Z3 )+2 : P is finite, co-connected and P = supp D, o for some D ∈ (Z3 )2 , dD = 0, D 6= 0 , n B = M ∈ (Z3 )+1 : M is finite, connected and M =

supp E,

o for some E ∈ (Z3 )1 , d∗ E = 0, E 6= 0 ,

where (Z3 )+1 (respectively (Z3 )+2 ) refers to the set of positively oriented bonds (plaquettes) of Z3 and the sets Ptotal = P ∈ (Z3 )+2 finite, so that P = supp D, for some D ∈ (Z3 )2 , dD = 0 , Btotal = M ∈ (Z3 )+1 finite, so that M = supp E, for some E ∈ (Z3 )1 , d∗ E = 0 . Note that the sets Ptotal and Btotal contain the empty set and that the non-empty elements of Ptotal and of Btotal are built up by unions of co-disjoint elements of P, respectively, by unions of disjoint elements of B. One has naturally P ⊂ Ptotal and B ⊂ Btotal . Each non-empty set P ∈ Ptotal and M ∈ Btotal can uniquely be decomposed into disjoint unions P = P1 + · · · + PAP , M = M1 + · · · + MBM (the symbol “+” indicates here disjoint union) where Pi ∈ P and Mj ∈ B. Then, if D ∈ (Z3 )2 is such that supp D = P , there is a unique decomposition D = D1 + · · · + DAP with Di ∈ (Z3 )2 , supp Di = Pi . Moreover if E ∈ (Z3 )1 is such that supp E = M , then there is a unique decomposition E = E1 + · · · + EBM with Ei ∈ (Z3 )1 , supp Ei = Mi . One can also decompose u = uD1 + · · · + uDAP with uDi ∈ (Z3 )1 , duDi = Di . For P ∈ Ptotal and M ∈ Btotal we define the sets D(P ) := {D ∈ (Z3 )2 so that

supp D

= P and dD = 0},

E(M ) := {E ∈ (Z ) so that

supp E

= M and d∗ E = 0}.

3 1

We consider now pairs (P, D) with P ∈ Ptotal and D ∈ D(P ) and pairs (M, E) with M ∈ Btotal and E ∈ D(M ) and define w((P, D), (M, E)) = w((M, E), (P, D)) ∈ {0, . . . , N − 1} as the “ZN -winding number” of (M, E) around (P, D): w((P, D), (M, E)) = w((M, E), (P, D)) := [D : E].

(A.2)

The pairs with P ∈ P and M ∈ B will be the building blocks of our polymers. With the help of w we can establish a connectivity relation between pairs (P, D) with P ∈ P, D ∈ D(P ) and pairs (M, E) with M ∈ B, E ∈ E(M ): we say that (P, D) and (M, E) are “w-connected” if w((P, D), (M, E)) 6= 1 and “w-disconnected” otherwise. We arrive then at the following Definition A.1.2. A polymer γ is formed by two pairs {(P γ , Dγ ), (M γ , E γ )} , with P γ ∈ Ptotal (V ), M γ ∈ Btotal (V ) and Dγ ∈ D(P γ ), E γ ∈ E(M γ ), so that the set


449

γ γ {(P1γ , D1γ ), . . . , (PAγ γ , DA ), (M1γ , E1γ ), . . . , (MBγ γ , EB )} γ γ

(A.3)

formed by the decompositions P γ = P1γ + · · · + PAγ γ , M γ = M1γ + · · · + MBγ γ with γ γ , E γ = E1γ + · · · + EB with Piγ ∈ P(V ), Mjγ ∈ B(V ) and Dγ = D1γ + · · · + DA γ γ γ γ γ γ Di ∈ D(Pi ), Ej ∈ E(Mj ) is a w-connected set. Below, when we write (M, E) ∈ γ and (P, D) ∈ γ we are intrinsically assuming that M ∈ B with E ∈ E(M ) and that P ∈ P with D ∈ D(P ). For a polymer γ = ((P γ , Dγ ), (M γ , E γ )) we call the pair γg := (P γ , M γ ) the geometrical part of γ and the pair γc := (Dγ , E γ ) is the “coloring” of γ. Of course the coloring determines uniquely the geometric part. Each pair (D, E), D ∈ D(P ), E ∈ E(M ) with P ∈ P, M ∈ B is a color for (P, M ). Another important definition is the “size” of a polymer. We define the size of γ by |γ| = |γg | := |P γ | + |M γ |, where |P γ | (respectively |M γ |) is the number of plaquettes (respectively bonds) making up P γ (respectively M γ ). The activity µ(γ) ∈ C of a polymer γ is defined to be       Aγ Bγ  Y  Y Y Y γ   g(Diγ (p)) h(Ejγ (b)) , (A.4) µ(γ) := D : E γ    γ γ i=1

j=1

p∈Pi

b∈Mj

with µ(∅) = 1. For a polymer model we need the notions of “compatibility” and “incompatibility” between pairs of polymers. This is defined in the following way. Two polymers γ and γ 0 are said to be incompatible, γ 6∼ γ 0 , if at least one of the following conditions hold: i)

0

0

There exist Maγ ∈ γg and Mbγ ∈ γg0 , so that Maγ and Mbγ are connected (i.e. there exists at least one lattice point x so that x ∈ ∂b and x ∈ ∂b0 for some bonds b ∈ Maγ 0 and b0 ∈ Mbγ ); 0

0

ii) There exists Paγ ∈ γg and Pbγ ∈ γg0 , so that Paγ and Pbγ are co-connected (i.e. there exists at least one cube c in the lattice so that p ∈ ∂c and p0 ∈ ∂c for some 0 plaquettes p ∈ Paγ and p0 ∈ Pbγ ); 0

0

0

0

iii) There exists (Maγ , Eaγ ) ∈ γ and (Pbγ , Dbγ ) ∈ γ 0 , so that (Maγ , Eaγ ) and (Pbγ , Dbγ ) are w-connected, or the same with γ and γ 0 interchanged. They are said to be compatible, γ ∼ γ 0 , otherwise. We will denote by G(V ) the set of all polymers in V ⊂ Z3 and by Gcom (V ) the set of all finite sets of compatible polymers. We want to express the vacuum expectation of classical observables in terms of our polymer expansion. We consider the following Definition A.1.3. Let α be a 1-cochain and β 2-cochain, both with finite support. Define the classical observable Y g((du + β)(p)) 2πi hα, ui . (A.5) B(α, β) := exp − N g(du(p)) p Any classical observable can be written as a linear combination of such functions.

450


For a finite volume V ⊂ Z3 (say, a cube) we have the following10 hB(α, β)iV =

1 ZV1

X



X

[D − β : E − α] 



Y

g(D(p)) 

p∈suppD

D∈V 2 E∈V 1 d(D−β)=0 d∗ (E−α)=0

Y

 h(E(b)) .

b∈suppE

(A.6) Here the normalization factor ZV1 is given by X ZV1 = µ0 , Y

in multi-index notation, i.e. µ0 :=

(A.7)

0∈Gcom (V )

µ(γ). We will often identify the elements of Gcom

γ∈0

with their characteristic functions. The cochains D appearing in the sums in (A.6) can uniquely be decomposed in such a way that D = D0 + D1 with d(D0 − β) = 0 and dD1 = 0 and so that supp D0 is co-connected and co-disconnected from supp D1 . If dβ = 0 we choose D0 = 0. Analogously, the cochains E appearing in the sums in (A.6) can be decomposed uniquely in such a way that E = E0 + E1 with d∗ (E0 − α) = 0 and d∗ E1 = 0 and so that supp E0 is connected and disconnected from supp E1 . If d∗ α = 0 we choose E0 = 0. We denote by C1 (α) the set of the supports of all such E0 ’s, for a given α and by C2 (β) the set of the supports of all such D0 ’s, for a given β. For d∗ α = 0 we have C1 (α) = ∅ and for dβ = 0 we have C2 (β) = ∅. We define the sets of pairs Conn1 (α, V ) := (M, E), so that M ∈ C1 (α) and E ∈ V 1 , (A.8) with supp E = M and d∗ E = d∗ α} , Conn2 (β, V ) := (P, D), so that P ∈ C2 (β) and D ∈ V 2 , (A.9) with supp D = P and dD = dβ} . We then write

X

hB(α, β)iV =

[D − β : E − α]

Conn1 (α, V ) (P, D)∈Conn2 (β, V )

(M, E)∈

 ×

Y p∈P

" g(D(p))

Y

# h(E(b))

X 0∈Gcom

a0(M, E), α b0(P, D), β µ0 X

b∈M

µ0

,

0∈Gcom

(A.10) for

a(M, E), α (γ) :=

0, if M γ is connected with M, , γ [D : E − α] , otherwise

and 10

For simplicity we will neglect some boundary terms, which can be controlled with more work.

(A.11)


b(P, D), β (γ) :=

451

0, if P γ is co-connected with P . γ [D − β : E ] , otherwise

(A.12)

It is for many purposes useful to write the normalization factor ZV1 in the form    X  c 0 µ0 . (A.13) ZV1 = exp   0∈Gclus (V )

Let us explain the symbols used above. Our notation is close to that of [2]. Gclus (V ) is the set of all finite clusters of polymers in V , i.e. an element 0 ∈ Gclus is a finite set of (not necessarily distinct) polymers building a connected “incompatibility graph”. An incompatibility graph is a graph which has polymers as vertices and where two vertices are connected by a line if the corresponding polymers are incompatible. We will often identify elements 0 ∈ Gclus with functions 0: G → N, where 0(γ) is the multiplicity of γ in 0 ∈ Gclus . The coefficients c0 are the “Ursell functions” and are of purely combinatorial nature. They are defined (see [2] and [14]) by c0 :=

∞ X (−1)n+1 n=1

n

Nn (0),

(A.14)

where Nn (0) is the number of ways of writing 0 in the form 0 = 01 + · · · + 0n where 0 6= 0i ∈ Gcom , i = 1, . . . , n. Relation (A.13) makes sense provided the sum over clusters is convergent. As discussed in [2] and [1] a sufficient condition for this is kµk ≤ kµkc , where kµk := sup |µ(γ)|1/|γ| , and kµkc is a constant defined in [2]. By (A.4), γ∈G

|γ| |µ(γ)| ≤ max{g(1), . . . , g(N − 1), h(1), . . . , h(N − 1)} ,

(A.15)

which justifies the conditions (2.23)–(2.24). Calling Conn1 (α) := Conn1 (α, Z3 ), Conn2 (β) := Conn2 (β, Z3 ) and Gclus := Gclus (Z3 ), we can also write the thermodynamic limit of hB(α, β)iV as

hB(α, β)i =



X

[D − β : E − α] 

Conn1 (α) Conn2 (β)

(M, E)∈ (P, D)∈

× exp

X

Y

"

Y

g(D(p))

p∈P

# h(E(b))

b∈M

c0 a0(M, E), α b0(P, D), β − 1 µ0

! .

(A.16)

0∈Gclus

The presence of the phases [D − β : E − α] is an important feature of this last expression and is responsible for the emergence of the anyonic statistics. Note that for α and β such that dβ = 0 and d∗ α = 0 the expectation hB(α, β)i is proportional to [β : α], i.e. to the winding number of β around α. Let 0 be a cluster of polymers. We say that a polymer γ is incompatible with 0, i.e. γ 6∼ 0, if there is at least one γ 0 ∈ 0 with γ 6∼ γ 0 . For two clusters 0, 00 we have 0 6∼ 00 if there is at least one γ ∈ 0 with γ 6∼ 00 . For the polymer system discussed in this work we have the following result:

452


Theorem A.1.4. There is a convex, differentiable, monotonically decreasing function F0 : (a0 , ∞) → R+ , for some a0 ≥ 0, with lima→∞ F0 (a) = 0 such that, for all sets of polymers 0, and for all a > a0 , X

e−a|γ| ≤ F0 (a) k0k,

(A.17)

γ6∼0

P where k0k = 0(γ 0 )|γ 0 |, 0(γ 0 ) being the multiplicity of γ 0 in 0. Once inequality (A.17) has been established, it has been proven in [2], Appendix A.1, that the two following results hold: X

|c0 | |µ0 | ≤ F1 (− ln kµk) k00 k,

(A.18)

0∈Gclus 06∼00

X

|c0 | |µ0 | ≤

0∈Gclus 06∼00 k0k≥n

kµk kµc k

n k00 kF0 (ac ),

(A.19)

where ac and kµc k > 0 are constants defined in [2], F1 : (ac + F0 (ac ), ∞) → R+ is the solution of F1 (a + F0 (a)) = F0 (a) and kµk := supγ |µ(γ)|1/|γ| . For a proof we refer the reader to [1] and [2]. The inequalities (A.18) and (A.19) are of central importance in the theory of cluster expansions and Pare often used for proving theorems. For instance, (A.19) tells us that the sums like 0 |c0 ||µ0 | involving only clusters with size larger than a certain n (and which are incompatible with some 00 fixed) decay exponentially with n. A.2. Expansions for the dyonic sectors. Let us now present the corresponding expansions for the states ωρ for ρ ∈ D0 . To each B ∈ F0 we can associate a classical observable Bcl, ρ = Bcl, ρ (dϕ − A) (see (3.31)). A possible but non-unique choice is (see [1] and [2]) Bcl, ρ

T rHV F(ϕ(0), A(0)), (ϕ(1), A(1)) B TV (ρ) , = T rHV F(ϕ(0), A(0)), (ϕ(1), A(1)) TV (ρ)

(A.20)

where F(ϕ(0), A(0)), (ϕ(1), A(1)) =

X

|ϕ(0), A(0)ihϕ(1), A(1)|,

(A.21)

(ϕ(0), A(0)), (ϕ(1), A(1))

and where ϕ(k), A(k) refers to the variables in the k th euclidean time plane. Since any such classical observable can be written as a finite linear combination of the functions B(α, β) previously introduced (with coefficients eventually depending on ρ) we concentrate on expectations of such functions. Proceeding as in the previous sections we can express hB(α, β)iρ defined in (3.31) as


hB(α, β)iρ = X Conn1 (α) Conn2 (β)

 [D − β : E − α] [D − β : −˜] [−µ˜ : E − α] 

"

×

Y

#

X

h(E(b)) exp

b∈M

Y

 g(D(p))

p∈P

(M, E)∈ (P, D)∈

453

c0 a0(M, E), α b0(P, D), β

−1

! a0 b0µ µ0

,

(A.22)

0∈Gclus

˜ n)) := µ(x) for all x ∈ (Z2 )2 and n ∈ Z where µ˜ is the 2-cochain on Z3 defined by µ((x, 3 and ˜ in the 1-cochain on Z defined by ˜((y, n + 1/2)) := (y), for all y ∈ Z2 and n ∈ Z. Here (y, n + 1/2) indicates the vertical bond in (Z3 )+1 whose projection onto Z2 is y and is located between the euclidean time-planes n and n + 1. Moreover, we defined a (γ) := [Dγ : −˜], bµ (γ) := [−µ˜ : E γ ].

(A.23) (A.24)

The cochains µ˜ and ˜ do not have finite support but, since the polymers are finite, the right hand side of the last two expressions can be defined using some limit procedure, for instance, by closing µ˜ and ˜ at infinity by adding, before the thermodynamic limit is taken, the cochains sµ and s to them. Since the polymers are finite, the limit does not depend on the particular sµ and s chosen.. The same can be said about the winding numbers [D − β : −˜] and [−µ˜ : E − α] in (A.22). Concerning the sum over clusters in (A.22), the following estimate can be established: Proposition A.2.1. For (M, E) ∈ Conn1 (α) and (P, D) ∈ Conn2 (β), one has X 0 0 0 0 0 c0 a(M, E), α b(P, D), β −1 a bµ µ ≤ c0 (|M |+|supp α|) + (|P |+|supp β|) , 0∈Gclus

(A.25) where c0 is a positive constant. Proof. One has X 0 0 0 0 0 c0 a(M, E), α b(P, D), β −1 a bµ µ ≤ 0∈Gclus

≤

X

|c0 | a0(M, E), α b0(P, D), β −1 |µ0 |

0∈Gclus

X

0∈Gclus 06∼γ1

|c0 | |µ0 | +

X

|c0 | |µ0 |

0∈Gclus 06∼γ2

≤ F1 (− ln kµk)(kγ1 k + kγ2 k),

(A.26)

by (A.18), where γ1 = (supp (E + α), E + α) and γ2 = (supp (D + β), D + β). Clearly, kγ1 k ≤ |M | + |sup α| and kγ2 k ≤ |P | + |sup β|. Proposition A.2.1 has a simple corollary

454


Proposition A.2.2. If gc and hc are small enough and if min{αl , βl } ≥ n, where αl := inf{|M |, (M, E) ∈ Conn1 (α)}, βl := inf{|P |, (P, D) ∈ Conn2 (β)},

(A.27) (A.28)

|hB(α, β)iρ | ≤ ca e−cb n ,

(A.29)

then

for positive constants ca and cb . Proof. Using the representation (A.22) of hB(α, β)iρ in terms of cluster expansions and the estimate (A.25), one gets X | gc|P | h|M exp c0 (|M | + |supp α|) + (|P | + |supp β|) . |hB(α, β)iρ | ≤ c Conn1 (α) Conn2 (β)

(M, E)∈ (P, D)∈

(A.30) By standard arguments one has X

(hc ec0 )|M | ≤ const. e−ca n/2 ,

(A.31)

(M, E)∈Conn1 (α)

for some positive ca , provided hc is small enough and, analogously, X (gc ec0 )|P | ≤ const. e−ca n/2 ,

(A.32)

(P, D)∈Conn2 (β)

provided gc is small enough. This proves the proposition.

An important particular case of (A.22) occurs when dβ = d∗ α = 0. In this case we get simply ! X 0 0 0 0 0 c0 a(∅, 0), α b(∅, 0), β − 1 a bµ µ . hB(α, β)iρ = [β : α] [β : ˜] [µ˜ : α] exp 0∈Gclus

(A.33) Notice the presence of the ZN -factors [β : α] [β : ˜] [µ˜ : α] related to winding numbers involving α, β and the background charges ρ.

B. The Remaining Proofs B.1. Proof of Proposition 3.1.1. Let EG be the projection Floc → Aloc . Since ωV, ρ is gauge invariant, it is enough to prove the existence of lim ωV, ρ Aloc . Expectations V ↑Z2

like ωV, ρ (A) for A ∈ Aloc can be written as finite linear combinations of the previously introduced classical expectations hB(α, β)iV, ρ , whose thermodynamic limit was described in Subsect. A.2. To show that ωρ Aloc is a ground state with respect to αρ we first notice that, by the representation of ωV, ρ (A), A ∈ Aloc in terms of cluster expansions we can write, in analogy to (3.25),


T rHV TV (ρ)n ATV (ρ)n+1 EVρ ωV, ρ (A) = lim , n→∞ T rHV TV (ρ)2n+1 EVρ

455

A ∈ F(V ),

(B.1)

and, hence, for V large enough, one has T rHV TV (ρ)n A∗ ATV (ρ)TV (ρ)n EVρ , n→∞ T rHV TV (ρ)2n+1 EVρ

ωV, ρ (A∗ αρ (A)) = lim

A ∈ F(V ).

(B.2) Now, by (3.26) and (2.30)-(2.31), EVρ is a positive operator and so, the numerator in (B.2) is clearly positive. This proves that ωρ (A∗ αρ (A)) ≥ 0. To show that ωρ (A∗ αρ (A)) ≤ ωρ (A∗ A) we can make use of Lemma 2.3.2 and show that ωρ fulfills the cluster property with respect to αρ . We can represent ωρ (A∗ αρn (A)) in terms of classical expectations of the classical functions associated to the operator A∗ αρn (A). These classical expectations can be written as a finite linear combination of expectations like hB(αn , βn )iρ , where the local cochains αn and βn can be written, for n large enough, as sums αn = α(0) + α(n) and βn = β(0) + β(n), where the local cochain α(n) (respectively β(n)) is the complex conjugate of the translate of α(0) (respectively, of β(0)) by n units in the euclidean time direction. Recalling now the representation (A.22) of hB(α, β)iρ in terms of cluster expansions we notice that, by Proposition A.2.2, the contributions of sets (M, E) ∈ Conn1 (α) connecting the support of α(0) to the support of α(n) decay exponentially with n, the same happening with the contribution of the sets (P, D) ∈ Conn2 (β) connecting the support of β(0) to the support of β(n). The only surviving terms, after taking the limit n → ∞ correspond to sets (M, E) ∈ Conn1 (α) and sets (P, D) ∈ Conn2 (β) connecting the supports of α(0), α(n), β(0) and β(n) with themselves. The contributions of these last terms converges to the product hB(α(0), β(0))iρ hB(α(n), β(n))iρ . This implies that ωρ (A∗ αρn (A)) → ωρ (A∗ )ωρ (A), n → ∞, thus proving the ground state property. The general cluster property ωρ (Aαρn (B)) → ωρ (A)ωρ (B), A, B ∈ Aloc , follows from the same arguments. B.2. Proof of Propositions 3.1.2 and 3.1.5 . In order to prove Proposition 3.1.2 we have to study lim hB(α, β)iρ−ρ0 a . We recall the representation (A.22) of hB(α, β)iρ−ρ0 a a→∞ and notice that, since c0 a0(M, E), α b0(P, D), β − 1 a0−0 a b0µ−µ0 a µ0 ≤ c0 a0(M, E), α b0(P, D), β − 1 µ0 , (B.3) which is summable, we can write X c0 a0(M, E), α b0(P, D), β − 1 a0−0 a b0µ−µ0 a µ0 = lim a→∞

X 0∈Gclus

0∈Gclus

c0 a0(M, E), α b0(P, D), β − 1 ( lim a0−0 a b0µ−µ0 a )µ0 . a→∞

(B.4)

But, clearly, lim a0−0 a b0µ−µ0 a = a0 b0µ for every cluster 0, since the polymers are a→∞ finite. The limit does not depend on the particular way as a → ∞. This shows that the representation (A.22) holds also for ρ ∈ Dq , q 6= 0, and can be used to describe ωρ (A), A ∈ Aloc with ρ ∈ Dq , q 6= 0. The cluster property, and consequently the ground state property for A ∈ Aloc , can be proven in the same way as in the previous case. The proof of Proposition 3.1.5 is analogous to the proof of Proposition 3.1.2 and does not need to

456


be repeated, but in the next subsection we present the proof of the more general Theorem 3.1.6. n (A0 τa−1 (B 0 )) for B.3. Proof of Theorem 3.1.6 . Let us consider ωρ(a) τa−1 (B)Aαρ(a) n (A0 ) = A, B, A0 and B 0 ∈ Aloc . According to (3.16) one has, for |a| large enough, αρ(a) n αρn1 (A0 ) and αρ(a) (τa−1 (B 0 )) = αρn2 a (τa−1 (B 0 )) = τa−1 (αρn2 (B 0 )). Hence, for |a| large enough, n (A0 τa−1 (B 0 )) = ωρ(a) Aαρn1 (A0 )τa−1 (Bαρn2 (B 0 )) . ωρ(a) τa−1 (B)Aαρ(a)

(B.5)

The representation of the last expectation in terms of classical expectations is given by finite sums of classical expectations like hB(α(a), β(a))iρ(a) , where α(a) = α1 + α2 a and β(a) = β1 + β2 a, for local cochains α1, 2 and β1, 2 , where the cochains α1 and β1 are related to the operators Aαρn1 (A0 ) and where the cochains α2 and β2 are related to the operators Bαρn2 (B 0 ). Let us now consider the representation of hB(α(a), β(a))iρ(a) in terms of cluster expansions. It is given by hB(α(a), β(a))iρ(a) = X

˜ [D − β(a) : E − α(a)] D − β(a) : −(a)

Conn1 (α(a)) Conn2 (β(a))

(M, E)∈ (P, D)∈

× [−µ(a) ˜ : E − α(a)] × exp

X

hQ p∈P

i Q

g(D(p))

b∈M

h(E(b))

c0 a0(M, E), α(a) b0(P, D), β(a) − 1 a (a)0 bµ (a)0 µ0

(B.6)

! ,

0∈Gclus

Let us assume |a| so large as to include the set sup α1 ∪ sup β1 ∪ sup α2 ∪ sup β2 in the ball of radius |a|/8 centered at the origin and let us consider two infinite cylinders C1 and C2 = C1 + a with radius |a|/4, parallel to the euclidean time axis and extending from +∞ to −∞. The cylinder C1 contains the set sup α1 ∪ sup β1 and C2 contains the set (sup α2 ∪ sup β2 ) + a. By construction, the sets Conn1 (α(a)) and Conn2 (β(a)) will contain some elements which are entirely contained in C1 ∪ C2 and some which are not. These last ones must have a size larger than |a|/4 and, therefore, by arguments analogous to those used in the proof of Proposition A.2.2, their contribution to (B.6) decay exponentially with |a|. So, up to an exponentially falling error, we can restrict the sums over Conn1 (α(a)) and Conn2 (β(a)) to elements contained only in C1 ∪ C2 . The next question is, what happens to the sums over clusters, provided the elements M and P are now contained in C1 ∪ C2 ? Let us denote Ma := M ∩ Ca and Pa := M ∩ Ca , for a = 1, 2, with M = M1 ∪ M2 and P = P1 ∪ P2 , as disjoint unions and let E = E1 + E2 and D = D1 + D2 with sup Ea = Ma and sup Da = Pa for a = 1, 2. We claim that the difference


X

457

c0 a0(M, E), α(a) b0(P, D), β(a) − 1 a (a)0 bµ (a)0 µ0 −

0∈Gclus

X

c0 a0(M1 , E1 ), α1 b0(P1 , D1 ), β1 − 1 a01 b0µ1 µ0

(B.7)

0∈Gclus

+

P

0 0 0 0 0 a c b − 1 a b µ 0 2 a µ 2 a (M2 , E2 ), α2 a (P2 , D2 ), β2 a 0∈Gclus

decays exponentially to zero with |a|. For, notice that the difference above is given by sums over clusters connecting C1 to C2 , having thus a size larger than |a|/2. Therefore, by (A.19), their contribution decay exponentially with |a|. Using now the exact factorization ˜ [D − β(a) : E − α(a)] D − β(a) : −(a) "  # Y Y g(D(p)) h(E(b)) = ˜ : E − α(a)]  [−µ(a) p∈P

b∈M

[D1 − β1 : E1 − α1 ] [D1 − β1 : −˜1 ] "  # Y Y g(D(p)) h(E(b)) × [−µ˜ 1 : E1 − α1 ]  p∈P1

(B.8)

b∈M1

[D2 − β2 a : E2 − α2 a] [D2 − β2 a : −˜2 a]  " # Y Y g(D(p)) h(E(b)) , [−µ˜ 2 a : E2 − α2 a]  p∈P2

b∈M2

valid in C1 ∪ C2 , we get using the translation invariance of the cluster expansions and taking |a| → ∞,

lim hB(α(a), β(a))iρ(a) = hB(α1 , β1 )iρ1 hB(α2 , β2 )iρ2 .

|a|→∞

With this, the proof of Theorem 3.1.6 is complete.

(B.9)

B.4. Proof of Proposition 3.4.5. Here we establish part i of Proposition 3.4.5. Let ρ = (, µ) and ρ0 = (0 , µ0 ) and define ρ0 = (0 , µ0 ) = ( − 0 , µ − µ0 ). We need (2ν+1) (2ν) first an expression in terms of the cluster expansions for the ratio ωρ (Xρ, ρ0 )/ωρ (Xρ, ρ0 ). Using a pictorial notation, this ratio can be written in terms of classical expectations as

458


........................................................................... . .. .. .. .. è or `m .. .. .. .. 6 .. .. .. .. . .. .. 2ν + 1... .. .. CC .. .. .. .. C .. .. C .. .. ? C .. .. 0 .. ρ−ρ C .. C ............................................................................... C ρ

C C

........................................................................... . . .. .. .. . è or `m .. .. .. .. 6 .. .. . .. .. .. .. 2ν ... . .. .. CC .. .. .. C .. .. .. C .. . ? C . .. 0 . ρ−ρ C .. . C ................................................................................ C ρ

C

(2ν+1) ωρ (Xρ, ρ0 ) (2ν) ωρ (Xρ, ρ0 )

=

C

C

C

C CC

.

(B.10)

C

C

C C

C

CC

The infinite vertical lines indicate the background charges ρ and the finite loops are constructed over the charge distribution ρ − ρ0 . Their horizontal lines represent the strings è and/or `m used in the definition (3.50) and their vertical lines have length 2ν + 1 in the numerator and 2ν in the denominator, respectively. Notice that ρ and ρ − ρ0 may have a non-empty overlap, a circumstance not shown in the figure for reasons of clarity. The next step is to find an expansion for the last expression in terms of our cluster expansions. The result is ( ) (2ν+1) X ωρ (Xρ, 0 0 0 ρ0 ) 0 0 0 0 = exp c0 a0 , 0, 2ν+1 bµ0 , 0, 2ν+1 − a0 , 0, 2ν bµ0 , 0, 2ν a bµ µ , (2ν) ωρ (Xρ, ρ0 ) 0 (B.11) where, in an almost self-explanatory notation, a0 , α, β (γ), α < β, represents the winding number of the magnetic part of the polymer γ with the electric loop built by the horizontal strings è located at euclidean times α and β ∈ Z and by the vertical electric lines located over the support of 0 with length β − α. The quantity bµ0 , α, β is defined analogously. The right-hand side of (B.11) can be written as ) ( X 0 0 0 0 0 0 0 c0 a0 , 0, 1 bµ0 , 0, 1 − 1 a0 , −2ν, 0 bµ0 , −2ν, 0 a bµ µ , (B.12) exp 0

where, above, we used the factorization properties


a0 , α, β (γ)a0 , β, δ (γ) = a0 , α, δ (γ), bµ0 , α, β (γ)bµ0 , β, δ (γ) = bµ0 , α, δ (γ),

459

(B.13) (B.14)

for α < β < δ ∈ Z and we used translation invariance. Taking the limit ν → ∞ of expression (B.12) is easy and gives ( ) X kTρ (ρ0 )k = exp c0 a00 , 0, 1 b0µ0 , 0, 1 − 1 a00 , −∞, 0 b0µ0 , −∞, 0 a0 b0µ µ0 , kTρ (ρ)k 0 (B.15) where a0 , −∞, 0 (γ) = lim a0 , −j, 0 (γ), etc., which is a well defined limit for each γ, j→∞

since the polymers are finite (for each γ, the limit is reached at finite j). kTρ1 (a) (ρ2 (a))k using its repreNext, we are interested in studying the limit lim |a|→∞ kTρ2 (a) (ρ2 (a))k sentation in terms of cluster expansions. The main technical problem we have to confront is the fact that, if ρ1 − ρ2 have a non-zero total charge, the strings è and `m have to connect elements of the support of ρ1 − ρ2 with elements of the support of (ρ1 − ρ2 ) · a and have, hence, a length which increases with |a|. The crucial observation is, however, that the left hand side of (B.15) does not depend on the strings è and `m , although this independence cannot apparently be seen from the representation in terms of cluster expansions. Let us consider two cylinders C1 (r) and C2 (r) = C1 (r)+a, such that C1 (r) is centered on the euclidean time axis, extending from −∞ to ∞ and has a radius r. Denote by r0 the largest distance from the set sup (ρ1 ) ∪ sup (ρ2 ) to the origin of the lattice and consider |a| large enough so that sup (ρ1 ) ∪ sup (ρ2 ) is contained in C1 (|a|/8) (by taking, say, |a| > 16r0 ). We first observe that the sum over clusters contained in Z3 \ (C1 (|a|/8) ∪ C2 (|a|/8)) does not contribute to (B.12). This can be seen at best in (B.11) by noticing that: 1) clusters contained in Z3 \(C1 (|a|/8)∪C2 (|a|/8)) crossing the t = 0 euclidean plane and having a side smaller than 2ν have a zero contribution (for them, one has a00 , 0, 2ν+1 b0µ0 , 0, 2ν+1 = a00 , 0, 2ν b0µ0 , 0, 2ν ); 2) clusters contained in Z3 \ (C1 (|a|/8) ∪ C2 (|a|/8)) crossing the t = 2ν euclidean plane, having a side smaller than 2ν and having a non-zero contribution cancel with their translates by one unit in euclidean time direction; 3) the only surviving clusters in Z3 \ (C1 (|a|/8) ∪ C2 (|a|/8)) must cross the planes t = 0 and t = 2ν, and therefore, their side is larger than 2ν and their contribution decays exponentially when the limit ν → ∞ is taken. It remains to consider two classes of clusters: a) those entirely contained in C1 (|a|/4) ∪ C2 (|a|/4) and having a non-empty intersection with C1 (d0 ) ∪ C2 (d0 ) for a fixed d0 with r0 < d0 < a/8 and b) those having a non-empty intersection with both C1 (d0 ) ∪ C2 (d0 ) and Z3 \ (C1 (|a|/4) ∪ C2 (|a|/4)). The contribution to the clusters belonging to class b decays exponentially with |a|. For, note that the clusters which give a non-zero contribution to (B.12) must either cross the t = 0 plane or the t = 1 plane (or eventually both). The clusters of this sort having a non-empty intersection with both C1 (d0 ) ∪ C2 (d0 ) and Z3 \ (C1 (|a|/4) ∪ C2 (|a|/4)) must have a size larger that |a|/8 and , hence, by (A.19), their contribution decays exponentially with |a|. It remains now to consider the clusters belonging to class a above. They are entirely contained inside one of the cylinders C1 (|a|/4) or C2 (|a|/4). Since we have freedom to choose the strings è and `m at will, we choose them depending on a such that, inside of C1 (|a|/4) \ C1 (d0 ) and C2 (|a|/4) \ C2 (d0 ) they run parallel to a fixed direction, say

460


to the positive x-axis of Z2 . Now, taking the limit |a| → ∞ is straightforward and gives cρ1 , ρ2 independent on the way the sequence a goes to infinity. The result is that cρ1 , ρ2 = dρ1 , ρ2 d−ρ1 , −ρ2 , where X dρ1 , ρ2 := exp c0 a012 , 0, 1; ∞ b0µ12 , 0, 1; ∞ − 1 0 0∩C1 (d0 )6=∅

a012 , −∞, 0; ∞ b0µ12 , −∞, 0; ∞ a01 b0µ1 µ0

(B.16)

,

for any sufficiently large d0 , where ρ1 − ρ2 = (12 , µ12 ) and where a12 , α, β; ∞ (γ) = lim a(1 −2 )−(1 −2 )·a, α, β (γ), |a|→∞

(B.17)

etc., where α < β ∈ Z and in a(1 −2 )−(1 −2 )·a, α, β (γ) the strings è depend on a in the way described above, i.e. such that, inside of C1 (|a|/4) \ C1 (d0 ) and C2 (|a|/4) \ C2 (d0 ) they point parallel to the positive x-axis of Z2 . Note that, for each γ, the limit above is reached at finite values of |a|. The condition 0 ∩ C1 (d0 ) 6= ∅ means that the geometrical part of the cluster 0 must have a non-empty intersection with the cylinder C1 (d0 ). Note also that the convergence of the sum over clusters in (B.16) can be shown using the fact that the contributing clusters have a non-empty intersection with C1 (d0 ) and with the t = 0 and/or t = 1 euclidean time slices together with the exponential decay provided by (A.19). We can say, for instance, that the sum over clusters in (B.16) can be bounded by const.

∞ X

X

t=−∞

0

0∩(C1 (d0 )∩Tt )6=∅ k0k≥t

|c0 | |µ0 | ≤ const.

∞ X

e−ca |t| < ∞

(B.18)

t=−∞

with some positive constant ca , where Tt is the euclidean time-plane at euclidean time t. Using the representation above in terms of cluster expansions one can also easily show that d−ρ1 , −ρ2 = dρ1 , ρ2 . The next problem is to prove the factorization property (3.68). The arguments used are analogous to those leading to (B.16). We can namely prove that lim dρ1 +ρ01 ·b, ρ2 +ρ02 ·b = b→∞

dρ1 , ρ2 dρ01 , ρ02 . This can be obtained using the representation (B.16) with d0 depending on b such that C1 (d0 ) contains sup ρ1 ∪ sup ρ2 ∪ sup (ρ01 · b) ∪ sup (ρ02 · b). We consider again cylinders D1 (|b|/4) = C1 (|b|/4) and D2 (|b|/4) = D1 (|b|/4) + b, both contained in C1 (d0 (b)), with D1 (|b|/4) containing the set sup ρ1 ∪ sup ρ2 and D2 (|b|/4) containing the set sup (ρ01 · b) ∪ sup (ρ02 · b) for some |b| large enough. Repeating the previous arguments, we can neglect contributions from clusters contained outside of Z3 \ (D1 (|b|/4) ∪ D2 (|b|/4)) and take the pieces of the strings è and `m which join the supports of ρ1 and ρ2 with the supports of ρ01 · b and ρ02 · b so that they again run parallel to the x-axis at sufficiently large distances. The desired relation will follow again from the usual clustering properties of the cluster expansions established above. B.5. Completing the proof of Proposition 3.3.2. We will here complete some missing points in the proof of Proposition 3.3.2. The ideas are actually contained in [2] and therefore we will concentrate only on the more relevant details.


461

6 l1

m

6

è or `m

n

n

l1

è or `m

− 21

?? 66

− 21 0

è or `m

? m

l2

? 1 0 0 0 Fig. 1. Schematic representation of the expression a0 l1 bl1 − 2 al1 , l2 bl1 , l2 − l2 (below)

1 2

and the loops l1 (above) and

.. .. .. è or `m

ρ

6 b

? a6 ?

ρ0

0 è or `m .. .. .. ..

Fig. 2. Schematic representation of the partial replacement of the infinite vertical line representing the background charge distribution ρ by ρ0 . The connections are performed at euclidean time planes b and −a, 0 ≤ a < b, through the strings è and/or `m

462


To show that the sequence of unit vectors 8nρ, ρ0 , n ∈ N is a Cauchy sequence it is enough to show that, for any > 0, one has |(8nρ, ρ0 , 8m ρ, ρ0 ) − 1| < , provided n is large enough, with m > n. The scalar product (8nρ, ρ0 , 8m ρ, ρ0 ) can be expressed as the exponential of a sum over clusters and, hence, it is enough to show that this sum is small enough provided n is large enough, with m > n. This sum can be written as X 1 0 1 0 0 0 a0m, m b0m, m a0ρ b0ρ µ0 . c 0 a l 1 bl 1 − a l 1 , l 2 b l 1 , l 2 − (B.19) 2 2 0 Above a0l1 , b0l1 , a0l1 , l2 and b0l1 , l2 are the electric and magnetic winding numbers on the loops l1 and l1 ∪ l2 schematically represented in Fig. 1 (where l2 = θl1 , θ meaning reflection on the t = 0 euclidean time plane). Also above a0a, b and b0a, b (with 0 ≤ a < b) are the electric and magnetic winding numbers around the infinite vertical lines schematically represented in Fig. 2. By a straightforward inspection we can verify that a cluster 0 with a size smaller than n with a non-trivial winding number with, say, the loop l1 are canceled in the sum (B.19) by the contribution of the reflected cluster θ0. The contribution of the clusters entirely contained between the time-slices n and m and the contribution of the clusters entirely contained between the time-slices −n and −m also cancel mutually. The only surviving clusters must have non-trivial winding numbers with both l1 and l2 simultaneously and must cross both planes at time n and −n. Therefore, they must have a size which increases with n. By estimate (2.31) their contributions disappear when n → ∞ uniformly m, completing thus the proof. B.6. Proof of Theorem 4.1.1. Let us start proving i). We will first consider the case where d∗ l1 6= 0 and d∗ l2 6= 0. Without loss, we will take l1 and l2 as having support on single lattice links. According to the definitions we have Eˆq (l1 + l2 )ρ = φel ρ1, 2 (l1 + l2 )

πρ1, 2 αρn Aρ1, 2 , ρ ((l1 + l2 ), 0) n→∞ N1 (n)

= lim with

ρ1, 2

N1 (n) := πρ1, 2 αρn Aρ1, 2 , ρ ((l1 + l2 ), 0) ρ1, 2

and

Eˆq (l1 )Eˆq (l2 )ρ = lim Eˆq (l1 )

πρ2 αρp Aρ2 , ρ (l2 , 0)

πρ2

where ρi = ρ + (d∗ li , 0), i = 1, 2 and ρ1, 2 = ρ + (d∗ (l1 + l2 ), 0). The vector in the right hand-side can be written as πρ1, 2 αρp Aρ2 , ρ (l2 , 0) αρq 2 Aρ1, 2 , ρ2 (l1 , 0) ρ1, 2 , lim lim p→∞ q→∞ N2 (p)N3 (q) where N1 (p) and N2 (q) are the normalization factors

(B.20)

(B.21)

ρ2

Aρ2 , ρ (l2 , 0) ρ2 πρ1, 2 αρp Aρ2 , ρ (l2 , 0) φel ρ (l1 ) 1, 2 , = lim p

p→∞ πρ2 αρ Aρ2 , ρ (l2 , 0) ρ2 p→∞

αρp

,

(B.22)

(B.23)


463

N2 (p) := πρ2 αρp Aρ2 , ρ (l2 , 0) ρ2 ,

(B.24)

N3 (q) := πρ1, 2 αρq 2 Aρ1, 2 , ρ2 (l1 , 0) ρ1, 2 .

(B.25)

and

The scalar product in (4.16) can now be written as ρ1, 2 , πρ1, 2 (A) ρ1, 2 , N1 (n)N2 (p)N3 (q)

lim lim lim

n→∞ p→∞ q→∞

(B.26)

with ∗

A := αρn Aρ1, 2 , ρ ((l1 + l2 ), 0)

αρp Aρ2 , ρ (l2 , 0) αρq 2 Aρ1, 2 , ρ2 (l1 , 0) .

(B.27)

After expressing the expectation values above in terms of classical expectations (which involve only closed loops) and these in terms of cluster expansions, we arrive at the following expression: ! X 0 1 0 0 0 0 0 0 0 a + a3 + a4 − a5 aρ1, 2 bρ1, 2 , c 0 µ a1 − lim lim lim exp n→∞ p→∞ q→∞ 2 2 0 (B.28) where ai (γ) represent winding numbers of γ with respect to the loops successively presented in Fig. 3. The quantities aρ1, 2 (γ) and bρ1, 2 (γ) are electric and magnetic winding numbers with respect to the background charge ρ1, 2 . q p n

− 21

l1 l 2

− 21

+ 21

− 21

0

−n l1 l 2

−p l2 l1 l1

l1

−q

1 0 0 0 0 Fig. 3. Pictorial representation of the expression a0 1 − 2 a2 + a3 + a4 − a5 appearing in (B.38). The ai ’s are winding numbers with respect to the sets of loops presented in the picture (counted from the left to the right and separated by the associated factor ±1/2). The vertical lines are parallel to the euclidean time-axis. The open loops cross d∗ l1 and extend to the euclidean time infinity. At the right we indicate the different time planes

We have to perform a detailed analysis of the sum over clusters appearing in (B.28). For the sake of brevity we will sketch the main arguments.

464


Define the time planes Ha := {(x, x0 ) ∈ Z3 with x0 = a} and denote by GB the set of all clusters 0 not crossing any of the planes H±n , H±p and H±q .It is easy to verify that for a cluster 0 ∈ GB one either has a01 − 21 a02 + a03 + a04 − a05 = 0 or it happens that its contribution cancels that of the other cluster in GB obtained by translating 0 in the time direction. This is, for instance, what happens for clusters located between Hp and Hq and translated clusters located between H−q and H−p . On the other hand, the size of clusters which cross at least two of the planes H±n , H±p or H±q is at least min{2n, p − n, q − p} (assuming q > p > n). After the limits q → ∞, p → ∞ and n → ∞ are taken the total contribution of such clusters is zero, which can be shown using the exponential decay given in (A.19) and noticing that the size of the loops of Fig. 3 grows only linearly in n, p or q. The remaining terms belong to clusters crossing one and only one of the planes H±n , H±p or H±q . Using the translation invariance of the sum over clusters, we may express these remaining terms (after the limits are taken) in the following form: 1 X c0 µ0 fl01 − (flt1 )0 a0ρ1, 2 b0ρ1, 2 + 2 06∼H0 P 1 0 fl02 − (flt2 )0 a0(d∗ l1 , 0) a0ρ1, 2 b0ρ1, 2 + 06∼H0 c0 µ 2

(B.29)

1 X c0 µ0 (flt1 )0 (flt2 )0 − fl01 fl02 a0ρ1, 2 b0ρ1, 2 , 2 06∼H0

where, with some abuse of notation, 0 6∼ H0 indicates that the geometric part of at least one polymer composing 0 crosses the plane H0 . Above, fl (γ) (respectively, flt (γ)) represents the winding number of the polymer γ with respect to the semi-infinite loops formed by l and by vertical lines starting at d∗ l and extending to the negative (positive) euclidean time infinity. See Fig. 4. Note that the second sum in (B.29) can be simplified, since a0(d∗ l1 , 0) a0ρ1, 2 b0ρ1, 2 = a0ρ2 b0ρ2 . .. .. .. ..

.. .. .. ..

flt :

l

fl :

l

.. .. .. ..

.. .. .. ..

Fig. 4. The semi-infinite loops for which fl (γ) and flt (γ) are defined. The horizontal lines represent the link l located at H0 . The vertical lines are parallel to the euclidean time axis and extend to the negative (left) or positive (right) euclidean time infinity

It is easy to show that each of the sums over clusters in (B.29) is absolutely convergent. Analogously, sums like


X

c0 µ0 fl01 − 1 a0ρ1, 2 b0ρ1, 2

465

(B.30)

06∼H0

are also absolutely convergent. It can be seen by reflecting polymers on the plane H0 X that the last expression is the complex conjugate of c0 µ0 (flt1 )0 − 1 a0ρ1, 2 b0ρ1, 2 . 06∼H0

This means that each of the sums over clusters in (B.29) is purely imaginary. Defining   1 X  c0 µ0 fl02 − (flt2 )0 a0ρ2 b0ρ2 , (B.31) zρel (l2 ) := exp 2  06∼H0

which is a pure phase, we conclude from (B.29) the proof of part i) of Theorem 4.1.1. Part ii) can be proven analogously, and we do not need to show the details. The proof of part iii) is also analogous but with an important difference. Since in this case l1 is a magnetic link and l2 an electric one, the closed loops formed by l1 and by l2 , appearing in the left Fig. 3, can have a nontrivial winding number, which can contribute to the classical expectations in the numerator of (B.26) with an additional ZN phase factor, as the phase factor [β : α] emerging from (A.33). This phase equals eihl1 , l2 i . In order to prove iv), consider that the support that l2 is, say, an elementary plaquette at H0 . Following the same steps of the proof of i) we would arrive at relations like (B.29) and (B.31), where both fl2 (γ) and flt2 (γ) represent the winding number of γ around this plaquette. Therefore, for any polymer γ, fl2 (γ) = flt2 (γ) and hence zρel (l2 ) = 1. The proof of v) is analogous. B.7. Proof of Proposition 4.2.3. We will prove only (4.41) since (4.42) is analogous. ihµ, si ρ k = 0, and to prove one has only to show Relation (4.41) means kφel ρ (le ) − e el ihµ, si that ρ , φρ (le ) = e . According to the definitions ρ , Pρ (ρ − (d∗ le , 0))πρ (Aρ, ρ−(d∗ le , 0) (le , 0))ρ el

. (B.32) ρ , φρ (le ) = Pρ (ρ − (d∗ le , 0))πρ Aρ, ρ−(d∗ le , 0) (le , 0) ρ Under the hypothesis d∗ le = 0 and, hence, we can write the right-hand side of (B.32) as ρ , πρ αρn (Aρ, ρ (le , 0)) ρ ρ , Tρ (ρ)n πρ Aρ, ρ (le , 0) ρ

= lim lim

Tρ (ρ)n πρ Aρ, ρ (le , 0) ρ

πρ αρn (Aρ, ρ (le , 0)) ρ . n→∞ n→∞ (B.33) We now expand the right hand side of (B.33) in terms of our cluster expansions and treat it with the same methods used in the proof of Theorem 4.1.1 above. We get el ihµ, si , (B.34) ρ , φel ρ (le ) = zρ (le ) e where the ZN phase factor eihµ, si emerges in this expression as the factor [µ˜ : α] emerges from (A.33): it represents the winding number of d∗ s in the background charge ρ. Actually eihµ, si = [µ˜ : s]. Since d∗ le = 0, one has zρel (le ) = 1 and the proposition is proven. Acknowledgement. We would like to thank K. Fredenhagen for stimulating interest and several useful discussions.

466


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Commun. Math. Phys. 191, 467 – 492 (1998)

Communications in


Coalgebra Bundles? ´ Tomasz Brzezinski, Shahn Majid?? Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, United Kingdom Received: 22 February 1996 / Accepted: 29 May 1997

Abstract: We develop a generalised theory of bundles and connections on them in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examples include ones in which the gauge groups are the braided line and the quantum plane.

1. Introduction In a recent paper [Brz96b] it was shown by the first author that a generalisation of the quantum group principal bundles introduced in [BM93] is needed if one wants to include certain “embeddable” quantum homogeneous spaces, such as the full family of quantum two-spheres of Podle´s [Pod87]. A one-parameter specialisation of this family was used in [BM93] in construction of the q-monopole, but the general members of the family do not have the required canonical fibering. The required generalised notion of quantum principal bundles proposed in [Brz96b], also termed a C-Galois extension (cf. [Sch92]), consists of an algebra P , a coalgebra C with a distinguished element e and a right action of P on P ⊗ C satisfying certain conditions. In the present paper we develop a version of such “coalgebra principal bundles” based on a map ψ : C ⊗ P → P ⊗ C and e ∈ C, and giving now a theory of connections on them. Another motivation for the paper is the search for a generalisation of gauge theory powerful enough to include braided groups [Maj91, Maj93b, Maj93a] as the gauge group. Although not quantum groups, braided groups do have at least a coalgebra and hence can be covered in our theory. We describe the main elements of such a braided ?

Research supported by the EPSRC grant GR/K02244 Royal Society University Research Fellow and Fellow of Pembroke College, Cambridge. On leave 1995 and 1996 at the Department of Mathematics, Harvard University, Cambridge MA 02138, USA ??

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principal bundle theory as arising in this way. This is a first step towards a theory of braided-Lie algebra valued gauge fields, Chern-Simons and Yang-Mills actions, to be considered elsewhere. As well as providing a unifying point of view which includes our previous quantum group gauge theory [BM93], the theory of embeddable homogeneous spaces [Brz96b] and braided group gauge theory, our coalgebra bundles have their own characteristic properties. In particular, the axioms obeyed by ψ involve the algebra and coalgebra in a symmetrical way, opening up the possibility of an interesting self-duality of the construction. This becomes manifest when we are given a character κ on P ; then we have also the possibility of a dual “algebra principal bundle”, corresponding in the finitedimensional case to a coalgebra principal bundle with the fibre P ∗ , total space C ∗ and the structure map ψ ∗ . This is a new phenomenon which is not possible within the realm of ordinary (non-Abelian) gauge theory. Moreover, the axioms obeyed by ψ correspond in the finite-dimensional case to the factorisation of an algebra into P op C ∗ , which is a common situation [Maj90]. Indeed, all bicrossproduct quantum groups [Maj90] provide a dual pair of examples. Finally, we note that some steps towards a theory of fibrations based on algebra factorisations have appeared independently in [CKM94], including topological considerations which may be useful in further work. However, we really need the present coalgebra treatment for our infinite-dimensional algebraic examples, for our treatment of differential calculus and in order to include quantum and braided group gauge theories. We demonstrate the various stages of our formalism on some concrete examples based on the braided line and quantum plane. Preliminaries. All vector spaces are taken over a field k of generic characteristic and all algebras have the unit denoted by 1. C denotes a coalgebra with the coproduct 1 : C → C ⊗ C and the counit : C → k which satisfy the standard axioms. For the coproduct we use the Sweedler notation 1c = c(1) ⊗ c(2) ,

12 c = (1 ⊗ id) ◦ 1c = c(1) ⊗ c(2) ⊗ c(3) ,

etc.,

where c ∈ C, and the summation sign and the indices are suppressed. A vector space P is a right C-comodule if there exists a map 1R : P → P ⊗ C, such that (1R ⊗ id) ◦ 1R = (id ⊗ 1) ◦ 1R , and (id ⊗ ) ◦ 1R = id. For 1R we use the explicit notation ¯ ¯ 1R u = u(0) ⊗ u(1) , ¯

¯

where u ∈ P and u(0) ⊗ u(1) ∈ P ⊗ C (summation understood). For e ∈ C, we denote by PecoC the vector subspace of P of all elements u ∈ P such that 1R u = u ⊗ e. H denotes a Hopf algebra with product µ : H ⊗ H → H, unit 1, coproduct 1 : H → H ⊗ H, counit : H → k and antipode S : H → H. We use Sweedler’s sigma notation as before. Similarly as for a coalgebra, we can define right H-comodules. We say that a right H-comodule P is a right H-comodule algebra if P is an algebra and 1R is an algebra map. If P is an algebra then by n P we denote the P -bimodule of universal n-forms on P , which is defined as n P = 1 P ⊗P · · · ⊗P 1 P (n-fold tensor product over P ). By the natural identification P ⊗P P = P we have [Con85, Kar87] n P = {ω ∈ P ⊗n+1 : ∀i ∈ {1, . . . , n}, µi ω = 0}, where µi denotes a multiplication in P acting on the i and i + 1 factors in P ⊗n+1 . P = L n 0 n=0 P , where P = P , is a differential algebra with the universal differential

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d : P → 1 P , du = 1 ⊗ u − u ⊗ 1. When extended to n P ⊂ P ⊗ n+1 , d explicitly reads: n+1 X (−1)k u0 ⊗ . . . ⊗ uk−1 ⊗ 1 ⊗ uk ⊗ . . . ⊗ un . (1) d u 0 ⊗ u1 ⊗ . . . ⊗ u n = k=0 n

(P ) denotes a bimodule of n-forms on P obtained from n P as an appropriate quotient. Finally, if C is a coalgebra and P is an algebra then we define a convolution product ∗ in the space of linear maps C → P by f ∗ g(c) = f (c(1) )g(c(2) ), where f, g : C → P and c ∈ C. The map f : C → P is said to be convolution invertible if there is a map f −1 : C → P such that f ∗ f −1 = f −1 ∗ f = η ◦ , where η : k → P is given by η : α 7→ α1. In addition, we will also discuss examples based on the theory of braided groups [Maj91, Maj93b, Maj93a] and the theory of bicrossproduct and double cross product and Hopf algebras [Maj90, Maj94b], due to the second author. Chapters 6.2,7.2,9 and 10 of the text [Maj95] contain full details on these topics. 2. Coalgebra ψ-Principal Bundles In this paper we will be dealing with a particular formulation of C-Galois extensions or generalised quantum principal bundles. This formulation is more tractable than the one in [Brz96b], allowing us to develop a theory of connections for it in the next section. Yet, it is general enough to include all our main examples of interest. Our data is the following: Definition 2.1. We say that a coalgebra C and an algebra P are entwined if there is a map ψ : C ⊗ P → P ⊗ C such that ψ ◦ (id ⊗ µ) = (µ ⊗ id) ◦ ψ23 ◦ ψ12 ,

ψ(c ⊗ 1) = 1 ⊗ c,

∀c ∈ C

(2)

(id ⊗ 1) ◦ ψ = ψ12 ◦ ψ23 ◦ (1 ⊗ id), (id ⊗ ) ◦ ψ = ⊗ id, (3) where µ denotes multiplication in P , and ψ23 = id ⊗ ψ etc. Explicitly, we require that the following diagrams commute: C ⊗P ⊗P

id ⊗ µ C ⊗P

ψ ⊗ id ψ ? ? P ⊗C ⊗P P ⊗C HH * id ⊗ ψ j H µ ⊗ id P ⊗P ⊗C id ⊗ 1 P ⊗C ⊗C 6 ψ ⊗ id

C ⊗k

id ⊗ η C ⊗P

k⊗C

ψ ? - P ⊗C

η ⊗ id

(4) id ⊗

P ⊗C 6 ψ

P ⊗k

C ⊗P ⊗C C ⊗P YHid ⊗ ψ H 1 ⊗ id H C ⊗C ⊗P

k⊗P

⊗ id

P ⊗C 6 ψ C ⊗P (5)

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where η is the unit map η : α 7→ α1. In the finite-dimensional case this is exactly equivalent by partial dualisation to the requirement that ψ˜ : C ∗ ⊗ P op → P op ⊗ C ∗ is an algebra factorisation structure (which is part of the theory of Hopf algebra double cross products [Maj90]). This is made precise at the end of the section, where it provides a natural way to obtain examples of such ψ. Proposition 2.2. Let C, P be entwined by ψ. For every group-like element e ∈ C we have the following: 1. For any positive n, P ⊗n is a right C-comodule with the coaction 1nR = ψnn+1 ◦ ← −n ψn−1n ◦ . . . ◦ ψ12 ◦ (ηC ⊗ idn ) ≡ ψ ◦ (ηC ⊗ idn ), where ηC : k → C, α 7→ αe. n 2. The coaction 1n+1 R restricts to a coaction on P . 3. M = PecoC = {u ∈ P ; 11R u = u ⊗ e} is a subalgebra of P . 4. The linear map χM : P ⊗M P → P ⊗ C, u ⊗M v 7→ uψ(e ⊗ v) is well-defined. If χM is a bijection we say that we have a ψ-principal bundle P (M, C, ψ, e). P Proof. We write ψ(c ⊗ u) = α uα ⊗ cα and henceforth we omit the summation sign. In this notation, the conditions (4) and (5) are (uv)α ⊗ cα = uα vβ ⊗ cαβ ,

1α ⊗ cα = 1 ⊗ c,

uα ⊗ cα (1) ⊗ cα (2) = uαβ ⊗ c(1) β ⊗ c(2) α ,

(cα )uα = (c)u,

(6) (7)

for all u, v ∈ P and c ∈ C. 1. The map 1nR is given explicitly by 1nR (u1 ⊗ . . . ⊗ un ) = u1α1 ⊗ . . . ⊗ unαn ⊗ eα1 ...αn . Hence (1nR ⊗ id)1nR (u1 ⊗ . . . ⊗ un ) = u1α1 β1 ⊗ . . . ⊗ unαn βn ⊗ eβ1 ...βn ⊗ eα1 ...αn = u1α1 β1 ⊗ . . . ⊗ unαn βn ⊗ e(1) β1 ...βn ⊗ e(2) α1 ...αn = u1α1 ⊗ u2α2 β2 ⊗ . . . ⊗ unαn βn ⊗ eα1 (1) β2 ...βn ⊗ eα1 (2) α2 ...αn = ... = u1α1 ⊗ . . . ⊗ unαn ⊗ eα1 ...αn (1) ⊗ eα1 ...αn (2) = (idn ⊗ 1)1nR (u1 ⊗ . . . ⊗ un ), where we used the group-like property of e to derive the second equality and then we used the condition (5) n times to obtain the penultimate one. We also have (idn ⊗ )1nR (u1 ⊗ . . . ⊗ un ) = u1α1 ⊗ . . . ⊗ unαn (eα1 ...αn ) n α1 ...αn−1 = u1α1 ⊗ . . . ⊗ un−1 ) αn−1 ⊗ u (e

= . . . = (e)u1 ⊗ . . . ⊗ un = u 1 ⊗ . . . ⊗ un , where we have first used the condition (5) n-times and then the group-like property of e. Hence 1nR is a coaction.

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P P P P 2. If i ui ⊗v i ∈ ker µ then (µ⊗id) i 12R (ui ⊗v i ) = i uiα vβi ⊗eαβ = i (ui v i )α ⊗ eα = 0, using (6). Hence the coaction preserves 1 P . Similarly for n P . 3. Here M = {u ∈ P |uα ⊗ eα = u ⊗ e}, and if u, v ∈ M , then (uv)α ⊗ eα = uα vβ ⊗ eαβ = uvβ ⊗ eβ = uv ⊗ e as well, using (6). 4. It is easy to see that χM is well-defined as a map from P ⊗M P . Thus, if x ∈ M we have χM (u, xv) = u(xv)α ⊗ eα = uxα vβ ⊗ eαβ = uxvβ ⊗ eβ = χM (ux, v), using (6). We remark that parts 3 and 4 also follow from the theory of C-Galois extensions of [Brz96b], for P (M, C, ψ, e) is such an extension. The required right action of P on P ⊗ C is given by (µ ⊗ id) ◦ ψ23 : P ⊗ C ⊗ P → P ⊗ C. Example 2.3. Let H be a Hopf algebra and P be a right H-comodule algebra. The linear ¯ ¯ map ψ : H ⊗ P → P ⊗ H defined by ψ : c ⊗ u → u(0) ⊗ cu(1) entwines H, P . Therefore a quantum group principal bundle P (M, H) with universal differential structure as in [BM93] is a ψ-principal bundle P (M, H, ψ, 1). Proof. For any c ∈ H and u ∈ P we have uα ⊗cα = u(0) ⊗cu(1) . Clearly 1α ⊗cα = 1⊗c. We compute ¯

¯

uα vβ ⊗ cαβ = u(0) vβ ⊗ (cu(1) )β = u(0) v (0) ⊗ cu(1) v (1) = (uv)(0) ⊗ c(uv)(1) = (uv)α ⊗ cα , ¯

¯

¯

¯

¯

¯

¯

¯

hence the condition (4) is satisfied. Furthermore, (cα )uα = (cu(1) )u(0) = (c)u and ¯

¯

uαβ ⊗ c(1) β ⊗ c(2) α = u(0) β ⊗ c(1) β ⊗ c(2) u(1) = u(0)(0) ⊗ c(1) u(0)(1) ⊗ c(2) u(1) ¯

¯

¯ ¯

¯ ¯

¯

= u(0) ⊗ (cu(1) )(1) ⊗ (cu(1) )(2) = uα ⊗ cα (1) ⊗ cα (2) , ¯

¯

¯

so that the condition (5) is also satisfied. Clearly the induced coaction in Proposition 2.2 coincides with the given coaction of H. We can easily replace H here by one of the braided groups introduced in [Maj91, Maj93b]. To be concrete, we suppose that our braided group B lives in a k-linear braided category with well-behaved direct sums, such as that of modules over a quasitriangular Hopf algebra or comodules over a dual-quasitriangular Hopf algebra. This background quantum group does not enter directly into the braided group formulae but rather via the braiding 9 which it induces between any objects in the category. We refer to [Maj93a] for an introduction to the theory and for further details. In particular, a right braided B-module algebra P means a coaction P → P ⊗B in the category which is an algebra homomorphism to the braided tensor product algebra [Maj91] (u ⊗ b)(v ⊗ c) = u9(b ⊗ v)c.

(8)

The coproduct 1 : B → B⊗B of a braided group is itself a homomorphism to such a braided tensor product. Example 2.4. Let B be a braided group with braiding 9 and P a right braided Bcomodule algebra. The linear map ψ : B ⊗ P → P ⊗ B defined by ψ : c ⊗ u → ¯ ¯ 9(c ⊗ u(0) )u(1) entwines B, P . If the induced map χM is a bijection we say that the associated ψ-principal bundle P (M, B, ψ, 1) is a braided group principal bundle, and denote it by P (M, B, 9).

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Proof. This is best done diagrammatically by the technique introduced in [Maj91]. Thus, we write 9 = and products by µ = . We denote coactions and coproducts by . The proof of the main part of (4) is then the diagram:

=

= P B

P B

B P P

B P P

B P P

B P P

=

P B

P B ,

where box is ψ as stated, in diagrammatic form. The first equality is the assumed homomorphism property of the braided coaction . The second equality is associativity of the product in B, and the third is functoriality of the braiding, which we use to push the diagram into the right form. The minor condition is immediate from the axioms of a braided comodule algebra and the properties of the unit map η : 1 → P . Here 1 denotes the trivial object for our tensor product and necessarily commutes with the braiding in an obvious way (such that 1 is denoted consistently by omission). For the proof of (5) we ask the reader to reflect the diagram in a mirror about a horizontal axis (i.e. view it up-side-down and from behind) and then reverse all braid crossings (restoring them all to ). The result is the diagrammatic proof for the main part of (5) if we relabel the product of P as the coproduct of B and relabel the product of B as the right coaction of B on P . The minor part of (5) is immediate from properties of the braided counit. Example 2.5. Let H be a Hopf algebra and π : H → C a coalgebra surjection. If ker π is a minimal right ideal containing {u − (u)|u ∈ M } then ψ : C ⊗ H → H ⊗ C defined by ψ(c ⊗ u) = u(1) ⊗ π(vu(2) ) entwines C, H, where u ∈ H, c ∈ C and v ∈ π −1 (c), and we have a ψ-principal bundle H(M, C, ψ, π(1)) in the setting of Proposition 2.2, denoted H(M, C, ψ, π). Hence the generalised bundles over embeddable quantum homogeneous spaces in [Brz96b] are examples of ψ-principal bundles. Proof. In this case uα ⊗ cα = u(1) ⊗ π(wu(2) ), for any u ∈ H, c ∈ C and w ∈ π −1 (c). Clearly 1α ⊗ cα = 1 ⊗ c. We compute uα vβ ⊗ cαβ = u(1) vα ⊗ π(wu(2) )α = u(1) v (1) ⊗ π(wu(2) v (2) ) = (uv)α ⊗ cα , where w ∈ π −1 (c). Hence condition (4) is satisfied. Furthermore, we have (cα )uα = (π(wu(2) ))u(1) = (c)u and uαβ ⊗ c(1) β ⊗ c(2) α = u(1) α ⊗ π(w(1) u(2) ) ⊗ c(2) α = u(1) ⊗ π(w(1) u(2) ) ⊗ π(w(2) u(3) ) = u(1) ⊗ π(wu(2) )(1) ⊗ π(wu(2) )(2) = uα ⊗ cα (1) ⊗ cα (2) , where again w(1) ∈ π −1 (c(1) ) and w(2) ∈ π −1 (c(2) ). Therefore condition (5) is also satisfied. Some concrete examples of coalgebra bundles over quantum embeddable homogeneous spaces may be found in [Brz96b] (cf. [DK94]).

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We note that diagrams (4) and (5) are dual to each other in the following sense. The diagrams (5) may be obtained from diagrams (4) by interchanging µ with 1, η with and P with C, and by reversing the arrows. With respect to this duality property the axioms for the map ψ are self-dual. Therefore we can dualise Proposition 2.2 to obtain the following: Proposition 2.6. Let C, P be entwined by ψ : C ⊗ P → P ⊗ C. For every algebra character κ : P → k we have the following: 1. For any positive integer n, C ⊗n is a right P -module with the action /n = (κ ⊗ idn ) ◦ − →n ψ12 ◦ ψ23 ◦ . . . ◦ ψnn+1 = (κ ⊗ idn ) ◦ ψ . 2. The action /n maps 1n (C) to itself. 3. The subspace Iκ = span{c/1 u−cκ(u)|c ∈ C, u ∈ P } is a coideal. Hence M = C/Iκ is a coalgebra. We denote the canonical surjection by πκ : C → M . 4. There is a map ζ M : C ⊗ P → C ⊗M C defined by ζ M (c ⊗ u) = c(1) ⊗M c(2) /1 u, where C ⊗M C = span{c ⊗ d ∈ C ⊗ C|c(1) ⊗ πκ (c(2) ) ⊗ d = c ⊗ πκ (d(1) ) ⊗ d(2) } is the cotensor product under M . If ζ M is a bijection, we say that C(M, P, ψ, κ) is a dual ψ-principal bundle. Proof. 1. The explicit action is α1 n (cn ⊗ · · · ⊗ c1 )/n u = cα n ⊗ · · · ⊗ c1 κ(uα1 ···αn ).

Then clearly 1 β1 n βn ((cn ⊗ · · · ⊗ c1 ) /n u)/n v = cα ⊗ · · · ⊗ cα κ(uα1 ···αn vβ1 ···βn ) n 1

α

β

n−1 n−1 1 β1 n = cα ⊗ · · · ⊗ cα κ((uα1 ···αn−1 vβ1 ···βn−1 )αn ) n ⊗ cn−1 1 α1 n n ⊗ · · · ⊗ c κ((uv) = · · · = cα α ···α n 1 n ) = (cn ⊗ · · · ⊗ c1 )/ (uv) 1

for all ci ∈ C and u, v ∈ P . We used (6) repeatedly. 2. We have (c(1) ⊗ c(2) )/2 u = c(1) β ⊗ c(2) α κ(uαβ ) = cα (1) ⊗ cα (2) κ(uα ) = 1(c/1 u) by (7), and similarly for higher 1n (C). 3. Explicitly, Iκ = span{cα κ(uα ) − cκ(u)|c ∈ C, u ∈ P }. But using (7) we have 1(cα κ(uα ) − cκ(u)) = c(1) β ⊗ c(2) α κ(uαβ ) − c(1) ⊗ c(2) κ(u) = c(1) ⊗(c(2) α κ(uα ) − c(2) κ(u))+(c(1) β κ(uαβ )−c(1) κ(uα )) ⊗ c(2) α ∈ C ⊗ Iκ +Iκ ⊗ C. Hence Iκ is a coideal. 4. The stated map ζ M (c ⊗ u) = c(1) ⊗ c(2) α κ(uα ) has its image in C ⊗M C since c(1) ⊗ πκ (c(2) ) ⊗ c(3) α κ(uα ) = c(1) ⊗ πκ (c(2) β )κ(uαβ ) ⊗ c(3) α using (7) and πκ (Iκ ) = 0. By dimensions in the finite-dimensional case, it is natural to require that this is an isomorphism. This is also an example of a dual version of the theory of C-Galois extensions. The proposition is dual to Proposition 2.2 in the sense that all arrows are reversed. In concrete terms, if P, C are finite-dimensional then ψ ∗ : P ∗ ⊗ C ∗ → C ∗ ⊗ P ∗ and κ ∈ P ∗ make C ∗ (M ∗ , P ∗ , ψ ∗ , κ) a ψ ∗ -principal bundle. Here M ∗ = {f ∈ C ∗ |(κ ⊗ f ) ◦ ψ = f ⊗ κ}. If C, P are entwined and we have both e ∈ C and κ : P → k, we can have both a ψprincipal bundle and a dual one at the same time. An obvious example, in the setting of Example 2.3, is P = C = H a Hopf algebra and ψ(c ⊗ u) = u(1) ⊗ cu(2) by the coproduct.

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Then Proposition 2.2 with e = 1 gives a quantum principal bundle with M = k and right coaction given by the coproduct. On the other hand, Proposition 2.6 with κ = gives a dual bundle with action by right multiplication. Finally, we note that there is a close connection with the theory of factorisation of (augmented) algebras introduced in [Maj90, Maj94b] as part of a factorisation theory of Hopf algebras. According to this theory, a factorisation of an algebra X into subalgebras A, B (so that the product A ⊗ B → X is a linear isomorphism) is equivalent to a factorisation structure ψ˜ : B ⊗ A → A ⊗ B with certain properties. It was also shown that when A, B are augmented by algebra characters then the factorisation structure induces a right action of A on B and a left action of B on A, respectively. Proposition 2.7. Let C be finite-dimensional. Then an entwining structure ψ : C ⊗ P → P ⊗ C is equivalent by partial dualisation to a factorisation structure ψ˜ : C ∗ ⊗ P op → P op ⊗ C ∗ . In the augmented case, the induced coaction 11R and action /1 in Propositions 2.2 and 2.6 are the dualisations of the actions induced by the factorisation. ˜ ⊗ u) = ui ⊗ f i say, for f ∈ C ∗ and u ∈ P . The Proof. We use the notation ψ(f equivalence with ψ is by ui hf i , ci = uα hf, cα i, where h , i denotes the evaluation pairing. It is easy to see that ψ entwines C, P iff ψ˜ obeys [Maj94b]cf. [Maj90] ψ˜ ◦ (µ ⊗ id) = (id ⊗ µ) ◦ ψ˜ 12 ◦ ψ˜ 23 ,

˜ ⊗ 1) = 1 ⊗ f, ψ(f

(9)

ψ˜ ◦ (id ⊗ µ) = (µ ⊗ id) ◦ ψ˜ 23 ◦ ψ˜ 12 ,

˜ ⊗ u) = u ⊗ 1 ψ(1

(10)

for all f ∈ C ∗ and u ∈ P op . Thus, the first of these is ui hc, (f g)i i = uα hcα , f gi = uα hcα (1) , f ihcα (2) , gi = uαβ hc(1) β , f ihc(2) α , gi = uαi hc(1) , f i ihc(2) α , gi = uji hc(1) , f i i hc(2) , g j i using (7). Similarly for (10) using (6), provided we remember to use the opposite product on P . Such data ψ˜ is equivalent by [Maj94b, Maj90] to the existence of an algebra ˜ ⊗ u) in X, and X factorising into P op C ∗ . Given such X we recover ψ˜ by uc = µ ◦ ψ(c op ∗ ˜ ˜ conversely, given ψ we define X = P ⊗C as in (8), but with ψ. Also from this theory, if we have κ an algebra character on P op (or on P ) then / = (κ ⊗ id) ◦ ψ˜ is a right action of P op on C ∗ , which clearly dualises to the right action of P on C in Proposition 2.6. Similarly, if e is a character on C ∗ then . = (id ⊗ e) ◦ ψ˜ is a left action of C ∗ on P op (or on P ) which clearly dualises to the right coaction of C in Proposition 2.2. An obvious setting in which factorisations arise is the braided tensor product (8) of algebras in braided categories [Maj91, Maj93b, Maj93a], with ψ˜ = 9 the braiding. Thus if A⊗B is a braided tensor product of algebras (e.g. of module algebras under a background quantum group) we can look for a suitable dual coalgebra B ∗ in the category and the corresponding entwining ψ of B ∗ , Aop . This provides a large class of entwining structures. Another source is the theory of double cross products G ./ H of Hopf algebras in [Maj90]. These factorise as Hopf algebras and hence, in particular, as algebras. In this context, Proposition 2.7 can be combined with the result in [Maj90, Sect. 3.2] that the double cross product is equivalent by partial dualisation to a bicrossproduct H ∗ I/G. These bicrossproduct Hopf algebras (also due to the second author) provided one of the first general constructions for non-commutative and non-cocommutative Hopf algebras, and many examples are known. Proposition 2.8. Let CI/P op be a bicrossproduct bialgebra [Maj90, Sect. 3.1], where P op , C are bialgebras suitably (co)acting on each other. Then C, P are entwined by

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475 ¯

¯

ψ(c ⊗ u) = u(1) (0) ⊗ u(1) (1) (u(2) .c). ¯

¯

Here . is the left action of P op on C and u(0) ⊗ u(1) is the right coaction of C on P op , as part of the bicrossproduct construction. Proof. We derive this result under the temporary assumption that C is finite-dimensional. Thus the bicrossproduct is equivalent to a double cross product P op ./ C ∗ with actions ¯ ¯ ., / defined by f .u = u(0) hf, u(1) i and hu.c, f i = hc, f /ui for all f ∈ C ∗ . Then ψ˜ ˜ ⊗ u) = f (1) .u(1) ⊗ f (2) /u(2) according to [Maj90, Maj94b]. for this factorisation is ψ(f The correspondence in Proposition 2.7 then gives ψ as stated. Once the formula for ψ is known, one may verify directly that it entwines C, P given the compatibility conditions between the action and coaction of a bicrossproduct in [Maj90, Sect. 3.1]. Now we describe trivial ψ-principal bundles and gauge transformations in them. Proposition 2.9. Let P and C be entwined by ψ as in Definition 2.1 and let e be a group-like element in C. Assume the following data: 1. A map ψ C : C ⊗ C → C ⊗ C such that C C ◦ ψ23 ◦ (1 ⊗ id), (id ⊗ 1) ◦ ψ C = ψ12

(id ⊗ ) ◦ ψ C = ⊗ id,

(11)

and ψ C (e ⊗ c) = 1c, for any c ∈ C; 2. A convolution invertible map 8 : C → P such that 8(e) = 1 and ψ ◦ (id ⊗ 8) = (8 ⊗ id) ◦ ψ C .

(12)

Then there is a ψ-principal bundle over M = PecoC with structure coalgebra C and total space P . We call it the trivial ψ-principal bundle P (M, C, 8, ψ, ψ C , e) associated to our data, with trivialisation 8. Proof. The proof of the proposition is similar to the proof that the trivial quantum principal bundle in [BM93, Example 4.2] is in fact a quantum principal bundle. First we observe that the map 2 : M ⊗ C → P,

x ⊗ c 7→ x8(c)

is an isomorphism of linear spaces. Explicitly the inverse is given by 2−1 : u 7→ u(0) 8−1 (u(1) (1) ) ⊗ u(1) (2) , ¯

¯

¯

where 8−1 : C → P is a convolution inverse of 8, i.e. 8−1 (c(1) )8(c(2) ) = 8(c(1) )8−1 (c(2) ) = (c)1. To see that the image of the above map is in M ⊗ C we first notice that (12) implies that 11R ◦ 8 = (8 ⊗ id) ◦ 1 and that ψ(c(1) ⊗ 8−1 (c(2) )) = 8−1 (c) ⊗ e.

(13)

Therefore for any u ∈ P , 11R (u(0) 8−1 (u(1) )) = u(0) ψ(u(1) (1) ⊗ 8−1 (u(1) (2) )) = u(0) 8−1 (u(1) ) ⊗ e, ¯

¯

¯

¯

¯

¯

¯

476


and thus u(0) 8−1 (u(1) ) ∈ M . Then it is easy to prove that the above maps are inverses to each other. We remark that 2 is in fact a left M -module and a right C-comodule map, where the coaction in M ⊗ C is given by x ⊗ c 7→ x ⊗ c(1) ⊗ c(2) . Moreover ψ ◦ (id ⊗ 2) = C ◦ ψ12 . (2 ⊗ id) ◦ ψ23 The proof that χM in this case is a bijection follows exactly the method used in the proof of [BM93, Example 4.2] and thus we do not repeat it here. ¯

¯

Next, we consider gauge transformations. Definition 2.10. Let P (M, C, 8, ψ, ψ C , e) be a trivial ψ-principal bundle as in Proposition 2.9. We say that a convolution invertible map γ : C → M such that γ(e) = 1 is a gauge transformation if C ◦ ψ12 ◦ (id ⊗ γ ⊗ id) ◦ (id ⊗ 1) = (γ ⊗ id ⊗ id) ◦ (1 ⊗ id) ◦ ψ C . ψ23

(14)

Proposition 2.11. If γ : C → M is a gauge transformation in P (M, C, 8, ψ, ψ C , e) then 80 = γ ∗ 8, where ∗ denotes the convolution product is a trivialisation of P (M, C, 8, ψ, ψ C , e). The set of all gauge transformations in P (M, C, 8, ψ, ψ C , e) is a group with respect to the convolution product. We say that two trivialisations 8 and 80 are gauge equivalent if there exists a gauge transformation γ such that 80 = γ ∗ 8. Proof. Clearly 80 is a convolution invertible map such that 80 (e) = 1. To prove that it satisfies (12) we first introduce the notation ψ C (b ⊗ c) = cA ⊗ bA

(summation assumed),

in which the condition (14) reads explicitly γ(c(1) )α ⊗ c(2) A ⊗ bαA = γ(cA(1) ) ⊗ cA(2) ⊗ bA , and then compute ψ ◦ (id ⊗ 80 )(b ⊗ c) = ψ(b ⊗ γ(c(1) )8(c(2) )) = γ(c(1) )α 8(c(2) )β ⊗ bαβ = γ(c(1) )α 8(c(2) A ) ⊗ bαA = γ(cA(1) )8(cA(2) ) ⊗ bA = (80 ⊗ id) ◦ ψ C (b ⊗ c).

This proves the first part of the proposition. Assume now that γ1 , γ2 are gauge transformations. Then (γ1 (c(1) )γ2 (c(2) ))α ⊗ c(3) A ⊗ bαA = γ1 (c(1) )α γ2 (c(2) )β ⊗ c(3) A ⊗ bαβA

= γ1 (c(1) )α γ2 (c(2) A (1) ) ⊗ c(2) A (2) ⊗ bαA = γ1 (cA(1) )γ2 (cA(2) ) ⊗ cA(3) ⊗ bA .

Therefore γ1 ∗ γ2 is a gauge transformation too. Clearly is a gauge transformation and thus provides the unit. Finally, to prove that if γ is a gauge transformation then so is γ −1 , we observe that if γ3 = γ1 ∗ γ2 and γ2 are gauge transformations then so is γ1 . Indeed, if γ1 ∗ γ2 is a gauge transformation then (γ1 (c(1) )γ2 (c(2) ))α ⊗ c(3) A ⊗ bαA = γ1 (cA(1) )γ2 (cA(2) ) ⊗ cA(3) ⊗ bA ,

Coalgebra Bundles

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but since γ2 is a gauge transformation, we obtain γ1 (c(1) )α γ2 (c(2) A (1) ) ⊗ c(2) A (2) ⊗ bαA = γ1 (cA(1) )γ2 (cA(2) ) ⊗ cA(3) ⊗ bA . Applying γ2−1 to the second factor in the tensor product and then multiplying the first two factors we obtain γ1 (c(1) )α ⊗ c(2) A ⊗ bαA = γ1 (cA(1) ) ⊗ cA(2) ⊗ bA , i.e. γ1 is a gauge transformation as stated. Now applying this result to γ3 = and γ2 = γ we deduce that γ −1 is a gauge transformation as required. This completes the proof of the proposition. Although the existence of the map ψ C as in Proposition 2.9 is not guaranteed for all coalgebras, the map ψ C exists in most of the examples discussed in this section: Example 2.12. For a quantum principal bundle P (M, H) as in Example 2.3, we define ψ H (b ⊗ c) = c(1) ⊗ bc(2) , for all b, c ∈ H. Then (2.9)–(2.11) reduces to the theory of trivial quantum principal bundles and their gauge transformations in [BM93]. Proof. It is easy to see by standard Hopf algebra calculations that (11) is satisfied by the bialgebra axiom for H = C in this case. Moreover, (12) reduces to 8 being an intertwiner of 1R with 1. The condition (14) is empty. This recovers the setting introduced in [BM93]. In the braided case we use the above theory to arrive at a natural definition of trivial braided principal bundle: Example 2.13. For a braided principal bundle P (M, B, 9) as in Example 2.4, we define a trivialisation as a convolution-invertible unital morphism 8 : B → P in the braided category such that 1R ◦ 8 = (8 ⊗ id) ◦ 1, where 1R is the braided right coaction of B on P . We define a gauge transformation as a convolution-invertible unital morphism γ : B → M , acting on trivialisations by the convolution product ∗. This is a trivial ψ-principal bundle with ψ B (b ⊗ c) = 9(b ⊗ c(1) )c(2) , where 1c = c(1) ⊗ c(2) is the braided group coproduct. Proof. This time, (11) follows from the braided-coproduct homomorphism property of a braided group [Maj91]. From this and the form of ψ, we see that (12) becomes 1R ◦ 8(c) = ((8 ⊗ id) ◦ 9(b ⊗ c(1) ))c(2) . Setting b = e gives the condition stated on 8 because the braiding with e = 1 is always trivial. Assuming the stated condition, (12) then becomes 8(c(1) ) ⊗ bc(2) = ((8 ⊗ id) ◦ 9(b ⊗ c(1) ))c(2) , which is equivalent (by replacing c(2) by c(2) ⊗ Sc(3) and multiplying, where S is the braided antipode) to (8 ⊗ id) ◦ 9 = 9 ◦ (id ⊗ 8). When all our constructions take place in a braided category, this is the functoriality property implied by requiring that 8 is a morphism in the category. The theory of trivial ψ-bundles only requires this functoriality condition itself. Similarly, we compute the gauge condition (14) using ψ(b ⊗ γ(c)) = 9(b ⊗ γ(c)) because γ(c) ∈ M , and operate

478


on it by replacing c(2) by c(2) ⊗ Sc(3) and multiplying. Then it reduces to 923 ◦ 912 ◦ (id ⊗ γ ◦ 1) = (γ ◦ 1 ⊗ id) ◦ 9. Since 1 is a morphism, we see (by applying the braided group counit) that the gauge condition (14) is equivalent to (γ ⊗ id) ◦ 9 = 9 ◦ (id ⊗ γ). As before, this is naturally implied by requiring that γ is a morphism in our braided category. It is clear that the convolution product ∗ preserves the property of being a morphism since 1 and 1R are assumed to be morphisms. For a ψ-principal bundle over a quantum homogeneous space as in Example 2.5, we can define a trivialisation if, for example, the map ψ C (b ⊗ c) = π(v (1) ) ⊗ π(uv (2) ),

(15)

where u ∈ π −1 (b), v ∈ π −1 (c) is well-defined. Then a trivialisation of the bundle is a convolution-invertible map 8 : C → H obeying 8 ◦ π(1) = 1 and 8(c)(1) ⊗ π(u8(c)(2) ) = 8 ◦ π(v (1) ) ⊗ π(uv (2) )

(16)

for all c ∈ C, u ∈ H, and v ∈ π −1 (c). Taking u = 1 requires, in particular, the natural intertwiner condition (8 ⊗ id) ◦ 1 = 1R ◦ 8. There is, similarly, a condition on gauge transformations γ obtained from (14). Hence our formulation of trivial ψ-principal bundles covers all the main sources of ψ-principal bundles discussed in this section. We conclude this section with some explicit examples of ψ-principal bundles. Example 2.14. Let H be a quantum cylinder Aq [x−1 ], i.e. a free associative algebra generated by x, x−1 and y subject to the relations yx = qxy, xx−1 = x−1 x = 1, with a natural Hopf algebra structure: 2|0

1x±1 = x±1 ⊗ x±1 ,

1y = 1 ⊗ y + y ⊗ x,

etc.

(17)

Consider a right ideal J in H generated by x − 1 and x−1 − 1. Clearly, J is a coideal and 2|0 therefore C = Aq [x−1 ]/J is a coalgebra and a canonical epimorphism π : H → C is a coalgebra map. C is spanned by the elements cn = π(y n ), n ∈ Z≥0 , and the coproduct and the counit are given by n X n c ⊗ cn−k , (cn ) = 0. (18) 1cn = k q k k=0

We are in the situation of Example 2.5 and thus we have the entwining structure ψ : C ⊗ H → H ⊗ C, which explicitly computed comes out as n X n m n q l(k+m) xm y k ⊗ cn+l−k , (19) ψ(cl ⊗ x y ) = k q k=0

where

and

[n]q ! n , = k q [n − k]q ![k]q ! [n]q ! = [n]q · · · [2]q [1]q ,

[0]q ! = 1,

[n]q = 1 + q + . . . + q n−1 .

Coalgebra Bundles

479

From this definition of ψ one easily computes the right coaction of C on H as well as the fixed point subalgebra M = k[x, x−1 ], i.e. the algebra of functions on a circle. By Example 2.5 we have just constructed a generalised quantum principal bundle 2|0 Aq [x−1 ](k[x, x−1 ], C, ψ, c0 ). Finally we note that the above bundle is trivial in the sense of Proposition 2.9. The 2|0 trivialisation 8 : C → Aq [x−1 ] and its inverse 8−1 are defined by 8(cn ) = y n ,

8−1 (cn ) = (−1)n q n(n−1)/2 y n .

(20)

One can easily check that the map 8 satisfies the required conditions. Explicitly, the map ψ C : C ⊗ C → C ⊗ C reads n X n q km ck ⊗ cm+n−k . ψ (cm ⊗ cn ) = k q C

k=0

Therefore n X n q km y k ⊗ cm+n−k ψ ◦ (id ⊗ 8)(cm ⊗ cn ) = ψ(cm ⊗ y ) = k q n

k=0

= (8 ⊗ id) ◦ ψ C (cm ⊗ cn ). Since the bundle discussed in this example is trivial, we can compute its gauge group. One easily finds that a convolution invertible map γ : C → k[x, x−1 ] satisfies condition (14) if and only if γ(cn ) = 0n xn (no summation), where n ∈ Z≥0 , 0n ∈ k and 00 = 1. Therefore the gauge group is equivalent to the group of sequences 0 = (1, 01 , 02 , ...) with the product given by n X n (0 · 0 )n = 0 00 . k q k n−k 0

k=0

For the simplest example of a braided principal bundle, one can simply take any braided group B and any algebra M in the same braided category. Then the braided tensor product algebra P = M ⊗B, along with the definitions 1R = id ⊗ 1,

8(b) = 1 ⊗ b,

8−1 (b) = 1 ⊗ Sb

(21)

put us in the setting of Examples 2.4 and 2.13. Note first that 1R is a coaction (the tensor product of the trivial coaction and the right coregular coaction) and makes P into a braided comodule algebra. Moreover, the induced map χM (m ⊗ b ⊗ n ⊗ c) = m9(b ⊗ n)c(1) ⊗ c(2) for m, n ∈ M , b, c ∈ B, is an isomorphism P ⊗M P → P ⊗ P ; it has inverse χ−1 M (m ⊗ b ⊗ c) = m ⊗ bSc(1) ⊗ 1 ⊗ c(2) . It is also clear that 8 is a trivialisation. This is truly a trivial braided principal bundle because P is just a (braided) tensor product algebra.

480


Example 2.15. Let B = k[c] be the braided line generated by c with braiding 9(c ⊗ c) = qc ⊗ c and the linear coproduct 1c = c ⊗ 1 + 1 ⊗ c. It lives in the braided category Vecq of Z-graded vector spaces with braiding q deg( ) deg( ) times the usual transposition. Here deg(c) = 1. Let M = k[x, x−1 ] be viewed as a Z-graded algebra as well, with deg(x) = 1. Then P = k[x, x−1 ]⊗k[c] is a trivial braided principal bundle with the coaction and trivialisation n X n m n xm ⊗ ck ⊗ cn−k , 8(cn ) = 1 ⊗ cn . (22) 1R (x ⊗ c ) = k q k=0

As a ψ-principal bundle, this example clearly coincides with the preceding one, albeit constructed quite differently: we identify cn = cn and y = 1 ⊗ c, and note that in the braided tensor product algebra k[x, x−1 ]⊗k[c] we have the product (1 ⊗ c)(x ⊗ 1) = 2|0 9(c ⊗ x) = q(x ⊗ 1)(1 ⊗ c), i.e. P = Aq [x−1 ]. It is also clear that the coproduct deduced in (18) can be identified with the braided line coproduct which is part of our initial data here. This particular braided tensor product algebra P is actually the algebra part of the bosonisation of B = k[c] viewed as living in the category of comodules over k[x, x−1 ] as a dual-quasitriangular Hopf algebra (see [Maj95, p. 510]), and becomes in this way a Hopf algebra. This bosonisation is the Hopf algebra H which was part of the initial data in the preceding example. Finally, gauge transformations γ from the braided point of view are arbitrary degree-preserving unital maps k[c] → k[x, x−1 ], i.e. given by the group of sequences 0 as found before. This example demonstrates the strength of braided group gauge theory; even the most trivial braided quantum principal bundles may be quite complicated when constructed by more usual Hopf algebraic means. On the other hand, the following embeddable quantum homogeneous space does not appear to be of the braided type, nor (as far as we know) a trivial bundle. Example 2.16. Let P be the algebra of functions on the quantum group GLq (2). This is generated by elements α, β, γ, δ and D subject to the relations αβ = qβα,

αγ = qγα,

βδ = qδβ,

γδ = qδγ,

αδ = δα + (q − q −1 )βγ,

βγ = γβ,

(αδ − qβγ)D = D(αδ − qβγ) = 1.

Let C be a vector space spanned by cm,n , m ∈ Z>0 , n ∈ Z with the coalgebra structure m X m q k(m−k) c ⊗ cm−k,n+k , (cm,n ) = δm0 . 1(ci,j ) = k q−2 k,n k=0

Let the linear map ψ : C ⊗ P → P ⊗ C be given by ψ(ci,j ⊗ αk γ l β m δ n Dr ) n m X X m n q (m−s)(s+t−l)+(n−t)t−i(k+l−t−s) × = s q−2 t q−2

(23)

s=0 t=0 k+m−s l+n−t s t

α

β δ Dr ⊗ ci+m+n−s−t,j−r+t+s .

γ

Then ψ entwines P with C. Furthermore if we take e = c0,0 then the fixed point sub2|0 algebra PeC is generated by 1, α, γ and hence it is isomorphic to A1/q and there is a 2|0

ψ-principal bundle P (A1/q , C, ψ, e).

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481

Proof. The algebra GLq (2) can be equipped with the standard Hopf algebra structure 1

α γ

β δ

=

α γ

β α ⊗ δ γ

β , δ

S

α γ

β δ

=D

δ −qγ

−q −1 β , α

(α) = (δ) = 1, (β) = (γ) = 0. We define a surjection π : GLq (2) → C by π(αk β l γ m δ n Dr ) = δm0 q ln cl,n−r . In Sect. 5 of [Brz96b] it is shown that π is a coalgebra map and that the data H = GLq (2), π, C satisfy requirements of Example 2.5. Therefore we have a ψ-principal bundle with ψ as in Example 2.5. Written explicitly this ψ is exactly as in Eq. (23). In [Brz96b] it is also noted that the coalgebra C can be equipped with the algebra 2|0 structure of Aq−2 [x−1 ] by setting cm,n = q −mn xn y m . The coproduct in C is then the same as in Example 2.14, Eq. (17).

3. Connections in the Universal Differential Calculus Case From the first assertion of Proposition 2.2 we know that the natural coaction 1R = 11R of C on P extends to the coaction of C on the tensor product algebra P ⊗n for any positive integer n. Still most importantly this coaction can be restricted to n P by the second assertion of Proposition 2.2. Therefore the coalgebra C coacts on the algebra of universal forms on P . The universal differential structure on P is covariant with respect to the coaction 1nR in the following sense: Proposition 3.1. Let P , C, ψ and e be as in Proposition 2.2. Let d : P → 1 P be the universal differential, du = 1 ⊗ u − u ⊗ 1 extended to the whole of P as in the ← −n−1 ← −n for any integer n > 1. Therefore Preliminaries. Then ψ ◦ (id ⊗ d) = (d ⊗ id) ◦ ψ 1nR ◦ (id ⊗ d) = (d ⊗ id) ◦ 1n−1 R . P Proof. We take υ = i u0,i ⊗ u1,i ⊗ . . . ⊗ un,i ∈ n P (i.e., any adjacent product vanishes). Using conditions (4), and the explicit form of dυ (1), for any c ∈ C we compute ← −n+2 (c ⊗ dυ) ψ =

n+1 X

(−1)k

i

k=0

=

n+1 X k=0

X

(−1)k

X i

k−1,i k,i n,i α0 ...αk−1 βαk ...αn u0,i α0 ⊗ . . . ⊗ uαk−1 ⊗ 1β ⊗ uαk ⊗ . . . ⊗ uαn ⊗ c

k−1,i k,i n,i α0 ......αn u0,i α0 ⊗ . . . ⊗ uαk−1 ⊗ 1 ⊗ uαk ⊗ . . . ⊗ uαn ⊗ c

← −n+1 =(d ⊗ id) ◦ ψ (c ⊗ υ).

To discuss a theory of connections in P (M, C, ψ, e) it is important that the horizontal ← −2 one forms P 1 M P be covariant under the action of 12R or, more properly, ψ . The following lemma gives a criterion for the covariance of horizontal one-forms.

482


Lemma 3.2. For a ψ-principal bundle P (M, C, ψ, e) assume that ψ(C ⊗M ) ⊂ M ⊗C. ← −2 Then ψ (C ⊗ P 1 M P ) ⊂ P 1 M P ⊗ C. Proof. Using (6) one easily finds that for any u, v ∈ P , x, y ∈ M and c ∈ C, ← −2 ψ (c ⊗ ux ⊗ yv) = uα xβ ⊗ yγ vδ ⊗ cαβγδ . If we assume further that xβ , yγ ∈ M then the result follows.

We will see later that the hypothesis of Lemma 3.2 is automatically satisfied for braided principal bundles of Example 2.4. In contrast, it is not necessarily satisfied for ψ-bundles on quantum embeddable homogeneous spaces of Example 2.5. For example, one can easily check that it is satisfied for the bundle discussed in Example 2.16. On the other hand the ψ-principal bundle over the quantum hyperboloid, which is an embeddable homogeneous space of Eq (2) [BCGST96] fails to fulfil requirements of Lemma 3.2. The covariance of P and P 1 M P enables us to define a connection in P (M, C, ψ, e) in a way similar to the definition of a connection in a quantum principal bundle P (M, H) (compare [BM93]). Definition 3.3. Let P (M, C, ψ, e) be a generalised quantum principal bundle such that ψ(C ⊗ M ) ⊂ M ⊗ C. A connection in P (M, C, ψ, e) is a left P -module projection ← −2 ← −2 Π : 1 P → 1 P such that ker Π = P 1 M P and ψ (id ⊗ Π) = (Π ⊗ id) ψ . It is clear that for a usual quantum principal bundle P (M, H), Definition 3.3 coincides with the definition of a connection given in [BM93]. Thus, the condition in ← −2 ← −2 Lemma 3.2 always holds for ψ as in Example 2.3, while ψ (id ⊗ Π) = (Π ⊗ id) ψ if and only if 12R Π = (Π ⊗ id)12R , which was the condition in [BM93]. In what follows we assume that the condition in Lemma 3.2 is satisfied. A connection Π in P (M, C, ψ, e) can be equivalently described as follows. First we define a map φ : C ⊗ P ⊗ ker → P ⊗ ker ⊗ C by the commutative diagram ← −2 ψ

C ⊗ 1 P id ⊗ χ ? C ⊗ P ⊗ ker

- 1 P ⊗ C χ ⊗ id

φ

? - P ⊗ ker ⊗ C

where χ(u ⊗ v) = uψ(e ⊗ v). The map φ is clearly well-defined. Indeed, because χM ← −2 is a bijection, ker χ = P 1 M P and then ψ (C ⊗ ker χ) ⊂ ker χ ⊗ C, by Lemma 3.2. Therefore φ(0) = 0. By definition of P (M, C, ψ, e) we have a short exact sequence of left P -module maps χ (24) 0 → P 1 M P → 1 P → P ⊗ ker → 0.

Coalgebra Bundles

483

The exactness of the above sequence is clear since the fact that χM is bijective implies ← −2 that χ is surjective and ker χ = P 1 M P . By definition, χ intertwines ψ with φ. Proposition 3.4. The existence of a connection Π in P (M, C, ψ, e) is equivalent to the existence of a left P -module splitting σ : P ⊗ ker → 1 P of the above sequence such ← −2 that ψ ◦ (id ⊗ σ) = (σ ⊗ id) ◦ φ. Proof. Clearly the existence of a left P -module projection is equivalent to the existence of a left P -module splitting. It remains to check the required covariance properties. Assume that σ has the required properties, then ← −2 ← −2 ψ ◦ (id ⊗ Π) = ψ ◦ (id ⊗ σ) ◦ (id ⊗ χ) ← −2 ← −2 = (σ ⊗ id) ◦ φ ◦ (id ⊗ χ) = (σ ◦ χ ⊗ id) ◦ ψ = (Π ⊗ id) ◦ ψ . Conversely, if Π has the required properties, then one easily finds that ← −2 ψ ◦ (id ⊗ σ ◦ χ) = (σ ⊗ id) ◦ φ ◦ (id ⊗ χ).

Since χ is a surjection the required property of σ follows.

To each connection we can associate its connection one-form ω : ker → 1 P by setting ω(c) = σ(1 ⊗ c). 1 Similarly to the quantum bundle case of [BM93] we have Proposition 3.5. Let Π be a connection on P (M, C, ψ, e). Then, for all c ∈ ker , the connection 1-form ω : ker → 1 P has the following properties: 1. χ ◦ ω(c) = 1 ⊗ c, ← −2 2. For any b ∈ C, ψ (b ⊗ ω(c)) = c(1) α c(2) βγ ω(eγ ) ⊗ bαβ , where c(1) ⊗M c(2) (summation understood) denotes the translation map τ (c) = χ−1 M (1 ⊗ c) in P (M, C, ψ, e). Conversely, if ω is any linear map ω : ker → 1 P obeying conditions 1-2, then there is a unique connection Π = µ ◦ (id ⊗ ω) ◦ χ in P (M, C, ψ, e) such that ω is its connection 1-form. Proof. For any b ⊗ u ⊗ c ∈ C ⊗ P ⊗ ker the map φ is explicitly given by φ(b ⊗ u ⊗ c) = uα c(1) β c(2) γδ ⊗ eδ ⊗ bαβγ . Therefore if ω is a connection one-form then ← −2 ← −2 ψ (b ⊗ ω(c)) = ψ ◦ (id ⊗ σ)(b ⊗ 1 ⊗ c) = σ(c(1) α c(2) βγ ⊗ eγ ) ⊗ bαβ = c(1) α c(2) βγ ω(eγ ) ⊗ bαβ . Conversely, if ω : ker → 1 P satisfies condition 1 then σ = (µ ⊗ id) ◦ (id ⊗ ω) gives a left P -module splitting of (24). Furthermore, Condition 2 implies We can equivalently think of a connection 1-form as a map C → P given by ω(c − e(c)). This was the point of view adopted in [BM93]. 1

1

484


(σ ⊗ id) ◦ φ(b ⊗ u ⊗ c) = σ(uα c(1) β c(2) γδ ⊗ eδ ) ⊗ cαβγ = uα c(1) β c(2) γδ ω(eδ ) ⊗ cαβγ ← −3 ← −2 = uα ψ (bα ⊗ ω(c)) = (µ ⊗ id) ◦ ψ (b ⊗ u ⊗ ω(c)) ← −2 ← −2 = ψ (b ⊗ uω(c)) = ψ ◦ (id ⊗ σ)(b ⊗ u ⊗ c). Example 3.6. For a quantum principal bundle P (M, H), Condition 2 in Proposition 3.5 is equivalent to the AdR -covariance of ω. Proof. Using the definition of ψ in Example 2.3 one finds c(1) α c(2) βγ ⊗ eγ ⊗ bαβ = c(1) α c(2) β (0) ⊗ c(2) β (1) ⊗ bαβ ¯

¯

= c(1)(0) c(2) β (0) ⊗ c(2) β (1) ⊗ bβ c(1)(1) ¯

¯

¯

¯ ¯

¯

¯

¯ ¯

¯

¯

= c(1)(0) c(2)(0)(0) ⊗ c(2)(0)(1) ⊗ bc(1)(1) c(2)(1) ¯ ¯ ¯ ¯ = χM (c(1)(0) ⊗M c(2)(0) ) ⊗ bc(1)(1) c(2)(1) . From the covariance properties of the translation map [Brz96a] it then follows that c(1) α c(2) βγ ⊗ eγ ⊗ bαβ = χM (τ (c(2) )) ⊗ b(Sc(1) )c(3) = 1 ⊗ c(2) ⊗ bS(c(1) )c(3) . This also follows from covariance of χM as intertwining 12R projected to P ⊗M P with the tensor product coaction 11R ⊗ AdR on P ⊗ H. Hence Condition 2 may be written as ← −2 ψ (b ⊗ ω(c)) = ω(c(2) ) ⊗ b(Sc(1) )c(3) which is equivalent to 12R ◦ ω = (ω ⊗ id) ◦ AdR .

Example 3.7. For a braided group principal bundle P (M, B, 9) in Example 2.4, Lemma 3.2 holds. Moreover, Condition 2 in Proposition 3.5 is equivalent to AdR -covariance of ω, where AdR is the braided adjoint coaction as in [Maj94a]. Proof. The braided group adjoint action is studied extensively in [Maj94a] as the basis of a theory of braided Lie algebras; we turn the diagrams up-side-down for the braided adjoint coaction and its properties (or see earlier works by the second author). Firstly, ψ(B ⊗ M ) ⊂ M ⊗ B is immediate since by properties of e = 1, 9(B ⊗ M ) ⊂ M ⊗ B. Also clear is that 11R coincides with the given braided coaction of B on P and 12R coincides with the braided tensor product coaction on P ⊗ P . 12R projects to a coaction on P ⊗M P by Lemma 3.2. We show first that χM : P ⊗M P → P ⊗ B intertwines this coaction with the braided tensor product coaction 11R ⊗ AdR . We work with representatives in P ⊗ P and use the notation [Maj93a] as in the proof of Example 2.4. Branches labelled 1 are the coproduct of B; otherwise they are the given coaction of B on P . Thus, P

P

P

P

P

P

P ∆

∆ ∆

P P B

P P B

P

=

=

S

S

P ∆

∆

=

=

P

S

P P B

P P B

P P B

Coalgebra Bundles

485

where the upper box on the left is χM and the lower box is the braided adjoint coaction AdR . S denotes the braided antipode of B. The tensor product 11R ⊗ AdR uses the braiding and the product of B according to the theory of braided groups [Maj93b]. The first equality uses the homomorphism property of the given coaction of B on P . The second uses the comodule axiom. The third identifies an “antipode loop” and cancels it (using associativity and coassociativity, and the braided antipode axioms). The fourth equality uses the comodule axiom in reverse and also pushes the diagram into the form where we recognise the braided tensor product coaction 12R followed by χM . Using this intertwining property of χM , we write the right hand side of Condition 2 in Proposition 3.5 as B

B

B

B

B

B

τ

τ =

Ad

χ ω P P

P B

B

B

χ

=

ω

B

τ =

Ad

P B

ω

? =

ω

ω P

B

P P

P B

P B

P B

← −2 where τ = χ−1 M (1 ⊗( )). The left hand side ψ (b ⊗ ω(c)) is shown on the right hand side of the diagram (using associativity of the product in B). Hence equality is equivalent to 12R ◦ ω = (ω ⊗ id) ◦ AdR . We remark that in the framework with C ∗ in place of C as explained in Proposition 2.7, we can use for C ∗ braided groups of enveloping algebra type, in particular U (L) associated to a braided-Lie algebra L in [Maj94a] with braided-Lie bracket based on the properties of the braided adjoint action. In this case one could take ω ∈ L ⊗ 1 P with the corresponding covariance properties. Using the braided Killing form also in [Maj94a] one has the possibility (for the first time) to write down scalar Lagrangians built functorially from ω and its curvature. On the other hand, for a theory of trivial bundles (in order to have familiar formulae for gauge fields on the base) one needs to restrict trivialisations and gauge transforms in such a way that ω retains its values in L. This aspect requires further work, to be developed elsewhere. Example 3.8. Consider H(M, C, π), the ψ-principal bundle associated to an embeddable quantum homogeneous space in Example 2.5. Assume that ψ(C ⊗ M ) ⊂ M ⊗ C. Condition 2 in Proposition 3.5 is equivalent to 12R ◦ ω ◦ π = (ω ⊗ id) ◦ (π ⊗ π) ◦ AdR .

(25)

In particular, this implies that any linear inclusion i : M → H such that π ◦ i = id and (c) = ◦ i(c) gives rise to the canonical connection 1-form ω(c) = (Si(c)(1) )di(c)(2) , provided that (id ⊗ π) ◦ AdR ◦ i = (i ⊗ id) ◦ (π ⊗ π) ◦ AdR ◦ i. Proof. In this case ψ(c ⊗ v) = v (1) ⊗ π(uv (2) ), and τ (c) = Su(1) ⊗M u(2) , for any c ∈ C, v ∈ H and u ∈ π −1 (c). Also e = π(1). The transformation property of ω now reads

486


← −2 ψ (b ⊗ ω(c)) = (Su(1) )α u(2) βγ ω(π(1)γ ) ⊗ π(v)αβ = (Su(2) )u(3) βγ ω(π(1)γ ) ⊗ π(vSu(1) )β

= (Su(2) )u(3) γ ω(π(1)γ ) ⊗ π(v(Su(1) )u(4) ) = (Su(2) )u(3) ω(π(u(4) )) ⊗ π(v(Su(1) )u(5) ) = ω(π(u(2) )) ⊗ π(v(Su(1) )u(3) ), where v ∈ π −1 (b). Choosing v = 1 we obtain property (25). The converse is obviously true. Before we describe some concrete examples of connections we construct connections in the trivial ψ-bundles of Proposition 2.9. Proposition 3.9. Let P (M, C, 8, ψ, ψ C , e) be a trivial coalgebra ψ-principal bundle such that ψ(C ⊗ M ) ⊂ M ⊗ C. Let β : C → 1 M be a linear map, β(e) = 0 and such that C ◦ ψ23 ◦ ψ12 ◦ (id ⊗ β ⊗ id) ◦ (id ⊗ 1) = (β ⊗ id ⊗ id) ◦ (1 ⊗ id) ◦ ψ C . ψ34

(26)

Then the map ω : ker → 1 P , ω = 8−1 ∗ d8 + 8−1 ∗ β ∗ 8

(27)

is a connection one-form in P (M, C, 8, ψ, ψ C , e). In particular for β = 0 we have a trivial connection in P (M, C, 8, ψ, ψ C , e). Proof. To prove the proposition we will show that ω satisfies conditions specified in Proposition 3.5. Firstly, however, we observe that the translation map in P (M, C, 8, ψ, ψ C , e) is given by (28) τ (c) = 8−1 (c(1) ) ⊗M 8(c(2) ). Indeed, a trivial computation shows that χM (τ (c)) = 1 ⊗ c, as required. The same computation shows that for any c ∈ ker , χ(8−1 (c(1) )d8(c(2) ) + 8−1 (c(1) )β(c(2) )8(c(3) )) = χ(8−1 (c(1) ) ⊗ 8(c(2) )) = 1 ⊗ c, and therefore Condition 1 of Proposition 3.5 is satisfied by ω. Now we prove that Condition 2 of Proposition 3.5 holds for 8−1 ∗d8 and 8−1 ∗β ∗8 separately. For the former the left hand side of Condition 2 reads ← −2 LHS = ψ (b ⊗ 8−1 (c(1) ) ⊗ 8(c(2) )) = 8−1 (c(1) )α ⊗ 8(c(2) )β ⊗ bαβ = 8−1 (c(1) )α ⊗ 8(c(2) A ) ⊗ bαA On the other hand we use the definition of τ (28) and the properties of 8 to write the right-hand side of condition 2 as follows: RHS = 8−1 (c(1) )α 8(c(2) )βγ 8−1 (eγ (1) ) ⊗ 8(eγ (2) ) ⊗ bαβ = 8−1 (c(1) )α 8(c(2) )βγδ 8−1 (eδ ) ⊗ 8(eγ ) ⊗ bαβ = 8−1 (c(1) )α 8(c(2) A )γδ 8−1 (eδ ) ⊗ 8(eγ ) ⊗ bαA

= 8−1 (c(1) )α 8(c(2) A (1) )8−1 (c(2) A (2) ) ⊗ 8(c(2) A (3) ) ⊗ bαA = 8−1 (c(1) )α ⊗ 8(c(2) A ) ⊗ bαA = LHS.

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487

← −2 To compute the action of ψ on the second part of ω we will use the shorthand notation ← −2 ψ (b ⊗ ρ) = ρα ⊗ bα , for any b ∈ C and ρ ∈ 1 P . In this notation Eq. (26) explicitly reads β(c(1) )α ⊗ c(2) A ⊗ bαA = β(cA(1) ) ⊗ cA(2) ⊗ bA . ← −2 Using the similar steps as in computation of the action of ψ on the first part of ω we find that the right hand side of Condition 2 reads 8−1 (c(1) )α β(c(2) A (1) )8(c(2) A (2) ) ⊗ bαA , while the left hand side is 8−1 (c(1) )α β(c(2) )α 8(c(3) A ) ⊗ bααA = 8−1 (c(1) )α β(c(2) A (1) )8(c(2) A (2) ) ⊗ bαA . From Proposition 3.5 we now deduce that ω is a connection one-form as stated.

Using similar arguments as in [BM93] we can easily show that the behaviour of β under gauge transformations is exactly the same as in the case of quantum principal bundles. For example, if we make a gauge transformation of 8, 8 7→ γ ∗ 8 and then view ω in this new trivialisation then the local connection one-from β will undergo the gauge transformation β 7→ γ −1 ∗ dγ + γ −1 ∗ β ∗ γ.

(29)

As before, we can specialise this theory to our various sources of ψ-principal bundles. For quantum principal bundles we recover the formalism in [BM93]. For braided principal bundles we make a computation similar to the one for γ in Example 2.13, finding that (26) is naturally ensured by requiring that β : B → 1 M is a morphism in our braided category. Then the same formulae (27) and transformation law (29) etc. apply in the braided case. Indeed, they do not involve any braiding directly. Now we construct explicit examples of connections in one of the bundles described at the end of Sect. 2. Example 3.10. Consider the quantum cylinder bundle Aq [x−1 ](k[x, x−1 ], k[c], ψ, 1) in Example 2.14. Then ψ(k[c] ⊗ k[x, x−1 ]) ⊂ k[x, x−1 ] ⊗ k[c]. The most general connection of the type described in Proposition 3.9 has the form 2|0

n q k(k−1)/2 y k dy n−k k q k=0 m n X XX n m k k((k−1)/2+i) + (−1) q 0 xi y k (dxm−k−i )y n−m(30) , m q k q i,m−k

ω(cn )=

i

n−1 X

(−1)k

m=0 k=0

where for all i ∈ Z, n ∈ Z≥0 , 0n,i ∈ k, 00,i = 0.

488


Proof. If we set n = 0 in formula (19) then we find ψ(cl ⊗ xm ) = q lm xm ⊗ cl and the first assertion holds. This assertion also follows from Example 3.7. We identify C = k[c] by cn = cn , as a certain (braided) coalgebra. It is an easy exercise to check that a map β : k[c] → 1 k[x, x−1 ] satisfies condition (26) if and only if X 0n,i xi dxn−i , (31) β(cn ) = i

where i ∈ Z, 0n,i ∈ k, 00,i = 0. Now writing the explicit definition of trivialisation 8 (20), and the coproduct of cn (18) we see that ω in (30) is as in (27) with β given by (31). From the braided bundle point of view in Example 2.15 on the same bundle, we work in the braided category of Z-graded spaces and are allowed for β any degree-preserving that vanishes on 1. This immediately fixes it in the form (31), and hence ω from (27).

4. Bundles with General Differential Structures Let P (M, C, ψ, e) be a ψ-principal bundle as in Proposition 2.2. Let N be a subbimodule ← −2 ← −2 ← −2 of 1 P such that ψ (C ⊗ N ) ⊂ N ⊗ C. The map ψ induces a map ψ N : C ⊗ 1 P/N → 1 P/N ⊗ C and N defines a right-covariant differential structure 1 (P ) = 1 P/N on P . We say that 1 (P ) is a differential structure on P (M, C, ψ, e). Definition 4.1. Let P (M, C, ψ, e) be a coalgebra ψ-principal bundle and let ψ(C ⊗ M ) ⊂ M ⊗ C. Assume that N ⊂ 1 P defines a differential structure 1 (P ) on P (M, C, ψ, e). A connection in P (M, C, ψ, e) is a left P -module projection Π : ← −2 ← −2 1 (P ) → 1 (P ) such that ker Π = P 1 (M )P and ψ N ◦ (id ⊗ Π) = (Π ⊗ id) ◦ ψ N . Similarly as for the universal differential calculus case, a connection in P (M, C, ψ, e) can be described by its connection one-form. First we consider the vector space M = (P ⊗ ker )/χ(N ) with a canonical surjection πM : P ⊗ ker → M. Since χ is a left P -module map, χ(N ) is a left P -sub-bimodule of P ⊗ ker . Therefore M is a left P -module and πM is a left P -module map. The action of P on M is defined by X πM (uvi ⊗ ci ), u·υ = i

P −1 (υ). We denote 3 = πM (1 ⊗ ker ). The left where u ∈ P , υ ∈ M and i ui ⊗ ci ∈ πM P -module structure ofP M implies that for every element υ ∈ M, there exist ui ∈ P and λi ∈ 3 such that υ = i ui · λi . Therefore there is a natural surjection P ⊗ 3 → M. We assume that ψ(C ⊗ M ) ⊂ M ⊗ C, and hence the map φ can be defined. For any u ∈ P , c ∈ M and b ∈ C we have ← −2 φ(b ⊗ u ⊗ c) = φ(b ⊗ χ(n)) = (χ ⊗ id) ◦ ψ (b ⊗ n) ∈ χ(N ) ⊗ C, ← −2 where n ∈ N is such that χ(n) = u ⊗ c. We used the fact that ψ (C ⊗ N ) ⊂ N ⊗ C. Therefore we can define a map φN : C ⊗ M → M ⊗ C by the diagram

Coalgebra Bundles

489

C ⊗ P ⊗ ker

id ⊗ πM

-

- 0

C ⊗M φN

φ ? P ⊗ ker ⊗ C

πM ⊗ id

? M⊗C

-

- 0

The map χ induces a map χN : 1 (P ) → M by the commutative diagram πN

1 P

-

- 0

1 (P )

χ

χN

? P ⊗ ker

πM

-

? 0

? M

- 0

? 0

Clearly, χN is a left P -module map, i.e., χN (udv) = u · χN (dv). We can use the map χN to obtain another description of φN . Lemma 4.2. The following diagram

C ⊗ 1 (P )

← −2 ψN

- 1 (P ) ⊗ C

id ⊗ χN ? C ⊗M

χN ⊗ id φN

-

? M⊗C

is commutative. −1 (υ) and compute Proof. We take any υ ∈ 1 (P ), c ∈ C and υ˜ ∈ πN

490


φN ◦ (id ⊗ χN )(c ⊗ υ) = φN ◦ (id ⊗ πM )(c ⊗ χ(υ)) ˜ = (πM ⊗ id) ◦ φ(c ⊗ χ(υ)) ˜ 2 ← − = (πM ⊗ id) ◦ (χ ⊗ id) ◦ ψ (c ⊗ υ) ˜ ← −2 ← −2 = (χN ⊗ id) ◦ (πN ⊗ id) ◦ ψ (c ⊗ υ) ˜ = (χN ⊗ id) ◦ ψ N (c ⊗ υ). Using arguments similar to the proof of Example 4.11 of [BM93] and the definition of a coalgebra ψ-principal bundle P (M, C, ψ, e) we deduce that χN

0 → P 1 (M )P → 1 (P ) → M → 0

(32)

is a short exact sequence of left P -module maps. Proposition 4.3. A connection in P (M, C, ψ, e) with differential structure induced by N is equivalent to a left P -module splitting σN of the sequence (32), such that ← −2 (σN ⊗ id) ◦ φN = ψ N ◦ (id ⊗ σN ). Proof. We use Lemma 4.2 to deduce the covariance properties of χN and then preform calculation similar to the proof of Proposition 3.4. To each connection Π we can associate its connection one form ω : 3 → 1 (P ) by ω(λ) = σN (λ). Similarly to the case of universal differential structure, one proves Proposition 4.4. Let Π be a connection in P (M, C, ψ, e) with differential structure defined by N ⊂ 1 P . Then, for all λ ∈ 3 the connection 1-form ω : 3 → 1 (P ) has the following properties: 1. χN ◦ ω(λ) = λ, ← −2 2. For any b ∈ C, ψ N (b ⊗ ω(λ)) = c˜(1) α c˜(2) βδ ω(πM (1 ⊗ eδ )) ⊗ bαβ , where c˜(1) ⊗M c˜(2) denotes the translation map χ−1 M (1 ⊗ c˜), and c˜ ∈ ker is such that πM (1 ⊗ c˜) = λ. Conversely, if M is isomorphic to P ⊗ 3 as a left P -module and ω is any linear map ω : 3 → 1 (P ) obeying Conditions 1–2, then there is a unique connection Π = µ ◦ (id ⊗ ω) ◦ χN in P (M, C, ψ, e) such that ω is its connection 1-form. In the setting of [BM93] the condition P ⊗ 3 = M is always satisfied for quantum principal bundles, and 3 = ker /Q, where Q is an AdR -invariant right ideal in ker that generates the bicovariant differential structure on the structure quantum group H as in [Wor89]. The detailed analysis of braided group principal bundles with general differential structures will be presented elsewhere. Here we remark only that it seems natural to assume that M = P ⊗3 and then choose 3 to be the space dual to the braided Lie algebra L as discussed in Sect. 3. This choice of 3 is justified by the fact that from the properties of the maps φ and φN it follows that the space 3 is invariant under the braided adjoint coaction (cf. Example 3.7). We complete this section with an explicit example of differential structures and connections on the quantum cylinder bundle in Example 2.14 (cf. Example 3.10). Example 4.5. We consider the quantum cylinder bundle of Example 2.14 (cf. Exam2|0 ple 2.15) and we work with differential structures on Aq classified in [BDR92]. Using the definition of ψ (19) one easily finds that there are two differential structures for which

Coalgebra Bundles

491

− →2 the covariance condition ψ (k[c] ⊗ N ) ⊂ N ⊗ k[c] is satisfied. The subbimodules N are generated by (1 + s)x ⊗ x − x2 ⊗ 1 − 1 ⊗ x2 , y ⊗ x − qxy ⊗ 1 − q1 ⊗ xy + qx ⊗ y (1 + q)y ⊗ y − y 2 ⊗ 1 − 1 ⊗ y 2 , where s ∈ k is a free parameter, in the first case, and by (1 + q)x ⊗ x − x2 ⊗ 1 − 1 ⊗ x2 , y ⊗ x − xy ⊗ 1 − q1 ⊗ xy + x ⊗ y, (1 + q)y ⊗ y − y 2 ⊗ 1 − 1 ⊗ y 2 , 2|0

in the second case. In both cases the modules of 1-forms 1 (Aq ) are generated by the exact one-forms dx and dy. Definitions of the N imply the following relations in 2|0 1 (Aq ) xdx = sdxx,

xdy = q −1 dyx,

ydx = qdxy,

ydy = qdyy,

in the first case, and xdx = qdxx,

ydx = qdxy + (q − 1)dxy,

xdy = dyx,

ydy = qdyy,

in the second one. In both cases (Aq [x−1 ] ⊗ ker )/χ(N ) = Aq [x−1 ] ⊗ 3, where 3 is a one-dimensional vector space spanned by λ = πM (1 ⊗ c) and can be therefore identified with a subspace of k[c] spanned by c. Also in both cases the most general connection is given by 2|0

Π(dx) = 0,

2|0

Π(dy) = dy + αdx,

where α ∈ k, and extended to the whole of 1 (Aq [x−1 ]) as a left Aq [x−1 ]-module map. The corresponding connection one form reads 2|0

2|0

ω(λ) = dy + αdx. The bundle is trivial and this connection can be described by the map β : k[c] → (k[x, x−1 ]) as in Proposition 3.9 (cf. Eq. (31)) with β(cn ) = 0 if n 6= 1 and β(c) = αdx. References [BCGST96] Bonechi, F., Ciccoli, N., Giachetti, R., Sorace, E. and Tarlini, M.: Free q-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group. Commun. Math. Phys. 175, 161–176 (1996) [BDR92] Brzezi`nski, T., Da¸browski, H. and Rembieliński, J.: On the quantum differential calculus and the quantum holomorphicity. J. Math. Phys. 33, 19–24 (1992) [BM93] Brzeziński, T. and Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157, 591–638 (1993) Erratum 167, 235 (1995) [Brz96a] Brzeziński, T.: Translation map in quantum principal bundles. J. Geom. Phys. 20, 349–370 (1996) [Brz96b] Brzeziński, T.: Quantum homogeneous spaces as quantum quotient spaces. J. Math. Phys. 37, 2388–2399 (1996)

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[CKM94] [Con85] [DK94] [Kar87] [Maj90] [Maj91] [Maj93a]

[Maj93b] [Maj94a] [Maj94b] [Maj95] [Pod87] [Sch92] [Wor89]


Cap, A., Karoubi, M. and Michor, P.: The Chern-Weil homomorphism in non commutative differential geometry. Preprint 1994 Connes, A.: Noncommutative differential geometry. Publ. IHEs 62, 257–360 (1985) Dijkhuizen, M.S. and Koornwinder, T.: Quantum homogeneous spaces, duality and quantum 2-spheres. Geom. Dedicata 52, 291–315 (1994) Karoubi, M.: Homologie cyclique et K-théorie. Astérisque, 149 (1987) Majid, S.: Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130, 17–64 (1990) /From PhD Thesis, Harvard, 1988) Majid, S.: Braided groups and algebraic quantum field theories. Lett. Math. Phys. 22, 167–176 (1991) Majid, S.: Beyond supersymmetry and quantum symmetry (an introduction to braided groups and braided matrices). In: M-L. Ge and H.J. de Vega, editors, Quantum Groups, Integrable Statistical Models and Knot Theory, Singapore: World Sci., 1993. pp. 231–282 Majid, S.: Braided groups. J. Pure and Applied Algebra 86, 187–221 (1993) Majid, S.: Quantum and braided Lie algebras. J. Geom. Phys. 13, 307–356 (1994) Majid, S.: Some remarks on the quantum double. Czech. J. Phys. 44, 1059–1071 (1994) Majid, S. Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 Podle´s, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987) Schneider, H.-J.: Normal basis and transitivity of crossed products for Hopf algebras. J. Algebra 152, 289–312 (1992) Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys., 122, 125–170 (1989)

Communicated by A. Connes

Commun. Math. Phys. 191, 493 – 500 (1998)

Communications in


Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity J. D. Maitland Wright Analysis and Combinatorics Research Centre, Mathematics Department PO Box 220, University of Reading, Whiteknights, Reading, RG6 6AX, United Kingdom Received: 6 December 1996 / Accepted: 18 May 1997

Abstract: Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I2 direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give: Let d be a countably additive decoherence functional defined on all pairs of projections in L(H). If H is infinite dimensional then d must be bounded. By contrast, when H is finite dimensional, unbounded (countably additive) decoherence functionals always exist for L(H). 1. Introduction As in Wright [15], one of the key tools applied here is the solution of the Mackey–Gleason Problem, see [1–3] and the references given there, particularly [4,10]. In addition, deep results of Dorofeev and Shertsnev are needed [5–7]. Isham, Linden and Schreckenberg [11] determined the structure of bounded (or, equivalently, continuous) decoherence functionals (see below for definitions) on L(H), where H is a finite dimensional Hilbert space of dimension greater than two. They point out that Gell–Mann and Hartle have proposed a significant generalisation of quantum theory with a scheme whose basic ingredients are “histories” and decoherence functionals. For an excellent account of the physical significance of decoherence functionals – and much more – the reader should consult [11–13] and the references given there. The key importance of boundedness for decoherence functionals was observed by Isham et al. [11] in their penetrating analysis of the situation in finite dimensions. In

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J. D. M. Wright

[15] the author extended their analysis to infinite dimensions, showing, in particular, that whenever d is a bounded, finitely additive, decoherence functional associated with a von Neumann algebra A (where A has no direct summand of Type I2 ) then d can be represented as the difference of two norm-continuous semi-innerproducts on A. It could be argued that the assumption of boundedness is not immediately plausible physically. However, as remarked by Isham et al. in [11], it is physically reasonable to suppose that d is countably additive (see below for precise definitions). Our first aim here is to clarify the relationship between countable additivity and boundedness for decoherence functionals. It is shown in Theorem 3.1 that whenever A is a von Neumann algebra which is not commutative there exists a decoherence functional d, taking finite values on each pair of projections from A, such that d is unbounded. When A is finite dimensional, each family of non-zero, pairwise orthogonal projections is finite. Thus, when A is finite dimensional it is always possible to associate a “countably additive” unbounded decoherence functional with it. It follows from the work of Isham et al. [11] that such a d fails to be continuous. By contrast, let A be a von Neumann algebra which has no direct summand of Type In (where 2 ≤ n < ∞) and let d be a countably additive decoherence functional defined on all pairs of projections from A. Further assume that A can be embedded as a (weakly closed) subalgebra of L(H) where H is separable. Then we shall see in Theorem 4.1 that d is necessarily bounded. The hypothesis that A “lives” on a separable space is physically reasonable and simplifies some of the arguments. However this hypothesis can be dispensed with (at the price of some dull complications) if instead of requiring d to be countably additive it is required to be completely additive.

2. Decoherence Functionals In all that follows A shall be a von Neumann algebra and P (A) the lattice of projections in A. Definition. A decoherence functional associated with A is a function d : P (A) × P (A) → C with the following properties: 1. Hermiticity. For each p and q in P (A) d(p, q) = d(q, p)∗ (Here * denotes complex conjugation.) 2. Additivity. Whenever p1 is perpendicular to p2 and q is an arbitrary projection d(p1 + p2 , q) = d(p1 , q) + d(p2 , q) 3. Positivity d(p, p) ≥ 0 for each p in P (A).

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495

4. Normalisation. d(1, 1) = 1. If d satisfies only the properties of Hermiticity and Additivity then d is said to be an hermitian quantum bimeasure. 2# . Countable Additivity. A decoherence functional (or hermitian quantum bimeasure) is said to be countably additive if, whenever {pi : i = 1, 2, . . .} is a countable collection of pairwise orthogonal projections, then, for each q in P (A), X X d(pi , q). d( pi , q) = Here the series on the right hand side is rearrangement invariant and hence is absolutely convergent. ## 2 . Complete Additivity. A decoherence functional (or hermitian quantum bimeasure) is said to be completely additive if, whenever {pi : i ∈ I} is an infinite collection of pairwise orthogonal projections, X X d(pi , q) d( pi , q) = for each q in P (A). Here all but countably many of the terms d(pi , q) are zero and the convergence is absolute. 3. Unbounded Decoherence Functionals In this section we shall see that for “almost every” von Neumann algebra A, it is possible to produce an unbounded decoherence functional d which takes only finite values on each pair of projections from A. Theorem 3.1. Let A be a von Neumann algebra which is either not commutative or which has an infinite set of non-zero pairwise orthogonal projections. Then there exists a finite valued decoherence functional d : P (A) × P (A) → C such that d is unbounded. Proof. Let us consider the self-adjoint part of A, Asa , as a vector space over Q, where Q is the field of rational numbers. Let P0 = I and let us assume, for the moment, that there exists a countably infinite family of projections {Pn : n = 0, 1, 2, . . .} which is linearly independent over Q, that is, each non-empty finite subset is linearly independent over Q. Using Zorn’s Lemma, we can extend this countable set to a maximal set {Xλ : λ ∈ 3} of self adjoint elements of A which are linearly independent over Q (i.e. a Hamel base for Asa over Q). Here we assume that ω ⊂ 3 and that Xn = Pn for each n ∈ ω. Then every self-adjoint element S in A has a unique representation X S= sλ Xλ , where each sλ is rational and all but finitely many of the sλ are zero.

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P It follows that each element of A has a unique representation of the form zλ Xλ where, for each λ, zλ is complex rational and all but finitely many of the zλ are zero. Let us define a “pseudo-innerproduct” on A by defining X X X wλ Xλ i to be zλ wλ∗ . h zλ Xλ , Let us also define a map T from A to A, where T is linear when A is regarded as a vector space over the complex rationals, by setting T Pn = (n + 1)Pn for each n ∈ w, and for all other λ, T Xλ = 0 and extending T in the obvious way. Let D(z, w) = hT z, T wi for each z and w in A. Then D is additive in both variables and D(z, w) = D(w, z)∗ . Let P x be an arbitrary element of A. Then x has a unique representation of the form each λ, zλ is complex rational and all but finitely many of the zλ zλ Xλ , where, for P are zero. Then T x = zn (n + 1)Pn . So X D(x, x) = |zn |2 (n + 1)2 ≥ 0. On putting d(p, q) = D(p, q) for each p and q in P (A) we see that d is a quantum bimeasure where d(p, p) ≥ 0 for every projection p. Also d(Pn , Pn ) = hT Pn , T Pn i = (n + 1)2 . Since P0 = I, it follows that d is a decoherence functional. Clearly d is unbounded. To complete this argument we must establish the initial assumption. If A has a countably infinite sequence of non-zero orthogonal projections (Pn )(n = 1, 2, . . .) then {I, P1 , P2 , . . .} is linearly independent over the real numbers and hence also over Q. Otherwise, we may suppose that A is not commutative and hence contains a pair of noncommuting projections. Thus, see Takesaki [p. 306, 14], there is a projection e in A such that M2 (C) is a subalgebra of eAe. It now suffices to establish our initial assumption for M2 (C). Let √ t t(1 − t) . P (t) = √ t(1 − t) 1 − t Then P (t) is a projection for 0 ≤ t ≤ 1. Let θ be a transcendental number in the interval (0, 1), e.g. e−1 . Then {θn : n = 1, 2, · · ·} is linearly independent over Q for, otherwise, q would satisfy an algebraic equation with non-zero rational coefficients. Let P0 = I and let Pn = P (θn ) for n = 1, 2, · · · . Clearly these projections are linearly independent over Q. The proof is now complete. Corollary 3.2. Let V be a finite dimensional Hilbert space of dimension two or greater. Then there exists a countably additive, finite valued decoherence functional d associated with L(V ) where d is unbounded. Proof. Let d be obtained as in the theorem above. Then, since each family of non-zero pairwise orthogonal projections is finite, d is countably additive.


497

4. Countably Additive Decoherence Functionals Unless stated otherwise we shall assume that the von Neumann algebra A has a faithful normal representation on a separable Hilbert space. Thus each family of non-zero pairwise orthogonal projections is countable. Hence each countably additive decoherence functional on A is, in fact, completely additive. Theorem 4.1. Let A be a von Neumann algebra with no Type In direct summand (2 ≤ n < ∞). Let A have a faithful normal representation on a separable Hilbert space. Let d be a countably additive decoherence functional associated with A (or, more generally, a countably additive hermitian quantum bimeasure). Then d is bounded. Furthermore, there exists D : A × A → C such that D(p, q) = d(p, q) for each p and q in P (A), where D is a bounded sesquilinear hermitian form and for each y, x → D(x, y) is a normal functional on A. Proof. For each fixed p in P (A), the map q → d(p, q) is a countably additive (and so, as remarked above, completely additive) quantum measure on the projections. By hypothesis, A = A1 ⊕ A2 , where A1 is abelian and where A2 has no direct summand of Type In for n < ∞. By classical measure theory, see Lemma III. 4.4 [p. 127, 8] the measure is bounded when restricted to the projections in A1 , indeed on any maximal abelian *-subalgebra. By deep results of Dorofeev and of Dorofeev and Shertsnev, see Theorem 2 [5] and [6,7], see also Theorem 3.2.20 [9], the measure is bounded on the projections of A2 . Thus this measure is bounded on P (A). Hence, by the Generalised Gleason Theorem for complex valued quantum measures, see [1–3], there exists a bounded linear functional φp such that φp (q) = d(p, q) for all q in P (A). Moreover φp is completely additive on orthogonal projections because of the hypotheses on d. Hence, see Corollary III.3.11 [14], φp is a normal functional on A. Thus φp may be identified with an element of A∗ , the predual of A. For each p ∈ P (A), let M (p) = φp . Whenever p1 and p2 are perpendicular we have M (p1 + p2 )(q) = d(p1 + p2 , q) = d(p1 , q) + d(p2 , q) = M (p1 )(q) + M (p2 )(q). Since finite linear combinations of projections are norm dense in A, M (p1 + p2 ) = M (p1 ) + M (p2 ) In order to be able to apply the vector valued Generalised Gleason Theorem we must show that M maps P (A) into a norm bounded subset of A∗ . By the Uniform Boundedness Theorem it suffices to show that, for each x in A, {φp (x) : p ∈ P (A)} is a bounded set. Clearly it is enough to establish the boundedness of this set when x = x∗ and 0 ≤ x ≤ I. Let us fix such an x and then let B be the von Neumann subalgebra of A generated by x and I. Then B is abelian and generated by its projections and may be identified with L∞ (µ) for an appropriate probability measure µ.

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Since x=

∞ X

2−n qn ,

1

where each qn is a projection in B, it suffices to show that 8 = {φp : p ∈ P (A)} is uniformly bounded on the projections of B. Suppose that this is false. Then, for each n, we can find fn ∈ 8 and en ∈ P (B) such that |fn (en )| ≥ n2 . Let ψn = n−1 fn so that |ψn (en )| ≥ n for each n. For each fixed q in P (B), |ψn (q)| = n−1 |fn (q)| = n−1 |φp(n) (q)| for some p(n) in P (A). Since, for each p in P (A), |φp (q)| = |d(p, q)| = |d(q, p)| = |φq (p)| ≤ kφq k, we have |ψn (q)| = n−1 |φq (p(n))| ≤ n−1 kφq k. So, on letting n → ∞, lim ψn (q) = 0 for each q. For each q in P (B) let F (q) = (ψr (q))(r = 1, 2 · · ·) and let Fn (q) = (ψ1 (q), · · · ψn (q), o, · · ·). Then F and Fn take their values in c0 and kF (q)−Fn (q)k → 0. On restricting ψj to B = L∞ (µ) = L∞ (X, B, µ), say, we obtain a countably additive measure on (X, B) which vanishes on µ-null sets and hence, by Lemma III.4.13 [8], is µ-continuous. It follows that each Fn corresponds to a µ-continuous (vector valued) measure on (X, B). So, by the Vitali–Hahn–Saks Theorem, see Corollary III.7.3 [8], F can be identified with a countably additive vector measure. Hence, by Corollary III.4.5 [8], its range is a bounded subset of c0 . But |ψn (en )| ≥ n for each n. This is a contradiction. So {φp : p ∈ P (A)} is uniformly bounded on P (B). Hence {φp (x) : p ∈ P (A)} is bounded. Thus M maps P (A) into a bounded subset of A∗ . Since A has no Type I2 direct summand we now apply the vectorial form of the Generalised Gleason Theorem [1,2] to deduce that M extends to a bounded linear operator (also denoted by “M ”) from A into A∗ . Following [15], let us define D : A × A → C by D(x, y) = M (x)(y ∗ ). Then D is a bounded sesquilinear form on A × A. Since the map (x, y) → M (x)(y) is a bilinear extension of d, it follows from Theorem 3 [15] that it is unique and that M (x)(y ∗ ) = M (y)(x∗ )∗ . Hence D is hermitian. We now fix y and set φ(x) = D(x, y) = D(y, x)∗ . Then φ(x∗ ) = M (y)(x)∗ . Since M (y) is normal so, also, is φ.


499

5. Countably Additive and Bounded Decoherence Functionals Throughout this section the von Neumann algebra A will be assumed to be embedded as a (weakly closed) subalgebra of L(K), where K is a separable Hilbert space. Furthermore, we shall also assume that A has no Type I2 direct summand. Let d : P (A) × P (A) → C be a countably additive decoherence functional which is also bounded. It follows from Theorem 4.1 that if A has no Type In direct summand ( for 2 ≤ n < ∞) then the countable additivity of d automatically implies boundedness. However, as we have also seen, boundedness is not a consequence of countable additivity for Type In algebras (2 ≤ n < ∞). Lemma 5.1. There exists a unique bounded sesquilinear form D : A × A → C such that D(p, q) = d(p, q) for each p and q in P (A). Furthermore D is hermitian and the map x → D(x, y) is normal for each y in A. Proof. Since d is bounded, Corollary 4 [15] and its proof gives the existence and hermitian property of D. The uniqueness of D follows from its norm continuity and the observation that finite linear combinations of projections are dense in A. The normality of the map x → D(x, y) can be established by arguing as in the proof of Theorem 4.1. Remark. The Haagerup–Pisier–Grothendieck inequality easily implies that D is jointly strong continuous on the unit ball of A. See, for example, Lemma 8 [16]. Theorem 5.2. Let d : P (A)×P (A) → C be a bounded, countably additive decoherence functional. Let A have no direct summand of Type I2 . Then d can be extended to a unique normal Hermitian form D on A. There exist normal functionals (φn )(n = 1, 2, . . .) and (ψn )(n = 1, 2, . . .) such that, for each x and y in A, D(x, y) =

∞ X

φn (x)φn (y)∗ −

n=1

where, for same constant K, both k 2 ||x||2 .

∞ X

ψn (x)ψn (y)∗ ,

n=1

P∞

n=1

|φn (x)|2 and

P∞

n=1

|ψn (x)|2 are bounded by

Proof. The existence and uniqueness of a normal hermitian form D which extends d follows from Lemma 5.1. The existence of (φn )(n = 1, 2, . . .) and (ψn )(n = 1, 2, . . .) with the required properties then follows from Corollary 11 [16]. See, also, the elegant representation theorem of Ylinen [17]. Corollary 5.3. Let d be a countably additive decoherence functional on A, where A has no Type I2 direct summand. Then there exist semi-innerproducts h , i1 and h , i2 on A which are jointly strong* continuous on the unit ball of A and such that d(p, q) = hp, qi1 − hp, qi2 for each p and each q in P (A). P∞ P∞ Proof. Let hx, yi1 = n=1 φn (x)φn (y)∗ and let hx, yi2 = n=1 ψn (x)ψn (y)∗ .

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References 1. Bunce, L.J. and Wright, J.D.M.: The Mackey–Gleason Problem. Bull. Am. Math. Soc. 26, 288–293 (1992) 2. Bunce, L.J. and Wright, J.D.M.: The Mackey–Gleason Problem for vector measures on projections in von Neumann algebras. J. London Math. Soc. 49, 133–149 (1994) 3. Bunce, L.J. and Wright, J.D.M.: Complex measures on projections in von Neumann algebras. J. London Math. Soc. 46, 269–279 (1992) 4. Christensen, E.: Measures on projections and physical states. Commun. Math. Phys. 86, 529–538 (1982) 5. Dorofeev, S.: On the problem of boundedness of a signed measure on projections of a von Neumann algebra. J. Funct. Anal. 103, 209–216 (1992) 6. Dorofeev, S.: On the problem of boundedness of a signed measure on projections of a Type I von Neumann algebra. Proceedings of Higher Educational Institutions, Mathematics 3, 67–69, 1990 7. Dorofeev, S. and Shertsnev, A.N.: Frame type functions and their applications. Proceedings of Higher Educational Institutions, Mathematics 4, 23–29, 1990 8. Dunford, N., and Schwartz, J.T.: Linear operators I. New York: Interscience, 1967 9. Dvurechenskij, A.: Gleason’s theorem and its applications. London: Kluwer, 1993 10. Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957) 11. Isham, C.J., Linden, N., and Schreckenberg, S.: The classification of decoherence functionals: An analogue of Gleason’s theorem. J. Math. Phys. 35, 6360–6370 (1994) 12. Isham, C.J., Linden, N.: Quantum temporal logic and decoherence functionals in the histories approach to generalised quantum theory. J. Math. Phys. 35, 5452–5476 (1994) 13. Isham, C.J.: Quantum logic and the histories approach to quantum theory. J. Math. Phys. 35, 2157–2185 (1994) 14. Takesaki, M.: Theory of operator algebras. New York: Springer 1979 15. Wright, J.D.M.: The structure of decoherence functionals for von Neumann quantum histories. J. Math. Phys. 36, 5409–5413 (1995) 16. Wright, J.D.M.: Linear representations of bilinear forms on operator algebras. Expos. Math. (to appear) 17. Ylinen, K.: The structure of bounded bilinear forms on products of C∗ -algebras. Proc. Am. Math. Soc. 102, 599–601 (1988) Communicated by A. Araki

Commun. Math. Phys. 191, 501 – 541 (1998)

Communications in


Algebra of Screening Operators for the Deformed Wn Algebra Boris Feigin1 , Michio Jimbo2 , Tetsuji Miwa3,? , Alexandr Odesskii1 , Yaroslav Pugai1 1 2 3

L. D. Landau Institute for Theoretical Sciences, Chernogolovka, 142432, Russia Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan Institut Henri Poincaré and Ecole Normale Superieure, France

Received: 3 March 1997 / Accepted: 20 May 1997

Abstract: We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed Wn algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the deformed W3 algebra.

1. Introduction The method of bosonization is known to be the most effective way of calculating the conformal blocks in conformal field theory. The basic idea in this approach is to realize the commutation relations for the symmetry algebra (such as the Virasoro or affine Lie algebras) and the chiral primary fields in terms of operators acting on some bosonic Fock spaces. Quite often, the physical Hilbert space of the theory is not the total Fock space itself, but only a subquotient of it. In this case, it is necessary to ‘project out’ the physical space from the Fock spaces by a cohomological method. In the case of the Virasoro minimal models, Felder [2] introduced a two-sided complex d

d

d

d

d

· · · −→F (−1) −→F (0) −→F (1) −→F (2) −→ · · · ,

(1.1)

consisting of Fock spaces F (i) . As it turns out, the cohomology of this complex vanishes except at the 0th degree, and the remaining non-trivial cohomology affords the irreducible representation of the Virasoro algebra. The primary fields realized on the Fock spaces commute with the coboundary operator d, and hence make sense as operators on the cohomology space. Similar resolutions have been described for representations of affine Lie algebras [3–5]. For an extensive review on this subject, the reader is referred to [6]. ?

On leave from Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan.

502

B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai

It has been recognized that the idea of bosonization is quite fruitful also in non-critical lattice models [7, 8] and massive field theory [9]. The present work is motivated by recent progress along this line, on the restricted solid-on-solid models [10, 11]. In [10], the Andrews-Baxter-Forrester (ABF) model was studied. Here the counterpart of the conformal primary fields are the vertex operators (half transfer matrices) which appear in the corner transfer matrix method. Just as in conformal field theory, the bosonization discussed in [10] consists of two steps. The first step is to introduce a family of bosonic Fock spaces and realize the commutation relations of the vertex operators in terms of bosons. The second step is to realize the physical space of states of the model as the 0th cohomology of a complex of the type (1.1). In fact, the analogy with conformal field theory goes further. Each Fock space has the structure of a module over the deformed Virasoro algebra (DVA) discovered in [12], where the deformation parameter x (0 < x < 1) is the one which enters the Boltzmann weights of the models. As was shown in [10, 13], the above complex is actually that of DVA modules, i.e., the operator d commutes with the action of DVA. Felder’s complex (1.1) is recovered in the limit x → 1 (we shall refer to this as the conformal limit). The ABF models have sln generalizations, the former being the case n = 2. In [11], the first step of the bosonization was carried over to the case of general n. However the second step was not addressed there. The aim of the present paper is to construct an analog of the complex (1.1) in the case n = 3. In this situation the role of DVA is played by the deformed W3 algebra introduced in [14, 15]. We shall also construct for general n a family of intertwiners of deformed Wn algebras (DWA), which we expect to be sufficient to construct the complex in the general case. b 3 was constructed in [5]. In the conformal case, such a Felder-type complex for sl Strictly speaking, [5] discusses representations of affine Lie algebras, while our case corresponds (in the limit) to those of Wn algebras. In other words we are dealing with a coset theory rather than a Wess-Zumino-Witten theory. However the construction of the complex is practically the same for both cases. In the case n ≥ 3, each component F (i) of the complex is itself a direct sum of an infinite number of Fock spaces. The coboundary operator d can be viewed as a collection of maps between various Fock spaces. We call these maps the screening operators. They are given in the form of an integral of a product of screening currents, multiplied by a certain kernel function expressed in terms of elliptic theta functions. The main result of this paper is the explicit construction of these screening operators. In comparison with the conformal case, a simplifying feature is that the screening operators can be expressed as products of more basic, mutually commuting operators. In the conformal case, such a multiplicative structure exists only “inside the contour integral” (see [5] and Sect. 6 below). It has been pointed out [15] that the screening currents satisfy the commutation relations of the elliptic algebra studied in [16]. This connection turns out to be quite helpful in finding the basic operators referred to above and their commutation relations. Let us mention some questions that remain open. In order to ensure the nilpotency property d2 = 0, the signs of the screening operators have to be chosen carefully. We have verified that this is possible for n = 3. In the general case there are additional complications which we have not settled yet. More importantly, in this paper we do not discuss the cohomology of the complex, though we expect the same result persists as in the conformal case. The construction of the complex in [5] is based on the one-to-one b n , and the singular correspondence between intertwiners of Fock space modules over sl vectors in the Verma modules of Uq (sln ) with q a root of unity. It would be interesting to search for an analog of the latter in the deformed situation.

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503

The outline of this paper is as follows. In Sect. 2 we prepare the notation and the setting. Also the form of the complex to be constructed is briefly explained. The construction of the screening operators for general n is rather technical. To ease the reading, we first discuss in Sect. 3 the case n = 3 in detail. In Sect. 4 we introduce the screening operators in general, and state their commutation relations. In Sect. 5 we show that they commute with the action of DWA. In Sect. 6 we briefly discuss the CFT limit of the basic operators. The text is followed by 3 appendices. In Appendix A we discuss the condition when we construct intertwiners between two Fock modules. In Appendix B we list the commutation relations for the basic operators that will be used to derive the quadratic relations of the screening operators. In Appendix C we outline the proof that the screening operators commute with DWA.

2. Preliminaries In this section we prepare the notation to be used in the text. Throughout this paper, we fix a positive integer r ≥ n + 2 and a real number x with 0 < x < 1. 2.1. Lie algebra sln . Let us fix the notation concerning the Lie algebra sln . Let εi (1 ≤ i ≤ n) be an orthonormal in Rn relative to the standard inner Pbasis n product ( , ). We set ε¯i = εi − ε,ε = (1/n) j=1 εj . We shall denote by: Pi • ωi = j=1 ε¯j the fundamental weights, • αi = εi − εi+1 the simple roots, Pn−1 • θ = i=1 αi the maximal root, Pn • P = i=1 Zε¯i the weight lattice, Pn−1 • Q = i=1 Zαi the root lattice, • 1+ = {εi − εj | 1 ≤ i < j ≤ n − 1} the set of positive roots, • W ' Sn the classical Weyl group, • W ' W |× Q the affine Weyl group. For an element γ ∈ Q, we set |γ| =

n−1 X i=1

ci

for γ =

n−1 X

ci αi .

i=1

For a root α = εi − εj , rα signifies the corresponding reflection (often identified with the transposition (ij) ∈ Sn ). We also write si = rαi . j 2.2. Bosons. We recall from [11] our convention about the bosons. Let βm be the oscillators (1 ≤ j ≤ n − 1, m ∈ Z\{0}) with the commutation relations

[(n − 1)m]x [(r − 1)m]x δm+m0 ,0 (j = k), [nm]x [rm]x [m]x [(r − 1)m]x δm+m0 ,0 (j 6= k). = −mxsgn(j−k)nm [nm]x [rm]x

j k , βm [βm 0] = m

(2.1)

(2.2)

504


n Here the symbol [a]x stands for (xa − x−a )/(x − x−1 ). Define βm by n X

j x−2jm βm = 0.

(2.3)

j=1

Then the commutation relations (2.1),(2.2) are valid for all 1 ≤ j, k ≤ n. We also introduce the zero mode operators Pλ , Qλ indexed by λ ∈ P . By definition they are Z-linear in λ and satisfy [iPλ , Qµ ] = (λ, µ)

(λ, µ ∈ P ).

j We shall deal with the bosonic Fock spaces Fl,k (l, k ∈ P ) generated by β−m (m > 0) over the vacuum vectors |l, ki: j j , β−2 , · · ·}1≤j≤n ]|l, ki, Fl,k = C[{β−1

where j |l, ki = 0, βm

(m > 0), r r r r−1 l− k)|l, ki, Pα |l, ki = (α, r−1 r √ r−1 √ r |l, ki = ei r−1 Ql −i r Qk |0, 0i. In what follows we set πˆ i =

p r(r − 1)Pαi .

It acts on Fl,k as an integer, πˆ i |Fl,k = (αi , rl − (r − 1)k). def

In this paper we work on Fλ = Fl,λ (λ ∈ P ) with a fixed value of l ∈ P . 2.3. Screening currents. We define the screening currents ξj (u) (j = 1, · · · , n − 1) by P √ r−1 1 r−1 1 (β j −β j+1 )(xj z)−m ξj (u) ≡ Uj (z) = ei r Qαj z r πˆ j + r : e m6=0 m m m :, (2.4) where the variable u is related to z via z = x2u . We shall need the following commutation relations between them. [u − v − 1] ξj (v)ξj (u), [u − v + 1] [u − v + 21 ] ξj±1 (v)ξj (u), ξj (u)ξj±1 (v) = − [u − v − 21 ] ξi (u)ξj (v) = ξj (v)ξi (u) (|i − j| > 1). ξj (u)ξj (v) =

Here the symbol [u] stands for the theta function satisfying [u + r] = −[u] = [−u], τ

[u + τ ] = −e2πi(u+ 2 )/r [u] where τ =

πi log x .

(2.5) (2.6) (2.7)


505

Explicitly it is given by 2

[u] = xu /r−u 2x2r (x2u ), 2q (z) = (z; q)∞ (qz −1 ; q)∞ (q; q)∞ , ∞ Y (1 − zq i ). (z; q)∞ =

(2.8) (2.9) (2.10)

i=0

Quite generally we say that an operator X has weight ν if XFλ ⊂ Fλ+ν for any λ. Then ξj (u) has weight −αj . This implies (2.11) πˆ i ξj (u) = ξj (u) πˆ i − (αi , αj )(1 − r) . 2.4. The complex. In this section we describe the form of the complex we are going to construct. Fix an integral weight 3 ∈ P satisfying (3, αi ) > 0

(i = 1, · · · , n − 1),

(3, θ) < r.

(2.12)

Note that 0 < (3, α) < r

(2.13)

for any positive root α. Consider the orbit of 3 under the action of the affine Weyl group W . An element of W 3 can be written uniquely as λ = tγ σ3 = σ3 + rγ,

(2.14)

where σ ∈ W and γ ∈ Q. We assign a degree deg(λ) ∈ Z to (2.14) by setting deg(λ) = l(σ) − 2|γ|.

(2.15)

Here l(σ) denotes the length of σ ∈ W . (The right hand side of (2.15) is known as the modified length of w = tγ σ ∈ W , see e.g. [4, 5].) We shall construct a complex of the form d d d d d (2.16) · · · −→F3(−1) −→F3(0) −→F3(1) −→F3(2) −→ · · · , where

F3(i) =

M

Fλ

(i ∈ Z).

(2.17)

λ∈W 3 deg(λ)=i

Except for n = 2, F3(i) is a direct sum of an infinite number of Fock spaces. The coboundary map d : F3(i) → F3(i+1) can be viewed as a collection of operators dλ0 ,λ : Fλ(i) → Fλ(i+1) 0

(2.18)

associated with each pair λ, λ0 ∈ W 3 satisfying deg(λ0 ) = deg(λ) + 1. We shall impose a restriction on the possible pair λ, λ0 as explained below. For a positive root α and an element λ ∈ W 3, we define an integer mα (λ) by 0 < mα (λ) < r, mα (λ) ≡ (λ, α) mod r.

(2.19) (2.20)

506


In other words, if λ = tγ σ3, then we have (σ3, α) mα (λ) = (σ3, α) + r

if (σ3, α) > 0; if (σ3, α) < 0.

(2.21)

Set λ = λ − mα (λ)α = α

tγ rα σ3 tγ−α rα σ3

if (σ3, α) > 0; if (σ3, α) < 0.

(2.22)

Clearly the weight λα also belongs to the orbit W 3. Definition 2.1. We say that an ordered pair (λ, λ0 ) is admissible if the following hold for some positive root α. λ 0 = λα , deg(λα ) = deg(λ) + 1.

(2.23) (2.24)

We set dλ0 ,λ = 0 if λ, λ0 is not admissible. Otherwise, write λ0 = λα and dλα ,λ = Xα (λ) : Fλ −→ Fλα .

(2.25)

The construction of the complex is equivalent to finding an operator (2.25) for each admissible pair λ, λα , so that we have d2 = 0. We shall refer to (2.25) as a screening operator. We also require that the screening operators commute with the DWA generators. In practice, we find it convenient to construct the screening operators in the form Xα (λ) = sα (λ)X α (λ), where sα (λ) = ±1 is a sign factor. In Section 4 we give both sα (λ) and X α (λ) so that we have d2 = 0 for the case n = 3. The general case is incomplete because we could not find a proper choice of the signs sα (λ). The construction of the screening operators (2.25) is based on the screening currents (2.4). Let us consider the case where α in (2.25) is a simple root αj . It turns out that the pair λ, λαj is admissible for any λ ∈ W 3 (see Lemma A.3). In this case the operator (2.25) can be found as follows: X αj (λ) = Xja (a = mαj (λ)), I [u + 21 − πˆ j ] dz ξj (u) Xj = (z = x2u ). 2πiz [u − 21 ]

(2.26) (2.27)

Here the integration is taken over the contour |z| = 1. Notice that the kernel function F (u) = [u + 21 − πˆ j ]/[u − 21 ] has the quasi-periodicity F (u + τ ) = e2πi(1−πˆ j )/r F (u),

(2.28)

which ensures that the integrand of (2.27) is a single valued function in z. For n = 2, (2.26) exhausts the possible screening operators. For n ≥ 3 we must also construct operators corresponding to non-simple roots. As we shall see, they are given by similar (but more complicated) integrals over products of the screening currents.


507

3. Case n = 3 Before embarking upon the construction of the complex in general, let us first elaborate on the case n = 3. Hopefully this will make clear the main points of the construction. The following figure shows the configuration of the weights in the orbit W 3 for n = 3.

A

C

B

Fig. 1. The orbit W 3 for n = 3. It forms a hexagonal lattice, consisting of three types of basic hexagons A, B, C and their translates by r times the root lattice Q

In the figure, each vertex represents a weight λ ∈ W 3. An arrow from λ to λ0 indicates that the pair (λ, λ0 ) is admissible. As was mentioned before, (λ, λα ) is always admissible for a simple root α = α1 , α2 . For n = 3 there is also the ‘third root’ θ = α1 +α2 . It turns out that (λ, λθ ) is admissible if and only if λ = tγ σ3

with σ = s1 , s2 , s1 s2 s1 .

The nilpotency d2 = 0 leads to two types of relations for the screening operators. The first type involves only screening operators corresponding to one simple root αj , and has the form Xja Xjr−a = 0. For n = 2, this relation was proved in [13]. The same argument applies to show Xjr = 0 for any j. The second type involves the root θ = α1 + α2 , and occurs for each square inside a (a) (a = mθ (λ)), indicating hexagon (see Fig.1). Let us write the operator X θ (λ) as X12 that it has weight −aθ. Let ai = (3, αi ) (i = 1, 2), a0 = r − a1 − a2 , and suppose λ = tγ σ3. If σ = s1 , then mθ (λ) = a2 , and the following relations must be satisfied: (a2 ) a0 X2r−a1 X1a2 ± X12 X2 = 0,

X2a2 X1r−a1 X1r−a0 X2a2 X1a2 X2r−a0

± ± ±

(a2 ) X1a0 X12 (a2 ) a1 X12 X1 a1 (a2 ) X2 X12

(3.1)

= 0,

(3.2)

= 0,

(3.3)

= 0.

(3.4)

508


(a) The operator X12 and the signs ± in the commutation relations will be given later. The relations in the other cases σ = s2 , s1 s2 s1 are obtained by permuting the upper indices (a0 , a1 , a2 ) → (a2 , a0 , a1 ) successively. In the conformal case x = 1 (and n = 3), the ‘third’ screening operator Xθ (λ) satisfying the relations (3.1)-(3.4) was found in [5]. The construction in [5] is based on the observation that the screening operators Xj for the simple roots satisfy the same Serre relations as do the Chevalley generators of the quantum group Uq (sln ), with q being a root of unity (q 2r = 1 in the present notation). With the aid of these relations, the operator Xθ (λ) was expressed as a (non-commutative) polynomial in X1 and X2 . This method does not easily generalize to the deformed case, since it appears that there is no analog of the Serre relations between X1 and X2 . Nevertheless there exists a family of operators, in terms of which the third screening operator can be written in a simple factorized form. Consider an operator of weight −θ of the form II dz1 dz2 ξ1 (u1 )ξ2 (u2 )F (u1 , u2 ), 2πiz1 2πiz2

with some function F (u1 , u2 ) which is periodic in ui with period r. The integration is taken over |z1 | = |z2 | = 1. Recall (2.11) and (2.28). In order that the integrand be single valued in zi , we demand that F (u1 + τ, u2 ) = e−2πiπˆ 1 /r F (u1 , u2 ),

F (u1 , u2 + τ ) = e−2πi(πˆ 2 −1)/r F (u1 , u2 ),

where τ = πi/ log x. Assume further that F (u1 , u2 ) is holomorphic except for possible simple poles at ui = 1/2 and u1 − u2 + 1/2 = 0. (As for the last pole, we have taken into account the commutation relation (2.6).) If we regard the πˆ i ’s as constants, then the space of functions satisfying these conditions is 3 dimensional. It is straightforward to find a spanning set of such functions. This motivates us to introduce the following family of operators parameterized by k: II dz1 dz2 k ξ1 (u1 )ξ2 (u2 ) X12 (k) = (−1) 2πiz1 2πiz2 [u1 + k + 21 − πˆ 1 ] [u2 − k − 21 − πˆ 2 ] [u1 − u2 − k − 21 ] . (3.5) × [u1 − 21 ] [u2 − 21 ] [u1 − u2 + 21 ] Proposition 3.1. [X12 (k), X12 (l)] = 0 for any k, l, X1 X2 = X12 (−1), X2 X1 = X12 (0), X1 X12 (k) = X12 (k − 1)X1 , X2 X12 (k − 1) = X12 (k)X2 .

(3.6) (3.7) (3.8) (3.9)

The proof of these statements will be given later in the context of general n. Notice the periodicity relation (3.10) X12 (k + r) = (−1)r−1 X12 (k).


Set (a) (k) = X12

a Y

509

X12 (k − b + 1).

(3.11)

b=1

Then, for any non-negative integers a, b, we have (b) X1a+b X2b = X12 (−a − 1)X1a ,

(3.12)

(a) X1a X2a+b = X2b X12 (−b − 1), a a+b b (a) X2 X1 = X1 X12 (a + b − 1), (b) (a + b − 1)X2a . X2a+b X1b = X12

(3.13) (3.14) (3.15)

As an example, let us verify the first relation. Consider first the case b = 1. From (3.7) and (3.8) we have X1a+1 X2 = X1a X12 (−1) = X1a−1 X12 (−2)X1 = ··· = X12 (−a − 1)X1a . The case of general b follows immediately from this and the definition (3.11). The other relations are derived in a similar manner. Now set (a) X α1 +α2 (λ) = X12 (k),

a = mα1 +α2 (λ), k = (λ, α1 ) − 1.

Comparing (3.12)-(3.15) with (3.1)-(3.4) and taking (3.10) into account, we see that the desired relations are satisfied up to sign. It remains to settle the issue about the signs sα (λ). From the definition, the screening operators have the periodicity if β ∈ rZα1 + 2rZα2 .

X α (λ + β) = X α (λ)

Thus the signs can also be chosen according to the same periodicity. A direct verification shows that the following is one possible solution for the sα (λ). B

A +

+

+

- -

C +

-a -a +

+ +

+

+ -b

D +

+

+

- -

-b

b

E

b +

+

+

+

+

-a -a a

+ k

+ k

F + + -b -b

+ a

+

+

+ +

+

Fig. 2. The choice of the signs sα (λ). We set a = εr 1 , b = εr 2 , εr = (−1)r−1 . The vertices at the top row correspond to the weights A = 3, B = r1 r2 3 + α1 , C = r2 r1 3 + α1 , D = 3 + α1 − α2 , E = r1 r2 3 + 2α1 − α2 , F = r2 r1 3 + 2α1 − α2 , with rj = rαj

510


4. Screening Operators 4.1. Basic operators. We are now in a position to introduce the operators which will play a basic role in the construction of screening operators for general n. Let α = αi + · · · + αi+m be a positive root. We often write it as αi···i+m . Define I Xα (k1 , . . . , km ) = ×

···

I i+m Y dzj ξi (ui ) ξi+m (ui+m ) ··· 1 2πizj [ui − 2 ] [ui+m − 21 ] j=i

i+m Y

[uj−1 − uj ] fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i , . . . , πˆ i+m ). 1 [u − u + ] j−1 j 2 j=i+1

(4.1)

Here fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i , . . . , πˆ i+m ) = (−1)k1 +···+km m m Y [ui+l−1 − ui+l − kl − 21 ] Y 1 [ui+l − kl + kl+1 − − πˆ i+l ], [ui+l−1 − ui+l ] 2 l=1

(4.2)

l=0

and k0 = −1, km+1 = 0 is implied. The integrand of (4.1) is a single-valued function in zj (i ≤ j ≤ i + m). To see this note that i+m Y

ξi (ui ) · · · ξi+m (ui+m )

− r1 πˆ j

zj

− r−1

zi+mr

(4.3)

j=i

is single-valued. When α = αj is a simple root, (4.1) reduces to (2.26). The basic property of (4.1) is the following commutativity. Theorem 4.1. For any k1 , · · · , km , p we have [Xα (k1 , . . . , km ), Xα (k1 + p, . . . , km + p)] = 0.

(4.4)

In view of this, we define for a non-negative integer a Definition 4.2. Xα(a) (k1 , . . . , km ) =

a Y

Xα (k1 − b + 1, . . . , km − b + 1).

(4.5)

b=1 (a) Sometimes, we abbreviate Xα(a)i···i+m (k1 , . . . , km ) to Xi···i+m (k1 , . . . , km ).

Proof of Theorem 4.1. Using the commutation relations (2.5)-(2.7) and (2.11), we can write the product Xα (k1 , . . . , km )Xα (k1 + p, . . . , km + p) in the form I

I i+m Y dzj dwj ξi (ui ) ξi (vi ) ξi+m (ui+m ) ξi+m (vi+m ) ··· ··· 1 1 2πiz 2πiw [ui+m − 21 ] [vi+m − 21 ] j j [ui − 2 ] [vi − 2 ] j=i ×F (ui , vi , · · · , ui+m , vi+m )

(zk = x2uk , wk = x2vk ),

and likewise for the product in the opposite order. Symmetrizing with respect to the integration variables and equating the integrand, we are led to prove the following equality:


511

S fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i − (1 − r), πˆ i+1 , . . . , πˆ i+m−1 , πˆ i+m − (1 − r))fα(k1 +p,...,km +p) (vi , . . . , vi+m ; πˆ i , . . . , πˆ i+m ) ×

i+m Y j=i

i+m−1 i+m [uj − vj−1 + 21 ] [uj − vj − 1] Y [uj − vj+1 + 21 ] Y (−1) [uj − vj ] [uj − vj+1 ] [uj − vj−1 ] j=i j=i+1

= S fα(k1 +p,...,km +p) (vi , . . . , vi+m ; πˆ i − (1 − r), πˆ i+1 , . . . , πˆ i+m−1 , πˆ i+m − (1 − r))fα(k1 ,...,km ) (ui , . . . , ui+m ; πˆ i , . . . , πˆ i+m ) ×

i+m Y j=i

i+m−1 i+m [uj − vj+1 − 21 ] Y [uj − vj−1 − 21 ] [uj − vj + 1] Y . (−1) [uj − vj ] [uj − vj+1 ] [uj − vj−1 ] j=i j=i+1

(4.6) Here the symbol S means the symmetrization of (uj , vj ) for each j = i, . . . , i + m. This is equivalent to Y m 1 3 [ui+l−1 − ui+l − kl − ][ui + k1 + − πˆ i ] A 2 2 l=1 m−1 Y 1 [ui+l − kl + kl+1 − − πˆ i+l ] × 2 l=1

Y 1 1 1 − πˆ i+m ] [vi+l−1 − vi+l − kl − p − ][vi + k1 + p + − πˆ i ] 2 2 2 l=1 m−1 Y 1 1 [vi+l − kl + kl+1 − − πˆ i+l ] [vi+m − km − p − − πˆ i+m ] 2 2 l=1 i+m i+m−1 i+m Y Y 1 Y 1 [uj − vj − 1] [uj − vj+1 + ] [uj − vj−1 + ] × 2 2 j=i j=i j=i+1 Y m 1 1 =A [ui+l−1 − ui+l − kl − ][ui + k1 + − πˆ i ] 2 2 l=1 m−1 Y 1 [ui+l − kl + kl+1 − − πˆ i+l ] × 2 m

×[ui+m − km +

l=1

Y 1 1 3 − πˆ i+m ] [vi+l−1 − vi+l − kl − p − ][vi + k1 + p + − πˆ i ] 2 2 2 l=1 m−1 Y 1 1 [vi+l − kl + kl+1 − − πˆ i+l ] [vi+m − km − p + − πˆ i+m ] × 2 2 l=1 i+m i+m−1 i+m Y Y 1 Y 1 [uj − vj + 1] [uj − vj+1 − ] [uj − vj−1 − ] . × 2 2 j=i j=i m

×[ui+m − km −

j=i+1

(4.7)

512


Here the symbol A means the anti-symmetrization of (ui+j , vi+j ) for each j = 0, . . . , m. We prove this equality by induction on m. The (LHS) − (RHS) is a theta function of order 4 in ui . Using the induction hypothesis, one can check that it vanishes at ui = vi , vi ± 1. Taking into account the quasi-periodicity, we conclude that it must have a factor 1 [ui − vi ][ui − vi − 1][ui − vi + 1][ui + 2vi − ui+1 − vi+1 + − πˆ i ]. (4.8) 2 This is a contradiction unless (LHS) − (RHS) = 0. 4.2. Definition of screening operators. Let us come to the definition of the screening operators. Let (λ, λα ) be an admissible pair, with α = αi···i+m . We define X α (λ) : Fλ → Fλα by the formula X α (λ) = Xα(a) (k1 , . . . , km ), (4.9) where a = mα (λ), kj = (λ, αi···i+j−1 ) − 1.

(4.10) (4.11)

Note that on Fλ the operator πˆ j has the fixed value πˆ j |Fλ = (rl + (1 − r)λ, αj ) = (λ, αj ) mod r.

(4.12)

More explicitly the operator (4.9) is given as follows. Proposition 4.3. Notations being as in (4.2) and (4.9)–(4.12), we set (a) (1) (a) fα(a) (u(1) i , . . . , ui , . . . , ui+m , . . . , ui+m ) Y a (b) fα(k1 −b+1,...,km −b+1) (u(b) ˆ i − (a − b)(1 − r), πˆ i+1 , . . . , =S i , . . . , ui+m ; π b=1

Y

. . . , πˆ i+m−1 , πˆ i+m − (a − b)(1 − r))

1≤b < l(σ) is equivalent to i > j. Since α is positive and rα = (ij), we have α = εi − εj if i < j and α = εj − εi if i > j. Noting that σ −1 εi = εσ−1 (i) −1 > and σ −1 (i) < σ −1 (j), we conclude that l(rα σ) > < l(σ) is equivalent to σ α < 0, and therefore to (σ3, α) = (3, σ −1 α) > < 0. Lemma A.2. l(rα ) = 2|α| − 1. Proof. If α = αi···i+m , a reduced expression of rα is given by rα = si · · · si+m−1 si+m si+m−1 · · · si .

We consider an operator Xα (λ) if and only if def

dα (λ) = deg(λα ) − deg(λ) = 1.

(A.3)

Note that dα (λ) =

l(rα σ) − l(σ) l(rα σ) − l(σ) + 2|α|

if (σ3, α) > 0; if (σ3, α) < 0.

(A.4)

In particular, we have 0 < dα (λ) < 2|α|.

(A.5)

Lemma A.3. A pair (λ, λα ) is admissible if and only if one of the following holds: (i) (σ3, α) > 0, and (σ3, β) < 0 or (σ3, γ) < 0 for any partition α = β + γ (β, γ ∈ 1+ ). (ii) (σ3, α) < 0, and (σ3, β) < 0 and (σ3, γ) < 0 for any partition α = β + γ (β, γ ∈ 1+ ). In particular, (λ, λα ) is always admissible for a simple root α = αj . Proof. We follow the argument in the proof of Lemma A.1. If l(rα σ) > l(σ), we set β = εi − εk and γ = εk − εj for k such that i < k < j. The condition dα (λ) = l(rα σ) − l(σ) = 1 is equivalent to σ −1 (k) < σ −1 (i) or σ −1 (j) < σ −1 (k) for any such k. This is equivalent to (σ3, β) < 0 or (σ3, γ) < 0, respectively. If l(rα σ) < l(σ), we set β = εj − εk and γ = εk − εi for k such that j < k < i. The condition dα (λ) = l(rα σ) − l(σ) + 2|α| = 1 is equivalent to σ −1 (i) < σ −1 (k) < σ −1 (j) for any such k. This is equivalent to (σ3, β) < 0 and (σ3, γ) < 0.


527

A.2. Commuting squares. In this subsection we prove Theorem 4.4. The assertion (i) follows immediately from dα (λ) + dα (λα ) = 2|α|.

(A.6)

Below we shall prove the assertion (ii) case-by-case. Case (α, β) = 0. Set m = mα (λ) and m0 = mβ (λ). Recall (2.22). We have (rα σ3, β) = (σ3, β).

(A.7)

This implies mβ (λα ) = m0 . Therefore, we have λ − λα,β = mα + m0 β.

(A.8)

λ − λβ,α = mα + m0 β.

(A.9)

Similarly, we have

This implies dβ (λ) + dα (λβ ) = deg(λβ,α ) − deg(λ) = deg(λα,β ) − deg(λ) = 2.

(A.10)

Since dβ (λ), dα (λβ ) > 0, we have dβ (λ) = dα (λβ ) = 1.

(A.11)

Namely, (λ, λβ , λβ,α ) is admissible. Let us show the uniqueness of α0 , β 0 . Suppose that α = εi − εj and β = εk − εl . We consider only the case when k < i < j < l and set γ1 = εk − εi and γ2 = εj − εl . The other cases are similar. If (λ, λγ , λ − mα − m0 β)

(A.12)

is admissible and γ 6= α, β, then we have γ = εk − εj = γ1 + α or γ = εi − εl = α + γ2

(A.13)

m = m0 = mγ1 +α (λ) = mα+γ2 (λ).

(A.14)

and

Since mα (λ) ≡ (σ3, α) and mγ1 +α (λ) ≡ (σ3, γ1 + α) modr, we have (σ3, γ1 ) ≡ 0 mod r. This is a contradiction. Case (α, β) = 1. Set m = mα (λ) and m0 = mβ (λα ). We have (A.8). The only way other than (A.8) to write λ − λα,β as a positive linear combination of two positive roots is m(α − β) + (m + m0 )β if α − β ∈ 1+ ; λ − λα,β = (A.15) (m + m0 )α + m0 (β − α) if β − α ∈ 1+ . The uniqueness is then obvious from (A.15). Note that

528


(rβ σ3, α − β) = (σ3, α) = (rα σ3, β) = (σ3, β − α) =

m m−r

if (σ3, α) > 0; if (σ3, α) < 0,

(A.16)

m0 m0 − r

if (σ3, α − β) < 0; if (σ3, α − β) > 0.

(A.17)

Let us show that (λ, λβ , λβ,α−β ) is admissible if α − β ∈ 1+ . From (A.16) follows mα−β (λβ ) = m. Let us prove mβ (λ) = m + m0 .

(A.18)

If (σ3, α) > 0 and (σ3, α − β) < 0, the statement (A.18) follows from (A.16) and (A.17). The case (σ3, α) < 0 and (σ3, α − β) > 0 contradicts Lemma A.3 because (λ, λα ) is admissible. In the remaining cases, we have (σ3, β) = m + m0 − r. From Lemma A.3 (applied to (λ, λα )), we have m + m0 − r < 0, and therefore (A.18). Now, we have λβ,α−β = λα,β . This implies (see (A.11)) dα (λ) = dα−β (λβ ) = 1.

(A.19)

Thus we proved the admissibility of (λ, λβ , λβ,α−β ) if α − β ∈ 1+ . Next, we show that (λ, λβ−α , λβ−α,α ) is admissible if β − α ∈ 1+ . From (A.17) follows mβ−α (λ) = m0 . Let us prove mα (λβ−α ) = m + m0 .

(A.20)

Note that (rβ−α σ3, α) = (σ3, β). If (σ3, α) > 0 and (σ3, β − α) > 0, the statement (A.20) follows from (A.16) and (A.17). The case (σ3, α) < 0 and (rα σ3, β) = (σ3, β− α) < 0 contradicts with Lemma A.3 because (λα , λα,β ) is admissible and (rα σ3, α) = −(σ3, α) > 0. In the remaining cases, we have (σ3, β) = m + m0 − r. From Lemma A.3 (applied to (λα , λα,β )) we have (rα σ3, β −α) = (σ3, β) < 0, and therefore (A.20). Thus we proved (ii) when (α, β) = 1. Case (α, β) = −1. Set m = mα (λ) and m0 = mβ (λα ). Because of (2.13) we have (σ3, β) 6≡ 0 mod r, and therefore m 6= m0 . We have (A.8). The only way other than (A.8) to write λ − λα,β as a positive linear combination of two positive roots is (m − m0 )α + m0 (α + β) if m > m0 ; α,β = (A.21) λ−λ m(α + β) + (m0 − m)β if m < m0 . Again the uniqueness is obvious from (A.21). Note that m if (σ3, α) > 0; (rβ σ3, α + β) = (σ3, α) = m − r if (σ3, α) < 0, if (σ3, α + β) > 0; m0 (rα σ3, β) = (σ3, α + β) = m0 − r if (σ3, α + β) < 0.

(A.22) (A.23)

Let us show that (λ, λα+β , λα+β,α ) is admissible if m > m0 . From (A.23) follows mα+β (λ) = m0 . Let us prove mα (λα+β ) = m − m0 .

(A.24)


529

> Note that (rα+β σ3, α) = −(σ3, β). If (σ3, α) > < 0 and (σ3, α + β) < 0, we have −(σ3, β) = m − m0 > 0, and therefore (A.24). If (σ3, α) < 0 and (σ3, α + β) > 0, we have −(σ3, β) = m − m0 − r < 0, and therefore (A.24). If (σ3, α) > 0 and (σ3, α + β) < 0, we have −(σ3, β) = m − m0 + r > r. This is a contradiction. Let us show that (λ, λβ , λβ,α+β ) is admissible if m < m0 . From (A.23) we have mα+β (λβ ) = m. Let us prove mβ (λ) = m0 − m.

(A.25)

> 0 If (σ3, α) > < 0 and (σ3, α + β) < 0, we have (σ3, β) = m − m > 0, and therefore (A.25). If (σ3, α) < 0 and (σ3, α + β) > 0, we have (σ3, β) = m0 − m + r > r. This is a contradiction. If (σ3, α) > 0 and (σ3, α+β) < 0, we have (σ3, β) = m0 −m−r < 0, and therefore (A.25). We have completed the proof of (ii) when (α, β) = −1. B. Generalized Serre Relations We modify the relations (2.5), (2.6) and (2.11) (keeping (2.7)) as follows. [u − v − δ] ξi (v)ξi (u), [u − v + δ] [u − v + δ2 ] ξj (v)ξi (u) if |i − j| = 1, ξi (u)ξj (v) = [u − v − δ2 ] πˆ i ξj (u) = ξj (u) πˆ i − (αi , αj )δ . ξi (u)ξi (v) =

(B.1) (B.2) (B.3)

Here, δ is a parameter. Note that if we set δ = 0 we have a commutative algebra. Fix (a1 , . . . , an−1 )

(ai ∈ Z≥0 ).

(B.4)

Consider a function f of the variables u(b) j (1 ≤ j ≤ n − 1, 1 ≤ b ≤ aj ) and (a )

j κj (1 ≤ j ≤ n − 1). We assume that f is symmetric in (u(1) j , . . . , uj ) for each 1 ≤ j ≤ n − 1. We call f a function of type (a1 , . . . , an−1 ). Suppose that f is of type (a1 , . . . , an−1 ) and g is of type (b1 , . . . , bn−1 ). We define the ∗-product f ∗ g of f and g to be the function of type (a1 + b1 , . . . , an−1 + bn−1 ) given by

(a

)

(b

)

(a1 ) (1) (b1 ) (1) (1) n−1 n−1 (f ∗ g)(u(1) 1 , . . . , u1 , v1 , . . . , v1 , . . . , un−1 , . . . , un−1 , vn−1 , . . . , vn−1 ; (an−1 ) (a1 ) (1) κ1 , . . . , κn−1 ) = S f (u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 ; κ1 + (−2b1 + b2 )δ,

κ2 + (b1 − 2b2 + b3 )δ, . . . , κn−1 + (bn−2 − 2bn−1 )δ) (b

)

(1) n−1 , . . . , vn−1 ; κ1 , κ2 , . . . , κn−1 ) ×g(v1(1) , . . . , v1(b1 ) , . . . , vn−1

Y 1≤j≤n−1 1≤a≤aj 1≤b≤bj

(b) [u(a) j − vj − δ] (b) [u(a) j − vj ]

Y 1≤j≤n−2 1≤a≤aj 1≤b≤bj+1

(b) δ [u(a) j − vj+1 + 2 ] (b) [u(a) j − vj+1 ]

Y 2≤j≤n−1 1≤a≤aj 1≤b≤bj−1

(b) δ [u(a) j − vj−1 + 2 ] (b) [u(a) j − vj−1 ]

.

(B.5)

530

B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai (a )

(b )

(1) j j Here the symbol S means the symmetrization of (u(1) j , . . . , uj , vj , . . . , vj ) for each 1 ≤ j ≤ n − 1. (k1 ,...,km ) be the following function of type (a1 , . . . , an−1 ) with Let fi···i+m

n aj =

1 0

if i ≤ j ≤ i + m; otherwise,

(k1 ,...,km ) (ui , . . . , ui+m ; κi , . . . , κi+m ) = fi···i+m

m Y [ui+j−1 − ui+j − (kj + 21 )δ] [ui+j−1 − ui+j ] j=1

×

m Y j=0

1 [ui+j − (kj − kj+1 + )δ − κi+j ] 2

(k0 = −1, km+1 = 0).

(B.6)

i ∨

δ If m = 0, we understand the function fi of type (0, . . . , 0, 1 , 0, . . . , 0), to be [u(1) i + 2 −κi ].

Theorem B.1. (k1 ,...,km ) (k1 +p,...,km +p) (k1 +p,...,km +p) (k1 ,...,km ) ∗ fi···i+m = fi···i+m ∗ fi···i+m . fi···i+m

Proof. This is similar to Theorem 4.4.

(B.7)

Set (k1 ,...,km ) , fi...i+m [k1 , . . . , km ] = fi···i+m a Y (a) [k1 . . . , km ] = ∗ fi···i+m [k1 − b + 1, . . . , km − b + 1], fi···i+m

(B.8) (B.9)

b=1

where the symbol ∗ in front of the usual product symbol means that this is a ∗-product. (a) [k1 , . . . , km ] satisfy a set of quadratic relations in ∗-product. They The functions fi···i+m are given below. By specialization δ = 1, we get the relations for the screening operators (a) (4.5) Xi···i+m (k1 , . . . , km ). For the proof of the quadratic relations we prepare a lemma. Let F be the algebra over C with the ∗-product, that is generated by elements fi···i+m [k1 , . . . , km ]. The algebra F is graded F = ⊕(a1 ,...,an−1 )∈Zn−1 Fa1 ,...,an−1 , ≥0

(B.10)

where Fa1 ,...,an−1 consists of the functions of type (a1 , . . . , an−1 ). Lemma B.2. If f, g ∈ F and f ∗ g = 0, then f = 0 or g = 0. Proof. Suppose that f is of type (a1 , . . . , an−1 ) and g is of type (b1 , . . . , bn−1 ). We expand f and g in power series of δ. If both f and g are non-zero, we have f = f0 δ m1 + o(δ m1 ), g = g0 δ m2 + o(δ m2 ),

f0 = 6 0, g0 6= 0,

for some m1 and m2 . From f ∗ g = 0 it follows that


531

(an−1 ) (a1 ) (1) S f0 (u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 ; κ1 , . . . , κn−1 ) (bn−1 ) (1) ·g0 (v1(1) , . . . , v1(b1 ) , . . . , vn−1 , . . . , vn−1 ; κ1 , . . . , κn−1 ) = 0. (B.11) We will show that f0 = 0 or g0 = 0. Choose w1 , . . . , wn−1 ∈ C so that f0 and g0 (b) are holomorphic at u(b) j = wj and vj = wj , respectively. Power series expansion in (b) u(b) j − wj and vj − wj reduces the problem to the case when f0 and g0 are symmetric (a )

(a )

(1) j j polynomials in (u(1) j , . . . , uj ) and (vj , . . . , vj ), respectively. Finally the following lemma reduces the problem to the case of the polynomial ring.

Lemma B.3. Let Ga1 ,...,an−1 be the C-linear space of polynomials in the variables (a

)

(a1 ) (1) n−1 u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 , (a )

j that are symmetric in (u(1) j , . . . , uj ) for each 1 ≤ j ≤ n − 1. Set

G = ⊕(a1 ,...,an−1 )∈Zn−1 Ga1 ,...,an−1 .

(B.12)

≥0

Define the ∗-product in G by f ∈ Ga1 ,...,an−1 , g ∈ Gb1 ,...,bn−1 → f ∗ g ∈ Ga1 +b1 ,...,an−1 +bn−1 , where (a

)

(b

)

(a1 ) (1) (b1 ) (1) (1) n−1 n−1 (f ∗ g)(u(1) 1 , . . . , u1 , v1 , . . . , v1 , . . . , un−1 , . . . , un−1 , vn−1 , . . . , vn−1 ) (an−1 ) (a1 ) (1) (1) (b1 ) (1) = S f (u(1) 1 , . . . , u1 , . . . , un−1 , . . . , un−1 )g(v1 , . . . , v1 , . . . , vn−1 , . . . , (bn−1 ) . . . , vn−1 ) .

There is a ring homomorphism between G and the polynomial ring of the variables (1) (2) (0) (1) (2) (x(0) 1 , x1 , x1 , . . . , ; . . . , ; xn−1 , xn−1 , xn−1 , . . .).

Proof. For simplicity, we consider the case n = 2. The isomorphism is such that the subspace Ga of G corresponds to the space of degree a homogeneous polynomials in (1) (2) (x(0) 1 , x1 , x1 , . . .). The isomorphism is given by m1 ma 1) a) · · · x(m 7→ S (u(1) · · · (u(a) . x(m 1 1 1 ) 1 ) The basic relations are

(B.13)

532


Lemma B.4. fi···i+l−1 [k1 , . . . , kl−1 ] ∗ fi+l···i+l+m [kl+1 , . . . , kl+m ] = fi···i+l+m [k1 , . . . , kl−1 , −1, kl+1 , . . . , kl+m ], (B.14) fi+l···i+l+m [kl+1 , . . . , kl+m ] ∗ fi···i+l−1 [k1 , . . . , kl−1 ] = fi···i+l+m [k1 , . . . , kl−1 , 0, kl+1 , . . . , kl+m ], (B.15) fi ∗ fi···i+m [k1 , k2 , . . . , km ] = fi···i+m [k1 − 1, k2 , . . . , km ] ∗ fi , (B.16) fi+m ∗ fi···i+m [k1 , . . . , km−1 , km − 1] = fi···i+m [k1 , . . . , km−1 , km ] ∗ fi+m , (B.17) 0 0 0 0 fi···i+m [k1 , . . . , km ] ∗ fj···j+l [k1 , . . . , kl ] = fj···j+l [k1 , . . . , kl ] if i + m + 1 < j. ∗fi···i+m [k1 , . . . , km ], (B.18) The proof is straightforward. Lemma B.5. fi···i+l [k1 , . . . , kl ] ∗ fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 , kl+2 , . . . , kl+m ] = fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 − 1, kl+2 , . . . , kl+m ] ∗fi···i+l [k1 , . . . , kl ], (B.19) fi+l···i+l+m [kl+1 , . . . , kl+m ] ∗ fi···i+l+m [k1 , . . . , kl−1 , kl − 1, kl+1 + kl , . . . , . . . , kl+m + kl ] = fi···i+l+m [k1 , . . . , kl−1 , kl , kl+1 + kl , . . . , . . . , kl+m + kl ] ∗ fi+l···i+l+m [kl+1 , . . . , kl+m ], (B.20) fi···i+l+m [k1 , . . . , kl−1 , kl , kl+1 + kl , . . . , kl+m + kl ] ∗fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl , kl+m+2 , . . . , kl+m+p ] = fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl − 1, kl+m+2 , . . . , kl+m+p ] ∗fi···i+l+m [k1 , . . . , kl−1 , kl − 1, kl+1 + kl , . . . , kl+m + kl ]. (B.21) The proof will be given later. The following are simple consequences of the above. Proposition B.6. (a) (b) [k1 , . . . , kl ] ∗ fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 + a − 1, kl+2 , . . . , fi···i+l (b) . . . , kl+m ] = fi···i+l+m [k1 + kl+1 , . . . , kl + kl+1 , kl+1 − 1, kl+2 , . . . , kl+m ] (a) [k1 , . . . , kl ], ∗fi···i+l

(B.22) (a) [kl+1 , . . . , kl+m ] fi+l···i+l+m

∗

(b) fi···i+l+m [k1 , . . . , kl−1 , kl

− 1, kl+1 + kl , . . . ,


533

(b) . . . , kl+m + kl ] = fi···i+l+m [k1 , . . . , kl−1 , kl + a − 1, kl+1 + kl , . . . , kl+m + kl ] (a) [kl+1 , . . . , kl+m ], ∗fi+l···i+l+m

(B.23) (a) fi···i+l+m [k1 , . . . , kl−1 , kl + b − 1, kl+1 + kl , . . . , kl+m + kl ] (b) ∗fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl + a − 1, kl+m+2 , . . . , kl+m+p ] (b) = fi+l···i+l+m+p [kl+1 , . . . , kl+m , −kl − 1, kl+m+2 , . . . , kl+m+p ] (a) [k1 , . . . , kl−1 , kl − 1, kl+1 + kl , . . . , kl+m + kl ]. ∗fi···i+l+m

(B.24) The following is also valid. However, the general case for a, b does not follow from the special case a = b = 1. Proposition B.7. (a) (b) [kl+1 , . . . , kl+m ] ∗ fi···i+l+m+p [k1 , . . . , fi+l···i+l+m

. . . , kl−1 , kl − 1, kl+1 + kl , . . . , kl+m + kl , kl + a − 1, kl+m+2 , . . . , kl+m+p ] (b) [k1 , . . . , kl−1 , kl + a − 1, kl+1 + kl , . . . , kl+m + kl , = fi···i+l+m+p (a) [kl+1 , . . . , kl+m ] kl − 1, kl+m+2 , . . . , kl+m+p ] ∗ fi+l···i+l+m

(B.25) We use these relations for integer parameters ki . However, they are valid without this restriction because the general case follows from the integer case. Let us derive (B.19). The other cases follow similarly. Without loss of generality one can assume that m = 1. Using (4.4) and (B.15) we have fi+l ∗ fi···i+l−1 [k1 , . . . , kl−1 ] ∗ fi···i+l [k1 + kl , . . . , kl−1 + kl , kl ] = fi···i+l [k1 + kl , . . . , kl−1 + kl , kl ] ∗ fi+l ∗ fi···i+l−1 [k1 , . . . , kl−1 ]. (B.26) Using (B.17) we have fi+l ∗ fi···i+l−1 [k1 , . . . , kl−1 ] ∗ fi···i+l [k1 + kl , . . . , kl−1 + kl , kl ] −fi···i+l [k1 + kl , . . . , kl−1 + kl , kl − 1] ∗ fi···i+l−1 [k1 , . . . , kl−1 ] = 0. (B.27) Since fi+l is not a zero divisor, we have (B.19). Combining all these, in particular (B.22) and (B.23), we arrive at the formulas which we need for the quadratic relations of the screening operators. Proposition B.8. (a+b) (b) [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] fi···i+l (b) (a) = fi···i+l+m [k1 − a, . . . , kl − a, −a − 1, kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ], (B.28) (a) (a+b) [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] fi···i+l

534

B. Feigin, M. Jimbo, T. Miwa, A. Odesskii, Ya. Pugai (b) (a) = fi+l+1···i+l+m [kl+2 , . . . , kl+m ] ∗ fi···i+l+m [k1 , . . . , kl , −b − 1, kl+2 − b, . . . , . . . , kl+m − b],

(B.29) (a) (a+b) [kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ] fi+l+1···i+l+m (b) (a) = fi···i+l [k1 − a, . . . , kl − a] ∗ fi···i+l+m [k1 , . . . , kl , a + b − 1, kl+2 , . . . , kl+m ], (B.30) (a+b) (b) [kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ] fi+l+1···i+l+m (b) (a) = fi···i+l+m [k1 , . . . , kl , a + b − 1, kl+2 , . . . , kl+m ] ∗ fi+l+1···i+l+m [kl+2 − b, . . . , . . . , kl+m − b]. (B.31)

Proof. Let us derive (B.28). The other cases can be proven similarly. Using (4.5), (B.14), (B.22), we have (a+b) (b) [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] fi···i+l (a+b−1) = fi···i+l [k1 , . . . , kl ] ∗ fi···i+l [k1 − a − b + 1, . . . , kl − a − b + 1] (b−1) ∗fi+l+1···i+l+m [kl+2 − b + 1, . . . , kl+m − b + 1] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] (a+b−1) = fi···i+l [k1 , . . . , kl ] ∗ fi···i+l+m [k1 − a − b + 1, . . . , kl − a − b + 1, −1, (b−1) kl+2 − b + 1, . . . , kl+m − b + 1] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] = fi···i+l+m [k1 − a − b + 1, . . . , kl − a − b + 1, −a − b, kl+2 − b + 1, . . . , (a+b−1) (b−1) . . . , kl+m − b + 1] ∗ fi···i+l [k1 , . . . , kl ] ∗ fi+l+1···i+l+m [kl+2 , . . . , kl+m ] (b) (a) = fi···i+l+m [k1 − a, . . . , kl − a, −a − 1, kl+2 , . . . , kl+m ] ∗ fi···i+l [k1 , . . . , kl ]. (B.32)

C. Commutativity with DWA In this section we prove Lemma 5.3. It would be more convenient for us to use here the “multiplicative” variable z instead of u. For this reason, let us define the theta function [[z]] ≡ [u]

z = x2u

,

(C.1)

having the periodicity property [[zx2r ]] = −[[z]]. Abusing the notations, let us use (a) (a) (b) the same symbol fα(a) (zi(1) , · · · , zi+m ) for the function fα(a) (u(1) i , · · · , ui+m ), where zj = (b)

x2uj . The screening operator X α (λ) in the notations (4.9)–(4.12) is given by (4.15), i.e., in the multiplicative variables, I X α (λ) = ×

I ···

Y 1≤b≤a i≤j≤i+m

Ui (zi(1) ) [[zi(1) /x]]

···

dzj(b) 2πizj(b)

Ui (zi(a) ) [[zi(a) /x]]

···

(1) ) Ui+m (zi+m (1) [[zi+m /x]]

···

(a) Ui+m (zi+m ) (a) [[zi+m /x]]


×

Y

[[zj(b) /zj(c) ]]

1≤b i|h(j) = m = h(j + 1) + 1}, and mark i0 . The migration is indicated by an arrow. The corresponding strip of length m has also been represented

(2m) Let us consider any walk a ∈ W2n . As shown in Fig. 13, the line h = m intersects the walk a at least once along an ascending slope (at some point j, where h(j) = m and h(j − 1) = m − 1 on a). Let i denote the position of the rightmost3 such intersection, namely i =max{j|h(j) = m = h(j − 1) + 1}. Cutting the walk a at the point (i, h(i) = m) separates the walk into a left part L ∈ Wi(m) and a right part R, which may be viewed (m) as an element of W2n−i (see Fig. 13). Indeed, from the definition of i, the walk R stays above the line h = m until its end: subtracting m from all its heights, and counting its (m) . steps from 0 to 2n − i (instead of from i to 2n) expresses R as an element of W2n ¯ i.e. describing it in the opposite direction (R¯ is a walk on the half-line Reflecting R → R, 3

The fact that we take the rightmost intersection here is responsible for the bijectivity of the mapping.

562

P. Di Francesco

starting at height m and ending at height 0 after 2n − i steps), and composing L and ¯ i.e. attaching the origin of R¯ to the end of L, we form a walk b = LR¯ ∈ W2n (see R, Fig. 13). In this walk, we have h(i) = m. If h(i + 1) = m − 1, i is an end of strip of height m, which we mark. If h(i + 1) = m + 1, i cannot be an end of strip. Nevertheless, we just have to consider the smallest point i0 > i such that h(i0 ) = m = h(i0 + 1) + 1, which always exists, as the walk a goes back to height 0 at position 2n. This point i0 is an end of strip at height m, which we mark. Conversely, let us start from some a ∈ W2n with a marked end of strip at position i and height m. By definition, this end of strip satisfies h(i) = m and h(i + 1) = m − 1. If i is a maximum of a, namely h(i − 1) = m − 1, it separates the walk a into a left part L and a right part R. The left part is a walk on the half-line, ending at height m after i steps, hence L ∈ Wi(m) . The right part R is a walk on the half-line starting at height m and ending at the origin, after 2n − i steps. The reflected walk R¯ is obtained by describing R in the opposite direction, namely starting from the origin, and ending at height m, after (m) . Now if we compose the walks L and 2n − i steps. Hence we can write that R¯ ∈ W2n−i (2m) R¯ (attach the origin of R¯ to the end of L), the resulting walk b = LR¯ ∈ W2n , and due to the fact that i was a maximum of a, we have h(2n − 1) = 2m − 1 and h(2n) = 2m in b. If i is not a maximum of a, we first migrate the marked point from i to the largest value i0 < i, such that h(i0 ) = m = h(i0 − 1) + 1 (the closest ascending slope at height m to the left of i). Then we apply the previous cutting, reflecting and pasting procedure (2m) , with the particular property that at the point i0 . This produces a walk b = LR¯ ∈ W2n h(2n − 1) = 2m + 1 and h(2n) = 2m on b. We have in fact established a more refined mapping between (i) the a ∈ W2n with a marked maximum, of height m (namely at a point i such that h(i) = m = h(i + 1) + 1 = (2m−1) , (ii) the a ∈ W2n with a marked descending h(i − 1) + 1) and the b ∈ W2n−1 slope at height m (i such that h(i) = m = h(i − 1) − 1 = h(i + 1) + 1) and the (2m+1) . This forms a bijection between the walks a ∈ W2n with a marked end of b ∈ W2n−1 (2m) strip (either a maximum or a descending slope) and the walks b ∈ W2n (with either h(2n − 1) = 2m − 1 or h(2n − 1) = 2m + 1). Hence we conclude that (2m) | = c2n,2m , s2n,m = |W2n

(3.50)

which proves Proposition 2. To translate the result (3.49) of Proposition 2 into the formula of Theorem 1, using (3.48), we simply have to reexpress the meander determinant in terms of the Chebyshev polynomials Um (q), using µm = Um−1 /Um . Equation (3.48) becomes det G2n (q) =

s2n,m n n Y Y s2n,m −s2n,m+1 Um = Um Um−1

m=1

by noting that s2n,n+1 =

2n −1

−

(3.51)

m=1

2n −2

= 0. This takes exactly the form of (2.6), with

a2n,2m = s2n,m − s2n,m+1 = c2n,2m − c2n,2m+2 , which completes the proof of Theorem 1.

(3.52)

Meander Determinants

563

4. The Semi-Meander Determinant: Proof of Theorem 2 The strategy of the proof of Theorem 2 is exactly the same as for Theorem 1. It is based on the representation of open arch configurations by a particular set of reduced elements of the Temperley-Lieb algebra, forming the basis (still called basis 1, but not to be confused with that of previous section) of a vector subspace thereof. The semi-meander determinant is then expressed in terms of the Gram determinant of this basis 1. The next step is the explicit Gram-Schmidt orthogonalization of this basis, defining another basis, called basis 2. The semi-meander determinant is then computed by using the change of basis 1 → 2. 4.1. Temperley-Lieb algebra and open arch configurations. The open arch configurations of A(h) n , with order n and with h open arches, can be represented by some particular reduced elements of the Temperley-Lieb algebra T Ln (q).

Fig. 14. The interpretation of an open arch configuration of order n = 15 and with h = 3 open arches (right diagram) as a reduced element of T L15 (q) (left diagram). Note that exactly h = 3 strings go across the domino, namely link three lower to (the three rightmost) upper ends. The linking of the upper ends of the domino is made through (n − h)/2 = 6 strings connecting consecutive ends by pairs

In Fig. 14, we have represented in the string-domino pictorial representation the domino corresponding to a reduced element of the Temperley-Lieb algebra, immediately interpretable as an open arch configuration. Starting from a ∈ A(h) n , let us construct an element, still denoted4 by (a)1 of T Ln (q): representing the corresponding domino as acting from bottom to top, the connection of its n lower ends of strings is realized through the closed arches of a, whereas the h open arches just go across the domino, and connect h of the lower ends to the h rightmost upper ends of strings. The remaining n − h ends are then connected by consecutive pairs like in the meander case. This construction establishes a bijection between A(h) n and the reduced elements of T Ln (q) with exactly h strings connecting lower ends to the h rightmost upper ends, and (n − h)/2 strings connecting the remaining n−h upper ends by consecutive pairs. Let us denote by In(h) (q) the vector space spanned by these reduced elements. From now on, we will refer to the basis {(a)1 |a ∈ A(h) n } as the basis 1. Like in the meander case, the basis 1 is best expressed in the equivalent language (h) of walk diagrams a ∈ Wn(h) . Let a(h) n be the fundamental element of Wn , with h(0) = h(2) = ... = h(n − h) = 0, h(1) = h(3) = ... = h(n − h − 1) = 1 and h(n − h + j) = j for j = 1, 2, ..., h. Any a ∈ Wn(h) may be viewed as the result of box additions on the (h) fundamental a(h) n . The construction of (a)1 , a ∈ Wn is performed recursively. We first set (4.1) (a(h) n )1 = e1 e3 ...en−h−1 , 4 Here we adopt the same notation for elements of T L (q) corresponding to open arch configurations as n that used before for closed arch configurations. These will correspond to another basis {(a)1 } for a ∈ A(h) n , which we will refer to again as the basis 1. This should not be confusing, as we are only dealing with the open arch case from now on.

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and then for a box addition at position i, we set (a + i )1 = ei (a)1 .

(4.2)

As an example, the basis 1 elements for I4(2) read

= e1

= e2 e 1

1

1

(4.3)

= e3 e 2 e 1 1

where we have represented the boxes added on the walk diagrams.

Fig. 15. The string-domino picture corresponding to the box decomposition of an open walk diagram (3) . Note that exactly 3 strings join upper and lower ends. The domino is rather read from top to a ∈ W11 bottom, as opposed to the case of Fig. 14, where it is read from bottom to top

To make direct contact with the string-domino pictorial representation, we may attach to the box decomposition of any walk diagram a ∈ Wn(h) a domino using the same rule as in Sect. 3.1, namely represent all the boxes corresponding to left multiplications by ei (including those of the fundamental element a(h) n ), and decorate them by a horizontal double line (string), as in (3.10). The picture is then completed by drawing vertical strings joining the string ends on the upper and lower borders of the domino. This is illustrated in Fig. 15, where the strings are represented in thick black lines. The main and new difficulty here, in comparison with the former meander case, is that these reduced elements of T Ln (q) do not form an ideal5 . For instance, we have listed in (4.3) the basis 1 elements for I4(2) (q). If we multiply the first (fundamental) element by the third one, we find (e1 )(e3 e2 e1 ) = e1 e3 , which does not belong to the space I4(2) (q) (there is no string connecting lower and upper ends in e1 e3 , whereas there must be 2 such strings in any element of I4(2) (q)), which is therefore not an ideal. Nevertheless, we can still form the Gram matrix 0(h) n (q) for the basis 1, by using the restriction to In(h) (q) of the bilinear form (3.12). This reads

0(h) n (q)

a,b

= (a)1 , (b)1

for a, b ∈ A(h) n .

(4.4)

5 This will be responsible for the absence of a generalization of the Lemma 1 of Sect. 3.3 for the present case.


565

t

(b)1

a

(a) 1

b

t

Fig. 16. Computation of (a)1 , (b)1 . We put the reflected domino (b)t1 on top of the domino (a)1 (here, (3) ). The upper ends are then identified one by one to the lower ends of strings. Counting the loops a, b ∈ W11 formed yields: (n−h)/2 = 4 central loops formed at the connection between the two dominos, plus κ(a|b) = 3 loops coming from the superposition of the open arch configurations a and bt (reflected w.r.t. the river). This gives finally (a)1 , (b)1 = q 7

As illustrated in Fig. 16, to compute (a)1 , (b)1 , we glue the dominos (a)1 and the reflected (b)t1 , identify the upper and lower string ends, and count the number of resulting connected components. The connection of the two dominos creates (n − h)/2 loops, from the strings connecting the upper ends by consecutive pairs on (a)1 and (b)1 . The remaining part simply creates κ(a|b) loops, from the superposition of the open arch configurations a and bt (reflected w.r.t. the river), and the connection of their h open arches (see Fig. 16). Hence the Gram matrix for the basis 1 of In(h) (q) is simply related to the semi-meander matrix (2.4), through n−h n−h 2 +κ(a|b) = q 2 (4.5) Gn(h) (q) a,b . [0(h) n (q) a,b = q The semi-meander determinant is therefore related to the Gram determinant of the basis 1 through n−h 2 cn,h

det Gn(h) (q) = µ1

det 0(h) n (q).

(4.6)

4.2. Orthogonalization of the basis 1. In this section, we introduce a basis 2 of In(h) (q), still indexed by a ∈ Wn(h) , which will be orthonormal with respect to the bilinear form (3.12). Like in the meander case, the basis 2 will be defined recursively through box additions. We start from the basic element n/2

(h) (4.7) (a(h) n )2 = µ1 (an )1 , n+h (h) −n n−h where the normalization ensures that (a(h) q 2 q 2 = 1, where we n )2 , (an )2 = q have counted the contributions of the (n − h)/2 loops formed by the strings pairing upper ends by consecutive pairs on (a)1 , and that of the κ(a|a) = (n + h)/2 loops created by the superposition of a with its own reflection at . To proceed, we need to define the concept of floor of a walk diagram a ∈ Wn(h) . Let us denote by h(i), i = 0, 1, 2, ..., n the heights of a, with h(0) = 0 and h(n) = h. The floor of a is yet another diagram f (a) ∈ Wn(h) , such that f (a) ⊂ a, and with heights h0 (i), i = 0, 1, 2, ..., n, defined as follows. Let us denote by J the set of integers

J = {j ∈ {0, 1, ..., n} such that h(k) ≥ h(j) , ∀ k ≥ j}.

(4.8)

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P. Di Francesco

J3 J4

} } }

J2

}

J1

}

J0 0

4 5 6

12 13 14 16 17 19 20

(6) Fig. 17. A diagram a ∈ W20 (thick black line) and the construction of its floor f (a) ⊂ a. The segments J0 , J1 , ..., J4 of positions forming J are indicated by dotted lines. The floor f (a) is represented filled with grey boxes. The boxes in between f (a) and a are represented in white. The floor-ends have positions 0, 4, 6, 12, 14, 16, 17, 19, 20

As illustrated in Fig. 17, this set J is clearly the union of ordered segments of positions, of the form j = J0 ∪ J1 ∪ ... ∪ Jk , with Ji = {ji , ji + 1, ji + 2, ..., ji + ni }, for some integers ni and ji , i = 0, ..., k. These segments correspond to the ascending slopes of a such that no point on their right has a lower height. With these notations, the floor f (a) of a is defined to have the heights h0 (j), j = 0, 1, ..., n, according to the following rules: h0 (j) = h(j) 0

∀ j ∈ J,

0

h (ji + ni + 2r) = h (ji + ni ) 0

∀ r ≥ 0 with 2r ≤ ji+1 − ji − ni ,

0

h (ji + ni + 2r − 1) = h (ji + ni ) − 1

∀ r ≥ 1 with 2r − 1 ≤ (ji+1 − ji − ni ). (4.9) This is valid for all i = 1, 2, ..., k. For i = 0, we have to be more careful, as the leftmost floor piece has a different status. If J0 6= {0} (this leftmost floor piece is empty), then (4.9) is valid for i = 0 as well. If J0 = {0} (this leftmost floor piece is not empty: this is the case in Fig. 17), we have to add the values h0 (0) = h0 (2) = · · · = h0 (j1 ) = 0, h0 (1) = h0 (3) = · · · = h0 (j1 − 1) = 1.

(4.10)

The floor diagram is represented filled with grey boxes in Fig. 17. The floor diagram f (a) is in fact a succession of horizontal broken lines, with heights alternating h(ji +ni ) = `+1, h(ji + ni + 1) = `, h(ji + ni + 2) = ` + 1,..., h(ji+1 ) = ` + 1, on the intermediate positions in between the segments Ji and Ji+1 . These are separated by ascending slopes (along the segments Ji ). For each such intermediate floor Fi , we define the floor height to be the number ` = h(ji + ni − 1) = h(ji + ni + 1) = ... = h(ji+1 ) − 1, for i ≥ 1. The leftmost floor F0 , of height 0 if J0 = {0}, is a little different as we have ` = 0 = h(j0 = 0) = h(2) = ... = h(j1 ) from (4.10). We will also refer to these intermediate floors as simply the floors of a, for which this decomposition is implied. The endpoints with positions ji + ni and ji+1 (and equal height h(ji + ni ) = h(ji+1 ) except maybe for the rightmost floor-end) of each of these floors will be called floor-ends in the following. To define the basis 2 of In(h) (q), we will need a pictorial representation of the walk diagrams a ∈ Wn(h) in which the floor f (a) is also represented. As in Sect. 3, we adopt the representation (3.23) by grey and white boxes ofpthe left mutliplications of a reduced element of In(h) (q) by respectively ei at position i or µm+1 /µm (ei −µm ) on a minimum of height m and position i. The basis 2 elements then correspond to


567

(i) grey box additions for all the boxes forming the floor f (a), including the basic boxes forming a(h) n (see below) (ii) white box additions for all the superstructures of a above its floor f (a). There is however a final subtlety with the height of these white boxes, which is counted along strips, w.r.t. the grey floor. In the case (4.7) of the fundamental diagram, the representation is simply a(h) n

n/2

2

= µ1

....

(4.11)

n/2

= µ1 e1 e3 ...en−h−1 (h) as the floor of this element is simply f (a(h) n ) = an , and we have represented the basic grey boxes under the floor. The other elements of the basis 2 are obtained by white box additions on a(h) n 2 . The novelty, when compared to the case of Sect. 3, is that some box additions may create a new floor, namely change previously added white boxes into grey ones. In general, the best way to construct the basis 2 elements, is to first list all the walk diagrams a ∈ Wn(h) , represent them together with their floor f (a) ⊂ a, and then write the corresponding products of grey and white boxes. This is illustrated now in the case of W6(2) . = µ31 = µ31 e1 e3 , 2 5/2 1/2 = µ31 = µ1 µ2 (e2 − µ1 )e1 e3 , 2 5/2 1/2 = µ31 = µ1 µ2 (e4 − µ1 )e1 e3 , 2 = µ31 = µ21 µ2 (e2 − µ1 )(e4 − µ1 )e1 e3

2 5/2 1/2

= µ31

= µ1 µ2 (e5 − µ1 )e4 e1 e3 ,

= µ31

= µ21 µ2 (e2 − µ1 )(e5 − µ1 )e4 e1 e3 ,

2

2

= µ31 2 1/2 1/2

= µ21 µ2 µ3 (e3 − µ2 )(e2 − µ1 )(e4 − µ1 )e1 e3 ,

= µ21 µ2 (e3 − µ1 )(e5 − µ1 )e2 e4 e1 e3 ,

= µ31 2

= µ31

,

2 1/2 1/2

= µ21 µ2 µ3 (e4 − µ2 )(e3 − µ1 )(e5 − µ1 )e2 e4 e1 e3 ,

(4.12)

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P. Di Francesco

where we have represented the grey and white boxes corresponding to each walk diagram. Note e.g. for the last element of (4.12) that the rightmost white box is counted to have height 1 (instead of 2) because this height is the relative height w.r.t. the grey floor on the same strip, which is already at height 1. This construction results in the following change of basis 1 → 2 X Pb,a (b)1 (4.13) (a)2 = f (a)⊂b⊂a

with possibly non-vanishing matrix elements Pb,a only for the walks b ∈ Wn(h) such that b is above the floor of a (f (a) ⊂ b) and below a (b ⊂ a). Like in the meander case of Sect. 3, we can arrange the walk diagrams by growing length (number of boxes, grey and white), and make the matrix P upper triangular. With this definition, the basis 2 satisfies the following Proposition 3. The basis 2 elements are orthonormal with respect to the bilinear form (3.12), namely for all a, b ∈ Wn(h) . (4.14) (a)2 , (b)2 = δa,b This result will be proved in the remainder of this section. Note first, in comparison with the meander case (Proposition 1), that no stronger statement (generalizing Lemma 1) will hold here for the products of elements of In(h) (q). This is because, as mentioned earlier, In(h) (q) is no longer an ideal, hence we have no good control of what the product of two elements of In(h) (q) can be. Thus, instead of resorting to the multiplication of elements, we will directly consider the bilinear form (3.12). The main forthcoming results (Lemmas 2, 3 and 4 below) will deal with reexpressions and simplifications of this bilinear form, when evaluated on two elements of In(h) (q). In particular, Lemma 3 will give a reexpression in terms of the form (3.12), evaluated respectively on elements (h) (q) and Ip (q), which will enable us to use the results of Sect. 3, namely the of In−2p proposition 1, to eventually compute (4.14). To prove Proposition 3, we need a few more definitions. As we are basically dealing with elements of the basis 2, it will be useful to trade the usual notion of walk diagram a ∈ Wn(h) for that of bicolored box diagram, namely the corresponding pictorial representation using grey and white box addition, i.e. the arrangement of grey and white boxes forming (a)2 . For convenience, we still denote by (a)2 the bicolored box diagram corresponding to (a)2 , with a ∈ Wn(h) .

(a)2

s (a) s (a) s (a) s4(a) s (a) 1

2

3

5

Fig. 18. The bicolored box diagram corresponding to an element a ∈ Wn(n−10) for all n ≥ 14. The width of the diagram is w = 5. It is decomposed into 5 strips sj (a), j = 1, 2, ..., 5

Such a bicolored box diagram may be viewed as the succession of strips s1 (a), s2 (a), ..., sw (a), made of a succession of grey, then white boxes of consecutive positions and heights. The number w stands for the number of these strips, namely the width of the base of (a)2 , i.e. the number of grey boxes of height 0 in (a)2 . Note that for all a ∈ Wn(h) ,


569

the element (a)2 has width w = (n − h)/2. Moreover, we have the following identity between elements of In(h) (q): n/2

(a)2 = µ1 s1 (a)s2 (a)...sw (a)

(4.15)

by considering the strips (i.e. successions of grey and white boxes) as elements of the Temperley-Lieb algebra. To proceed with the proof of Proposition 3, we will compute the quantity (a)2 , (b)2 = Tr (a)t2 (b)2 . The strategy is the following. We will start by comparing the rightmost strips sw (a) and sw (b) of (a)2 and (b)2 . Both are a succession of grey boxes, topped by one white box, in the form r µ2 (e2w+j−2 − µ1 )e2w+j−3 e2w+j−4 ...e2w e2w−1 (4.16) s = µ1 with possibly different values of j = ja or jb , the total size (total number of boxes) of the strip. Note that if ja = 1, sw (a) is reduced to a single grey box, without white box on top (this is the case when (a)2 only has one floor of height 0). We have the first result Lemma 2. For all a, b ∈ Wn(h) , and w = (n − h)/2, if sw (a) 6= sw (b), then (a)2 , (b)2 = 0. If sw (a) 6= sw (b), have different size. Let us assume that ja < jb . then these strips Writing (a)2 , (b)2 = Tr (b)2 (a)t2 , and (b)2 = Bsw (b), (a)2 = Asw (a), we have (4.17) (a)2 , (b)2 = Tr Bsw (b)sw (a)t At . In this expression, we now transfer the boxes of sw (b) onto (a)t2 , starting from the lowest one, up to the top of sw (b). These boxes now act on sw (a)t from below. Thanks to the relation ei ei−1 ei = ei , the first ja − 2 grey boxes of the strip sw (a)t are annihilated by the action of the first ja − 1 grey boxes of sw (b), namely µ2 (e2w+jb −2 − µ1 )e2w+jb −3 ...e2w e22w−1 µ1 × e2w ...e2w+ja −3 (e2w+ja −2 − µ1 ) µ2 = 2 (e2w+jb −2 − µ1 )e2w+jb −3 ...e2w+1 e2w e2w+1 µ1 × ...e2w+ja −3 (e2w+ja −2 − µ1 ) µ2 = 2 (e2w+jb −2 − µ1 )e2w+jb −3 ...e2w+ja −3 (e2w+ja −2 − µ1 ). µ1

sw (b)sw (a)t =

(4.18)

(Note that the last factor (e2w+ja −2 − µ1 ) must be replaced by e2w+ja −2 = e2w−1 in the case ja = 1, but this does not alter the following discussion.) Let us now transfer in the same way all the boxes of sw−1 (b), sw−2 (b), ..., s1 (b) onto (a)t2 . But these occupy only positions k ≤ 2w + jb − 3, and the largest position k = 2w + jb − 3 may only be occupied by a white box. Hence, after the transfer of (b)2 onto (a)t2 is complete, the resulting element is a linear combination of the form (b)2 (a)t2 = α C 0 (e2w+jb −2 − µ1 )e2w+jb −3 C + β D0 e2w+jb −3 (e2w+jb −2 − µ1 )e2w+jb −3 D,

(4.19)

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P. Di Francesco

where C, D, C 0 , D0 are elements of the Temperley-Lieb algebra only involving the generators ek , k < 2w + jb − 3, and α and β two complex coefficients, coming from the various normalization factors. The second term in (4.19) vanishes identically, thanks to the identity ei (ei+1 − µ1 )ei = 0. We are therefore left with (a)2 , (b)2 = α Tr C 0 (e2w+jb −2 − µ1 )e2w+jb −3 C (4.20) = α Tr (e2w+jb −2 − µ1 )e2w+jb −3 CC 0 . To show that this expression vanishes, let us use the string representation of the ei , and the definition of the trace as computing q L , where L is the number of loops of the string representation of the element, after identification of the upper and lower ends of its strings. In this picture (setting i = 2w + jb − 2, CC 0 = E, and taking the adjoint of the expression in the trace, which does not change its value), we have Tr E(e1 , e2 , ..., ei−2 )ei−1 (ei − µ1 )

E

=

− µ1

E

(4.21)

=

X

i-1 i

αr (q − µ1 q L

i-1 i

L+1

)

r

= 0, where we have expanded E(e1 , e2 , ..., ei−2 ) as a linear combination of diagrams involving only grey boxes with positions k ≤ i − 2. In each of these diagrams, the second term has always one more loop than the first one, hence the cancellation, with the factor 2. µ1 = q −1 . This completes the proof of Lemma Lemma 2 guarantees that (a)2 , (b)2 = 0 as soon as the last strips sw (a) and sw (b) are distinct. In the latter case, Proposition 3 is therefore proved. Let us assume now that (a)2 and (b)2 have the same last strip, say with j boxes. Then both a and b have a rightmost floor of height H = j −1. Let pa and pb denote their respective widths, namely the respective numbers of grey boxes of height j − 1 forming this floor in a and b. Two situations may occur for these floors: (i) they have the same width pa = pb . In this case, we will show that (a)2 , (b)2 is factored into the bilinear form (3.12) evaluated on smaller diagrams, obtained by cutting (a)2 and (b)2 into two pieces (Lemma 3 below). (ii) The width of the rightmost floor of a is strictly smaller that that of the rightmost floor of b pa < pb . In this case, we will show that (a)2 , (b)2 = 0 (Lemma 4 below). Let us treat these cases separately. CASE (i). The two rightmost floors of a and b have the same width pa = pb = p. We will simply grind the j − 2 consecutive layers of grey boxes underlying the floor of height j − 1, and detach the corresponding portions of a and b, so that the quantity (a)2 , (b)2 will factorize into a product of analogous terms, for smaller diagrams (see Lemma 3 below). More precisely, let us compute the quantity S(a)S(b)t = sw−p+1 (a)sw−p+2 (a)...sw (a)sw (b)t ...sw−p+1 (b)t

(4.22)


571

involved in the computation of (a)2 (b)t2 . In S of (4.22), all the strips involved have a floor of height j − 1, i.e. have the form sw−m+1 = ... e2w−2m+j−1 e2w−2m+j−2 ...e2w−2m+1 = s˜w−m+1 e2w−2m+j−2 ...e2w−2m+1 ,

(4.23)

where each strip s˜ has a floor of only one grey box, topped by white boxes. The idea is to transfer the grey boxes from sw (a) to sw (b), from below, just like we did in (4.18), and do it again for sw−1 (a) and sw−1 (b), etc..., until we are left only with the amputated strips s. ˜ The final result simply reads ˜ S(b) ˜ t = s˜w−p+1 (a)s˜w−p+2 (a)...s˜w (a)s˜w (b)t ...s˜w−p+1 (b)t . (4.24) S(a)S(b)t = S(a) This result implies the following Lemma 3. If a and b ∈ Wn(h) have identical rightmost floors of width p, then (a)2 , (b)2 = (a0 )2 , (b0 )2 (a00 )2 , (b00 )2 , where

n−2p 2

(a0 )2 = µ1

n−2p 2

0

(b )2 = µ1

(4.25)

s1 (a)s2 (a)...sw−p (a), s1 (b)s2 (b)...sw−p (b),

(a00 )2 = µp1 s˜w−p+1 (a)...s˜w (a),

(4.26)

(b00 )2 = µp1 s˜w−p+1 (b)...s˜w (b). The normalizations in (4.26) are chosen to guarantee that all the elements (a0 )2 , (a00 )2 , ... have norm 1, as we will see below. Llemma 3 will follow from the application of (4.24) to the computation of (a)2 , (b)2 = Tr (a)2 (b)t2 . Indeed, we simply write 0 t 0 t (a)2 , (b)2 = µ2p 1 Tr (a )2 S(a)S(b) (b )2 0 ˜ ˜t 0 t = µ2p 1 Tr (a )2 S S (b )2 (4.27) = Tr (a00 )2 (b00 )t2 (b0 )t2 (a0 )2 = (a00 )2 , (b00 )2 × (a0 )2 , (b0 )2 In the last step, we have noted that (a00 )2 (b00 )t2 involves only generators ek with positions ˜ k ≥ 2w − 2p + j − 1 (position of the leftmost grey box in S(a) = (a00 )2 ), whereas 0 0 t (a )2 (b )2 involves only generators ek with k ≤ 2w − 2p + j − 3 (maximum position of the rightmost (white) box in (a0 )2 ). The last line of (4.27) follows then from the locality of the trace, namely that for any two sets of positions I, J, with i < j − 1 for all i ∈ I, j ∈ J, Y Y Y Y ej ) = Tr( ei ) Tr( ej ), (4.28) Tr( ei i∈I

j∈J

i∈I

j∈J

which follows from the definition of the trace (the loops arising from the two terms are independent). In (4.25), the bilinear forms (a)2 , (b)2 , (a0 )2 , (b0 )2 and (a00 )2 , (b00 )2 , are respec(h) (1) tively evaluated in the spaces In(h) (q), In−2p (q) and I2p+1 (q). Let us concentrate on the (1) 00 00 (q) and last term (a )2 , (b )2 . There is a simple morphism ϕ of algebras between I2p+1 (0) I2p+2 (q) = Ip (q),

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P. Di Francesco

ϕ(E) =

√ µ1 E e2p+1 ∈ Ip (q)

(1) ∀ E ∈ I2p+1 (q).

(4.29)

The morphism ϕ consists simply in adding the missing rightmost grey box (e2p+1 ) to complete the floor of E into that of the ideal Ip (q). Moreover, we have added an ad-hoc multiplicative normalization factor µ1 . With this normalization, we have the following simple correspondence between traces over the two spaces: t Tr ϕ(E)ϕ(F )t = µ1 µ−1 1 Tr E e2p+1 F

=

=

(4.30)

= Tr EF t , t where we have used e22p+1 = µ−1 1 e2p+1 , then transferred all the boxes of F onto E, and represented the result in the pictorial string-domino representation of the trace (see Fig. 15), to show that the presence of the grey box does not change the value of the trace (it does not affect the structure of the loops). Using this fact, we can now apply 00 00 ) , (b ) of (4.25), by the result of Proposition 1 of Sect. 3.3 above to the factor (a 2 2 00 00 simply interpreting ϕ (a )2 and ϕ (b )2 as elements of Ip (q). We conclude that 00 00 00 00 (a )2 , (b )2 = ϕ (a )2 , ϕ (b )2 (4.31) = δa00 ,b00 .

Hence, if a00 6= b00 , the bilinear form vanishes, and Proposition 3 follows. If a00 = b00 , we go back to the beginning of our study, with now (a)2 and (b)2 replaced with (a0 )2 and (h) (b0 )2 ∈ In−2p (q). If only the case (i) occurs, we will dispose successively of each portion of a and b above their common successive floors (as above), and get an expression Y δa00 ,b00 . (4.32) (a)2 , (b)2 = portions a00 ,b00 above successive floors

If case (ii) occurs, the result will vanish, as we will see now. CASE (ii). The diagrams (a)2 and (b)2 have a rightmost floor, of same height j − 1, but with different widths p = pa < pb . As in the case (i), we concentrate on the portions of a and b above this rightmost floor, over a width p, namely consider S(a) = sw−p+1 (a)sw−p+2 (a)...sw (a), (4.33) S(b) = sw−p+1 (b)sw−p+2 (b)...sw (b). To compute the quantity (a)2 , (b)2 , we now write (a)2 = (a0 )2 S(a) and (b)2 = (b0 )2 S(b), and get (a)2 , (b)2 = Tr (a0 )2 S(a)S(b)t (b0 )t2 (4.34) ˜ S(b) ˜ t . = Tr (b0 )t2 (a0 )2 S(a) Using the cyclicity of the trace and the symmetry of (3.12), we may also write


573

(a)2 , (b)2

˜ ˜ t (a0 )t2 (b0 )2 S(b) = Tr S(a) ˜ t (b0 )2 S(b) ˜ . = Tr (a0 )t2 S(a)

(4.35)

˜ We have used the commutation of S(a), which involves only boxes of positions ≥ α = 0 2w−2p+j−1, with (a )2 , which only involves boxes of positions ≤ α−2 (as its rightmost floor has now an height < j − 1). Let us now compute (4.35) by transferring the white ˜ ˜ ˜ t onto (b0 )2 S(b). Once this transfer is complete, (b0 )2 S(b) is replaced by boxes of S(a) ˜ with all possible box additions/subtractions a linear combination of diagrams (c0 )2 S(c) induced by the process of transfer. Note that the left portion (b0 )2 of (b)2 is also affected, ˜ For notational simplicity, we have used the as these boxes act on both (b0 )2 and S(b). denomination c0 for the left part of c, so that we still have (c)2 = (c0 )2 S(c). We are then ˜ t , namely those of height j −1. left with the transfer of the first layer of grey boxes of S(a) ˜ (the action To get a non-zero result, those must only hit minima or maxima on (c0 )2 S(c) of a grey box on a white slope vanishes, according to (3.32), (3.33)). Concentrating ˜ above the position α = 2w − 2p + j − 1 (namely the on the configuration of (c0 )2 S(c) ˜ only two situations may yield configuration of (c0 )2 above the leftmost grey box in S(c)), a non-zero answer (a) (c0 )2 has no white box above the position α. (b) (c0 )2 has a white maximum at α. In this case, this maximum is necessarily at height j + 2 (white box of height j + 1, hence of relative height 2 w.r.t. the floor), because no white slope is allowed at any of the positions α, α + 2, ..., α + 2p − 2 = 2w + j − 3, ˜ and the rightmost white box of S(c) has an height ≤ j, hence a relative height ˜ t= ˜ t (call it S(d) ≤ 1. Let us transfer this white box back onto what is left of S(a) eα eα+2 ...eα+2p−2 ). Actually, this diagram has now a grey maximum at the position α (this is the position of the leftmost The white p grey box in the floor of S(a)). √ box acts on this grey maximum as µ3 /µ2 (eα − µ2 )eα = eα / µ2 µ3 , hence is eliminated up to some multiplicative constant. We therefore end up in a situation where (c0 )2 → (c0 − α )2 has no white box above the position α hence in the case (a) above. ˜ In either case, we end up in a situation where α is a floor-end on both diagrams (a0 )2 S(d) 00 ˜ 00 0 00 0 and (c )2 S(c), where c = c in the case (a) and c = c − α in the case (b). So we can reexpress X ˜ S(d) ˜ t (a)2 , (b)2 = λc Tr (a0 )t2 (c00 )2 S(c) c

=

X

λc

c

(4.36) = µ1

X

λc

c

= µ1

X

˜ S(d) ˜ t , λc (a0 )2 , (c00 )2 Tr S(c)

c

where, by using the string-domino picture, we have removed the grey box linking the left and right parts of the operator in the trace, at the expense of creating a new loop,

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hence the extra factor of µ1 = q −1 . Note that in this argument it was crucial that there should be no white box above the grey box we have removed, further linking the left and right parts: this is why we had to go through case (b) above and modify c0 → c00 = c0 − to get back to the situation (a). Now the main feature of (c00 )2 is that it has still a rightmost grey floor of height j − 1, whereas by definition (a0 )2 has a rightmost grey floor of height < j − 1. Hence the rightmost strips in both diagrams are distinct: sw−p (a0 ) 6= sw−p (c00 ). We can therefore apply Lemma 2, to conclude that (4.37) (a)2 , (c00 )2 = 0 in (4.36), so that finally (a)2 , (b)2 = 0. Hence we deduce the Lemma 4. For any two bicolored box diagrams (a)2 and (b)2 , with rightmost floors of same height j − 1, but of differents widths pa < pb , we have (4.38) (a)2 , (b)2 = 0. The proof of Proposition 3 is now straightforward. We start with the two bicolored box diagrams (a)2 and (b)2 . If their rightmost strips are distinct, then (a)2 , (b)2 = 0 by Lemma 2. Otherwise, we focus our attention to their rightmost floors, which have the same height j − 1. If they have different widths, Lemma 4 above implies Q that (a)2 , (b)2 = 0. If they have the samewidth, Lemma 3 expresses (a)2 , (b)2 = δa00 ,b00 , hence we finally get that (a)2 , (b)2 = 0 unless a and b are identical, in which case (a)2 , (a)2 = 1. This completes the proof of Proposition 3. 4.3. The semi-meander determinant: A preliminary formula. The semi-meander determinant (4.6) follows from the Gram determinant of the basis 1. The latter is best expressed through the change of basis 1 → 2, in which the Gram matrix is sent to the cn,h × cn,h identity matrix I. With the upper triangular matrix P defined in (4.13), this reads t (4.39) P 0(h) n (q)P = I. −2 Hence det 0(h) n (q) = (det P ) . The diagonal elements of P are linked by the recursion relation r µ`+1 Pa+i,` ,a+i,` = Pa,a , (4.40) µ`

where the box addition i,` is performed at the point i, and at relative height `, with n/2 (4.7) for respect to the grey box floor in a. With the initial condition Pa(h) (h) = µ 1 n ,an the fundamental walk diagram of Wn(h) , this gives Y µ`+1 2 Pa,a = µn1 , (4.41) µ` white boxes of a

where ` denotes the height of the white box addition, relative to the grey floor in a.


575

Fig. 19. The strips of white boxes on a walk a ∈ Wn(h) with n = 20 and h = 6. The walk is represented in a thick black line. We have also represented the floor of grey boxes for this walk. We have (n − h)/2 = 7 strips of white boxes, of respective lengths 2, 1, 3, 2, 2, 2, 2

Like in the meander case, let us arrange the white boxes of any bicolored box diagram corresponding to an a ∈ Wn(h) into strips of white boxes, namely sequences of white boxes with consecutive positions and heights, added on top of the grey floor of a (see Fig. 19 for an illustration). There are exactly (n−h)/2 such white strips. The strip length is now defined as the relative height of the top of the white box sitting on top of the strip (hence an empty strip has length 1). With this definition, we simply have Y n+h 2 = µ1 2 µ` , (4.42) Pa,a white strips of a

where, in the product over the (n − h)/2 strips of a ∈ Wn(h) , ` stands for the strip length (all denominators have been cancelled along the strips, except for the µ1 ones, which have rebuilt the prefactor). This yields the determinant of the basis 1 Y Y −(n+h)cn,h /2 −2 Pa,a = µ1 µ−1 (4.43) det 0(h) n (q) = ` , white strips of (h) all a∈Wn

a∈Wn(h)

and thanks to (4.6), the semi-meander determinant Y

−hcn,h

det Gn(h) (q) = µ1

µ−1 ` .

(4.44)

white strips of (h) all a∈Wn

The latter can be recast into det Gn(h) (q)

=

−hc µ1 n,h

n−h 2 +1

Y

µm

−s(h) n,m

,

(4.45)

m=1

where s(h) n,m denotes the total number of white strips of length m in all the bicolored box diagrams corresponding to the walk diagrams of Wn(h) (the notation is such that s(0) 2n,m = s2n,m (3.50)). Note also that the strips all have length ≤ (n − h)/2 + 1, hence the upper bound in the product in (4.45). The formula of Theorem 2 will follow from the explicit computation of the numbers s(h) n,m . This will be done in two steps. The first step (Proposition 4, Sect. 4.4 below) consists in arranging the s(h) n,m walks above according to their floor configuration (namely their configuration of grey boxes). The second step (Proposition 5, Sect. 4.5 below) consists in enumerating the walks with minimal floor configurations (namely made of only one layer of grey boxes). Finally in Proposition 6, Sect. 4.6, the combination of these two results will eventually lead to a formula for s(h) n,m , which will complete the proof of Theorem 2. 4.4. Enumeration of the floor configurations. In this first step, we note that many different diagrams a ∈ Wn(h) have the same contribution to (4.44), namely those with identical white strips, but different floors of grey boxes. Assembling all these contributions leads to the following formula for s(h) n,m

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P. Di Francesco

Proposition 4. s(h) n,m

=

X h + k − 1 k≥0

k

fn−h+2,m,k ,

(4.46)

where f2n,m,k denotes the total number of walk diagrams a ∈ W2n with k + 1 floors of height 0, and with a marked top of strip of length m.

Fig. 20. A typical walk diagram a ∈ Wn(h) is represented in thick black line on the upper diagram. We have also represented its floor of grey boxes, and the white boxes topping it. The floor of grey boxes in a is a succession of a number k + 1 of horizontal floors, F0 , F1 , ..., Fk , with respective heights H0 = 0, H1 , H2 , ..., Hk ≥ 0. The conjugates of a are obtained by varying these heights, without changing the white strips of a (this is done by letting the floors slide along the dashed lines separating them). The minimal conjugate aˆ ∈ Wn−h+2 of a is represented below it: it has H1 = H2 = ... = Hk = 0. The floor-ends are indicated by arrows

Indeed, as illustrated in Fig. 20, in any walk diagram a ∈ Wn(h) , the floor of grey boxes may be viewed as a succession of a number, say k + 1 of consecutive horizontal floors of grey boxes F0 , F1 , ..., Fk , with respective heights H0 = 0, H1 , ..., Hk , and Hj ≥ 0 for all j ≥ 1. The leftmost floor F0 , of height H0 = 0, is made of one layer of grey boxes of the form e1 e3 e5 ..., and occupies a segment I0 = {i0 = 0, 1, 2, ..., i1 − 1} of positions (we include here the case when F0 = ∅, i.e. I0 = {0}, corresponding to the case J0 6= {0} of (4.8)). It is topped by white strips of arbitrary lengths. Any horizontal floor Fj , j = 1, 2, ..., k, of height Hj ≥ 0, is a parallelogram made of Hj + 1 horizontal layers of grey boxes, whose base occupies a segment of positions Ij = {ij , ij +1, ij +2, ..., ij+1 }, with ik+1 = n −h +1. What distinguishes these floors from F0 is that they are necessarily topped with at least two layers of white boxes, resulting in white strips of lengths m ≥ 2 only. The separation between two consecutive floors of height Hj ≥ 0 is formed by the strips of length 2 (with one white box), as illustrated in the upper diagram of Fig. 20, where the floor separations are indicated by dashed lines. The various floor heights are subject to the constraint 0 ≤ H1 ≤ H2 ≤ ... ≤ Hk ≤ h − 1,

(4.47)

arising from the original definition of the floor of a walk a ∈ Wn(h) (the floor Fj is always of lesser or equal height than Fj+1 ).


577

By varying only the heights H1 , H2 , ..., Hk subject to (4.47), and by keeping the white strips fixed, we describe the set of all conjugates of a given walk diagram a ∈ Wn(h) . There are therefore h+k−1 (4.48) |{(H1 , ..., Hk ) ∈ N s.t. 0 ≤ H1 ≤ H2 ≤ ... ≤ Hk ≤ h − 1}| = k such conjugates for each diagram a ∈ Wn(h) with k + 1 floors. We now choose among the conjugates of a, the minimal one, namely that with H1 = H2 = ... = Hk = 0, which we denote by aˆ (the bottom diagram of Fig. 20). We may amputate this diagram from the final slope with positions n − h + 2, n − h + 3, ..., n, and view it as a diagram aˆ ∈ Wn−h+2 . Indeed, the diagram aˆ has h(n − h + 1) = 1, the height of the rightmost floor-end, hence we may complete it by h(n − h + 2) = 0 into an element of Wn−h+2 . Denoting by f2n,k,m the total number of walks of W2n with k + 1 floors of height 0, and with a marked top of strip of length m, Proposition 4 follows, by enumerating these fn−h+2,k,m walk diagrams with a marked top of strip of length m, and weighing each of them by the number of its conjugates (4.48). 4.5. The mapping of walk diagrams. The second step of the calculation of s(h) n,m is the computation of the numbers f2n,k,m appearing in (4.46). The result reads Proposition 5. The total number f2n+2,k,m of walks in W2n+2 , with k + 1 floors of height 0, and with a marked top of strip of length m reads for m ≥ 2, n ≥ 1, f2n+2,k,m = c2n−k,2m+k + kc2n−k,2m+k−4 for m = 1, n ≥ 1, f2n+2,k,1 = c2n−k,k+2

(4.49)

where the numbers cn,h are defined in (2.3). Here we have excluded the trivial case n = 0, for which no strip appears, hence f2,k,m = 0

for all k and m.

(4.50)

To prove this proposition, we will construct a bijection from the set of walk diagrams of W2n+2 with k + 1 floors of height 0, and with a marked top of strip of length m ≥ 2 (2m+k) (2m+k−4) to (i) the set W2n−k (ii) k copies of the set W2n−k , which will prove (4.49), as (h) |Wn | = cn,h . (In the case m = 1, only part (i) will apply, namely, we will construct a bijection between the walks of W2n+2 with a marked end of empty strip (length 1) and (k+2) W2n−k .) We start from a ∈ W2n+2 , with k + 1 floors, all of height 0, and with a marked top of strip of length m. Two cases may occur: (i) The marked top of strip lies above the leftmost floor (F0 ). In this case, we will con(2m+k) struct a walk b ∈ W2n−k by a cutting-reflecting-pasting procedure on a, analogous to that used in the meander case. This will produce the first term in (4.49). (ii) The marked top of strip lies above one of the k other floors (F1 , F2 , ..., Fk ). This is possible only if m ≥ 2, as there is no empty strip above these floors, by definition. By a circular permutation of the k floors, we can always bring the block containing the marked point to the right. We therefore have a k-to-one mapping to the situation where the marked strip is above the rightmost floor. This k-fold circular permutation symmetry is responsible for the factor k in the second term of (4.49). The diagrams with the marked top of strip of length m above the rightmost (Fk ) floor are then

578

P. Di Francesco

mapped to the walk diagrams with a marked top of strip at height m − 2 above (2(m−2)+k) the leftmost (F0 ) floor considered in the case (i), hence to the set W2n−k = (2m+k−4) W2n−k (the case m = 2 will have to be treated separately). Together with the multiplicity factor k this will produce the second term of (4.49). Let us now construct the maps for the cases (i) and (ii) above. CASE (i). We start from a walk diagram a ∈ W2n+2 , with k + 1 floors of height 0, and with a marked top of strip of length m above its leftmost floor F0 , say at position i. The point (i, h(i) = m) separates a into a left L and right R parts, respectively such (m) . Reflecting L and pasting it again at the left end of that L ∈ Wi(m) and R¯ ∈ W2n+2−i (2m) R, we create a walk diagram b0 , whose reflection b¯0 ∈ W2n+2 . To construct the eventual (2m+k) image b ∈ W2n−k of a, we perform the following amputations of the walk b0 . We will suppress some pieces of b0 at each separation of floor, according to the following rules: (1) (2) (3)

j

i

i i

j j

j

(h → h + 1, o → o − 1),

j

(h → h − 2, o → o − 2),

i

i

i j

(h → h + 3, o → o − 1), (4.51)

where we have represented the floor end by an empty circle, and where we indicate the change in final height (h) and in the order (o) resulting from the amputation. Considering that the rules in (4.51) apply respectively (1) to the first floor separation only (between F0 and F1 ), (2) to the k − 1 intermediate floor separations (between Fj and Fj+1 , j = 1, 2, ..., k − 1), and (3) to the rightmost floor end (right end of Fk ), we get an overall change from the initial values (h = 2m, o = 2n + 2) of the height and order of b0 to the amputed b00 with h → h + 3 + (k − 1) − 2 = 2m + k

o → o − 1 − (k − 1) − 2 = 2n − k. (4.52)

(2m+k) . Hence taking b = b¯00 , we get an element of W2n−k To prove that this mapping is bijective, let us compute its inverse. Starting from (2m+k) , let i be the position of the rightmost intersection between b and the b ∈ W2n−k line h = m + k at an ascending slope (h(i − 1) + 1 = h(i) = h(i + 1) − 1). This point separates the walk b into a left part L and a right part R. Let us reflect R and (k) . As before, if paste it again to the right end of L. This produces a walk a0 ∈ W2n−k h(i + 1) − 1 = h(i) = m + k, we mark the point i, which will be an end of strip (in the eventually reflected walk). Otherwise, h(i + 1) = h(i) − 1, and we migrate the mark to the point i0 = max{j < i|h(j + 1) = h(j) − 1 = m + k}). Let us now mark (by black dots) the rightmost intersections between a0 and the lines h = k, h = k − 1, ..., h = 1 at ascending slopes of a0 , and also the left end (i = 0, h = 0) of a0 . We reconstruct the k + 1 separations of floors using the following rules (inverse of (4.51)):

(1)

(h → h + 2, o → o + 2),

(2)

(h → h − 1, o → o + 1),

(3)

(h → h − 3, o → o + 1).

(4.53)

The corresponding separations have been represented by empty circles. They all lie at height h = 1 in the resulting final walk a00 . The three rules of (4.53) apply respectively


579

(1) to the left end of a0 , (2) to any of the k − 1 intermediate points of intersection with the lines h = 1, ..., h = k − 1, and (3) to the rightmost intersection with the line h = k. The rules (4.53) therefore result in a change of final height and order (h = k, o = 2n − k) → (h + 2 − (k − 1) − 3 = 0, o + 2 + (k − 1) + 1 = 2n + 2), hence a00 ∈ W2n+2 . The last step consists simply in reflecting a00 , to produce a = a¯00 , with a marked top of strip of height m + k − k = m above the leftmost (F0 ) floor, and a has a total of k + 1 floors, all of height 0. As before, the bijectivity of the map follows from the fact that we considered rightmost points of intersection, which makes the construction unique. This bijection yields the number c2n−k,2m+k of walks in W2n+2 with k + 1 floors of height 0, and with a marked top of strip of length m above F0 . This is the first term of (4.49). CASE (ii). We start from a walk a ∈ W2n+2 , with k + 1 floors F0 , F1 ,...Fk , all of height zero, and with a marked top of strip of length m above its rightmost floor Fk . By definition, we necessarily have m ≥ 2, and in fact there is one and only one strip of length 2 above the floor Fk (the one just above the right floor-end), and all other strips have length ≥ 3. We now construct a bijection between these walks and the b ∈ W2n+2 with k + 1 floors F00 , F10 , ..., Fk0 , all of height 0, and with a marked top of strip of length m − 2 above their leftmost floor F00 . If m = 2, the above remark shows that the number of walks a with k + 1 floors of height 0, and with a marked top of strip of length 2 above Fk is equal to the number of such walks, without marked top of strip (there is exactly one such strip of length 2 per walk). Skipping the cutting-reflecting-pasting procedure of case (i) (we have no more marked top of strip), we can still apply the amputation rules (4.51) on the walk (k) . Conversely, starting from any b0 = a¯ ∈ W2n+2 : this results in a walk b ∈ W2n−k (k) b ∈ W2n−k , let us apply to it the inverse of the amputation rules (4.53), after marking the rightmost intersections at ascending slopes with the lines h = k, h = k − 1, ..., h = 1. This produces a walk a0 ∈ W2n+2 , and finally a = a¯0 ∈ W2n+2 has k + 1 floors of height 0. This bijection yields the number c2n−k,k of walks in W2n+2 with k + 1 floors of height 0, and a marked top of strip of height m = 2 above Fk . Together with the k-fold cyclic degeneracy of the case (ii) this gives the second term of (4.49), for m = 2.

Fig. 21. The exchange map on walk diagrams of Wn(h) , maps the walks with a marked strip of length m above their rightmost floor onto those with a marked strip of length m−2 above the leftmost floor (the corresponding strip of length m = 3 is marked with a black dot on the figure). We have indicated by a thick broken line the portions exchanged. The double-layer of white boxes on the rightmost floor is adapted to fit the exchange

580

P. Di Francesco

If m ≥ 3, we simply exchange the floors F0 and Fk in the following way. The floor Fk is by definition topped by at least two layers of white boxes (see Fig. 21). Let ik , ik +1, ..., ik+1 denote the positions occupied by Fk , the ends ik and ik+1 being at height 1. Let us cut out the portion ak of a in between the positions ik + 2 and ik+1 − 2, both at height 3 (the level of the second layer of white boxes). Let us also cut the portion a0 of a above the leftmost floor F0 , in between the positions i0 = 0 and i1 − 1, both at height 0. We form a walk b ∈ W2n+2 by simply exchanging the portions a0 and ak in a, as depicted in Fig. 21. The marked top of strip on ak has been therefore transferred above the leftmost floor of b, but as two layers of white boxes have been suppressed, all the lengths of strips have been decreased by 2. Hence the walk b ∈ W2n+2 has k + 1 floors of height 0, and a marked top of strip of length m − 2 above its leftmost floor F00 . This construction is clearly invertible, by just exchanging again ak and a0 . From case (i) above, we learn that (2(m−2)+k) (2m+k−4) the walk b can be mapped onto an element of W2n−k = W2n−k , in a bijective way. This bijection yields the number c2n−k,2m+k−4 of walks a ∈ W2n+2 with k + 1 floors of height 0, and with a marked top of strip of length m above its rightmost floor Fk . With the overall k-fold cyclic degeneracy mentioned above, this gives the second term in (4.49) for all m ≥ 3. The mappings of the cases (i) and (ii) above complete the proof of Proposition 5, with the understanding that the case m = 1 only gives rise to case (i), hence the different answer. 4.6. The semi-meander determinant: The final formula. Combining the results of Propositions 4 and 5, namely Eqs. (4.46) and (4.49), we get the following formula for the (h) numbers s(h) n,m of walk diagrams in Wn with a marked top of strip of length m above its floor: X h + k − 1 = for m ≥ 2, (cn−h−k,2m+k + kcn−h−k,2m+k−4 ) s(h) n,m k k≥0 (4.54) X h + k − 1 (h) for m = 1. sn,1 = cn−h−k,k+2 k k≥0

This is valid for h ≤ n − 1. If h = n, (4.50) yields s(n) n,m = 0 for all m. By a direct calculation, we find Proposition 6. The numbers of walks in W2n+2 with k+1 floors of height 0 and a marked end of strip of length m read s(h) n,m = cn,h+2m + hcn,h+2m−2 s(h) n,1 (n) sn,m

= cn,h+2

for m ≥ 2, h ≤ n − 1,

for m = 1, h ≤ n − 1,

(4.55)

for h = n and all m ≥ 1.

= 0

The proof relies on the following classical identity for binomial coefficients: c−d X k+a c−k a+c+1 = b d b+d+1

(4.56)

k=b−a

for all integers a, b, c, d. This is easily proved by use of generating functions. We now simply have to apply (4.56) to the various sums appearing on the r.h.s. of (4.54):


581

X k + h − 1n − h − k

n n+h 2 +m

n

= = n−(h+2m) , n−h h−1 2 +m 2 (4.57) X k + h − 1 n − h − k n n = n+h = n−(h+2m) , n−h −1 h−1 2 +m+1 2 +m+1 2 k≥0

k≥0

X k + h − 1

hence

and, noting that k

k+h−1 k

cn−h−k,k+2m = cn,h+2m

k

k≥0

=h

(4.58)

k+h−1 k−1

, we also get X k+h−1 k cn−h−k,k+2m−4 = hcn,h+2m−2 . k

(4.59)

k≥0

Proposition 6 follows from (4.58) and (4.59). Substituting the result (4.55) above into (4.45), we finally get the semi-meander determinant n−h 2 +1 Y −(cn,h+2m +hcn,h+2m−2 ) (h) , (4.60) µm det Gn (q) = m=1 −hc

where we have absorbed the prefactor µ1 n,h of (4.45) into the m = 1 term of the product. Finally, using the fact that µm = Um−1 (q)/Um (q), for m ≥ 1, Theorem 2 follows. 5. Conclusion In this paper, we have proved two determinant formulas for meanders and semimeanders. This has been done by the Gram-Schmidt orthogonalization of the corresponding bases 1 of the Temperley-Lieb algebra or some of its subspaces. The main philosophy of the construction leading to the Gram-Schmidt orthogonalization of these bases 1 lies in the concept of box addition, the building block of the definition of the bases 2 elements. We believe that this type of construction should be much more general and apply to many other cases of algebra-related Gram-Schmidt orthogonalization. An important remark about Theorems+1 and+2 above is that they implicitly give the structure (including multiplicity) of the zeros of the Gram determinants, considered as functions of the variable q. Due to the definition of the Chebyshev polynomials, the zeros of the Gram determinant always take the form q = 2 cos(π

m ) p+1

(5.1)

with 1 ≤ m ≤ p ≤ n−h 2 + 1 in the semi-meander case. These zeros actually correspond to the cases when the corresponding subspace of the Temperley-Lieb algebra is reducible (there are linear combinations of the basis 1 elements which are orthogonal to all the basis 1 elements: i.e. there may be vanishing linear combinations of the basis 1 elements, the basis 1 being therefore no longer a basis at these values of q). The multiplicity of these zeros is linked to the degree of reducibility (namely to how many such independent linear combinations exist).

582

P. Di Francesco

Unfortunately, the information we obtain from these determinant formulas on the meanders and the semi-meanders themselves is very difficult to exploit. Indeed, quantities such as asymptotics (for large order) of the meander and semi-meander numbers and distributions are only indirectly related to the Gram determinants, as they would rather involve the exact knowledge of the asymptotics of the Gram matrices, or at least of their eigenvalue spectra. However, the exact orthogonalization performed above is useful to derive new asymptotic formulas for the meander numbers, as sums over walk diagrams (see [9] for the meander example). We hope to return to this question in a later publication. Theorems 1 and 2 above can probably be generalized in many directions. A first possibility relies on the fact that there exists a canonical Temperley-Lieb algebra attached to any non-oriented, connected graph (see [12] and references therein), which may still be interpreted as the image of a walk diagram on that graph. The only constraint is that the number q must be an eigenvalue of the adjacency matrix of the graph (a matrix Ga,b made of 1’s and 0’s according to whether the couple (a, b) of vertices of the graph is joined by a link or not). In the examples treated here, this graph is simply the set of heights, namely the integer points on the (infinite) half-line, linked by segments between consecutive pairs (hence Gi,j = δj,i+1 + δj,i−1 for i, j > 0 and G0,j = δj,1 , with the eigenvalue q for the (infinite) eigenvector v = (U0 (q), U1 (q), U2 (q), ...)). But nothing prevents us from considering more complicated graphs. We believe that there exists a general determinant formula, associated to each such graph, expressing the result in terms of features of the graph only (with c2n,2m replaced by a corresponding number of paths of given length and given origin and end on the graph, and Um (q) replaced by the components of the eigenvector of the adjacency matrix for the eigenvalue q). Another direction of generalization has to do with replacing the Temperley-Lieb algebra by a more general quotient of the Hecke algebra. Indeed, recall that the TemperleyLieb algebra T Ln (q) is nothing but a simple quotient of the Hecke algebra, defined as follows. The Hecke algebra Hn (q) is defined by generators 1, e1 , e2 , ..., en−1 satisfying the following relations (i) e2i = q ei , (ii) [ei , ej ] = 0 if |i − j| > 1, (iii) ei ei+1 ei − ei = ei+1 ei ei+1 − ei+1 ,

(5.2)

hence the Temperley-Lieb algebra is the quotient of the Hecke algebra by the ideal generated by the elements ei ei±1 ei − ei . This quotient was identified as the commutant of the quantum enveloping algebra Uqˆ (sl2 ) acting on the fundamental representation of Hn (q), with q = qˆ + qˆ−1 . More quotients are found by considering the commutants of other quantum enveloping algebras (such as Uqˆ (slk ) for instance) [13]. These quotients await a good combinatorial interpretation, but should lead to a natural generalization of meanders and semi-meanders. We believe that many Gram determinants can still be computed exactly in this framework. References 1. 2. 3.

Hoffman, K., Mehlhorn, K., Rosenstiehl, P. and Tarjan, R.: Sorting Jordan sequences in linear time using level-linked search trees. Information and Control 68, 170–184 (1986) Touchard, J.: Contributions a` l’étude du problème des timbres poste. Canad. J. Math. 2, 385–398 (1950) Lunnon, W.: A map–folding problem. Math. of Computation 22, 193–199 (1968)


4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14.

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Phillips, A.: La topologia dei labirinti. In: M. Emmer, ed. L’ occhio di Horus: itinerario nell’immaginario matematico. Roma: Istituto della Enciclopedia Italia, 1989, pp. 57–67 Arnold, V.: The branched covering of CP2 → S4 , hyperbolicity and projective topology. Siberian Math. Jour. 29, 717–726 (1988) Ko, K.H., Smolinsky, L.: A combinatorial matrix in 3-manifold theory. Pacific. J. Math 149) 319–336 (1991) Lando, S. and Zvonkin, A.: Plane and Projective Meanders. Theor. Comp. Science 117, 227–241 (1993) and Meanders. Selecta Math. Sov. 11, 117–144 (1992) Di Francesco, P., Golinelli, O. and Guitter, E.: Meander, folding and arch statistics. To appear in Mathematical and Computer Modelling (1996), hep-th/950630 Makeenko, Y.: Strings, Matrix Models and Meanders. Proceedings of the 29th Inter. Ahrenshoop Symp., Germany (1995) Di Francesco, P., Golinelli, O. and Guitter, E.: Meanders and the Temperley-Lieb algebra. To appear in Commun. Math. Phys. (1996), hep-th/9602025 Temperley, H. and Lieb, E.: Relations between the percolation and coloring problem and other graphtheoretical problems associated with regular planar lattices: some exact results for the percolation problem. Proc. Roy. Soc. A322, 251–280 (1971) Martin, P.: Potts models and related problems in statistical mechanics. Singapore: World Scientific, 1991 Di Francesco, P.: Integrable lattice models, graphs and modular invariant conformal field theories. Int. Jour. Mod. Phys. A7, 407–500 (1992) Reshetikhin, N.: Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links 1 and 2. LOMI preprints E-4-87 and E-17-87 (1988)

Communicated by D. C. Brydges

Commun. Math. Phys. 191, 585 – 601 (1998)

Communications in


The sh Lie Structure of Poisson Brackets in Field Theory G. Barnich1,4,? , R. Fulp2 , T. Lada2 , J. Stasheff 3,?? 1 Center for Gravitational Physics and Geometry, The Pennsylvania State University, 104 Davey Laboratory, University Park, PA 16802, USA 2 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA 3 Department of Mathematics, The University of North Carolina at Chapel Hill, Phillips Hall, Chapel Hill, NC 27599-3250, USA 4 Freie Universit¨ at Berlin, Institut für Theoretische Physik, Arnimallee 14, D-14195 Berlin, Germany

Received: 5 March 1997 / Accepted: 21 May 1997

Abstract: A general construction of an sh Lie algebra (L∞ -algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel’fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the BatalinFradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.

1. Introduction In field theories, an important class of physically interesting quantities, such as the action or the Hamiltonian, are described by local functionals, which are the integral over some region of spacetime (or just of space) of local functions, i.e., functions which depend on the fields and a finite number of their derivatives. It is often more convenient to work with these integrands instead of the functionals, because they live on finite dimensional spaces. The price to pay is that one has to consider equivalence classes of such integrands modulo total divergences in order to have a one-to-one correspondence with the local functionals. The approach to Poisson brackets in this context, pioneered by Gel’fand, Dickey and Dorfman, is to consider the Poisson brackets for local functionals as being induced by brackets for local functions, which are not necessarily strictly Poisson. We will analyze here in detail the structure of the brackets for local functions corresponding to the Poisson ? Research supported by grants from the Fonds National Belge de la Recherche Scientifique and the Alexander-von-Humboldt foundation. New address: Université Libre de Bruxelles, Faculté des Sciences, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium. ?? Research supported in part by NSF grant DMS-9504871.

586

G. Barnich, R. Fulp, T. Lada, J. Stasheff

brackets for local functionals. More precisely, we will show that these brackets will imply higher order brackets combining into a strong homotopy Lie algebra. The paper is organized as follows: In the case of a homological resolution of a Lie algebra, it is shown that a skewsymmetric bilinear map on the resolution space inducing the Lie bracket of the algebra extends to higher order “multi-brackets” on the resolution space which combine to form an L∞ -algebra (strong homotopy Lie algebra or sh Lie algebra). For completeness, the definition of these algebras is briefly recalled. For a highly connected complex which is not a resolution, the same procedure yields part of an sh Lie algebra on the complex with corresponding multi-brackets on the homology . This general construction is then applied in the following case. If the horizontal complex of the variational bicomplex is used as a resolution for local functionals equipped with a Poisson bracket, we can construct an sh Lie algebra (of order the dimension of the base space plus two) on the graded differential algebra of horizontal forms. The construction applies not only for brackets in Darboux coordinates as well as for non-canonical brackets such as those of the KDV equation, but also in the presence of Grassmann odd fields for graded even brackets, such as the extended Poisson bracket appearing in the Hamiltonian formulation of the BRST theory, and for graded odd brackets, such as the antibracket of the Batalin-Vilkovisky formalism. 2. Sh Lie Algebras from Homological Resolution of Lie Algebras 2.1. Construction. Let F be a vector space and (X∗ , l1 ) a homological resolution thereof, i.e., X∗ is a graded vector space, l1 is a differential and lowers the grading by one with F ' H0 (l1 ) and Hk (l1 ) = 0 for k > 0. The complex (X∗ , l1 ) is called the resolution space. (We are not using the term “resolution” in a categorical sense.) Let C∗ and B∗ denote the l1 cycles (respectively, boundaries) of X∗ . Recall that by convention X0 consists entirely of cycles, equivalently X−1 = 0. Hence, we have a decomposition X0 = B0 ⊕ K,

(1)

with K ' F. We may rephrase the above situation in terms of the existence of a contracting homotopy on (X∗ , l1 ) specifying a homotopy inverse for the canonical homomorphism η : X0 −→ H0 (X∗ ) ' F. We may regard F as a differential graded vector space F∗ with F0 = F and Fk = 0 for k > 0; the differential is given by the trivial map. We then consider the chain map η : X∗ −→ F∗ with homotopy inverse λ : F∗ −→ X∗ ; i.e., we have that η ◦ λ = 1F∗ and that λ ◦ η ∼ 1X∗ via a chain homotopy s : X∗ −→ X∗ with λ◦η −1X∗ = l1 ◦s+s◦l1 . Observe that this equation takes on the form λ◦η −1X∗ = l1 ◦s on X0 . We may summarize all of the above with the commutative diagram s ←− · · · −→ X2 −→ X1 x  l1 x    λyη λyη

s ←− −→ l1

· · · −→

−→ H0 = F .

0

−→

0

X0 x  λyη

The sh Lie Structure of Poisson Brackets in Field Theory

587

It is clear that η(b) = 0 for b ∈ B0 . Let (−1)σ be the signature of a permutation σ and unsh(k, p) the set of permutations σ satisfying σ(1) < . . . < σ(k) {z } |

and

first σ hand

σ(k + 1) < . . . < σ(k + p). | {z } second σ hand

We will be concerned with skew-symmetric linear maps l˜2 : X0 ⊗ X0 −→ X0

(2)

that satisfy the properties (i) l˜2 (c, b1 ) = b2 , X (−1)σ l˜2 (l˜2 (cσ(1) , cσ(2) ), cσ(3) ) = b3 , (ii)

(3) (4)

σ∈unsh(2,1)

where c, ci ∈ X0 , bi ∈ B0 and with the additional structures on X∗ as well as on F that such maps will yield. We begin with N Lemma 1. The existence of a skew-symmetric linear map l˜2 : X0 X0 −→ X0 that satisfies N condition (i) is equivalent to the existence of a skew-symmetric linear map [·, ·] : F F −→ F. N F via the diagram Proof. l˜2 induces a linear mapping on F N l˜2 X0 X0 X0 −→ λ⊗λ↑ ↓η N [·,·] F F −→ F . The fact that l˜2 satisfies condition (i) guarantees that [·, ·] is well-defined on the homology classes. N Conversely, given [·, ·] : F F −→ F , define l˜2 = λ ◦ [·, ·] ◦ η ⊗ η. It is clear that l˜2 is skew-symmetric because [·, ·] is, and condition (i) is satisfied in the strong sense that l˜2 (c, b) = 0. N Lemma 2. Assume that l˜2 : X0 X0 −→ X0 satisfies condition (i). Then the induced bracket on F is a Lie bracket if and only if l˜2 satisfies condition (ii). Proof. Assume that the induced bracket on F is a Lie bracket; recall that the bracket is given by the composition η ◦ l˜2 ◦ λ ⊗ λ. The Jacobi identity takes on the form, for arbitrary fi ∈ F, X (−1)σ (η ◦ l˜2 ◦ λ ⊗ λ)(η ◦ (l˜2 ⊗ 1) ◦ (λ ⊗ λ ⊗ 1) σ∈unsh(2,1)

X

⇔

(fσ(1) ⊗ fσ(2) ⊗ fσ(3) ) = 0 σ

(−1) (η ◦ l˜2 ◦ λ ⊗ λ)(η ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ fσ(3) ) = 0

σ∈unsh(2,1)

⇔

X

(−1)σ η ◦ l˜2 (λ ◦ η ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = 0

σ∈unsh(2,1)

588


X

⇔

(−1)σ η ◦ l˜2 ((1 + l1 ◦ s) ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = 0

σ∈unsh(2,1)

X

⇔

σ∈unsh(2,1)

X

+

(−1)σ η ◦ l˜2 (l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) ))

(−1)σ η ◦ l˜2 (l1 ◦ s ◦ l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = 0

σ∈unsh(2,1)

X

⇔ η(

(−1)σ l˜2 (l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) ))) + η(b) = 0.

σ∈unsh(2,1)

But η(b) = 0 and so X

(−1)σ l˜2 (l˜2 (λ(fσ(1) ) ⊗ λ(fσ(2) )) ⊗ λ(fσ(3) )) = b0 ∈ B0 .

(5)

σ∈unsh(2,1)

The converse follows from a similar calculation.

Remark. The interesting case here is when F is only known as X0 /B0 and the only characterization of the Lie bracket [·, ·] in F is as the bracket induced by l˜2 . An important particular case, to be considered elsewhere, occurs when X0 is a Lie algebra G with Lie bracket L2 and B0 a Lie ideal. The bracket l˜2 is defined by choosing a vector space complement K of the ideal B0 in X0 and then projecting the Lie bracket L2 onto K. Hence, L2 (c1 , c2 ) = l˜2 (c1 , c2 ) + b(c1 , c2 ), where b(c1 , c2 ) is a well-defined element of B0 . Indeed, by definition, property (i) holds with zero on the right hand side. Property (ii) follows from the Jacobi identity for L2 : 0= =

X

X

(−1)σ L2 (L2 (cσ(1) , cσ(2) ), cσ(3) )

σ∈unsh(2,1)

(−1)σ [l˜2 (L2 (cσ(1) , cσ(2) ), cσ(3) ) + b(L2 (cσ(1) , cσ(2) ), cσ(3) )]

σ∈unsh(2,1)

=

X

(−1)σ [l˜2 (l˜2 (cσ(1) , cσ(2) ), cσ(3) ) + b(L2 (cσ(1) , cσ(2) ), cσ(3) )].

(6)

σ∈unsh(2,1)

We now turn our attention to the maps that l˜2 induces on the complex X∗ . N Lemma 3. A skew-symmetric linear map l˜2 : X0 X0 −→ XN 0 that satisfies condition (i) extends to a degree zero skew-symmetric chain map l2 : X∗ X∗ −→ X∗ . N ˜ Proof. We first Nextend l2 to a linear map l2 : X1 X0 −→ X1 by the following: N let x1 ⊗ x0 ∈ X1 X0 . Then l1 (x1 ⊗ x0 ) = l1 x1 ⊗ x0 + x1 ⊗ l1 x0 = l1 x1 ⊗ x0 ∈ X0 X0 . So we have that l2 l1 (x1 ⊗ x0 ) = l˜2 (l1 x1 ⊗ x0 ) = b by condition N (i). Write b = l1 z1 for z1 ∈ X1 and define l2 (x1 ⊗ x0 ) = z1 . Also extend l2 to X0 X1 by skew-symmetry: l2 (x0 ⊗ x1 ) = −l2 (x1 ⊗ x0 ). Note that l2 is a chain map by construction. Now assume thatN l2 is defined and is a chainNmap on elements of degree less than or equal to n in X∗ X∗ . Let xp ⊗ xq ∈ Xp Xq , where p + q = n + 1. Because l1 (xp ⊗ xq ) has degree n, l2 l1 (xp ⊗ xq ) is defined. We have that


589

l1 l2 l1 (xp ⊗ xq ) = = l1 l2 [l1 xp ⊗ xq + (−1)p xp ⊗ l1 xq ] = l2 l1 [l1 xp ⊗ xq + (−1)p xp ⊗ l1 xq ] because l2 is a chain map = l2 [l1 l1 xp ⊗ xq + (−1)p−1 l1 xp ⊗ l1 xq +(−1)p l1 xp ⊗ l1 xq + xp ⊗ l1 l1 xq ] = 0

(7)

because l12 = 0 and (−1)p and (−1)p−1 have opposite parity. Thus l2 l1 (xp ⊗ xq ) is a cycle in Xn and so there is an element zn+1 ∈ XN n+1 with l1 zn+1 = l2 l1 (xp ⊗ xq ). Define l2 (xp ⊗ xq ) = zn+1 . As before, extend l2 to Xq Xp by skew-symmetry and note that l2 is a chain map by construction. Nn X∗ −→ Remark. We will be concerned with (graded) skew-symmetric maps fn : Nn+k Nk X∗ −→ X∗ via the equation X∗ that have been extended to maps fn : fn (x1 ⊗ . . . ⊗ xn+k ) = X (−1)σ e(σ)fn (xσ(1) ⊗ . . . ⊗ xσ(n) ) ⊗ xσ(n+1) ⊗ . . . ⊗ xσ(n+k) ,

(8)

unsh(n,k)

where e(σ) is the Koszul sign (see e.g. [16]). This extension arises from the skewsymmetrization of the extension of a linear map as a skew coderivation on the tensor coalgebra on the graded vector space X∗ [15]. The extension of the differential l1 assumed in the previous lemma is equivalent to the one given by this construction. We assume for the remainder of this section that all maps have been extended in this fashion when necessary; moreover, we will use the uniqueness of such extensions when needed. When the original skew-symmetric map l˜2 satisfies both conditions (i) and (ii), there is a very rich algebraic structure on the complex X∗ . N Proposition 4. A skew-symmetric linear map l˜2 :N X0 X0 −→ X0 that satisfies conditions (i) and (ii) extendsNto a chain N map l2 : X∗ X∗ −→ X∗ ; moreover, there exists a degree one map l3 : X∗ X∗ X∗ −→ X∗ with the property that l1 l3 +l3 l1 +l2 l2 = 0. Here, we have suppressed the notation that is used to indicate the indexing of the summands over the appropriate unshuffles as well as the corresponding signs. They are given explicitly in Definition 5 below. N Proof. We extend l˜2 to l2 : X∗ X∗ −→ X∗ as in the previous lemma. In degree zero, l2 l2 (x1 ⊗ x2 ⊗ x3 ) is equal to a boundary b in X0 by condition (ii). There exists an element Nz ∈ X N1 with l1 z = b and so we define l3 (x1 ⊗ x2 ⊗ x3 ) = −z. Because l1 = 0 on X0 X0 X0 , we have that l1 l3 + l2 l2 + l3 l1 = 0. N N Now assume that l3 is defined up to degree p in X∗ X∗ X∗ and satisfies the relation l1 l3 + l2 l2 + l3 l1 = 0. Consider the map l2 l2 + l3 l1 which is inductively defined on N N elements of degree p+1 in X∗ X∗ X∗ . We have that l1 [l2 l2 +l3 l1 ] = l1 l2 l2 +l1 l3 l1 = l2 l1 l2 + l1 l3 l1 = l2 l2 l1 + l1 l3 l1 = [l2 l2 + l1 l3 ]l1 = −l3 l1 l1 = 0. Thus the image of l2 l2 + l3 l1 is a boundary in Xp which is then the image of an element, p+2 ∈ Xp+2 . Define now Nsay zN l3 applied to the original element of degree p + 1 in X∗ X∗ X∗ to be this element zp+1 . In the proof of the proposition above, we made repeated use of the relation l1 l2 −l2 l1 = 0 when extended to an arbitrary number of variables. We may justify this by observing that the map l1 l2 − l2 l1 is the commutator of the skew coderivations l1 and l2 and is thus

590


a coderivation; it follows that the extension of this map must equal the extension of the 0 map. The relations among the maps li that were generated in the previous results are the first relations that one encounters in an sh Lie algebra (L∞ algebra). Let us recall the definition [17, 16]. Definition 5. An sh Lie structure on a graded vector space X∗ is a collection of linear, Nk X∗ −→ X∗ of degree k − 2 that satisfy the relation skew symmetric maps lk : X

X

e(σ)(−1)σ (−1)i(j−1) lj (li (xσ(1) , . . . , xσ(i) ), . . . , xσ(n) ) = 0,

i+j=n+1 unsh(i,n−i)

(9) where 1 ≤ i, j. Remark. Recall that the suspension of a graded vector space X∗ , denoted by ↑ X∗ , is the graded vector space defined by (↑ X∗ )n = Xn−1 while the desuspension of X∗ is given by (↓ X∗ )n = Xn+1 . It can be shown, [17, 16], Theorem 2.3, that the data in the definition is equivalent to V∗ ↑ X, the cocommutative coalgebra a) the existence of a degree −1 coderivation D on on the graded vector space ↑ X, with D2 = 0. and to V∗ ↑ X ∗ , the exterior algebra on the b) the existence of a degree +1 derivation δ on 2 suspension of the degree-wise dual of X∗ , with δ = 0. In this case, we require that X∗ be of finite type. Let us examine the relations in the above definition independently of the underlying vector space X∗ and write them in the form X

(−1)i(j−1) li lj = 0,

i+j=n+1

where we are assuming that the sums over the appropriate unshuffles with the corresponding signs are incorporated into the definition of the extended maps lk . We will require the fact that the map X

(−1)i(j−1) lj li :

n O

X∗ −→ X∗

i,j>1

is a chain map in the following sense: Lemma 6. Let {lk } be an sh Lie structure on the graded vector space X∗ . Then l1

X

(−1)(j−1)i lj li = (−1)n−1 (

i,j>1

where i + j = n + 1.

X

(−1)(j−1)i lj li )l1 ,

i,j>1

(10)


591

Proof. Let us reindex the left hand side of the above equation and write it as l1

n−1 X

(−1)(n−i)i ln−i+1 li

i=2

which, after applying the sh Lie relation to the composition l1 ln−i+1 , is equal to n−i n−1 XX

(−1)(n−i)i (−1)(k−1)(n−i−k)+1 lk ln−i−k+2 li .

i=2 k=2

On the other hand, the right hand side may be written as (−1)n−1

n−1 X

(−1)(i−1)(n−i+1) li ln−i+1 l1

i=2

which in turn is equal to (−1)n−1

n−i n−1 XX

(−1)(i−1)(n−i+1) (−1)(n−i−k+1)k (−1)n−i+1 li ln−i+2−k lk .

i=2 k=2

It is clear that the two resulting expressions have identical summands and a straightforward calculation yields that the signs as well are identical. The argument in the previous proposition may be extended to construct higher order maps lk so that we have N Theorem 7. A skew-symmetric linear map l˜2 : X0 X0 −→ X0 that satisfies conditions (i) and (ii) extends to an sh Lie structure on the resolution space X∗ . Proof. We already have the required maps l1 , l2 and l3 from our previous work. We use induction to assume that we have the maps lk for 1 ≤ k < n that satisfy the relation in the definition of an sh Lie structure. To construct the map ln , we begin with the map X

(−1)(j−1)i lj li :

n O

X0 −→ Xn−3

i+j=n+1

with i, j > 1. Apply the differential l1 to this map to get X X (−1)(j−1)i lj li = (−1)n−1 (−1)(j−1)i lj li l1 = 0 l1 i+j=n+1

(11)

i+j=n+1

where the first equality follows from the lemma and the second equality from the fact Nn X0 . The acyclicity of the complex that l1 is 0 on Nn X∗ will then yield, with care X∗ and it will satisfy the sh Lie to preserve the desired symmetry, the map ln on relations by construction. Finally, assume that all of the maps lk for k < n have been constructed so as to satisfy the sh relations and that ln has been constructed in a similar fashion through NLie n X∗ . We have the map degree p in

592


X

(−1)(j−1)i lj li : (

n O

X∗ )p −→ Xp−3

i+j=n+1

to which we may apply the differential l1 . This results in X X (−1)(j−1)i l1 lj li = (−1)(j−1)i l1 lj li + (−1)n−1 l1 ln l1 i+j=n+1

i,j>1

=

X

(−1)(j−1)i (−1)n−1 lj li l1 + (−1)n−1 l1 ln l1

i,j>1

=

X

(−1)(j−1)i (−1)n−1 lj li l1 + (−1)n−1 (−

i,j>1

X

(−1)(j−1)i lj li l1 ) = 0

i,j>1

and so again, we have the existence of ln together with the appropriate sh Lie relations. Remarks. (i) It may be the case in practice that the complex X∗ is truncated at height n, i.e. we have that X∗ is not a resolution but rather may have non-trivial homology in degree n as well as in degree 0. In such a case, our construction of the maps lk may be terminated by degree n obstructions. More precisely, we have that the vanishing of Hk X∗ for k different from 0 and n will then only guarantee the existence of the requisite Nk X∗ )p −→ Xp+k−2 for p + k − 2 ≤ n. maps lk : ( (ii) If property (i) holds with zero on the right hand side, i.e. so that l2 vanishes if one of the xi ’s is in B0 , then l2 can be extended trivially (to be a chain map) as zero on N2 ( X∗ )p for p > 0. It is easy to see that in the recursive construction, one can choose similarly trivial extensions of the maps lk for k > 2 to all of the resolution complex, i.e. they are defined to vanish whenever one of their arguments belongs to B0 or Xp Nk for p > 0. Hence they induce well-defined maps lˆk on F . With these choices, each of the defining equations of the sh Lie algebra on X∗ involves only two terms, namely l1 lk + lk−1 l2 = 0 for k ≥ 3. (iii) If Hk X∗ = 0 for 0 < k < n and property (i) holds with zero on the right hand side, Nn+2 we have defined a map on F which may be non-zero. If so, the only non-trivial defining equation of the induced sh Lie algebra on F reduces to lˆn+2 lˆ2 = 0. For example, if n = 1, Σ lˆ3 (lˆ2 (xi , xj ), xk , xm ) = 0 where the sum is over all permutations of (1234) such that i < j and k < m. In Sect. 3, we apply this construction in the context of Poisson brackets in field theory. 2.2. Generalization to the graded case. The above construction of an L∞ -structure on the resolution of a Lie algebra can be extended in a straightforward way to the graded case, i.e. when the Lie algebra is graded (either by Z or by Z/2, the super case) and the bracket is of a fixed degree, even or odd, satisfying the appropriate graded version of skew-symmetry and the Jacobi identity. We will refer to all of these possibilities as graded Lie algebras although the older mathematical literature uses that term only for the case of a degree 0 bracket. In these situations, the resolution X∗ is bigraded and the inductive steps proceed with respect to the resolution degree (see [20, 15] for carefully worked out examples).


593

The graded case occurs in the Batalin-Fradkin-Vilkovisky approach to constrained Hamiltonian field theories [8, 2, 7, 14] where their bracket is of degree 0 and in the Batalin-Vilkovisky anti-field formalism for mechanical systems or field theories [3, 4, 14] with their anti-bracket of degree 1. In all these cases, one need only take care of the signs by the usual rule: when interchanging two things (operators, fields, ghosts, etc.), be sure to include the sign of the interchange. 3. Local Field Theory with a Poisson Bracket We first review the result that the cohomological resolution of local functionals is provided by the horizontal complex. Then, we give the definition of a Poisson bracket for local functionals. The existence of such a Poisson bracket will assure us that the conditions of the previous section hold. Hence, we show that to the Poisson bracket for local functionals corresponds an sh Lie algebra on the graded differential algebra of horizontal forms. 3.1. The horizontal complex as a resolution for local functionals. In this subsection, we introduce some basic elements from jet-bundles and the variational bicomplex relevant for our purpose. More details and references to the original literature can be found in [18, 19, 5, 1]. For the most part, we will follow the definitions and the notations of [18]. Although much of what we do is valid for general vector bundles, we will not be concerned with global properties. We will use local coordinates most of the time, though we will set things up initially in the global setting. Let M be an n-dimensional manifold and π : E → M a vector bundle of fiber ∞ : dimension k over M . Let J ∞ E denote the infinite jet bundle of E over M with πE ∞ ∞ ∞ J E → E and πM : J E → M the canonical projections. The vector space of smooth sections of E with compact support will be denoted 0E. For each (local) section φ of E, let j ∞ φ denote the induced (local) section of the infinite jet bundle J ∞ E. The restriction of the infinite jet bundle over an appropriate open U ⊂ M is trivial with fibre an infinite dimensional vector space V ∞ . The bundle π ∞ : J ∞ EU = U × V ∞ → U

(12)

then has induced coordinates given by (xi , ua , uai , uai1 i2 , . . . , ).

(13)

We use multi-index notation and the summation convention throughout the paper. If j ∞ φ is the section of J ∞ E induced by a section φ of the bundle E, then ua ◦ j ∞ φ = ua ◦ φ and uaI ◦ j ∞ φ = (∂i1 ∂i2 ...∂ir )(ua ◦ j ∞ φ), where r is the order of the symmetric multi-index I = {i1 , i2 , ..., ir }, with the convention that, for r = 0, there are no derivatives. The de Rham complex of differential forms ∗ (J ∞ E, d) on J ∞ E possesses a differential ideal, the ideal C of contact forms θ which satisfy (j ∞ φ)∗ θ = 0 for all sections φ with compact support. This ideal is generated by the contact one-forms, which in local coordinates assume the form θJa = duaJ − uaiJ dxi . Contact one-forms of order 0 satisfy (j 1 φ)∗ (θ) = 0. In local coordinates, contact forms of order zero assume the form θa = dua − uai dxi .

594


Remarkably, using the contact forms, we see that the complex ∗ (J ∞ E, d) splits as a bicomplex (though the finite level complexes ∗ (J p E) do not). The bigrading is described by writing a differential p-form α as an element of r,s (J ∞ E), with p = r + s J (θJA ∧ dxI ), where when α = αIA dxI = dxi1 ∧ ... ∧ dxir

and

θJA = θJa11 ∧ ... ∧ θJass .

(14)

J be local We intend to restrict the complex ∗ by requiring that the functions αIA functions in the following sense.

Definition 8. A local function on J ∞ E is the pullback of a smooth function on some finite jet bundle J p E, i.e. a composite J ∞ E → J p E → R. In local coordinates, a local function L(x, u(p) ) is a smooth function in the coordinates xi and the coordinates uaI , where the order |I| = r of the multi-index I is less than or equal to some integer p. The space of local functions will be denoted Loc(E), while the subspace consisting ∞ ∗ ) f for f ∈ C ∞ M is denoted by LocM . of functions (πM Henceforth, the coefficients of all differential forms in the complex ∗ (J ∞ E, d) are required to be local functions, i.e., for each such form α there exists a positive integer p such that α is the pullback of a form of ∗ (J p E, d) under the canonical projection of J ∞ E onto J p E. In this context, the horizontal differential is obtained by noting that dα is in r+1,s ⊕ r,s+1 and then denoting the two pieces by, respectively, dH α and dV α. One can then write J J A θJA ∧ dxi ∧ dxI + αIA θJi ∧ dxi ∧ dxI }, dH α = (−1)s {Di αIA

where A = θJi

s X r=1

(15)

(θJa11 ∧ ...θJarr i ... ∧ θJass ),

and where Di =

∂ ∂ + uaiJ a ∂xi ∂uJ

(16)

is the total differential operator acting on local functions. We will work primarily with the dH subcomplex, the algebra of horizontal forms ∗,0 , which is the exterior algebra in the dxi with coefficients that are local functions. In this case we often use Olver’s notation D for the horizontal differential dH = dxi Di where Di is defined above. In addition to this notation, we also utilize the operation which is defined as follows. Given any differential r-form α on a manifold N and a vector field X on N , X α denotes that (r-1)-form whose value at any x ∈ N and (v1 , ..., vr−1 ) ∈ (Tx N × · · · × Tx N ) is αx (Xx , v1 , ..., vr−1 ). We will sometimes use the notation X(α) in place of X α. ∞ ∗ Let ν denote a fixed volume form on M and let ν also denote its pullback (πE ) (ν) to J ∞ E so that ν may be regarded either as a top form on M or as defining elements P ν of n,0 E for each P ∈ C ∞ (J ∞ E). We will almost invariably assume that ν = dn x = dx1 ∧ · · · ∧ dxn . It is useful to observe that for Ri ∈ Loc(E) and α = (−1)i−1 Ri (

∂ ∂xi

dn x),

(17)


595

then dH α = (−1)i−1 Dj Ri [dxj ∧ (

∂ ∂xi

dn x)] = Dj Rj dn x.

(18)

Hence, a total divergence Dj Rj may be represented (up to the insertion of a volume dn x) as the horizontal differential of an element of n−1,0 (J ∞ E). It is easy to see that the converse is true so, that, in local coordinates, one has that total divergences are in one-to-one correspondence with D-exact n-forms. Definition 9. A local functional Z Z L(x, φ(p) (x))dvolM = (j ∞ φ)∗ L(x, u(p) )dvolM L[φ] = M

(19)

M

is the integral over M of a local function evaluated for sections φ of E of compact support. The space of local functionals F is the vector space of equivalence classes of local functionals, where two local functionals are equivalent if they agree for all sections of compact support. If one does not want to restrict oneself to the case where the base space is a subset of Rn , one has to take the transformation properties of the integrands under coordinate transformations into account and one has to integrate a horizontal n-form rather than a multiple of dxn by an element of Loc(E). Lemma 10. The vector space of local functionals F is isomorphic to the cohomology group H n (∗,0 , D). Proof. Recall that one has a natural mapping ηˆ from n,0 (J ∞ E) onto F defined by Z (j ∞ φ)∗ (P )ν ∀φ ∈ 0E. (20) η(P ˆ ν)(φ) = M

It is well-known (see e.g. [18]) that η(P ˆ ν)[φ] = 0 for all φ of compact support if and only if in coordinates P may be represented as a divergence, i.e., iff P = Di Ri for ˆ ν) = 0 if and only if there exists a form some set of local functions {Ri }. Hence, η(P n−1,0 β∈ such that the horizontal differential dH maps β to P ν. So the kernel of ηˆ is precisely dH n−1,0 and ηˆ induces an isomorphism from H n (dH ) = n,0 /dH n−1,0 onto the space F of local functionals. For later use, we also note that the kernel of ηˆ coincides with the kernel of the Euler-Lagrange operator: for 1 ≤ a ≤ m, let Ea denote the ath component of the Euler-Lagrange operator defined for P ∈ Loc(E) by Ea (P ) =

∂P ∂P ∂P ∂P − ∂i a + ∂i ∂j a − ... = (−D)I ( a ). a ∂u ∂ui ∂uij ∂uI

(21)

The set of components {Ea (P )} are in fact the components of a covector density with respect to the generating set {θa } for C0 , the ideal generated by the contact one-forms of order zero. Consequently, the Euler operator E(P dn x) = Ea (P )(θa ∧ dn x),

(22)

596


for {θa } a basis of C0 , gives a well-defined element of n,1 . We have E(P ν) = 0 iff P ν = dH β. For convenience we will also extend the operator E to map local functions to n,1 , so that E(P ) is defined to be E(P dn x) for each P ∈ Loc(E). In Sect. 2, we have assumed that we have a resolution of F ' H n (∗,0 , dH ). In the case where M is contractible, such a resolution necessarily exists and is provided by the following (exact) extension of the horizontal complex ∗,0 (Loc(E), dH ): dH dH dH dH R −→ 0,0 −→ . . . −→ n−2,0 −→ n−1,0 −→ n,0 ↓η ↓η ↓η ↓η ↓η 0 −→ 0 −→ . . . −→ 0 −→ 0 −→ H0 = F Alternatively, we can achieve a resolution by taking out the constants: the space X∗ is given by Xi = n−i,0 , for 0 ≤ i < n, Xn = 0,0 /R, and Xi = 0, for i > n + 1. Either way, we have a resolution of F and can proceed with the construction of an sh Lie structure. (For general vector bundles E, the assumption that such a resolution exists imposes topological restrictions on E which can be shown to depend only on topological properties of M [1].) 3.2. Poisson brackets for local functionals. To begin to apply the results of Sect. 2, we must have a bilinear skew-symmetric mapping l˜2 from n,0 × n,0 to n,0 such that: (i) l˜2 (α, dH β) belongs to dH n−1,0 for all α ∈ n,0 and β ∈ n,0 , and P n−1,0 |σ| ˜ ˜ for all α1 , α2 , (ii) σ∈unsh(2,1) (−1) l2 (l2 (ασ(1) , ασ(2) ), ασ(3) ) belongs to dH n,0 α3 ∈ . To introduce a candidate l˜2 , we define additional concepts. We say that X is a generalized vector field over M iff X is a mapping which factors through the differential of the projection of J ∞ E to J r E for some non-negative integer r and which assigns to each ∞ (w). Similarly Y is a generalized vector field w ∈ J ∞ E a tangent vector to M at πM r over E iff Y also factors through J E for some r and assigns to each w ∈ J ∞ E a ∞ (w). In local coordinates one has vector tangent to E at πE X = X i ∂/∂xi ,

Y = Y j ∂/∂xj + Y a ∂/∂ua ,

(23)

where X i , Y j , Y a ∈ Loc(E). A generalized vector field Q on E is called an evolutionary vector field iff (dπ)(Qw ) = 0 for all w ∈ J ∞ E. In adapted coordinates, an evolutionary vector field assumes the form Q = Qa (w)∂/∂ua . Given a generalized vector field X on M , there exists a unique vector field denoted ∞ )(T ot(X)) = X and θ(T ot(X)) = 0 for every contact one-form θ. Tot(X) such that (dπM In the special case that X = X i ∂/∂xi , it is easy to show that T ot(X) = X i Di . We say that Z is a first order total differential operator iff there exists a generalized vector field X on M such that Z = T ot(X). More generally, a total differential operator Z is by definition the sum of a finite number of finite order operators Zα for which there exists functions ZαJ ∈ Loc(E) and first order total differential operators W1 , W2 , ...Wp on M such that Zα = ZαJ (Wj1 ◦ Wj2 ◦ ... ◦ W jp ),

(24)

where J = {j1 , j2 , ..., jp } is a fixed set of multi-indices depending on α (p = 0 is allowed).


597

In particular, in adapted coordinates, a total differential operator assumes the form Z = Z J DJ , where Z J ∈ Loc(E) for each multi-index J, and the sum over the multiindex J is restricted to a finite number of terms. In an analogous manner, for every evolutionary vector field Q on E, there exists its ∞ )(pr(Q)) = prolongation, the unique vector field denoted pr(Q) on J ∞ E such that (dπE Q and Lpr(Q) (C) ⊆ C, where Lpr(Q) denotes the Lie derivative operator with respect to the vector field pr(Q) and C is the ideal of contact forms on J ∞ E. In local adapted coordinates, the prolongation of an evolutionary vector field Q = Qa ∂/∂ua assumes the form pr(Q) = (DJ Qa )∂/∂uaJ . The set of all total differential operators will be denoted by T DO(E) and the set of all evolutionary vector fields by Ev(E). Both T DO(E) and Ev(E) are left Loc(E) modules. One may define a new total differential operator Z + called the adjoint of Z by Z Z (j ∞ φ)∗ [SZ(T )]ν = (j ∞ φ)∗ [Z + (S)T ]ν (25) M

M

for all sections φ ∈ 0E and all S, T ∈ Loc(E). It follows that [SZ(T )]ν = [Z + (S)T ]ν + dH ζ

(26)

for some ζ ∈ n−1,0 (E). If Z = Z J DJ in local coordinates, then Z + (S) = (−D)J (Z J S). This follows from an integration by parts in (26) and the fact that (26) must hold for all T (see e.g. [18] Corollary 5.52). Assume that ω is a mapping from C0 × C0 to T DO(E) which is a module homomorphism in each variable separately. The adjoint of ω denoted ω + is the mapping from C0 × C0 to T DO(E) defined by ω + (θ1 , θ2 ) = ω(θ2 , θ1 )+ . In particular ω is skew-adjoint iff ω + = −ω. Using the module basis {θa } for C0 , we define the total differential operators ω ab = a b ω(θ , θ ). From these operators, we construct a bracket on the set of local functionals [9–13] (see e.g. [18, 5] for reviews) by Z ω(θa , θb )(Eb (P ))Ea (Q)dn x, (27) {P, Q} = M

n

n

where P = P d x and Q = Qd x for local functions P and Q. As in other formulas of this type, it is understood that the local functional {P, Q} is to be evaluated at a section φ of the bundle E → M and that the integrand is pulled back to M via j ∞ φ before being integrated. We find it useful to introduce the condensed notation ω(E(P ))E(Q) = ω(θa , θb ) (Eb (P ))Ea (Q) throughout the remainder of the paper. In order to express ω(E(P ))E(Q) in a coordinate invariant notation, note that pr( ∂u∂ a ) E(L) = Ea (L)dn x for each local function L. Consequently, if ∗ is the operator from n,0 E to Loc(E) defined by ∗(P ν) = P , then ω(E(P ))E(Q) = ω(θa , θb )(∗[pr(

∂ ∂ ) E(P )])(∗[pr( a ) E(Q)]). b ∂u ∂u

If coordinates on M are chosen such that ν = dn x, then it follows that Z ω(θa , θb )(∗[pr(Yb ) E(P )])(∗[pr(Ya ) E(Q)])ν, {P, Q} = M

(28)

(29)

598


where {Ya } and {θb } are required to be local bases of Ev(E) and C0 , respectively, such that θb (Ya ) = δab . It is easy to show that the integral is independent of the choices of bases and consequently, one has a coordinate-invariant description of the bracket for local functionals. 3.3. Associated sh Lie algebra on the horizontal complex. This bracket for functionals provides us with some insight as to how l˜2 may be defined; namely for α1 = P1 ν and α2 = P2 ν ∈ n,0 , define l˜2 (α1 , α2 ) to be the skew-symmetrization of the integrand of {P1 , P2 } : 1 l˜2 (α1 , α2 ) = [ω(E(P1 ))E(P2 ) − ω(E(P2 ))E(P1 )]ν. 2

(30)

By construction, l˜2 is skew-symmetric and, from E(dH β) = 0 for β ∈ n−1,0 , it follows that l˜2 (α, dH β) = 0. Thus a strong form of property (i) required above for l˜2 holds. The symmetry properties of ω may be used to simplify the equation for l˜2 (α1 , α2 ). Skew-adjointness of ω implies ω(E(P1 ))E(P2 )ν = −ω(E(P2 ))E(P1 )ν + dH γ

(31)

n−1,0

, which depends on α1 = P1 ν and α2 = P2 ν. In fact, since for some γ ∈ E(dH β) = 0, the element γ depends only on the cohomology classes P1 , P2 of α1 and α2 . A specific formula for γ can be given by straightforward integrations by parts. Hence, from (30) and (31), we get 1 l˜2 (α1 , α2 ) = ω(E(P1 ))E(P2 )ν − dH γ(P1 , P2 ). 2 R ∞ ∗ Furthermore, since M (j φ) dH γ = 0 for all φ ∈ 0E, we see that Z Z ∞ ∗ {P1 , P2 }(φ) = (j φ) [ω(E(P1 ))E(P2 )]ν = (j ∞ φ)∗ l˜2 (α1 , α2 ). M

(32)

(33)

M

In order to explain the conditions necessary for l˜2 to satisfy the required “Jacobi” condition, we formulate the problem in terms of “Hamiltonian” vector fields (see e.g. [18] Chapter 7.1 or [5] Chapter 2.5) and their corresponding Lie brackets. Given a local function Q, one defines an evolutionary vector field vωEQ by vωEQ = ω ab (Eb (Q))∂/∂ua = ω(θa , θb )(∗[pr(∂/∂ub ) E(Q)])∂/∂ua .

(34)

Again, the vector field vωEQ depends only on the functional Q and not on which representative Q one chooses in the cohomology class Q ∼ Qν + dH n−1,0 . Thus, for a given functional Q, letR vˆ Q = vωEQ for any representative Q. Since {P1 , P2 } = M l˜2 (α1 , α2 ), we see that vˆ {P1 ,P2 } = vωE(l˜2 (α1 ,α2 )) .

(35)

ω(E(P1 ))E(P2 ) = ω ab (Eb (P1 ))Ea (P2 ) = pr[ω ab Eb (P1 )∂/∂ua ] E(P2 ) = pr(vωE(P1 ) ) E(P2 ) = pr(vˆ P1 ) E(P2 ).

(36)

Note also that

Moreover, integration by parts allows us to show that


pr(Q)(P )ν = pr(Q) d(P ν) = pr(Q) E(P ν) + dH (pr(Q) σ),

599

(37)

for arbitrary evolutionary vector fields Q and local functions P , and for some form σ ∈ n−1,0 depending on P . For every such Q, let IQ denote a mapping from n,0 to n−1,0 such that pr(Q)(P )ν = pr(Q) E(P ) + dH (IQ (P ν))

(38)

for all P ν ∈ n,0 . Explicit coordinate expressions for IQ can be found in [18] chapter 5.4 or in [5] chapter 17.5 . It follows from the identities (32), (36) and (38) that 1 l˜2 (α1 , α2 ) = pr(vˆ P1 )(P2 )ν − dH Ivˆ P1 (P2 ν) − dH γ(P1 , P2 )). 2

(39)

Thus, for α1 , α2 , α3 ∈ n,0 , we see that l˜2 (l˜2 (α1 , α2 ), α3 ) = −l˜2 (α3 , l˜2 (α1 , α2 )) = = −l˜2 (α3 , pr(vˆ P1 )(P2 )ν − dH (·)) = −l˜2 (α3 , pr(vˆ P1 )(P2 )ν) (40) and l˜2 (l˜2 (α1 , α2 ), α3 ) = −pr(vˆ P3 )(pr(vˆ P1 )(P2 ))ν + dH ζ,

(41)

where ζ is given by 1 ζ(P1 , P2 , P3 ) = Ivˆ P3 (pr(vˆ P1 )(P2 )ν) + γ(P3 , {P1 , P2 }). 2

(42)

Rewriting the left hand side of the Jacobi identity in Leibnitz form and using (35), (39) and (41), we find X (−1)|σ| l˜2 (l˜2 (ασ(1) , ασ(2) ), ασ(3) ) = σ∈unsh(2,1)

= −l˜2 (α3 , l˜2 (α1 , α2 )) − l˜2 (l˜2 (α1 , α3 ), α2 ) + l˜2 (α1 , l˜2 (α3 , α2 )) = = [−pr(vˆ P3 )(pr(vˆ P1 )(P2 )) + pr(vˆ P1 )(pr(vˆ P3 )(P1 )) −pr(vˆ {P1 ,P3 } )(P2 )]ν + dH η,

(43)

with η(P1 , P2 , P3 ) = ζ(P1 , P2 , P3 ) − ζ(P3 , P2 , P1 ) 1 +Ivˆ {P1 ,P3 } (P2 ν) + dH γ({P1 , P3 }, P2 ). 2

(44)

Although ζ depends on the representative P2 and not its cohomology class, η depends only on the cohomology classes Pi because it is completely skew-symmetric. It follows from this identity that if pr(vˆ {P1 ,P2 } ) = [pr(vˆ P1 ), pr(vˆ P2 )]

(45)

for all P1 , P2 , then {·, ·} satisfies the Jacobi condition. Under these conditions, the mapping H : F −→ Ev(E) defined by H(P) = vˆ P is said to be Hamiltonian. Equivalent conditions on the mapping H alone for the bracket {·, ·} to be a Lie bracket can be found

600


in [18, 5]. The derivation given here allows us to give, in local coordinates, an explicit form for the exact term (44) violating the Jacobi identity. If H is Hamiltonian, the bracket l˜2 satisfies condition (ii) and the construction of Sect. 2 applies. Because the resolution stops with the horizontal zero-forms, we get a possibly non-trivial L(n + 2) algebra on the horizontal complex. If we remove the constants, we can then extend to a full L∞ −algebra by defining the further li to be 0. In addition, because property (i) holds without any l1 exact term on the right hand side, remark (ii) at the end of Sect. 2.1 applies, i.e., we need only two terms in the defining equations of the sh Lie algebra and the maps lk induce well-defined higher order maps on the space of local functionals. On the other hand, if we do not remove the constants, the operation ln+2 takes values in Xn = 0,0 = Loc(E) and induces a multi-bracket on H n (∗,0 , dH ) ' F , the space of local functionals, with values in Hn (X∗ , l1 ) = H 0 (∗,0 , dH ) ' HDR (C ∞ (M )) = R. We have thus proved the following main theorem. Theorem 11. Suppose that the horizontal complex without the constants (∗,0 /R, dH ) is a resolution of the space of local functionals F equipped with a Poisson bracket as above. If the mapping H from F to evolutionary vector fields is Hamiltonian, then to the Lie algebra F equipped with the induced bracket lˆ2 = {·, ·}, there correspond maps li : (∗,0 /R)⊗i → ∗−i+2,0 /R for 1 ≤ i ≤ n + 2 satisfying the sh Lie identities l1 lk + lk−1 l2 = 0. The corresponding map lˆn+2 on F ⊗n+2 with values in H 0 (∗,0 , dH ) = R satisfies lˆn+2 lˆ2 = 0. Specific examples for n = 1 are worked out in careful detail by Dickey [6].

4. Conclusion The approach of Gel’fand, Dickey and Dorfman to functionals and Poisson brackets in field theory has the advantage of being completely algebraic. In this paper, we have kept explicitly the boundary terms violating the Jacobi identity for the bracket of functions, instead of throwing them away by going over to functionals at the end of the computation. In this way, we can work consistently with functions alone, at the price of deforming the Lie algebra into an sh Lie algebra. Our hope is that this approach will be useful for a completely algebraic study of deformations of Poisson brackets in field theory. Note: After this paper had been submitted for publication, it was pointed out to us by M. Markl, that if condition (i), (Eq.(3)), holds with zero on the right hand side, the maps lk with k > 3 can be chosen to vanish. This means in the case of the Gelfand-DickeyDorfman bracket that only the homotopy for the Jacobi identity l3 is in fact non-trivial. It will be interesting to explore its physical significance. Acknowledgement. The authors want to thank I. Anderson, L.A. Dickey and M. Henneaux for useful discussions.


601

References 1. Anderson, I.: The Variational Bicomplex. Preprint, Utah State University, 1996 2. Batalin, I.A. and Vilkovisky, G.S.: Relativistic s-matrix of dynamical systems with boson and fermion constraints. Phys. Lett. 309–312 (1977) 3. Batalin, I.A. and Vilkovisky, G.S.: Gauge algebra and quantization. Phys. Lett. 102 B, 27–31 (1981) 4. Batalin, I.A. and Vilkovisky, G.S.: Quantization of gauge theories with linearly dependent generators. Phys. Rev. D 28, 2567–2582 (1983); Erratum: Phys. Rev. D30, 508 (1984) 5. Dickey, L.A.: Soliton Equations and Hamiltonian Systems. Advanced Series in Mathematical Physics, vol. 12, Singapore: World Scientific, 1991 6. Dickey, L.A.: Poisson brackets with divergence terms in field theories: two examples. Preprint, University of Oklahoma, 1997 7. Fradkin, E.S. and Fradkina, T.E.: Quantization of relativistic systems with boson and fermion first and second class constraints. Phys. Lett. 72B, 343–348 (1978) 8. Fradkin, E.S. and Vilkovisky, G.S.: Quantization of relativistic systems with constraints. Phys. Lett. 55B, 224–226 (1975) 9. Gel’fand, I.M. and Dickey, L.A.: Lie algebra structure in the formal variational calculus. Funkz. Anal. Priloz. 10 no 1, 18–25 (1976) 10. Gel’fand, I.M. and Dickey, L.A.: Fractional powers of operators and hamiltonian systems. Funkz. Anal. Priloz. 10 no 4, 13–29 (1976) 11. Gel’fand, I.M. and Dorfman, I.Ya.: Hamiltonain operators and associated algebraic structures. Funkz. Anal. Priloz. 13 no 3, 13–30 (1979) 12. Gel’fand, I.M. and Dorfman, I.Ya.: Schouten bracket and Hamiltonian operators. Funkz. Anal. Priloz. 14 no 3, 71–74 (1980) 13. Gel’fand, I.M. and Dorfman, I.Ya.: Hamiltonian operators and infinite dimensional Lie algebras. Funkz. Anal. Priloz. 15 no 3, 23–40 (1981) 14. Henneaux, M. and Teitelboim, C.: Quantization of Gauge Systems. Princeton, W: Princeton Univ. Press, 1992 15. Kjeseth, L. BRST cohomology and homotopy Lie-Rinehart pairs. Dissertation, UNC-CH, 1996 16. Lada, T. and Markl, M.: Strongly homotopy Lie algebras. Comm. in Algebra 2147–2161 (1995) 17. Lada, T. and Stasheff, J.D.: Introduction to sh Lie algebras for physicists. Intern’l J. Theor. Phys. 32, 1087–1103 (1993) 18. Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107, Berlin–Heidelberg–New York: Springer-Verlag, 1986 19. Saunders, D.J.: The Geometry of Jet Bundles. London Mathematical Lecture Notes, vol. 142, Cambridge: Cambridge Univ. Press, 1989 20. Stasheff, J.D.: Homological reduction of constrained Poisson algebras. J. Diff. Geom. 45, 221–240 (1997) Communicated by T. Miwa

Commun. Math. Phys. 191, 603 – 611 (1998)

Communications in


Differentiation of Operator Functions and Perturbation Bounds Rajendra Bhatia, Dinesh Singh, Kalyan B. Sinha Indian Statistical Institute, New Delhi-110016, India. E-mail: [email protected] Received: 4 April 1997 / Accepted: 28 May 1997

Abstract: Given a smooth real function f on the positive half line consider the induced map A → f (A) on the set of positive Hilbert space operators. Let f (k) be the k th derivative map f . We of the real function f and Dk f the k th Fréchet

of the operator

derivative identify large classes of functions for which Dk f (A) = f (k) (A) , for k = 1, 2, .... This reduction of a noncommutative problem to a commutative one makes it easy to obtain perturbation bounds for several operator maps. Our techniques serve to illustrate the use of a formalism for “quantum analysis” that is like the one recently developed by M. Suzuki.

1. Introduction In several problems of quantum and statistical physics, one has to study various functions on the space of operators in a Hilbert space. Calculus of such functions, therefore, is of great interest. In a recent paper [12] Suzuki has developed a formalism for such a calculus and called it “quantum analysis”. Several references where such analysis is useful are given in this paper. Motivated by our interest in perturbation inequalities for operator functions, we have used a somewhat different, but essentially equivalent, formalism in some of our work. A summary of this approach is given in [2] ; see especially Sect. X.4. One attractive feature of this approach is the ease it affords in calculating norms of derivatives. In [4, 5, 6], it was observed that the norms of certain noncommutative derivatives are, rather surprisingly, equal to those of their commutative analogues. In the present paper this approach is taken further. We will evaluate precisely norms of higher order derivatives of some operator functions. At the same time this will illustrate the simplicity and the power of our methods which should be useful in other problems. Let B(H) denote the space of all bounded linear operators on a Hilbert space H. Let Bs (H) be the set of all self-adjoint operators and B+ (H) the set of all positive definite

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R. Bhatia, D. Singh, K. B. Sinha

operators. Let f be a real valued measurable function on (0, ∞). This induces (via the spectral theorem) a map from B+ (H) to Bs (H), which we denote again by f. One of the central problems in perturbation theory is to find bounds for kf (A) − f (B)k in terms of kA − Bk. All functions considered here are smooth and have derivatives of all orders. Note that Bs (H) is a real linear space and B+ (H) is an open subset of it. For k = 1, 2, ..., we denote by D k f (A) the k th order Fréchet derivative of f at A. This is a symmetric multilinear map from the k-fold product Bs (H) × · · · × Bs (H) into Bs (H). Its action can be described as [Dk f (A)](B1 , ..., Bk ) =

∂k f (A + t1 B1 + · · · + tk Bk ) |t1 =···=tk =0 . ∂t1 · · · ∂tk

(1.1)

For basic facts about the Fréchet differential calculus, the reader may see [1, 2, 8 and 10]. The norm of Dk f (A) is defined as

k

k

D f (A) =

[D f (A)](B1 , ..., Bk ) . sup (1.2) kB1 k=···=kBk k=1

The Taylor theorem says that for all B sufficiently close to A, we have f (B) = f (A) + [Df (A)](B − A) + · · · +

1 [Dk f (A)](B − A, ..., B − A) + · · ·. (1.3) k!

From this we have kf (B) − f (A)k ≤

N X

1

Dk f (A) kB − Akk + O kB − AkN +1 . k!

(1.4)

k=1

We call this an N th order perturbation bound for f . First order perturbation bounds for several matrix and operator functions(not restricted to the class we have delimited above) have been obtained by many authors. See, e.g. [11] for several references on such works for the exponential function and [3] for works on various matrix factorisations. Higher order bounds are rarer to find. One reason for this is the relatively more complex nature of the expression (1.1) for the higher Fréchet derivatives. In this paper, we will obtain such bounds for two large families of functions of positive operators. These include the exponential function and the power functions Ap , −∞ < p < ∞ . Let f (k) be the (ordinary) derivative of the real valued function f . Then for each A ∈ B+ (H),

(k) k

f (A) = [D f (A)](I, ..., I) . Consequently,

k

D f (A) ≥ f (k) (A) , for all A.

For k = 1, 2, ..., let

Dk = f : Dk f (A) = f (k) (A) for all A ∈ B+ (H) .

(1.5)

The classes Dk turn out to be nonempty, and have several unexpected properties, some of which have been studied in [4, 5, and 6]. In [4] it was shown that all operator monotone functions on (0, ∞) are in D1 and D2 . It is well known that the power function f (t) = tp is operator monotone if and only if 0 ≤ p ≤ 1. In [5] it was shown that every operator

Differentiation of Operator Functions and Perturbation Bounds

605

monotone function is in Dk for all k = 1, 2, ... . The major application of these results in 1 the two papers was to find nth order perturbation bounds for the function |A| = (A∗ A) 2 . This function has been of great interest to physicists as well as numerical analysts ; see [5] for references. It was also shown in [4] that the functions f (t) = tn , n = 2, 3, ... , and f (t) = exp t are also in D1 . None of these are operator monotone. In [6] it was shown p that the √ power function f (t) = t is in D1 if p is in (−∞, 1] or in [2, ∞), but not if p is in (1, 2). It will follow from results proved below that the functions f (t) = exp t and f (t) = tp , −∞ < p ≤ 1, are in the class Dk for all k = 1, 2, ..., and that for p > 1 the function f (t) = tp is in the class Dk for all k ≥ [p + 1] . In the earlier papers mentioned above we relied heavily on integral representations for the functions under consideration. The approach here is somewhat different: we use more of power series. A recent paper [7] follows the ideas in [6] to go up to second order derivatives. Much of this is subsumed here in our analysis. Of course once a function f is shown to be in the class Dk , the problem of finding the norm of Dk f (A) is reduced to that of finding the norm of f (k) (A). A noncommutative problem is thus reduced to its commutative version.

2. The Main Results We want to study the derivatives Dk f (A) for functions f represented by power series. First consider the function f (A) = An , A ∈ B(H) , where n is any natural number. For 1 ≤ k ≤ n , the derivative Dk f (A)(B1 , ..., Bk ) is given by an expression that is linear and symmetric in B1 , ..., Bk , and when dim H = 1 it reduces to the expression f (k) (x) = n(n − 1) · · · (n − k + 1)xn−k =

n! xn−k , (n − k)!

(2.1)

for the k th derivative of the function f (x) = xn . These three requirements dictate that X X Aj1 Bσ(1) · · · Ajk Bσ(k) Ajk+1 , (2.2) Dk f (A)(B1 , ....Bk ) = σ∈Sk

ji ≥0 j1 +···+jk+1 =n−k

n! terms where Sk is the set of permutations on {1, 2, ..., k}. Note that this is a sum of (n−k)! , each of which is a word of length n in which n − k letters are A and the remaining k letters are B1 , ..., Bk , each occurring exactly once. When k = 2, for instance, this reduces to X Aj1 B1 Aj2 B2 Aj3 + Aj1 B2 Aj2 B1 Aj3 . D2 f (A)(B1 , B2 ) = j1 +j2 +j3 =n−2

The reader may find it instructive to compare our formula (2.2) with (4.25b) in [12], and see how one can be derived from the other. Both are noncommutative analogues of (2.1) but the noncommutativity is expressed in different ways by them. Now if f is an analytic function with power series f (z) =

∞ X n=0

an (z − α)n ,

606


then we have Dk f (A)(B1 , ..., Bk ) =

X

an Dk (A − α)n (B1 , ..., Bk ),

(2.3)

where Dk (A − α)n is the k th derivative of (A − α)n given by the expression (2.2) with (A − α) in place of A. The only point that needs a little elaboration here is whether the term by term differentiation is justified. An easy way to see this is via the Cauchy integral formula. Let f (z) be analytic in the open right half plane C + and let A be a positive operator whose spectrum is contained in the positive real line . Let γ be a simple closed curve in C + such that the spectrum of A is enclosed by γ in its interior. In that case, by the Riesz functional calculus Z f (z) 1 dz, f (A) = 2πi γ (z − A) so that using (1.1) we see Df (A)(B) =

1 2πi

Z f (z) γ

1 1 B dz, (z − A) (z − A)

and in general Dk f (A)(B1 , ..., Bk ) Z X 1 1 1 1 1 f (z) [ = Bσ(1) ··· Bσ(k) ]dz. (z − A) (z − A) (z − A) (z − A) 2πi γ σ∈Sk

Using this formula, it is easy to justify the term by term differentiation in (2.3). The formula (2.3) should be compared with TheoremVII in [12]. We are now in a position to state and prove our main results. P Theorem 2.1. Let f have a power series representation, f (t) = an tn with an ≥ 0 for all n. Then ∞ \ Dk . f∈ k=1

Proof. Let A ≥ 0. By what has been said above Dk f (A)(B1 , ..., Bk ) =

∞ X

an (Dk An )(B1 , ..., Bk ),

n=k

where the summands (Dk An )(B1 , ..., Bk ) are given by (2.2). So ∞

k

X

D f (A) ≤ an n=k

From (2.1) above f (k) (A) =

∞ X n=k

Since A ≥ 0 and an ≥ 0,

an

n! n−k kAk . (n − k)!

n! An−k . (n − k)!


(k)

f (A) =

607

∞

X

n!

n−k A an

(n − k)! n=k

=

∞ X

an

n=k

Thus

n! n−k kAk . (n − k)!

k

D f (A) = f (k) (A) for all k.

(2.4)

Examples. Given below are some functions that satisfy the conditions of Theorem 2.1. Let f (t) be any polynomial all of whose coefficients are non-negative. Let f (t) = exp t. Let f (t) be sinh(t) or cosh(t). The argument presented in the proof of Theorem 2.1 works equally well if the power series representing the function has a finite circle of convergence. Of course one then considers only those operators A which have spectra inside the circle of convergence of the given power series. Thus f (t) = arcsin t, |t| ≤ 1 is in Dk for all k. (v) Hansen [9] has shown that functions that satisfy the hypothesis of Theorem 2.1 are exactly the ones that are monotone with respect to the entry-wise order relation on the space of Hermitian matrices with real entries. This makes Theorem 2.1 more interesting, since we already know [5] that operator monotone functions are in the class ∩∞ k=1 Dk . We are thankful to the referee for pointing this out. P Remark 2.2. If the function f (t) = an tn is such that an ≥ 0 for n ≥ k then the same proof shows that f is in the class Dn for all n ≥ k.

(i) (ii) (iii) (iv)

Our next theorem concerns functions whose Taylor series have real coefficients that are alternately positive and negative. Theorem 2.3. Let f be a map of the positive half line into R such that f has an analytic extension into the right half plane. Further, suppose that the Taylor series of f around each positive α is given by f (z) =

∞ X

an (z − α)n ,

(|z − α| < α)

n=0

with an an+1 < 0 for all n. Then f ∈ Dk for all k ≥ 1. Proof. Let A be a positive operator with inf σ(A) = a > 0 and sup σ(A) = b. Choose a positive real number α such that α > b and assume that f has the Taylor expansion around α as given by the statement of the theorem.We also assume, without loss of generality, that an > 0 for even n and an < 0 for odd n. Let an = b2m when n is even and an = −b2m+1 when n is odd, and where bm is positive for all m. Then f (z) =

∞ X n=0

so that,

b2n (z − α)

2n

−

∞ X n=0

b2n+1 (z − α)2n+1 ,

608


f (A) =

∞ X

b2n (A − α)2n −

n=0

∞ X

b2n+1 (A − α)2n+1 .

n=0

Then, as in the proof of Theorem 2.1, we have

k

D f (A)(B1 , ..., Bk ) ≤

X 2n≥k

X (2n + 1)! (2n)! 2n−k 2n+1−k b2n kA − αk b2n+1 kA − αk + . (2n − k)! (2n + 1 − k)! 2n≥k

Now suppose that k is even and k = 2m (the argument for odd k is similar). Then the right hand side of the above inequality is ∞ X

∞ X (2n)! (2n + 1)! 2n−k b2n (a − α) b2n+1 (α − a)2n+1−k + = (2n − k)! (2n + 1 − k)! n=m n=m ∞ X (2n)! (2n + 1)! b2n (a − α)2n−k − b2n+1 (a − α)2n+1−k . (2n − k)! (2n + 1 − k)! n=m n=m

(k)

Since a − α < 0 , this is also equal to f (A) . ( See the definition of a and α at the beginning of the proof and use the spectral resolution of A.) This shows that

k

D f (A) ≤ f (k) (A) .

=

Thus

∞ X

k

D f (A) = f (k) (A) .

Corollary 2.4. Let f be a completely monotone function on (0, ∞) ; i.e., let Z ∞ f (t) = e−λt dµ (λ) , 0

where µ is a positive measure on (0, ∞). Then f ∈ Dk for all k ≥ 1. [For k = 1 this is Theorem 2.1 of [6] and for k = 2 this is Theorem 1 of [7]. Proof. From the nature of the function f , it is easy to check that in the Taylor series expansion about any positive number α, the Taylor coefficients satisfy an an+1 < 0 for all n. Hence f is in Dk for all k. Remark 2.5. It is known that any function that maps the positive real line into itself and satisfies f (n) (x)f (n+1) (x) < 0 for all n and all x is completely monotone. See Widder [13, p. 145, 161]. Remark 2.6. It must be mentioned that Theorem 2.3 is valid in a much more general situation than the one described above. In effect the same proof works to give the following: Let f map the positive real line into the set of real numbers and let it be analytic in C + . Further suppose that there exists a sequence of positive numbers {αn } such that αn → ∞ and such that for each n and for each positive integer k, f (k) (αn )f (k+1) (αn ) < 0. Then f is in Dk for all k. An example of a function that satisfies these criteria but does not satisfy the conditions of Theorem 2.3 is given by f (t) = e−t sin t.


609

Remark 2.7. We also note that the same proof leads to the following: Let f map the positive half line into R and suppose that there exists a positive integer k such that f (n) (x)f (n+1) (x) < 0 for all n ≥ k and for all x ≥ δ > 0 for some δ. Then f is in Dn for all n ≥ k. Corollary 2.8. Let f and g be two functions whose Taylor expansions as given by Theorem 2.3 satisfy an an+1 < 0, for all n. Then f + g and f g are also in Dk for all k. Proof. This is a simple consequence of the fact that the Taylor series of the sum and the product is also of the type whose coefficients satisfy an an+1 < 0 for all n. Corollary 2.9. Let f (t) = tp , p ≤ 1. Then f ∈ Dk for all k. Proof. The proof is obvious when we look at the Taylor coefficients of tp .

Corollary 2.10. Let f (t) = tp , 1 < p. Then f ∈ Dk for all k ≥ [p + 1]. Proof. The Taylor coefficients of tp , for the above values of p, satisfy an an+1 < 0 for all n ≥ [p + 1]. So the assertion follows from Remark 2.7. Examples. Let f (t) = e−t . It is easy to see ( by looking at the Taylor series) that f is completely monotone. (ii) Let f (z) be analytic in the entire complex plane except in a disk in the open left half plane centred at a point a of the real line. Assume that its Laurent series representation is given by

(i)

f (z) =

∞ X n=1

αn , (z − a)n

where each αn ≥ 0. Then f (t) is in the class Dn for all n ≥ 0. This is again a consequence of the nature of the Taylor series of f (z) around any point α of the positive real line. It can be easily seen that the Taylor coefficients satisfy the condition of complete monotonicity around any point of the positive half line. 1

(iii) Let f (z) = e z . Though this function has a singularity at zero and not on the negative half line, the logic of the previous example applies to show that the Taylor coefficients of the function (around any point on the positive half line ) satisfy the condition of Theorem 2.3 and so it is in Dk for all k. (iv) It is known that the function f (t) = t + 1/t is not in D, see [6] . However it now follows from Remark 2.7 by looking at the Taylor series of f about any positive α, that the function is in Dk for all k ≥ 2. (v) As a final illustration we consider the function Z ∞ 1 1 − e−λ dλ. e−λt log(1 + ) = t λ 0 This function is in Dk , for all k, being completely monotone. It is interesting to note this because of its similarity to the function log(1 + t). This latter function is in Dk for all k since it is operator monotone.

610


3. Perturbation Bounds We have already explained in Sect. 1 that once we know how to evaluate, or estimate,

the norms Dk f (A) , we can use the inequality (1.4) to find perturbation bounds for f in a neighbourhood of A. Thus, for all f in ∩∞ k=1 Dk we can find such bounds up to any desired order. In Sect. 2 we have also shown that certain functions are in the classes Dk for all k larger than some integer m. For such functions too we can obtain perturbation bounds to any desired order. This is explained below. Suppose f ∈ Dk for k ≥ m. Let p be the polynomial of degree m − 1 obtained by keeping the first m terms in the Taylor expansion of f . Let g = f − p. Then, it is easy to see that g is in Dk for all k ≥ 1. If B is close to A, we can write kf (B) − f (A)k ≤ kp(B) − p(A)k + kg(B) − g(A)k . The quantity kg(B) − g(A)k can be estimated by the method explained above. The quantity kp(B) − p(A)k is easy to estimate. Note that kAr − B r k =

r−1 X

B j (A − B)Ar−j−1 .

j=0

From this it follows that kAr − B r k ≤ rM r−1 kA − Bk , where M = max(kAk , kBk). Since p(B) − p(A) is a finite linear combination of such terms, its norm can be easily estimated. Acknowledgement. The second author is on leave from the University of Delhi and is grateful to the Indian Statistical Institute, New Delhi for a visiting appointment and to the University Grants Commission for a Career Award. K.B.S. acknowledges the support of Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore, India.

References 1. Ambrosetti, A. and Prodi, G.: A Primer of Non-Linear Analysis. Cambridge: Cambridge Univ. Press, 1993 2. Bhatia, R.: Matrix Analysis. New York: Springer-Verlag, 1997 3. Bhatia, R.: Matrix factorisations and their perturbations. Linear Algebra Appl. 197/198, 245–276 (1994) 4. Bhatia, R.: First and second order perturbation bounds for the operator absolute value. Linear Algebra Appl. 208, 367–376 (1994) 5. Bhatia, R.: Perturbation bounds for the operator absolute value. Linear Algebra Appl. 226, 539–545 (1995) 6. Bhatia, R. and Sinha, K.B.: Variation of real powers of positive operators. Indiana Univ. Math. J. 43, 913–925 (1994) 7. Bist, V. and Vasudeva, H.L.: Second order perturbation bounds. Publ. RIMS Kyoto Univ (To appear) 8. Dieudonné, J.: Foundations of Modern Analysis. London–New York: Academic Press, 1969 9. Hansen, : Functions of matrices with nonnegative entries. Linear Algebra Appl. 166, 29–43 (1992) 10. Hille, E. and Phillips, R.S.: Functional Analysis and Semigroups. American Mathematical Society Colloquium Publ., 1975


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11. Najfeld, I. and Havel, T.F.: Derivatives of the matrix exponential and their computation. Advances Appl. Math. 16, 321–375 (1995) 12. Suzuki, M.: Quantum analysis – non-commutative differential and integral calculi. Commun. Math. Phys. 183, 339–363 (1997) 13. Widder, D.V.: The Laplace Transform. Princeton: Princeton Univ. Press, 1946 Communicated by H. Araki

Commun. Math. Phys. 191, 613 – 626 (1998)

Communications in


On Zero-Mass Ground States in Super-Membrane Matrix Models ¨ Fröhlich, Jens Hoppe? Jurg Theoretical Physics, ETH-Hönggerberg, CH–8093 Zürich, Switzerland Received: 2 April 1997 / Accepted: 28 May 1997

Abstract: We recall a formulation of super-membrane theory in terms of certain matrix models. These models are known to have a mass spectrum given by the positive halfaxis. We show that, for the simplest such matrix model, a normalizable zero-mass ground state does not exist. 1. Introduction and Summary of Results Some time ago [1], super-membranes in D space-time dimensions were related to supersymmetric matrix models where, in a Hamiltonian light-cone formulation, the D − 2 transverse space coordinates appear as non-commuting matrices [2]. It has been proven in [3] that the mass spectrum of any one of these matrix models, which is given by the (energy) spectrum of some supersymmetric quantum-mechanical Hamilton operator [4], fills the positive half-axis of the real line. This property of the mass spectrum in super-membrane models is in contrast to the properties of mass spectra in bosonic membrane matrix models [2] which are purely discrete, see [5]. One of the important open questions concerning super-membrane matrix models is whether they have a normalizable zero-mass ground state. Such states would describe multiplets of zero-mass one-particle states, including the graviton, (see [1]). A new interpretation of the mass spectrum of super-membrane matrix models (in terms of multi-membrane configurations) has been proposed in [6]. A first step towards answering the question of whether there are normalizable zeromass ground states in super-membrane matrix models has been undertaken in [1]. In this note, we continue the line of thought described in [7] and show that, in the simplest matrix model, a normalizable zero-mass ground state does not exist. Let us recall the definition of super-membrane matrix models. The configuration space of the bosonic degrees of freedom in such models consists of D − 2 copies of ?

Heisenberg Fellow. On leave of absence from Karlsruhe University

614

J. Fröhlich, J. Hoppe

the Lie-algebra of SU (N ), for some N < ∞, where D is the dimension of space-time, with D = 4 (5,7) 11. A point in this configuration space is denoted by X = (Xj ) with Xj =

2 N −1 X

XjA TA , j = 1, · · · , D − 2,

(1.1)

A=1

where {TA } is a basis of su(N ), the Lie algebra of SU (N ). In order to describe the quantum-mechanical dynamics of these degrees of freedom, we make use of the Heisenberg algebra generated by the configuration space coordinates XjA and the canonically conjugate momenta PjA satisfying canonical commutation relations A (1.2) Xj , XkB = PjA , PkB = 0, A B AB δjk . Xj , Pk = i δ To describe the quantum dynamics of the fermionic degrees of freedom, wemake D−2

use of the Clifford algebra with generators ΘαA , A = 1, · · · , N 2 − 1, α = 1, · · · , 2 2 (times 2, if D = 5 or 7), and commutation relations A (1.3) Θα , ΘβB = δαβ δ AB . The generators ΘαA can be expressed in terms of fermionic creation- and annihilation operators: A bA α + cα √ , 2 A i bA α − cα A √ Θ2α = , 2 D−2 ∗ 1 A 2 = c , α = 1, · · · , 2 with bA , and α α 2 A B A B bα , bβ = cα , cβ = 0, A B bα , cβ = δαβ δ AB . A = Θ2α−1

(1.4)

(1.5)

The Hilbert space, H, of state vectors (in the Schrödinger representation) is a direct sum of subspaces Hk , k = 0, · · · , K := (N 2 − 1) 9 =

1 2

2

D−2 2

. A vector 9 ∈ Hk is given by

K X 1 A1 α1 ···αk k b · · · bA αk ψA1 ···Ak (X), k! α1

(1.6)

k=0

where X = XjA , j = 1, · · · , D − 2, A = 1, · · · , N 2 − 1. We require that cA α 9 = 0,

for all

9 ∈ H0 .

(1.7)

The scalar product of two vectors, 9 and 8, in H is given by h9, 8i =

K X 1 k! k=0

X Z Y α1 ···αk A1 ···Ak

j,A

α1 ···αk 1 ···αk dXjA ψA (X) × φα A1 ···Ak (X). 1 ···Ak

(1.8)

On Zero-Mass Ground States in Super-Membrane Matrix Models

615

The Hilbert space H carries unitary representations of the groups SU (N ) and SO(D−2). Let H(0) denote the subspace of H carrying the trivial representation of SU (N ). D−2

One can define supercharges, Qα and Q†α , α = 1, · · · , 21 2 2 , with the properties that, on the subspace H(0) , o n † † {Qα , Qβ } = Qα , Qβ =0, (1.9) H(0)

and

n

Qα , Q†β

o

H(0)

H(0)

= δαβ H

,

(1.10)

H(0)

where H = M 2 , and M is the mass operator of the super-membrane matrix model. Precise definitions of the supercharges and of the operator H can be found in [1] (formulas (4.7) through (4.12)). In [3] it is shown that the spectrum of H |H(0) consists of the positive half-axis [0, ∞). The problem addressed in this note is to determine whether O is an eigenvalue of H corresponding to a normalizable eigenvector 90 ∈ H(0) . Using Eqs. (1.9) and (1.10), one can show that 90 must be a solution of the equations Qα 9 = Q†α 9 = 0,

for some α, 9 ∈ H(0) .

(1.11)

If Eqs. (1.11) have a solution, 90 = 9α0 , for α = α0 , they have a solution for all values of α, (by SO(D − 2) covariance). The problem to determine whether Eqs. (1.11) have a solution, or not, can be understood as a problem about the cohomology groups determined by the supercharges Qα . We define (0) , H+ := ⊕ H2l l≥0

(0) H− := ⊕ H2l+1 . l≥0

We define the cohomology groups o n (0) , 9 | 9 = Qα 8, 8 ∈ H−σ Hσ,α := 9 ∈ Hσ(0) | Qα 9 = 0 σ = ±1. If Hσ,α is non-trivial, for some σ and some α, then Eqs. (1.11) have a solution. 2. The (D = 4, N = 2) Model The goal of this note is a very modest one: We show that, for D = 4 and N = 2, Eqs. (1.11) do not have any normalizable solutions. Our proof is not conceptual; it relies on explicit calculations and estimates and does therefore not extend to the general case in any straightforward way. When D = 4 and N = 2 we use the following notations: ~qj := Xj1 , Xj2 , Xj3 , j = 1, 2, ~λ = λ1 , λ2 , λ3 := b1 , b2 , b3 , (2.1) α

and

∂ = ∂~λ

∂ ∂ ∂ , , 1 2 ∂λ ∂λ ∂λ3

α

α

:= c1α , c2α , c3α ,

α = 1. The operators representing the generators of su(2) on H are given by

616


~ = −i L

∂ ~ ~ ~ ~q1 ∧ ∇1 + ~q2 ∧ ∇2 + λ ∧ . ∂~λ

(2.2)

The supercharges are given by (see [1], Eq. (4.20)) ~ 2 · ∂ + i ~q · ~λ , ~ 1 − i∇ Q = ∇ ∂~λ and ~ 1 + i∇ ~ 2 · ~λ − i ~q · ∂ , Q† = − ∇ ∂~λ where

(2.3)

~q = ~q1 ∧ ~q2 ,

(2.4)

and ∧ denotes the vector product. We then have that ~ , (Q† )2 = i (~q1 + i ~q2 ) · L, ~ Q2 = i (~q1 − i ~q2 ) · L and H =

Q, Q†

.

(2.5)

A vector 9 ∈ H+ can be written as 1 ~ ~ ~ λ∧λ · ψ 2 1 = ψ + εABC λA λB ψ C . 2

9=ψ+

(2.6)

~ = 0), Eqs. (1.11) imply the following system (∗ ) For 9 ∈ H+(0) (i.e., 9 ∈ H+ with L9 of first-order differential equations: ~, ~ 1 − i∇ ~2 ∧ψ i ~q ψ = ∇ (2.7)

and

~ =0, ~q · ψ

(2.8)

~, ~ 2 ψ = i ~q ∧ ψ ~ 1 + i∇ ∇ ~ =0. ~ 1 + i∇ ~2 · ψ ∇

(2.9)

~ = 0 yields Moreover, the equation L9 ~ 1 + ~q2 ∧ ∇ ~2 ψ = 0, ~q1 ∧ ∇ X ~ 1 + ~q2 ∧ ∇ ~2 ~q1 ∧ ∇ ψB + εABC ψc = 0 , A

(2.10)

(2.11) ∀ A, B .

(2.12)

C

(0) It is straightforward to verify that, for 9 ∈ H− , Eqs. (1.11) imply a system of equation equivalent to (2.7) through (2.12). This can be interpreted as a consequence of Poincaré duality. The formal expression for the Hamiltonian H = Q, Q† is given by


617

H = HB + H F ,

(2.13)

where

~2 − ∇ ~ 2 + ~q 2 HB = − ∇ 1 2

~λ ∧ ~λ − (~q1 − i ~q2 ) · ∂ ∧ ∂ . ∂~λ ∂~λ As shown in [5], the spectrum of HB is discrete, with and

HF = (~q1 + i ~q2 ) ·

inf spec HB = E0 > 0 .

(2.14)

(2.15)

The representation of the group SO(D − 2) ' U (1), (D = 4) on H is generated by the operator ~ 2 − ~q2 · ∇ ~ 1 − 1 ~λ · ∂ . (2.16) J = − i ~q1 · ∇ 2 ∂~λ While J does not commute with Q or Q† , it does commute with QQ† and Q† Q and hence with H. It is therefore sufficient to look for solutions of Eqs. (2.7) through (2.12) transforming under an irreducible representation of U (1), i.e. solutions that are eigenvectors of J corresponding to eigenvalues j ∈ 21 Z . Thespectrum of the restriction of J to the subspace H+ is the integers, while spec J |H− consists of half-integers.

3. Analysis of Equations (∗ ) In this section, we assume that Q9 = Q† 9 = 0 has a solution 9 ∈ H+(0) and then show that 9 = 0. The assumption that Q9 = Q† 9 = 0 implies that hQ9, Q9i + hQ† 9, Q† 9i = 0 .

(3.1)

6 Let ξ := (~q1 , ~q2 ) ∈ R6 . Let p gn (ξ) ≡ gn (|ξ|), n = 1, 2, 3, · · · , be a function on R 2 2 only depending on |ξ| := ~q1 + ~q2 with the properties that gn is smooth, d monotonic decreasing, gn (|ξ|) = 1, for |ξ| ≤ n, gn (|ξ|) = 0 for |ξ| ≥ 3n, and dt gn (t) ≤ n1 . Let hk (ξ), k = 1, 2, 3, · · · R, be an approximate δ-function at ξ = 0 with the properties that hk is smooth, hk ≥ 0, hk (ξ)d6 ξ = 1, and 1 supp hk ⊆ ξ |ξ| ≤ 2 . (3.2) k

We define a bounded operator, Rn,k , on H by setting Z Rn,k 8 (ξ) = gn (ξ) hk ξ − ξ 0 8 ξ 0 d6 ξ 0 , for any 8 ∈ H. Clearly

s n→∞ − lim Rn,k 8 = 8 ,

(3.3)

(3.4)

k→∞

for any 8 ∈ H. Next, we note that, for a vector 8 in the domain of the operator Q, (3.5) Q, Rn,k 8 (ξ) = In,k (ξ) + IIn,k (ξ) ,

618


where

Z ~ 1 − i∇ ~ 2 gn (ξ) · hk ξ − ξ 0 ∂ 8 ξ 0 d6 ξ 0 , ∇ ∂~λ

In,k (ξ) =

Z

and IIn,k (ξ) = i gn (ξ)

hk ξ − ξ 0

~q (ξ) − ~q ξ 0

· ~λ8 ξ 0 d6 ξ 0 .

(3.6)

(3.7)

bounded by 1. Since The operator norm of the operators ∂λ∂A and λA , A = 1, 2, 3, is d ~ 1 − i∇ ~ 2 gn (·) gn (t) ≤ 1 , the operator norm of the multiplication operator ∇ dt

n

1 is by 6n 0 . The operator norm of the convolution operator 8 (ξ) 7→ R bounded 0above 0 hn ξ − ξ 8 ξ d ξ is equal to 1. This implies

18 ∂ 6

kIn,k k ≤ k8k . (3.8) 8 ≤ n ∂~λ n

Next, we note that, for ξ in the support of the function gn , hk ξ − ξ 0 ~q (ξ) − ~q ξ 0 ≤ 7n hk ξ − ξ 0 . 2 k Thus, for k ≥ n, kIIn,k k ≤

21 k8k . n

(3.9)

In conclusion

40 k8k , (3.10) k Q, Rn,k 8k ≤ n for k ≥ n. A similar chain of arguments shows that, for 8 in the domain of Q† , 40 k8k , k Q† , Rn,k 8k ≤ n

(3.11)

for k ≥ n. Next, we suppose that 9 solves (3.1). We claim that, given ε > 0, there is some finite n(ε) such that, for 9n,k := Rn,k 9 ,

and

k9k ≥ k9n,k k ≥ (1 − ε) k9k ,

(3.12)

hQ9n,k , Q9n,k i + hQ† 9n,k , Q† 9n,k i ≤ εk9k2 ,

(3.13)

for all k ≥ n ≥ n(ε). Inequality (3.12) follows directly from (3.4) and the fact that the operator norm of Rn,k is = 1. To prove (3.13), we note that, for k ≥ n, 2 # # 40 # # h9, 9i , (3.14) hQ 9n,k , Q 9n,k i = h Q , Rn,k 9, Q , Rn,k 9i ≤ n where Q# = Q or Q† . This follows from the equations Q9 = Q† 9 = 0 and inequalities (3.10) and (3.11). We now observe that, by the definition of Rn,k , 9n,k = Rn,k 9 is a smooth function of compact support in R6 , for all n ≤ k < ∞. It therefore belongs to the domain of definition of the operators Q Q† and Q† Q. Thus, for all n ≤ k < ∞,


619

hQ9n,k , Q9n,k i + hQ† 9n,k , Q† 9n,k i = h9n,k , Q, Q† 9n,k i = h9n,k , HB 9n,k i + h9n,k , HF 9n,k i ,

(3.15)

where HB and HF are given in Eq. (2.14), (and it is obvious from (2.14) that 9n,k belongs to the domains of definition of HB and HF ). As proven in [5], (3.16) h8, HB 8i ≥ E0 k8k2 , for some strictly positive constant E0 (= inf spec HB ), for all vectors 8 in the domain of HB . Thus, for k ≥ n ≥ n(ε), and using (3.13), we have that ε k9k2 ≥ h9n,k , HB 9n,k i + h9n,k , HF 9n,k i ≥ (1 − ε)2 E0 k9k2 + h9n,k , HF 9n,k i .

(3.17)

Our next task is to analyze h9n,k , HF 9n,k i. If 8 = (ϕ, ϕ ~ ) ∈ H+ belongs to the domain of definition of HF then Z ~ (ξ) d6 ξ + c.c. ϕ (ξ) (~q1 − i ~q2 ) · ϕ (3.18) h8, HF 8i = 2 Note that 9n := lim 9n,k , where 9n,k = Rn,k 9 and 9 solves (3.1), belongs to the k→∞ ~ solves the equations Q9 = Q† 9 = 0, domain of definition of HF . Since 9 = ψ, ψ ~ For ~q 6= 0, we find that see (3.1), we can use Eqs. (2.8) and (2.9) to eliminate ψ: ~ (ξ) = i ~q ∧ ∇ ~ 1 + i∇ ~ 2 ψ(ξ) ψ q2

(3.19)

(recall that ~q = ~q1 ∧ ~q2 ). Inserting (3.19) on the R.S. of (3.18), for 8 = 9n , we arrive at the equation Z (gn ψ) (ξ) (~q1 − i ~q2 ) h9n , HF 9n i = 2 i ~q ~ ~ × gn (ξ) ∧ ∇1 + i ∇2 ψ (ξ) d6 ξ + c.c. (3.20) q2

Let T := 2 (~q1 − i ~q2 ) ·

i ~q ~ ~ . ∧ ∇1 + i ∇ 2 q2

Then h9n , HF 9n i = h9n , T 9n i + c.c. Z − | ψ (ξ) |2 gn (ξ) [T, gn ] (ξ) d6 ξ + c.c.

(3.21)

Next, we make use of the fact that 9 must be SU(2)–invariant. This is expressed in Eq. (2.11), which implies that ψ(ξ) only depends on SU(2)–invariant combinations of the variables ~q1 and ~q2 , i.e., on r1 := |~q1 |, r2 := |~q2 |, x :=

~q1 · ~q2 . r1 r2

(3.22)

620


Instead, we may use variables q, p and ϕ defined by 1 1 2 (~q1 + i ~q2 )2 = r1 − r22 + i r1 r2 x, 2 2

p eiϕ := with

0 ≤ p < ∞,

0 ≤ ϕ < 4π,

q := |~q1 ∧ ~q2 |

0≤q < ∞.

(3.23)

(3.24)

If F is an SU(2)–invariant function then Z∞

Z 6

d ξ F (ξ) = c

Z∞ dq

0

Z4π dp

0

0

qp dϕ p F (q, p, ϕ) , q 2 + p2

(3.25)

where c is some positive constant. If ϕ is SU(2)–invariant then p q 2 + p2 ∂ ϕ, Tϕ=c q ∂q 0

(3.26)

where c0 is a positive constant. Using (3.26) in (3.21), we find that h9n , HF 9n i = c

00

Z∞

Z4π dp

0

− c00

Z∞ 0

dϕ 0

Z4π dp

Z∞ dq p

∂ |ψn (q, p, ϕ) |2 ∂q

0

Z∞ dq p |ψ (q, p, ϕ) |2

dϕ 0

∂ ∂q

p 2 g n 2 q 2 + p2 , (3.27)

0

with c00 = c.c0 > 0. By the definition of gn , ∂ ∂q

p 2 2 2 gn 2 q + p ≤0,

pointwise. Therefore h9n , HF 9n i ≥ − c

00

Z∞

Z4π dϕ p |ψn (q = 0, p, ϕ) |2 .

dp 0

In passing from (3.27) to (3.28), we have used that with respect to the measure p dp dϕ dq and that Z∞ 0

∂ ∂q |ψn

(q, p, ϕ) |2 is an L1 –function

Z4π dϕ p |ψn (q, p, ϕ) |2

dp

(3.28)

0

0

is right-continuous at q = 0. These facts will be proven below. Combining Eqs. (3.15), (3.17) and (3.28), we conclude that


εk9k2 ≥ lim

621

hQ9n,k , Q9n,k i + hQ† 9n,k , Q† 9n,k i

k→∞

≥ (1 − ε) E0 k9k2 Z∞ Z4π − c00 dp dϕ p |ψn (q = 0, p, ϕ) |2 , 0

(3.29)

0

for all n ≥ n(ε). Choosing ε sufficiently small, we conclude that either 9 = 0, or there is a constant β > 0 such that Z∞

Z4π dϕ p |ψn (q = 0, p, ϕ) |2 ≥ β ,

dp 0

(3.30)

0

for all sufficiently large n. Next, we explore the consequences of (3.30). Since 9 solves (3.1), we can use (3.19) to conclude that ~ 2 ∞ > k9k2 = kψk2 + kψk   Z ~ 1 + i∇ ~ 2 ψ(ξ)|2   | ∇ = d6 ξ |ψ(ξ)|2 + .   |~q|2 Using that 9 is SU(2)–invariant and passing to the variables q, p and ϕ, one finds that Z∞

Z∞ dp p

2 0

where ψ,x :=

dq q

Z4π

0

0

∂ψ ∂x

, and

iψ,ϕ dϕ ψ,p + p

Z∞

Z∞ dp

0

Z4π dq

0

0

! 2 2 e + | ψ,q | (q, p, ϕ) < K,

pq 2 e dϕ p |ψ (q, p, ϕ)| < K, 2 2 p +q

(3.31)

(3.32)

e = k9k < ∞ (with the constant c appearing in (3.25)). with K c Inequalities (3.31) and (3.32) also hold for ψn , instead of ψ, with a constant K that 2 ∂ is uniform in n → ∞. These inequalities prove that ∂q |ψn (q, p, ϕ)| is an L1 –function with respect to the measure p dp dϕ dq and that 2

Z∞

Z4π dϕ p |ψn (q, p, ϕ)|

dp

fn (q) := 0

2

0

is right-continuous at q = 0, properties that were used in our derivation of (3.28). By the Schwarz inequality and (3.31),

622


Z∞

Z4π dp

0

dϕ 0



≤ 2  

Z∞

∂ 2 dq p |ψn (q, p, ϕ)| ∂q

0

Z∞

Z∞

dq q

dp p 0

0

Z∞ 0

Z4π q dq

0

1/2 2 dϕ |ψn,q (q, p, ϕ)| 

0

Z∞ dp p

Z4π

1/2

2 dϕ |ψn (q, p, ϕ)| 

≤ K 0 n4 ,

0

for some finite constant K 0 ! To prove continuity of fn (q) in q, we note that, for q1 > q2 , Z∞ |fn (q1 ) − fn (q2 )| ≤

Z4π dp

0

Zq1 dϕ

∂ 2 dq p |ψn (q, p, ϕ)| ∂q

q2

0

2

∂ |ψn (q, p, ϕ)| is an L1 –function. which tends to 0, as (q1 − q2 ) → 0, because ∂q Next, we make use of the SO(D − 2) ' U (1) symmetry with generator J given in Eq. (2.16). We have noted below (2.16) that J commutes with QQ† and Q† Q, and hence that 9 ∈ H+ can be chosen to be an eigenvector of J corresponding to some eigenvalue m ∈ Z. In the variables q, p, ϕ,

J = −2i Hence we may write ψ (q, p, ϕ) = ei

m 2

∂ . ∂ϕ ϕ

m

p 2 φ (q, p) ,

(3.33)

for some function φ independent of ϕ. Eqs. (3.31) and (3.32) then simply Z∞

Z∞ dp p

α

0

and

dq 2 2 |φ,p | + |φ,q | (q, p) < ∞ q

(3.34)

pα q 2 dq p |φ (q, p)| < ∞, p2 + q 2

(3.35)

0

Z∞

Z∞ dp

0

0

where α = m + 1. Furthermore, inequality (3.30), in the limit as n → ∞, yields Z∞ 2

dp pα |φ (q = 0, p)| ≥

β . 4π

(3.36)

0

Let IN := N1 , N . Then inequality (3.34) implies that there exists a set ⊆ [0, ∞) with the property that ∩ [0, δ] has Lebesgue measures δ2 , for any δ > 0, and such that


1 N

|α| Z

623

2

dp |φ,p (q, p)| ≤ Kδ

(3.37)

IN

for some constant Kδ independent of N and all q ∈ ∩ [0, δ]. Moreover, Z 2 lim dp |φ,p (q, p)| = 0, q→0 q∈

(3.38)

IN

for all N . It follows that, for q ∈ ∩ [0, δ] , p1 , p2 ∈ IN , N < ∞, p Z 2 dp φ,p (q, p) |φ (q, p1 ) − φ (q, p2 )| = |p1 − p2 | |p − p | 1 2 p1 1/2 1/2 ≤ |p1 − p2 | N |α| Kδ .

(3.39)

Thus, for q ∈ ∩ [0, δ] and p1 , p2 ∈ IN , φ (q, p) is uniformly Hölder–continuous with exponent 21 . Thus φ0 (p) := lim φ(q, p) is uniformly continuous in p ∈ IN , for all q→0 q∈

N < ∞, and it then follows from (3.38) that φ0 (p) = φ0 = const.

(3.40)

Inequality (3.36) then implies that |φ0 | must be positive! Without loss of generality, we may then assume that φ0 > 0. Thus the function φ introduced in (3.33) has the following properties: (A)

lim φ (q, p) = φ0 > 0, q→0 q∈

Z∞

Z∞ dp

(B) 0

0

Z∞

Z∞ dp

(C) 0

qpα 2 dq p |φ (q, p)| < ∞, p2 + q 2 dq

pα 2 2 |φ,p (q, p)| + |φ,q (q, p)| < ∞. q

0

We now show that such a function φ (q, p) does not exist. Let us first consider the case α ≥ 0. We choose an arbitrary, but fixed p ∈ (0, ∞). Using the Schwarz inequality, we find that, for 0 < q0 < ∞, Zq0 0

1 dq 2 |φ,q (q, p)| ≥ q q0 

Zq0 dq |φ,q (q, p)| 0

1 ≥ q0

Zq0

2

2 dq |φ,q (q, p)|

0

2 q∗ (p) Z 1 ≥ dq φ,q (q, p) , q0 0

(3.41)

624


where q ∗ (p) ∈ [0, q0 ] is the point at which |φ(q, p)| takes its minimum in the interval [0, q0 ]. Note that φ(q, p) is continuous in q ∈ [0, q0 ], for almost every p ∈ [0, ∞). The R.S. of (3.41) is equal to 2 1 ∗ . (p), p − φ φ q 0 q02 Thus

1 |χ (p) − φ0 | q0

2

Zq0 ≤

dq 2 |φ,q (q, p)| , q

(3.42)

0 ∗

where χ(p) = φ(q (p), p). By property (C), Z∞

Zq0 dp p

dq 2 |φ,q (q, p)| ≤ ε(q0 ), q

α

0

0

for some finite ε(q0 ), with ε(q0 ) → 0, as q0 → 0. Hence Z∞ 2

dp pα |χ(p) − φ0 | < q02 ε (q0 ) .

(3.43)

0

We define a subset Mδ ⊆ [0, ∞) by Mδ := p |χ (p)| ≤ φ0 − δ . Then

Z dp p

α

1 ≤ 2 δ

Mδ

Z∞ 2

dp pα |χ (p) − φ0 |

Z∞

0

0

Z∞

Zq0

≥

dp 0

q0/2

≥

It follows that

q02 4

q0 α 2 p

p + q0

Zq0 dp

Mδc

p2 + q 2

dq

Z ≥

qpα

dq p

dp

p + q0

q0/2

(φ0 − δ)

|φ (q, p)|

q0 α 2 p

dq

Z 2 Mδc

|φ (q, p)|

dp

2

|φ (q, p)| pα . p + q0

2

2

(3.44)


Z dp

Z

pα + p + q0

dp Mδc

Mδ

pα 1 ≤ p + q0 q0

Z

625

Z dp pα +

dp Mδc

Mδ

pα < ∞. p + q0

R pα diverges. This is a contradiction, since Mδ ∪ Mδc = [0, ∞), and dp p+q 0 Next, we consider the case α ≤ −1. We change variables, k := p1 , dp = −

1 ∂ ∂ = − k2 . dk, 2 k ∂p ∂k

Then conditions (A)–(C) take the form (A0 )

lim φ (q, k) = φ0 > 0, q→0 q∈

Z∞

0

Z∞ dq q

dk

(B )

(C 0 )

0

0

Z∞

Z∞ dk

0

dq

q k γ−2 2 |φ (q, k)| < ∞, 1 2 + q2 k

k γ−2 4 2 2 k |φ,k (q, k)| + |φ,q (q, k)| < ∞, q

0

where γ := −α > 0. Repeating the same arguments as above, we get again

1 |χ (k) − φ0 | q0

2

Z∞ ≤

dq 2 |φ,q (q, k)| , q

(3.45)

0

where χ(k) is the value of φ(q, k) at the minimum of |φ(q, k)|, for q ∈ [0, q0 ]. By (C’), Z∞

Zq0 dk

0

dq

k γ−2 2 |φ,q (q, k)| < ε0 (q0 ) < ∞, q

0

0

with ε (q0 ) → 0, as q0 → 0. Hence Z∞

dk k γ−2 |χ(k) − φ0 | ≤ q02 ε0 (q0 ) . 2

(3.46)

0

Let Lδ ⊆ [0, ∞) be the set defined by Lδ := k |χ(k)| ≤ φ0 − δ . Then we have that Z dk k

γ−2

Lδ

Condition (B’) implies that

1 ≤ 2 δ

Z∞ 2

dk k γ−2 |χ(k) − φ0 | ≤ 0

q02 ε0 (q0 ) . δ2

(3.47)

626


Z∞ ∞>

Z∞ dk k

0

Zq0 dk k

q0/2

γ−2

1 k

q0/2

0

Zq0

Z ≥

dk k Lcδ

≥

1 k

0

Z∞ ≥

0

2

(φ0 − δ)

+ q0

|φ (q, k)|

2

q0/2 (φ0 − δ)2 1 + k q0

γ−1 q0/2

q 2

q 2 |φ (q, k)| +q

γ−2

Z 2

dk Lcδ

k γ−1 . 1 + k q0

(3.48)

Combining (3.47) and (3.48) we find that Z Z k γ−1 k γ−1 dk + dk 1 + k q0 1 + k q0 Lδ

≤

1 q0

Z

Lcδ

Z

dk k γ−2 + Lδ

dk Lcδ

k γ−1 < ∞. 1 + k q0

R kγ−1 This is a contradiction, because Lδ ∪ Lcδ = [0, ∞) and dk 1+k q0 diverges for γ ≥ 1. This completes the proof that functions satisfying properties (A), (B) and (C) do not exist. We have thus proven that Eq. (3.1) only has the trivial solution 9 = 0. Acknowledgement. We would like to thank H. Kalf for useful discussions.

References 1. de Wit, B., Hoppe, J., Nicolai, H.: Nuclear Physics B 305, [FS23] 545 (1988) 2. Goldstone, J.: Unpublished. Hoppe, J.: MIT Ph.D. Thesis, 1982, and Quantum Theory of a Relativistic Surface. In: Constraint’s Theory and Relativistic Dynamics. (eds. G. Longhi, L. Lusanna) Arcetri, Florence 1986, World Scientific 1987 3. de Wit, B., Lüscher, M., Nicolai, H.: Nuclear Physics B 320, 135 (1989) 4. Baake, M., Reinicke, P., Rittenberg, V.: J. Math. Physics 26, 1070 (1985) Claudson, M., Halpern, M.: Nucl. Phys. B 250, 689–715 (1985) Flume, R.: Annals of Physics 164, 189 (1985) 5. Lüscher, M.: Nuclear Physics B 219, 233 (1983) Simon, B.: Annals of Physics 146, 209 (1983) 6. Banks, T., Fischler, W., Shenker, S.H., Susskind, L.: M Theory as a Matrix Model: A Conjecture. hep-th/9610043 7. Hoppe, J.: On Zero-Mass Bound-States in Super-Membrane Models. hep-th/9609232 Communicated by A. Jaffe

Commun. Math. Phys. 191, 627 – 639 (1998)

Communications in


A Variational Principle Associated to Positive Tilt Maps Sen Hu Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544-1000, USA. E-mail: [email protected] Received: 14 May 1996 / Accepted: 30 May 1997

Abstract: In this paper we establish a variational principle for positive tilt maps. This implies that the Aubry–Mather set exists for positive tilt maps and most of the theory developed by Mather for twist maps now applies for positive tilt maps.

1. Introduction Let f be a C1 area preserving, exact, orientation preserving and end preserving diffeomorphism of the cylinder R/Z × R. For composition of positive twist maps Mather [7] established a variational principle h satisfying (H0 ) − (H6θ ). Then from Bangert’s work [1] for any h satisfying those conditions one can construct an Aubry–Mather set for each rotation number. One knows that compositions of twist maps are also positive tilt maps. It is not known whether those two sets coincide (cf. [14]). In this paper we establish a variational principle h for positive tilt maps satisfying (H0 ) − (H6θ ). Since most of Mather’s work on twist maps are based on a variational principle h satisfying (H0 ) − (H6θ ), this work implies that much of his work applies for positive tilt maps too. For example, Mather’s work on connecting orbits ([11, 13]), destruction of invariant curves ([8]), and differentiability of the average action functions ([10]), etc., are also true for positive tilt maps. We will not state all theorems. The interested reader may look at the cited papers. The work to establish such a variational principle for positive tilt maps relies on proving an elementary lemma suggested by Mather (Main Lemma in Sect. 4). It basically asserts that the tilt where the action function takes minimum for various points (x, y) with 0 fixed (x, x ) must be less than or equal to π. The action function h admits a geometrical interpretation as an area of some region (see Sect. 3). The proof of the main lemma involves counting areas for various situations. The proof is entirely elementary. It only uses the basic fact on the Jordan curve in plane topology to analyze positive tilt diagrams.

628

S. Hu

Once we have the main lemma it is not difficult to show that h satisfies (H0 ) − (H6θ ). For completeness we also give the proof of those. It is interesting to note that if we don’t have the Main Lemma we even cannot prove h satisfies (H0 ), i.e. h is continuous. 2. Statement of the Theorem 2.1. Positive tilt maps. Let f : R/Z × R → R/Z × R be a diffeomorphism of the cylinder. Let γ be a C1 embedded curve defined on R in R/Z × R, such that, γ(t)2 → −∞ as t → −∞, γ(t)2 → +∞ as t → +∞, where γ(t)2 means the second component of the coordinates. Then we define the tilt of γ as the real valued function on R satisfying the following: 0

1) tilt(γ)(t) := angle which γ (t) makes with the vertical (mod2π); 2) If γ(t)2 > γ(s)2 for all s < t, then −π/2 ≤ tilt(γ)(t) ≤ π/2. A C1 area preserving, exact, orientation preserving and end preserving map f : R/Z × R → R/Z × R is said to be a positive tilt map if for each vertical line l, tilt(f ol) > 0. 0

0

Let f (x, y) = (x , y ). To say that f is area-preserving means that f preserves the 0 0 area form dxdy. This implies that the 1-form y dx − ydx is a closed form. To say that f is exact means that this 1-form is exact. Remark. A positive twist map is a map satisfying 0 < tilt(f ol) < π for every vertical l. Remark. If f and g are positive tilt maps, then so is gof . Notice that if f and g are positive twist maps, gof may not be a positive twist map. 2.2. Action function h and conditions (H0 ) − (H6θ ). Let f¯ be a lift of f to R2 , f¯ : R2 → R2 . Since f¯ is area-preserving we can define h0 as a function of R2 , such that 0

0

dh0 (x, y) = y dx − ydx. Now define

0 0 h(x, x ) = min{h0 (x, y)|π1 f¯(x, y) = x }.

y

Here π1 is the projection to the first factor. The minimum is taken over the set 0 Σx,x0 = {y|π1 f¯(x, y) = x }.

We say that h satisfies conditions (H0 ) − (H6θ ) if it satisfies the following. H0 : h is a continuous function. 0 0 0 H1 : h(x + 1, x + 1) = h(x, x ), for all x, x ∈ R. H2 : lim|ξ|→∞ h(x, x + ξ) = ∞, uniformly in x. 0

0

0

0

0

0

H3 : h(x, x ) + h(ξ, ξ ) < h(x, ξ ) + h(ξ, x ) if x < ξ, x < ξ .

Variational Principle Associated to Positive Tilt Maps 0

629

0

0

¯ x, ξ ) are both minimal segments, and are distinct, then (x− ¯ H4 : If (x, ¯ x, x ), (ξ, ¯ ξ)(x − 0 ξ ) < 0. 0

0

H5 : There exists a positive continuous function ρ on R2 , such that h(ξ, x ) + h(x, ξ ) − R ξ R ξ0 0 0 0 0 h(x, x ) − h(ξ, ξ ) ≥ x x0 ρ, if x < ξ and x < ξ . 0

0

0

0

0

H6θ : x → θx2 /2 − h(x, x ) is convex, for any x , and x → θx 2 /2 − h(x, x ) is convex for any x. Here θ = cot α, α is a lower bound of tilt of f . 2.3. Statement of the theorem. Theorem. Let f : R/Z × R → R/Z × R be a C1 area preserving, exact, orientation preserving and end preserving positive tilt diffeomorphism. Let h be the action function defined above. Then h satisfies (H0 ) − (H6θ ). Remark. Again from Bangert [1] we have Aubry–Mather sets that exist for h. ∞ h(xi , xi+1 ) be the funcLet x = (..., xi , ...) be a configuration and W (x) = Σi=−∞ tional. We say that (..., xi , ...) is minimal if for any j < k, (xj , ..., xk ) is minimal k−1 k−1 h(xi , xi+1 ) ≤ Σi=j h(yi , yi+1 ) for any subject to constraint yj = xj , yk = xk , i.e. Σi=j (yj , ..., yk ) satisfying yj = xj , yk = xk . In [1] (see also [5]) it is shown that a minimal configuration corresponds to an orbit of f and has a well defined rotation number, i.e. lim xi /i = ρ(x) exists. (xj , ..., xk ) is called a minimal segment.

Corollary. For a positive tilt diffeomorphism f there exists minimal configurations for each rotation number. Remark. Existence of Aubry-Mather sets can also be established by topological method, see [4, 3, 2]. 3. Geometrical Interpretation of h0 0

0

Let f (x, y) = (x , y ). The generating function h0 (x, y) can be obtained by integrating 0 0 the 1-form y dx − ydx along the vertical segment from (x, 0) to (x, y) and along the horizontal segment from (x0 , 0) to (x, 0). See also [2]. The integral along the vertical 0 segment is the algebraic area of the region bounded by the vertical line through (x , 0), the image of the vertical line segment joining (x, 0) and (x, y), the vertical line through f (x, 0) and the horizontal line y = 0. The integral along the horizontal segment is a function of x, say U (x), only. Since the mapping is exact, U (x) is periodic. We see for our concern U (x) does not contribute. For conditions (H3 ) − (H5 ), U (x) cancelled in the expression. For condition (H6θ ), U (x) does not contribute because it is periodic. Because of this we will drop the U (x) part and consider h0 (x, y) to be the area we described above. Notice that the image of the vertical line through (x, 0) may have several intersections 0 with the vertical line through (x , 0). In this case we take the image of the vertical line through (x, 0) up to its intersection whose image is (x, y). Here the points in the vertical line are given the same order as the order of the vertical line. See Fig. 1. In Fig. 2 we illustrate this fact by describing a special case where the vertical line 0 through (x , 0) intersect with the image of the vertical line through (x, 0) at three points.

630

S. Hu

(x0,0)

(x, 0)

Fig. 1.

A B

Fig. 2.

We see that h0 (x, y2 ) = h0 (x, y1 )+ area (A) and h0 (x, y3 ) = h0 (x, y2 )− area (B). The sign before the areas depends on the orientation of the curve f (l) at points where the curve intersect with the vertical line. 4. The Main lemma 4.1. Approximate a diagram of infinite many intersections with a finite one. It may 0 happen that there are infinitely many number of intersections of f (l) with l . Since the mapping is smooth and we only restrict ourself to a compact region we can only have the following picture: either there are only finitely many transverse intersections or there are infinitely many intersections yet all intersections except finitely many are almost vertical (up to a given precision). This is true because the mapping is smooth. Because of the smoothness of f, f (l) locally looks like a line. And we restrict ourself to a compact region so we can divide the finite part with the part near the ends breaking down into several small pieces, each approximated by its tangent very well.

Fig. 3.

At the points of intersection if the tangent is quite different from the vertical direction, the curve and the line intersect transversely. If the tangent is almost parallel to the vertical

Variational Principle Associated to Positive Tilt Maps

631

line we will just replace this part by a vertical segment. Then we get a new diagram with finitely many transverse intersections plus some vertical segments. Our main lemma applies to those kinds of diagrams. The difference of actions of corresponding points is very small, i.e. by the given precision number times the length of the replaced segment. We will see that for our purpose, i.e. to prove (H0 ) − (H6θ ), this approximation is good enough. In the following we only consider approximated diagrams, i.e., diagrams with finitely many intersections or finitely many vertical segments. 4.2. The main lemma. 0

Main Lemma. If h(x, x ) = h0 (x, y), then 0 < tilt(x,y) (f ) ≤ π, 0

0

and f (x, .) crosses x ×R from left to right, i.e., the two unbounded components of f (l)−l 0 0 0 0 in R2 − l belong to two different components of R2 − l . Here l = x × R, l = x × R . This lemma follows from the following proposition. In the proposition we do not deal with the mapping of the cylinder directly. We will deal with a curve intersecting with a line. For a precise definition see Sect. 6. 0

Proposition. Let P be any point of intersection of f (l) with the line l . Suppose that 0 f (l) does not cross l from left to right, or that its tilt is greater than π at P , then h0 (P ) >

min

0 0 on a compact boundaryless manifold. For any ε > 0, e−εB is trace class and B has a well defined µ-regularised determinant.

646

M. Arnaudon, S. Paycha

Proof. We shall show that the assumptions of Lemma 1.2 are fulfilled. Condition 1) in Lemma 1.2 follows from the fact that a strictly positive s.a elliptic operator on a compact boundaryless manifold has purely discrete spectrum (λn )n∈N , α > 0 ( see e.g [G], Lemma 1.6.3). Indeed, from λn > 0, λn ' Cnα , for some C > 0,P this fact easily follows that tre−εB = n e−ελn is finite. Conditions 2) of Lemma 1.1 follow from the fact that for a s.a elliptic operator B of order m on a compact manifold of dimension d without boundary, tre−tB '0 PK j m for any K > 0 (this follows for example from Lemma 1.7.4 in [G]). j=−d aj t Applying Lemma 1.1, we can therefore define the heat-kernel regularised determinant of B. The above definition extends to a class of positive self-adjoint operators which satisfy requirements 1) and 2) of Lemma 1.2 and have possibly non zero kernel. Requirement 1) of the lemma implies that this kernel is finite dimensional. Let PB be the orthogonal projection onto the kernel of the operator B acting on H and let us set H ⊥ ≡ (I −PB )H. Let us consider the restriction B 0 ≡ B/H ⊥ . It is easily seen that the operator B 0 satisfies requirements of Lemma 1.2 with coefficients b0j = bj for j 6= 0 and b00 = b0 −dim(KerB). Formula (1.6) extends to B 0 with adapted changes in the coefficients. Let us at this stage see how the zeta-function regularised determinant fits into this picture. We refer the reader to [BGV, G] for a precise description of the zeta-function regularisation procedure and only describe the main lines of this procedure here. Recall that for a strictly positive self adjoint operator B acting on a separable Hilbert space with purely discrete spectrum given by the eigenvalues (λn , n ∈ N) with the property λn ≥ Cnα , C > 0, α > 0 for large enough n, we can define the zeta function of B by: X 1 λ−s s ∈ C, Res > . ζB (s) ≡ n , α n Furthermore, ζB (s) admits a meromorphic continuation on the whole plane (see e.g [G], Lemma 1.10.1) which is regular at s = 0 and one can define the zeta function regularised determinant of A by 0 (1.9) detζ (B) = e−ζB (0) . Remark. From the definition, easily follows that in the finite dimensional case the zetafunction regularised and the ordinary determinants coincide. The following lemma compares the zeta function and µ-regularisations. Lemma 1.4. Let B be a strictly positive self-adjoint densely defined operator on a Hilbert space H such that 1) B has purely discrete spectrum (λn )n∈N with λn ≥ Cnα ,C > 0, α > 0 for large enough n, 2) The function ε → tre−εB lies in C0 . Then for µ ∈ R, eζB (0)(γ−µ) detζ B = detµreg (B),

(1.10)

where γ = limn→∞ (1 + 21 + · · · + n1 − log n) is the Euler constant. In particular, detζ B = detγreg B.

Regularisable and Minimal Orbits for Group Actions in Infinite Dimensions

647

Remark. A proof of this result for µ = 0 and the Laplace operator on a compact Riemannian surface without boundary can be found in [AJPS]. Proof. Before starting the proof, let us recall that the function Gamma is defined by R ∞ −t 0(z) = 0 e t tz dt for 0 < Rez. Moreover 0(z)−1 is an entire function and we have ∞ Y z −z (1 + )e n , where γ is the Euler constant. From this follows that in 0(z)−1 = zeγz n n=1

−1 = s + γs2 + O(s3 ). a neighborhood of zero, we have the asymptotic expansion 0(s) Z +∞

Using the Mellin transform of the function λ−s = 0(s)−1

ts−1 e−tλ dt we can

0

write:

Z 0(s)ζB (s) =

1

ts−1 tre−tB dt +

Z

0

∞

ts−1 tre−tB dt.

(1.11)

1

Notice that the last expression on the r.h.s converges 0 for, setR ∞ for Res ≤ R, RR > 1 1 ∞ ting CR = supn supt≥1 tR−1 e− 2 tλn , we have 1 tR−1 e−tλn ≤ CR 1 e− 2 tλn = − 21 λn 2CR λ−1 , which is the general term of a convergent series. n e As before we set m−1 X j bj t m . (1.12) F (t) ≡ tre−tB − j=−J

Using (1.11 ) and (1.12), we can write for s ∈ C with large enough real part, Res >   Z 1 Z ∞ m−1 X bj ζB (s) = 0(s)−1  ts−1 tre−tB dt + ts−1 F (t)dt . + j + s 1 0 j=−J m

J m:

This equality then extends to an equality of meromorphic functions on Res > 0 with poles s = −j m . Using the asymptotic expansion of the inverse of the Gamma function −1 0(s) around zero, we have   Z 1 Z ∞ m−1 X bj 0 (s) = (s + γs2 + O(s3 ))  ts−1 tre−tB dt + ts−1 F (t)dt , + ζB j + s 1 0 j=−J m which yields b0 = ζB (0). Moreover  m−1 X 0 (s) = (1 + 2γs + O(s2 ))  ζB 

bj + +s

j j=−J m

m−1 X

+(s + γs2 + O(s3 )) −

j=−J

bj + j (m + s)2

Z

Z

∞

ts−1 tre−tB dt +

1

1

1 0

Z

∞

ts−1 F (t)dt 

ln(t)ts−1 tre−tB dt .

ts−1 F (t)ln(t)dt+ 0



Z

1

Letting s tend to zero, s > 0, since the divergent terms bs0 and −s sb02 arising in each of the terms of this last sum compensate, we get   Z ∞ −tB m−1 X mbj Z 1 F (t) tre 0 + dt + dt . (0) = b0 γ +  ζB j t t 0 1 j=−J,j6=0

648


Hence, comparing with the expression of detµreg B given in (1.5), for any µ ∈ R we find 0 log detζ (B) = −ζB (0) = −ζB (0)γ + log det µreg (B) + µζB (0) and hence the equality of the lemma. Remarks. 1) In the finite dimensional case with dimH = d, since limε→0 tre−εB = d = ζB (0), from the result of Lemma 1.4 and the fact that the zeta function regularised determinant coincides with the ordinary one, it follows that for µ ∈ R: detµreg B = ed(γ−µ) detζ B = ed(γ−µ) detB,

(1.13)

where detB denotes the ordinary determinant of B. For µ = γ, detγreg B = detB = detζ B. 2) Let M be a Riemannian manifold of dimension d and B a positive self-adjoint elliptic operator with smooth coefficients acting on sections of a vector bundle V on M with finite dimensional fibres of dimension k. We know by [G] Theorem 1.7.6 (a) that ζB (0) = 0 if n is odd. However, in general the coefficient ζB (0) is a complicated expression given in terms of the jets of the symbol of the operator B. In the following we shall be concerned with the dependence of ζB (0) on the geometric data given on that manifold.

2. Regularisable Principal Fibre Bundles The aim of this section is to describe a class of principal fibre bundles for which we can define a notion of regularised volume of the fibres and for which these regularised volumes have differentiability properties. Let P be a Hilbert manifold equipped with a (possibly weak) right invariant Riemannian structure. The scalar product induced on Tp P by this Riemannian structure will be denoted by < ·, · >p . We shall assume this Riemannian structure induces a Riemannian connection denoted by ∇ and an exponential map with the usual properties. In particular, for all p0 , expp0 yields a diffeomorphism of a neighborhood of 0 in the tangent space Tp0 P onto a neighborhood of p0 in the manifold P. Let G be a Hilbert Lie group ( in fact a Hilbert manifold with smooth right multiplication is enough here, see e.g. [T]) acting smoothly on P on the right by an isometric action Θ : G × P → P, (2.0) (g, p) → p · g. Let for p ∈ P,

τp : G → Tp P, d u 7→ (p · etu ) , dt t=0

(2.0bis)

where G denotes the Lie algebra of G. We shall assume that the action Θ is free (so that τp is injective on G) and that it induces a smooth manifold structure on the quotient space P/G and a smooth principal fibre bundle structure given by the canonical projection π : P → P/G. Let us furthermore equip the group G with a smooth family of equivalent (possibly weak) Adg invariant Riemannian metrics indexed by p ∈ P. The scalar product induced on G by the Riemannian metric on G indexed by p ∈ P will be denoted by (·, ·)p . Since


649

the metrics are all equivalent, the closure of G w.r.t (·, ·)p does not depend on p and we shall denote it by H. Since G is dense in H, τp is a densily defined operator on H and we can define its adjoint operator τp∗ w.r. to the scalar products (·, ·)p and < ·, · >p . We shall assume that τp∗ τp has a self-adjoint extension on a dense domain D(τp∗ τp ) of H. Definition. The orbit of a point p0 is volume preregularisable if the following assumptions 1) and 2) on the operator τp∗ τp are satisfied: ∗

1) Assumption on the spectral properties of τp∗0 τp0 . The operator e−ετp0 τp0 is trace class for any ε > 0 and for any vector X at point p0 , there is a neighborhood I0 of p0 on ∗ the geodesic pκ = expp0 κX such that for all p ∈ I0 , e−ετp τp is trace class. 2) Regularity assumptions. We shall assume that the maps p 7→ τp and p 7→ τp∗ τp are ∗ Gâteaux differentiable and that for any t > 0, the function p 7→ tre−tτp τp is Gâteaux differentiable at point p0 . We furthermore assume that the Gâteaux-differentials at point p0 in the direction X of these operators are related as follows: ∗

∗

δX (tre−ετp τp ) = −εtr(δX (τp∗ τp )e−ετp τp ).

(2.1)

Moreover, for any vector X at point p0 , there are constants C > 0, u > 0 and a neighborhood I0 of p0 on the geodesic pκ = expp0 κX such that for any p ∈ I0 : ∗

tre−tτp τp ≤ Ce−tu

(2.2)

and ∗

(τp∗ τp )e−tτp τp |||∞ MI0 (t) ≡ supp∈I0 |||δX(p) ¯

(2.3)

is finite and a decreasing function in t. Here ||| · |||∞ denotes the operator norm on G induced by (·, ·)p , X¯ is a local vector ¯ κ ) = expp ∗ (κX)(X). field defined in a neighborhood of p0 by X(p κ The orbit Op0 is called volume-regularisable if dim Kerτp∗ τp is constant on some neighborhood of p0 on any geodesic containing p0 and if the following assumption is satisfied: 3) Assumption on the asymptotic behavior of the heat-kernel traces. Both the functions ∗ ∗ t 7→ tre−tτp τp and t 7→ δX tre−tτp τp lie in the class C0 (see Sect. 1). There is an integer m > 0 and a family of maps p 7→ bj (p), j ∈ {−J, · · · , m − 1} which are Gâteaux differentiable in the direction X at point p0 such that ∗

tre−ετp τp '0

m−1 X

j

bj (p)ε m

(2.4)

j=−J

in a neighborhood I0 of p0 on the geodesic p = expp0 κX, and ∗

δX tre−ετp τp '0

m−1 X j=−J

j

δX bj (p)ε m .

(2.5)

650


Furthermore, setting Fp (t) ≡ tre

−tτp∗ τp

−

m−1 X

j

bj (p)t m , for any vector X at point

j=−J

p0 , there is a constant K > 0, and a neighborhood I0 of p0 on the geodesic κ → pκ = expp0 κX such that: supp∈I0 kδX(p) Fp (t)k∞ ≤ Kt. ¯

(2.5bis)

A principal bundle as described above with all its orbits volume-preregularisable (resp. volume- regularisable) will be called preregularisable (resp. regularisable). Remark. Since the Riemannian structure on P is right invariant and the one on G is Adg invariant, the above assumptions do not depend on the point chosen in the orbit for we have τp·g = Rg∗ τp Adg . Most fibre bundles we shall come across are not only preregularisable but also regularisable so that the notion of preregularisabiblity might seem somewhat artificial. However, in the case of the coadjoint action of loop groups mentioned in the introduction, it is sufficient to verify the conditions required for preregularisability in order to prove a certain minimality of the orbits, namely strong minimality, a notion which will be defined in the following and which implies minimality. Natural examples of regularisable fibre bundles arise in gauge field theories (YangMills, string theory). In gauge field theories, P and G are modelled on spaces of sections of vector bundles E and F based on a compact finite dimensional manifold M and the operators τp∗ τp arise as smooth families of Laplace operators on forms. As elliptic operators on a compact boundaryless manifold, they have purely discrete spectrum which satisfies condition 1) (see [G] Lemma 1.6.3) and (2.4) (see [G], Lemma 1.7.4.b)). By classical results concerning one parameter families of heat-kernel operators, they satisfy (2.1) (see [RS], Proposition 6.1) and (2.2) (see proof of Theorem 5.1 in [RS]). Since δX Bp is also a partial differential operator, by [G], Lemma 1.7.7, δX tre−εBp satisfies (2.5). Assumptions on the Gâteaux-differentiability and assumptions (2.3 ), (2.5 bis) are fulfilled in applications. Indeed, the parameter p is a geometric object such as a connection, a metric on M and choosing these objects regular enough (of class H k for ∗ k large enough) ensures that the maps p 7→ τp , p 7→ τp∗ τp , p 7→ tre−tτp τp , etc., are regular enough for they involve these geometric quantities and their derivatives, but no derivative of higher order. Remark. In the context of gauge field theories, the underlying Riemannian structure w.r.to which the traces (arising in (2.2)-(2.5 bis)) are taken are weak L2 Riemannian structures, the ones that also underlie the theory of elliptic operators on compact manifolds. In [AP2], we discuss how far this weak Riemannian structure could be replaced by a strong Riemannian structure, in order to set up a link between this geometric picture and a stochastic one developed in [AP2]. Notation. We shall set with the notations of Sect. 1, for ε > 0 and p ∈ P detε (Bp ) = exptr(hε (Bp )). Proposition 2.1. Let Op0 be a volume-preregularisable orbit such that for any geodesic containing p0 , there is a neighborhood of p0 on this geodesic on which τp∗ τp is injective. 1) detε (τp∗ τp ) is well defined for any ε > 0 and for p in a neighborhood of p0 on any geodesic containing p0 .


651

2) The map p 7→ detε (τp∗ τp ) is Gâteaux-differentiable at point p0 , the operator Z +∞ ∗ δX (τp∗ τp )e−tτp τp dt is trace class for any p in a neighborhood of p0 on any ε

geodesic of p0 . For any tangent vector X at point p0 , we have: Z ∞ ∗ tr (δX τp∗ τp )e−tτp0 τp0 dt δX log detε (τp∗ τp ) = ε Z +∞ ∗ (δX τp∗ τp )e−tτp0 τp0 dt. = tr

(2.6a)

ε

3) If the orbit Op0 is moreover volume-regularisable, for any µ ∈ R, the map p 7→ detµreg (τp∗ τp ) is Gâteaux differentiable in all directions at point p0 , and for p in a geodesic neighborhood of p0 , the map ε 7→ δX log det ε (τp∗ τp ) lies in the class C. For µ ∈ R, Limµε→0 δX log detε (τp∗ τp ) = δX log detµreg τp∗ τp ! ∗ Z 1 Z ∞ m−1 X m e−tτp τp δX Fp (t) δX bj − dt − dt − µδX b0 . =− δX tr j t t 1 0 j=−J,j6=0

(2.6b) Proof. We set Bp =

τp∗ τp

and as before, detε (Bp ) = exptrhε (Bp ).

1) By the first assumption for volume-preregularisable orbits, we know that e−εBp is p trace class so that by Lemma 1.1 so is Apε ≡ log hε (Bp ). Hence detε (Bp ) = etrAε is well defined. 2) Let us show the first equality in (2.6 a). Assumption 2) for volume-preregularisability t t Bp e−tBp )| ≤ CMI0 ( )e− 2 u . yields that for any p ∈ I0 and any t > ε > 0 |tr(δX(p) ¯ 2 Here, we have used the fact that |tr(U V )| ≤ |||U |||trV | for any bounded opt erator U and any trace class operator V applied to U = δX(p) Bp e− 2 Bp0 and ¯ t V = Re− 2 Bp . Hence, by the Lebesgue dominated convergence theorem, the map ∞ p 7→ ε t−1 tre−tBp dt is Gâteaux-differentiable in the direction X at point p0 and Z ∞ Z ∞ −1 −tBp δX t tre dt = t−1 δX tre−tBp dt ε ε Z ∞ tr((δX Bp )e−tBp0 )dt, =− ε

Z

+∞

using (2.2). Using the fact that log detε (Bp ) = −

t−1 tre−tBp dt then yields the

ε

first equality in (2.6 a). The second equality in (2.6 a) and the fact that we can swap the trace and the integral follow from the estimate: |||δX Bp e−tBp0 |||1 ≤ |||δX Bp e

−ε 2 Bp 0

≤ C|||δX Bp e

|||∞ k|e− 2 tBp0 k|1

−ε 2 Bp 0

1

|||∞ e−tu ,

(∗)

valid for t ≥ ε, using Assumption Z +∞ Z +∞ (2.2). We finally obtain by dominated convergence: tr δX Bp e−tBp0 dt = trδX Bp e−tBp0 dt. ε

ε

652


3) Let us first check that the map p 7→ detµreg Bp is Gâteaux differentiable at point p0 in the direction X. By (1.5), we have m−1 X

log det reg Bp = −

j=−J,j6=0

bj (p) − j

Z

∞

tr 1

e−tBp dt − t

Z

1 0

Fp (t) dt. t

The first term on the r.h.s. is Gâteaux differentiable in the direction X by the assumption on the maps p 7→ bj (p). The second term on the r.h.s. is Gâteaux differentiable by the result (applied to ε = 1) of part 2 of this proposition which tells us that p 7→ detε (Bp ) is Gâteaux differentiable. The Gâteaux differentiability of the last term follows from the local uniform upper bound (2.5 bis). Pm−1 j mb We now check (2.6 b). The map p 7→ log detε (Bp ) − j=−J,j6=0 j j ε m − b0 log ε is Gâteaux differentiable in the direction X and we can write δX ( log detε (Bp ) − ∞

= δX (−

tr ε

e−tBp dt − t

m−1 X

= δX −

j=−J,j6=0

=−

j

j=−J,j6=0

Z 

m−1 X

m−1 X j=−J,j6=0

δ X bj

bj m − j m − j

Z

mbj ε m − b0 log ε) j m−1 X

j=−J,j6=0

Z

∞ 1

e−tBp dt − tr t

∞

δX tr 1

j

mbj ε m − b0 log ε) j Z

e−tBp dt − t

1 ε

Z

 Fp (t)  dt t 1

δX ε

as in (1.8)

Fp (t) dt, t

which tends to δX log detreg Bp by (1.6) and dominated convergence. Here we have used Z ∞ −tBp e dt = tr the results of point 2) of the proposition applied to ε = 1 to write δX t 1 Z ∞ R 1 δ F (t) R 1 F (t) δX tre−tBp dt and (2.5 bis) to write δX ε pt dt = ε X tp dt. 1

Remark. These results extend to the case when instead of assuming that τp∗ τp is injective locally around p0 , one considers orbits of an action at points p0 for which the dimension of the kernel of τp is constant on some neighborhood of p0 on each geodesic starting at point p0 . For this, one should replace detε τp∗ τp and detreg τp∗ τp by det0ε τp∗ τp and det0reg τp∗ τp . This extension is useful for the applications mentioned in the introduction. A naive generalisation of the finite dimensional notion of volume to volume of infinite dimensional orbits would give infinite quantities. But for volume-preregularisable or regularisable orbits, one can define a notion of preregularised or µ-regularised volume (µ ∈ R), which justifies a posteriori the term “volume-preregularisable or volumeregularisable orbits” for these orbits. Since τp·g = Rg∗ τp Adg and since the metric on G ∗ is Adg and that on P right invariant, for any ε > 0, we have detε (τp·g τp·g ) = detε (τp∗ τp ) so that it makes sense to set the following definitions:


653

Definition. 1) Let Op be a volume-preregularisable orbit, then volε (Op ) ≡ p det ε (τp∗ τp )0 defines a one parameter family of preregularised volumes of Op . q 2) Let Op be a volume-regularisable orbit, then for µ ∈ R, volµreg (Op ) = det µreg (τp∗ τp )0 defines the µ-regularised volume of Op . q 3) Let Op be a volume-regularisable orbit, then volζ (Op ) = det ζ (τp∗ τp )0 defines the zeta function regularised volume of Op . From Lemma 1.4 it follows that 1

0

volζ (Op ) = e 2 (−γ+µ)b0 (p) volµreg (Op ),

(2.7)

where γ is the Euler constant and b00 (p) = ζτp∗ τp (0) − dim Ker(τp∗ τp ) is the coefficient arising from the heat-kernel asymptotic expansion of τp∗ τp given by (2.4). In finite dimensions, when dimH = d and G is a compact LieZ group equipped with the Haar µ (Op ) = e(µ−γ)d |detτp | measure dvol, this yields volreg

As a consequence of Proposition 2.1:

|detAdg dvol(g)|. G

Proposition 2.2. For any µ ∈ R, the µ-(resp. pre)-regularised volume of a volume(pre)regularisable orbit Op is Gâteaux-differentiable at the point p. Let us now introduce a notion of extremality of orbits which generalises the corresponding finite dimensional notion [H]. Definition. A strongly extremal orbit is a volume-preregularisable orbit, the preregularised volume of which is extremal, i.e. Op is strongly extremal if δX volε (Op ) = 0 for any horizontal vector X at point p and any ε > 0. For a given µ ∈ R, a µ- extremal orbit of a preregularisable bundle is an orbit, the µ-regularised volume of which is extremal, i.e. δX volµreg (Op ) = 0 for any horizontal vector X at point p. Notice that whenever ζτp∗ τp (0) − dim(Ker(τp∗ τp )) does not depend on p, the extremality of the volume of an orbit does not depend on the parameter µ. From (2.7) it also follows that this notion generalises the finite dimensional notion of extremality of the volume of the fibre.

3. Minimal Orbits as Orbits with Extremal Volume We shall consider a preregularisable principal fibre bundle P → P/G. By assumption, the bundle is equipped with a Riemannian connection given by a family of horizontal spaces Hp , p ∈ P such that Tp P = Hp ⊕ Vp , where Vp is the tangent space to the orbit at point p and the sum is an orthogonal one. For a horizontal vector X at point p, we define the shape operator H X : V p → Vp ¯ v (p), Y 7→ −(∇Y X)

654


where the subscript v denotes the orthogonal projection onto Vp and X¯ is a horizontal vector field with value X at p. Similarly, we define the second fundamental form: S p : V p × V p → Hp (Y, Y 0 ) 7→ (∇Y¯ Y¯ 0 )h (p), where Y¯ , Y¯ 0 are vertical vector fields such that Y¯ (p) = Y , Y¯ 0 (p) = Y 0 . These definitions are independent of the choice of the extensions of X,Y and Y 0 . An easy computation shows that the shape operator and the second fundamental form are related as follows: < HX (Y ), Y 0 >p =< S p (Y, Y 0 ), X >p .

(3.1)

Note that this explicitly shows that HX only depends on X and not on the extension X¯ of X. Since S p is symmetric, so is HX . As in the finite dimensional case, one can define the notion of totally geodesic orbit, an orbit Op being totally geodesic whenever the second fundamental form S p vanishes. Definition. The orbit Op of a point p ∈ P will be called preregularisable if for any horizontal vector X at p, ∀ε > 0, ∗

∗

ε ≡ e− 2 ετp τp HX e− 2 ετp τp HX 1

1

(3.2)

is trace class. A preregularisable orbit Op will be called strongly minimal if moreover ε = 0 ∀ε > 0. for any q ∈ Op and X a horizontal vector at point q, trHX ε trace class) is autoRemarks. 1) The preregularisability of the orbits ( namely HX matically satisfied if the manifold P is equipped with a strong smooth Riemannian structure, since in that case the second fundamental form is a bounded bilinear form and its weighted trace is well defined (see also [AP2] where this is discussed in further detail). 2) Since on a preregularisable bundle, the Riemannian structure on P is right invariant and the one on G is Adg invariant, the notion of (pre)regularisability and (strong) minimality of the orbit does not depend on the point chosen on the orbit. 3) Notice that if HX is trace class, as in the finite dimensional case, strong minimality implies that trHX = 0 and hence ordinary minimality. The fact that strong minimality implies minimality in the finite dimensional case motivates the choice of the adjective “strong”. ε and the second fundamental form are related 4) This preregularised shape operator HX as follows: ∗

∗

ε (Y ), Y 0 >p =< S p (e− 2 ετp τp Y, e− 2 ετp τp Y 0 ), X >p < HX 1

1

ε Since τp τp∗ is an isomorphism of the tangent space to the fibre Tp Op , HX vanishes whenever the second fundamental form vanishes and an orbit is totally geodesic whenever this regularised shape operator vanishes on the orbit for some ε > 0.

Definition. A preregularisable orbit Op will be said to be regularisable if furthermore, ε , ε ∈]0, 1] admits a regularised limit-trace (as defined in the one parameter family HX Sect. 1). For µ ∈ R, we denote by trµreg HX its µ-regularised limite trace.


655

Definition. For a given µ ∈ R, a regularisable orbit Op will be called µ− minimal if trµreg HX = 0 for any horizontal vector X at point p. Remarks. 1) As we shall see later on, for different values of µ, the notions of µminimality do not coincide in general. ε admits a regularised limit 2) In the finite dimensional case, the one parameter family HX trace given by the ordinary trace trµreg HX = trHX and µ- minimality is equivalent to the finite dimensional notion of minimality. 3) A strongly minimal preregularisable orbit Op is µ-regularisable and µ-minimal for any µ ∈ R. The notion of minimality of orbits for group actions in the infinite dimensional case has been discussed in the literature before. King and Terng in [KT] introduced a notion of regularisability and minimality for submanifolds of path spaces using zeta-function regularisation methods. They show zeta function regularisability and minimality for the orbits of the coadjoint action of a (based) loop group on a space of loops in the corresponding Lie algebra. One can check that these orbits are also regularisable and strongly minimal (hence minimal) within our framework . A notion of zeta function regularisability and minimality was discussed by Maeda, Rosenberg and Tondeur in [MRT1] (see also [MRT2]) in the case of orbits of the gauge action in Yang-Mills theory. In fact, it can be seen as a particular example of µ-minimality for µ = γ, the Euler constant. Let us introduce some notations. Let P → P/G be a preregularisable principal fibre bundle and let (Tnp )n∈N be a set of eigenvectors of τp∗ τp in G corresponding to the eigenvalues (λpn )n∈N counted with multiplicity and in increasing order. Let p0 be a fixed point in P and let Ipp0 be the isometry from (G, (·, ·)p0 ) into (G, (·, ·)p ) which takes the orthonormal set (Tnp0 )n of eigenvectors of τp∗0 τp0 to the orthonormal set of eigenvectors (Tnp )n of τp∗ τp . Notice that Ipp00 = I. Lemma 3.1. Let P → P/G be a preregularisable principal fibre bundle. Let p0 ∈ P be a point at which the map p 7→ Ipp0 u is Gâteaux-differentiable for any u ∈ G. Let X be a horizontal vector at p0 . We shall consider eigenvalues λpn that correspond to eigenvectors that do not belong to Ipp0 Kerτp∗0 τpo . p 1) The maps p → λpn are Gâteaux-differentiable Z in the direction X at point p0 , δX λn = +∞

(δX (τp∗ τp )Tnp0 , Tnp0 )p0 and δX log hε (λpn ) =

∗

(δX (τp∗ τp )e−tτp0 τp0 Tnp0 , Tnp0 )p0 dt.

ε

2) Furthermore,we have p0

ε ˜p ˜p Un , Un >p0 +e−ελn (δX Ipp0 Tnp0 , Tnp0 )p0 = − < HX

1 δX log hε (λpn ), 2

(3.5)

where we have set U˜ np = kτp Tnp k−1 τp Tnp . 3) If the Riemannian structure on G is fixed (independent of p), then δX Ipp0 is antisymmetric and Z ∗ 1 +∞ ε ˜p ˜p U n , Un > p 0 (δX (τp∗ τp )e−tτp τp Tnp , Tnp )p0 dt = − < HX 2 ε (3.6) p0 1 1 −1 = δX log hε (λpn ) = λpn0 δX λpn e−ελn . 2 2

656


Proof. As before, we shall set Bp = τp∗ τp . Since p0 is fixed, we drop the index p0 in Ipp0 and denote this isometry by I p . Notice that I p0 = I. As before, we denote by (Tnp )n∈N the orthonormal set of eigenvectors of τp∗ τp which correspond to the eigenvalues (λpn )n∈N in increasing order and counted with multiplicity. We shall set Tñp = τp Tnp , T¯np = τp Tnp0 . 1) Using the relations (I p ·, I p ·)p = (·, ·)p0 , I p (Tnp0 ) = Tnp , I p ∗ I p = I, we can write λpn = (Bp Tnp , Tnp )p = (Bp I p Tnp0 , I p Tnp0 )p0 and the map p 7→ λpn is Gâteaux differentiable in all directions at point p0 since p 7→ Bp , p 7→ I p are Gâteaux-differentiable by assumption on the bundle. Furthermore δX (Bp Tnp , Tnp )p = δX (I p ∗ Bp I p Tnp0 , Tnp0 )p0 = ((δX Bp )Tnp0 , Tnp0 )p0 + (δX (I p ∗ )Bp0 Tnp0 , Tnp0 )p0 + + (I p0 ∗ Bp0 (δX I p )Tnp0 , Tnp0 )p0 = ((δX Bp )Tnp0 , Tnp0 )p + λpn0 ([I p0 ∗ δX (I p ) + (δX I p ∗ )I p0 ]Tnp0 , Tnp0 )p0 . Since I p ∗ I p = I, we have δX I p ∗ I p0 + I p0 ∗ δX I p = 0 so that finally λpn is Gâteauxdifferentiable and δX λpn = ((δX Bp )Tnp0 , Tnp0 )p0 . Using the local uniform estimate (2.3), and with the same notations, we have for t > ε p0 1 1 (Bp )e−tBp0 Tnp0 , Tnp0 )p0 k ≤ MI0 ( t)e− 2 tλn so that the map p 7→ log hε (λpn ) k(δX(p) ¯ 2 is Gâteaux-differentiable at point p0 in the direction X and Z ∞ t−1 (e−tBp Tnp , Tnp )dt δX log hε (λpn ) = −δX ε Z +∞ δX (Bp )e−tBp0 Tnp0 , Tnp0 )p0 dt. =( ε

2) By definition of hε we have: δX log hε (λpn ) = (log hε )0 (λpn )δX λpn p0

= (λpn0 )−1 e−ελn δX λpn . On the other hand δX λpn = δX < Tñp , Tñp >p = 2 < δX (τp I p )Tnp0 , T¯np0 >p0 = 2 < δX T¯np , T¯np0 >p0 +2 < τp δX I p Tnp0 , T¯np0 >p0 ¯ T¯np0 >p0 +2 < τp δX I p Tnp0 , T¯np0 >p0 = −2 < ∇T¯np0 X, ¯ Tñp0 >p0 +2 < τp δX I p Tnp0 , T¯np0 >p0 = −2 < ∇Tñp0 X, = −2λpn0 < HX U˜ np0 , U˜ np0 >p0 +2λpn0 (δX I p Tnp0 , Tnp0 )p0 , where for the third equality, we have used the fact that, X¯ being right invarip ¯ np0 , U¯ np0 >p0 ¯ = 0. Hence δX loghε (λpn ) = −2e−ελn0 < HX(p ant, [T¯np , X] ¯ 0)U p0

+2e−ελn (δX I p Tnp0 , Tnp0 )p0 , which yields 2). 3) On one hand, since the scalar product on the Lie algebra is fixed, we have δX I p∗ ⊂ (δX I p )∗ . On the other hand, since I p∗ I p = I, we have −δX I p ⊂ δX I p∗ so that the second term in the l.h.s of (3.5) vanishes.


657

Definition. We shall call an orbit Op0 of a preregularised bundle an orbit of type (T ) whenever the following conditions are satisfied: 1) The map p 7→ Ipp0 is Gâteaux-differentiable at point p0 . ∗

2) The operator δX Ipp0 e−ετp0 τp0 is trace class for any p0 ∈ P and ε > 0. ∗ 3) For any p ∈ P, tr Ipp0 e−ετp0 τp0 is Gâteaux-differentiable at point p0 ∈ P and ∗

∗

δX tr(Ipp0 e−ετp0 τp0 ) = tr(δX Ipp0 e−ετp0 τp0 ). Whenever the Riemannian structure on G is independent of p, any orbit satisfying condition 1) is of type (T ), for in that case the traces involved in 2) and 3) vanish, δX Ipp0 being an antisymmetric operator. Proposition 3.2. Let P → P/G be a preregularisable principal fibre bundle. Then 1) Any orbit of type (T ) is preregularisable. More precisely, if Op0 is an orbit of type ε is trace class, the map (T ), for any horizontal vector X at point p0 , the operator HX p 7→ volε (Op ) is Gâteaux differentiable in the direction X at point p0 and ∗

ε trHX − δX tr(Ipp0 e−ετp0 τp0 ) = −δX log vol0ε (Op ) Z ∗ 1 +∞ 0 tr [δX (τp∗ τp )e−tτp0 τp0 ]dt. =− 2 ε

(3.7)

2) If the Riemannian structure on G is independent of p, the orbit of any point p0 is a preregularisable orbit and Z ∗ 1 +∞ 0 ε = −δX log vol0ε (Op ) = − tr [δX (τp∗ τp )e−tτp0 τp0 ]dt, (3.7bis) trHX 2 ε where tr0 means we have restricted to the orthogonal of the kernel of τp∗0 τp0 and volε0 means that we only consider eigenvalues λpn that correspond to eigenvectors that do not belong to Ipp0 Kerτp∗0 τpo . Remarks. 1) In finite dimensions, for a compact connected Lie group acting via isometries on a Riemannian manifold P of dimension d, we have for any ε > 0 and using the various definitions of the volumes, including the µ-volume, µ ∈ R: lim δX log volε (Op ) = δX log volµreg Op

ε→0

= δX log volOp . Hence going to the limit ε → 0 on either side of (3.7 bis) we find: trHX = −δX log volOp . If the Gâteaux-differentiability involved is a C 1 - Gâteaux-differentiability, this yields trS p = −grad log volOp This leads to a well known result, namely (Hsiang’s theorem [H]) that the orbits of G whose volume are extremal among nearby orbits is a minimal submanifold of M .

658


2) Equality (3.7) tells us that whenever the Riemannian structure on G is independent of p (as in the case of Yang-Mills theory), strongly minimal orbits of a preregularisable principal fibre bundle are pre-extremal orbits. This gives a weak (in the sense that we only get a sufficient condition for strong minimality and not for minimality) infinite dimensional version of Hsiang’s [H] theorem. 3) If both the spectrum of τp∗ τp and the Riemannian structure on G are independent of p, as in the case of Yang-Mills theory in the abelian case (where the spectrum only depends on a fixed Riemannian structure on the manifold M ), the orbits are strongly minimal (see also [MRT 1] par.5). Proof of Proposition 3.2. We set Bp = τp∗ τp . For the sake of simplicity, we assume that Bp is injective on its domain, the general case then easily follows. 1) From the preregularisability of the principal bundle follows (see Proposition 2.1) that ) is Gâteaux-differentiable in the direction X at point p0 and the map p 7→ detε (B Zp +∞

δX log det ε (Bp ) =

dttr(δX Bp e−tBp ). On the other hand, by Lemma 3.1

ε

Z +∞ p0 1 h dt(δX Bp e−tBp )Tnp , Tnp ip − e−ελn (δX I p Tnp0 , Tnp0 )p0 2 ε ε ˜p ˜p Un , Un ip . = −hHX

(∗)

The fibre bundle being preregularisable, by the results of Proposition 2.0, the first term on the left-hand side is the general term of an absolutely convergent series. On the other hand, the orbit being of type (T ), the series with general term given p0 by e−ελn (δX I p Tnp0 , Tnp0 )p0 is also absolutely convergent. Hence the right-hand side ε of (*) is absolutely convergent and HX is trace class since (U˜ n )n∈N is a complete orthonormal basis of Imτp , Z +∞ ε dttr(δX Bp e−εBp ) = trHX − δX log Volpε 0 (Bp ) = −δX log det ε (Bp ), − ε

which then yields (3.7). 2) This follows from the above and point 3) of Lemma 3.1 and holds for any orbit Op of a regularisable fibre bundle since it does not involve δX Ip . The following proposition gives an interpretation of trµreg HX in terms of the variation of the regularised volume of the orbit. Proposition 3.3. The fibres of a regularisable principal fibre bundle with structure group equipped with a fixed (p-independent) Riemannian metric are regularisable. 1) For a given µ ∈ R, orbits are µ- minimal whenever they are µ- extremal. More precisely, for any point p0 ∈ P and any horizontal vector X at point p0 , the ε has a limit trace trµreg HX and one parameter family HX trµreg HX = −δX log volµreg (Op )   Z 1 Z ∞ m−1 . ∗ 1 X bj (p) δX Fp (t) + dt + δX t−1 δX tre−tτp τp dt − µδX b00  = 2 j t 0 1 j=−J,j6=0

(3.8)


659

For µ0 ∈ R, 0

trµ HX = trµ HX + γ(µ0 − µ)δX b00 ,

(3.9)

where as before b00 = b0 − dimKerB. 2) Orbits are µ-minimal whenever   Z ∞ X p 1 0(s)−1 ts−1 e−tλn δX λpn dt + (s − 1)−1 δX b00 (p)dt lim − s→1 2 0 λn 6=0

exists for any horizontal field X at point p. Furthermore, setting µ = γ, the Euler constant, we have   Z ∞ X p 1 γ HX = − lim 0(s)−1 ts−1 e−tλn δX λpn dt + (s − 1)−1 δX b00 (p) . trreg 2 s→1 0 λn 6=0

(3.10) If moreover δX b00 = 0 for any horizontal vector X at point p0 , if an orbit is µ-minimal for one value of µ, it is for any value of µ. Remarks. 1) From (3.9) follows that unless δX b0 = 0, µ-minimality depends on the choice of the parameter µ. 2) In the case of a compact connected Lie group acting via isometries on a finite dimensional Riemannian manifold P of dimension d, the various notions of minimality coincide since b0 = d, volµreg (Op ) = vol(Op ) (this being the ordinary volume) and (1.10) yields: trS p = −grad log vol(Op ), where S p is the second fundamental form. It tells us that the orbits of G, the volume of which are extremal among nearby orbits is a minimal submanifold of P. This proposition therefore gives an infinite dimensional version of Hsiang’s theorem [H]. 2) A zeta function formulation of Hsiang’s theorem in infinite dimensions was already discussed in [MRT1] in the context of Yang-Mill’s theory. However, there was an obstruction due to the factor b0 (p) in the zeta-function regularisation procedure which does not appear here (see also [MRT2]). A formula similar to (3.10) (but using zeta function regularisation) can be found in [GP] (see in [GP] formula (3.17) combined with formula (A.3)). Proof of Proposition 3.3. As before, we set Bp = τp∗ τp . and we shall assume for simplicity that Bp is injective; the proof then easily extends to the case when the dimension of the kernel is locally constant on each geodesic containing p0 . 1) Since the fibre bundle is regularisable, we know by Proposition 2.1 that the map p 7→ detreg (Bp ) is Gâteaux-differentiable in the direction X. Let us now check that ε HX has a regularized limit trace, applying Lemma 1.1. For this, we first investigate ε . By the result of Proposition 3.2, we the differentiability of the map ε 7→ trHX Z ∞ −tBp 1 1 e ε have trHX = − δX log detε (Bp ). The differentiability in ε = dtδX tr 2 ε t 2 easily follows from the shape of the middle expression.

660


Setting as before Fp (t) = tre−tBp −

Pm−1

j

j=−J

bj t m , we have furthermore

∂ 1 ε trHX = − ε−1 δX tre−εBp ∂ε 2 m−1 j−m 1 Fp (ε) 1 X − = − δX δ X bj ε m . 2 ε 2 j=−J

F (ε)

From the regularisability of the fibre bundle follows that |δX pε | ≤ K for some K > 0 and 0 < ε < 1 (see assumption (2.5 bis)) which in turn implies that 1 ∂ ε trHX '0 − ∂ε 2

−1 X

j

δX bj+m ε m .

j=−J−m

ε ) in Lemma 1.1, we can define the regularised limit trace Setting f (ε) ≡ tr(HX −1

1 1 X δ X bj j 1 ε trµreg HX + µδX b0 = lim (trHX ε m + δX b0 log ε) + m ε→0 2 2 j 2 j=−J   −1 X 1 bj j δX log detε (Bp ) − mδX ε m − δX b0 log ε by (3.7 bis ) = lim − ε→0 2 j j=−J   −1 X 1  mbj j = lim − δX log detε (Bp ) − ε m − b0 log ε ε→0 2 j j=−J

1 = − δX log det µreg (Bp ) by (1.5) 2 Z 1 Z ∞ m−1 ∗ δX Fp (t) 1 X mδX bj + + dt t−1 δX tre−tτp τp dt] by (2.6 b). = [ 2 j t 0 1 j=−J,j6=0

R∞ P 2) It is well known that the expression 0(s)−1 0 ts−1 n e−tλn is finite for Res large enough and that it has a meromorphic continuation to the whole plane. Since 0(s) = (s − 1)0(s − 1), we have for s with large enough real part:

0(s)−1

Z

∞ s−1

t

0

X p e−tλn δX λpn dt = (s − 1)−1 n

1 = −(s − 1)−1 0(s − 1)

Z

∞ s−2

t

1 0(s − 1)

Z

∞

ts−1

0

X

p

e−tλnδX λpn dt

n

δX tre−tBp dt

0

see assumption (2.2) and Lemma 3.1


661

 m−1 XZ 1 j 1  = −(s − 1)−1 t m +s−2 δX bj dt + 0(s − 1) 0 j=−J ! Z Z ∞

+ 1

δX Fp (t)ts−2 dt

by (2.5)

0

= −(s − 1) Z

1

ts−2 δX tre−tBp dt +

∞

+

−1

 m−1 X 1  0(s − 1)

j j=−J m

ts−2 δX tr e−tBp dt +

1

Z

1 δ X bj +s−1

1

#

ts−2 δX Fp (t)dt , 0

Pm−1 j where we have set Fp (t) = tre−εBp − j=−J bj (p)t m . Hence, since 0(s)−1 = s + γs2 + O(s3 ) around s = 0, going to the limit s → 1, we find: Z ∞ X p −1 lim [0(s) ts−1 e−tλn δX λpn dt + (s − 1)−1 δX b0 (p)] = s→1

0

n



= lim (−1 − γs + O(s2 ))  s→0

Z

m−1 X j=−J,j6=0

#

1

t

+

s−1

1 δ X bj + j m +s

Z

∞

ts−1 δX tr e−tBp dt

1

δX Fp (t)dt − γδX b0

0

= δX det0reg (Bp ) − γδX b0 = −2tr

0 reg HX

by formula (1.6) (with µ = 0) and (2.6 b)

− γδX b0 ,

R∞ R∞ where lims→0 1 ts−1 δX tr e−tBp dt = 1 t−1 δX tr e−tBp dt holds using estimate R1 (*) arising in the proof of Proposition 2.0 and lims→0 0 ts−1 δX Fp (t)dt+s−1 δX b0 = R 1 −1 t δX Fp (t)dt by (2.5 bis) and using dominated convergence. 0 The rest of the assertions of 2) then easily follow.

Acknowledgement. We would like to thank Steve Rosenberg most warmly for very valuable critical comments he made on a previous version of this paper. We would also like to thank David Elworthy for his generous hospitality at the Mathematics Department of Warwick University where part of this paper was completed.

References [AMT]

[AP1] [AP2]

Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, tensor analysis and applications, Global Analysis, Pure and Applied. Modern Methods for the study of non linear phenomena in engineering, Reading, MA: Addison Wesley, 1983 Arnaudon, M., Paycha, S.: Factorization of semi-martingales on infinite dimensional principal bundles. Stochastics and Stochastic Reports, Vol. 53, 81–107 (1995) Arnaudon, M., Paycha, S.: Stochastic tools on Hilbert manifolds, interplay with geometry and physics. Commun. Math. Phys. 187, 243–260 (1997)

662

[AJPS]


Albeverio, S., Jost, J., Paycha, S., Scarlatti, S.: A mathematical introduction to string theoryvariational problems, geometric and probabilistic methods. To appear, Cambridge: Cambridge University Press [BF] Bismut, J.M. et Freed, D.S: The Analysis of elliptic families I. Commun.Math.Phys. 106, 159–176 (1986) [BGV] Berline, N., Getzler, E., Vergne, M.: Heat-Kernels and Dirac Operators. Second edition, Berlin– Heidelberg–New York: Springer Verlag 1996 [FT] Fischer, A.E., Tromba, A.J.: On a purely Riemannian proof of the structure and dimension of the ramified moduli space of a compact Riemann surface. Mathematische Annalen 267, 311–345 (1984) [FU] Freed, D.S, Uhlenbeck, K.K.: Instantons on four manifolds. Berlin–Heidelberg–New York: Springer Verlag (1984) [G] Gilkey, P.B.: Invariance Theory, The heat equation and the Atiyah-Singer index theorem. Wilmington, DE: Publish or Perish, 1984 [GP] Groisser, D., Parker, T.: Semi-classical Yang-Mills theory I:Instantons. Commun. Math. Phys. 135, 101–140 (1990) [H] Hsiang, W.Y.; On compact homogeneous minimal submanifolds. Proc.Nat. Acad. Sci. USA 56, 5–6 (1966) [KR] Kondracki, W., Rogulski, J.. On the stratification of the orbit space. Dissertatione Mathematicae Polish Acad. of Sci. 250, 1–62 (1986) [KT] King, C., Terng, C.L.: Volume and minimality of submanifolds in path space. In: Global Analysis and Modern Mathematics, ed. K. Uhlenbeck, Wilmington, DE: Publish or Perish ( 1994) [MRT1] Maeda, Y., Rosenberg, S., Tondeur, P.: The mean curvature of gauge orbits. In: Global Analysis and Modern Mathematics. Ed. K. Uhlenbeck, Wilmington, DE: Publish or Perish ( 1994) [MRT2] Maeda, Y., Rosenberg, S., Tondeur, P.: Minimal orbits of metrics and Elliptic Operators. To appear in J. Geom. Phys. [MV] Mitter, P.K., Viallet, C.M.: On the bundle of connections and the gauge orbit manifold in Yang-Mills theory. Commun. Math. Phys. 79, 457–472 (1981) [P] Paycha, S.: Gauge orbits and determinant bundles: Confronting two geometric approaches. Proceedings of conference on Infinite dimensional Kähler manifolds. Ed. A. Huckleberry, Bael, Boston: Birkhäuser Verlag (to appear) [PR] Paycha, S., Rosenberg, S.: Work in progress [RS] Ray, D.S., Singer, I.M.: R-torsion and the Laplacian on Riemannian manifolds. Advances in Mathematics 7, 145–210 (1971) [T] Tromba, A.J.: Teichmüller theory in Riemannian geometry. Basel–Boston: Birkhäuser Verlag, 1992 Communicated by A. Jaffe

Commun. Math. Phys. 191, 663 – 696 (1998)

Communications in


Yangian Gelfand-Zetlin Bases, glN -Jack Polynomials and Computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model Denis Uglov Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan. E-mail: [email protected] Received: 1 April 1997 / Accepted: 1 June 1997

Abstract: We consider the glN -invariant Calogero-Sutherland Models with N = 1, 2, 3, . . . in the framework of Symmetric Polynomials. In this framework it becomes apparent that all these models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters q and t. The Hamiltonian of the glN -invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both q and t approach an N th elementary root of unity. This is a generalization of the well-known situation in the case of the Scalar Calogero-Sutherland Model (N = 1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the glN -invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call glN Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The glN -Jack Polynomials describe the orthogonal eigenbasis of the glN -invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N = 1). For each known property of Macdonald Polynomials there is a corresponding property of glN -Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl2 -invariant Calogero-Sutherland Model at integer values of the coupling constant. 1. Introduction In this paper we study the spin generalization of the Calogero-Sutherland Model [23] which was proposed in [4] and [1]. This model describes n quantum particles with coordinates y1 , . . . , yn moving along a circle of length L (0 ≤ yi ≤ L). Each particle carries a spin with N possible values, and the dynamics of the model are governed by the Hamiltonian

664

D. Uglov

1 X ∂2 π2 + 2 2 ∂yi 2L2 n

H β,N = −

i=1

X

β(β + Pij ) , π sin2 L (yi − yj ) 1≤i6=j≤n

(1.1)

where integer β > 0 is a coupling constant and the Pij is the spin exchange operator for particles i and j. As pointed Q out in [1] itπ is convenient to make a gauge transformation (yi − yj ) and defining the gauge-transformed of (1.1) by taking W = 1≤i<j≤n sin L Hamiltonian Hβ,N by Hβ,N =

L2 −β W H β,N W β . 2π 2

(1.2)

If we set zj = exp( 2πi L yj ), then Hβ,N acts on the vector space

FN,n := C[z1±1 , . . . , zn±1 ] ⊗ (⊗n CN )

antisymm

,

(1.3)

where antisymm means total antisymmetrization. In this paper we always will be working with the gauge-transformed model which has FN,n as its space of quantum states. This space of states is a Hilbert space with a β-dependent scalar product ( · , · )β,N which we describe in Sect. 2. It is well-known that the scalar version of the model (N = 1) is best understood in the framework of symmetric polynomials. In this case the space of states F1,n is naturally isomorphic – by multiplication with the Vandermonde determinant – to the vector space of symmetric Laurent polynomials, and the orthogonal eigenbasis of Hβ,1 1 is described by Jack Polynomials with parameter β+1 in notations of Macdonald’s book [17]. Properties of Jack Polynomials known in mathematical literature (e.g. [17, 22]) are of paramount importance for the scalar Calogero-Sutherland Model, for a rather straightforward application of these properties allows to compute two-point Dynamical Correlation Functions [9, 15, 14]. The main observation of the present paper is that, from our viewpoint, the Spin Model with arbitrary N is best understood in the framework of symmetric polynomials as well. At the first glance the model with N ≥ 2 seems to have little to do with symmetric polynomials since it has two disparate types of degrees of freedom: coordinates and spins. It turns out, however, that there is an isomorphism between the Hilbert space FN,n and the space of symmetric Laurent polynomials such that an orthogonal eigenbasis of Hβ,N is described by symmetric polynomials which are natural generalizations of Jack Polynomials, and which we call glN -Jack Polynomials. These symmetric polynomials are generalizations of Jack Polynomials in the sense that they are also limiting cases of Macdonald Polynomials [17]. The Jack Polynomial is a limit of the Macdonald Polynomial when both parameters in the latter approach 1 [17]. And the glN -Jack Polynomial is a limit of the Macdonald Polynomial when both parameters in the latter approach the N th root of unity ωN = exp( 2πi N ). It is clear that for the glN -Jack Polynomials one can derive analogues of all properties known for Macdonald polynomials just by taking the limit. In precisely the same way as in the case N = 1 these properties are straightforward to apply in order to compute two-point Dynamical Correlation Functions in the Spin Calogero-Sutherland Model with arbitrary N. As an example, in this paper we give a derivation of Spin-Density and Density two-point Dynamical Correlation Functions for the case of N = 2 and non-negative integer β. Let us now outline our approach and results in a more detailed manner.

Yangian Gelfand-Zetlin Bases, Jack Polynomials and Correlation Functions

665

1.1. Main result. As we have already mentioned, we treat the Spin Calogero-Sutherland Model with non-negative integer β by mapping it isomorphically onto a problem formulated in the language of symmetric polynomials. ±1 ±1 Sn be the vector space of symmetric To be more precise, let 3± n = (C[x1 , . . . , xn ]) Laurent polynomials in variables x1 , . . . , xn with complex coefficients. For any Laurent polynomial f = f (x1 , . . . , xn ) we will denote by [f ]1 the constant term in f. And we will use the bar to denote the complex conjugation. The vector space 3± n is equipped with a scalar product which we denote by h · , · iβ,N . This scalar product is defined in terms of the weight function Y −N β (1 − xN ) (1 − xi x−1 (1.4) 1(β, N ; x1 , . . . , xn ) := i xj j ), 1≤i6=j≤n

and its value on any two symmetric Laurent polynomials f and g is given by the formula h f , g iβ,N =

i 1 h −1 f (x−1 1 , . . . , xn )1(β, N ; x1 , . . . , xn )g(x1 , . . . , xn ) . n! 1

(1.5)

Thus the Hilbert space structure on 3± n is defined. On the other hand the space of states FN,n is also a Hilbert space with the scalar product ( · , · )β,N which we define in Sect. 2. The Hilbert space 3± n has an orthogonal basis whose elements are parameterized by an integer number r and a partition λ with length less or equal to n − 1. The elements of this basis are (1.6) (x1 · · · xn )r Pλ(N β+1,N ) (x1 , . . . , xn ), where for any positive real number γ the symmetric polynomial Pλ(γ,N ) (x1 , . . . , xn ) is defined as the following limit of the Macdonald Polynomial Pλ (q, t; x1 , . . . , xn ) (cf. [17]): Pλ(γ,N ) = Pλ(γ,N ) (x1 , . . . , xn ) = lim Pλ (ωN p, ωN pγ ; x1 , . . . , xn ). (1.7) p→1

Here ωN is an N th elementary root of unity. We will call the polynomial Pλ(γ,N ) a glN Jack Polynomial since when N = 1 this polynomial is nothing but the Jack Polynomial −1 Pλ(γ ) in notations of Macdonald [17]. Here it may be useful to observe that with γ = N β + 1 the scalar product h · , · iβ,N is just the limit of the scalar product for Macdonald Polynomials [17]. Now we formulate our main result. Main result. In the Hilbert space of states FN,n of the Spin Calogero-Sutherland Model with β > 0 there exists an orthogonal eigenbasis of the Hamiltonian Hβ,N . The elements (β,N ) are parameterized by an integer r and a partition λ with length of this eigenbasis Xr,λ less or equal to n − 1. Moreover for integer positive β there exists an isomorphism of Hilbert spaces FN,n and 3± n such that (β,N ) ) = (x1 · · · xn )r Pλ(N β+1,N ) (x1 , . . . , xn ), (Xr,λ

where Pλ(N β+1,N ) (x1 , . . . , xn ) is the glN -Jack Polynomial.

(1.8)

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D. Uglov

Let us make two comments. When N = 1 the statement above is well-known. In this case acting with the isomorphism amounts simply to dividing a skew-symmetric Laurent polynomial by the Vandermonde determinant so that the result is a symmetric Laurent polynomial. Another comment is about the severity of the restriction that β be an integer. We need this restriction in order to be able to carry out all our proofs in a completely algebraic manner – so as not to deal with questions of convergence of various integrals. We conjecture that all our results and formulas, in particular the result above, are valid for all real positive β as well, modulo some evident modifications. We do not however have necessary proofs which would require analytical considerations. 1.2. Yangian Gelfand-Zetlin bases in the spin Calogero-Sutherland Model. Let us now (β,N ) } in the previous section. explain how we specify the eigenbasis {Xr,λ It is known from the work [1] that the space of states FN,n admits an action of the algebra Y (glN ) – the Yangian of glN [6, 20], such that this action commutes with the Hamiltonian Hβ,N . The Yangian Y (glN ) has a maximal commutative sub-algebra A(glN ) [2, 19, 20] which is generated by centers of all sub-algebras in the chain Y (gl1 ) ⊂ Y (gl2 ) ⊂ . . . ⊂ Y (glN ),

(1.9)

where Yangians Y (gl1 ) ⊂ Y (gl2 ) ⊂ . . . ⊂ Y (glN −1 ) are realized inside Y (glN ) by the standard embeddings [20] (see Sect. 3). The Hamiltonian Hβ,N is known [1] to belong to the center of the Y (glN )-action on FN,n which is generated by the Quantum Determinant. This means that the Hamiltonian belongs to the commutative family of operators A(glN ; β) which give action of the subalgebra A(glN ) on FN,n . Because of this it is natural to concentrate our attention on this commutative family rather than on the Hamiltonian alone. Doing this has two advantages. The first is that the spectrum of the commutative family A(glN ; β) is simple [24] unlike the spectrum of the Hamiltonian which has degeneracy coming from the Yangian symmetry. Therefore an eigenbasis of A(glN ; β) is defined uniquely up to normalization (β,N ) }. of eigenvectors. It is precisely the eigenbasis {Xr,λ The second advantage is that since each operator from the family A(glN ; β) is selfadjoint relative to the scalar product ( · , · )β,N [24], the elements of the eigenbasis (β,N ) {Xr,λ } are mutually orthogonal automatically because of the simplicity of the spectrum. According to [20] Yangian representations where the action of the sub-algebra A(glN ) is diagonalizable are called tame, and A(glN )-eigenbases in irreducible tame representations are called Yangian Gelfand-Zetlin bases. As was established in [24] (see also Sects.3.3 and 3.4 of the present paper) the space of states FN,n is a tame and com(β,N ) } is just a direct sum of pletely reducible Yangian representation. The basis {Xr,λ Yangian Gelfand-Zetlin bases of the irreducible components of FN,n . From the main result given in Sect. 1.1 and the above discussion it is apparent that the image of the commutative family A(glN ; β) under the isomorphism is nothing but the degeneration of the commutative family of Macdonald Operators of which Macdonald Polynomials are eigenvectors [17]. 1.3. Computation of the spin-density and density dynamical correlation functions for N = 2 . One of the consequences of our main result is that we may compute SpinDensity and Density Dynamical Correlation Functions in the Spin Calogero-Sutherland


667

Model (N ≥ 2) in practically the same way as in the Scalar Calogero-Sutherland Model (N = 1) [9, 15, 18]. The only extra work which has to be done in the case N ≥ 2 is to identify the ground state, and identify the operators on the space of symmetric Laurent polynomials that are obtained when we twist the Spin-Density and Density operators with the isomorphism . The identification of the ground state is a painstaking process of finding the state with the lowest energy eigenvalue which, however, is easy to do when N = 2 and the number of particles in the Model n is even such that n/2 is odd. In this case we find that the ground state is a basis in a one-dimensional Yangian representation and is, in particular, a spin singlet. It is mapped by the into the Laurent polynomial 1. Images of the Spin-Density and Density operators under the isomorphism are just power-sums acting as multiplication operators on the space of symmetric Laurent polynomials. In particular, when N = 2 the Spin-Density is mapped into a sum of odd power sums, and the Density is mapped into a sum of even power sums. This being established, our computation of the Correlation Functions for N = 2 follows exactly the computation for the scalar case [9, 15] with the only difference that we use the gl2 -Jack Polynomials instead of Jack Polynomials. As in the scalar case our result for e.g. the Spin-Density Correlation Function is represented as a sum over all partitions λ of length less or equal to n such that for a non-negative integer β the summand vanishes if the diagram of λ contains the square with leg-colength 1 and arm-colength 2β + 1 [17]. In the present paper we do not consider the thermodynamic limit of these Correlation Functions. Another problem which we do not consider in this paper is a computation of Green’s functions [9, 15]. This problem, however, also appears to be tractable in our approach if we take into account the Cauchy formula for glN -Jack Polynomials: X λ

Pλ(γ,N ) (x1 , . . . , xn )Pλ(γ0

−1

,N )

(y1 , . . . , yn ) =

n Y

(1 + xi yj ),

(1.10)

i,j=1

where λ0 is the partition conjugated to λ. This Cauchy formula is obviously the limit of the corresponding formula for Macdonald Polynomials [17]. 1.4. Plan of the paper. Let us now outline the plan of the present paper. In Sect. 2.1 we summarize some notational conventions to be used troughout the paper. We also define here the wedge vectors, or simply, wedges which form a basis of the space of states FN,n . The isomorphism mentioned in Sect. 1.1 will map these wedges into Schur polynomials [17], or, more precisely, into natural extensions of Schur polynomials which form a basis in the space of symmetric Laurent polynomials. In Sect. 2.2 we define an appropriate scalar product on the space FN,n . In Sect. 2.3 we describe the gauge-transformed Hamiltonian Hβ,N and recall its relation to the Cherednik-Dunkl operators following [1]. In Sect. 2.4 we construct a certain eigenbasis of Hβ,N . This eigenbasis is not orthogonal, but it plays an important role in subsequent considerations. Sections 3.1–3.4 deal with the Yangian symmetry of the Spin Calogero-Sutherland Model and the Yangian Gelfand-Zetlin bases. In Sect. 3.1 we summarize properties of the Yangian algebra giving particular attention to the maximal commutative subalgebra A(glN ). In Sect. 3.2 we recall the definition of the Yangian action in the Spin CalogeroSutherland Model following [1].

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Section 3.3 is the summary of the results of the paper [24] concerning the decomposition of the space of states FN,n into irreducible Yangian subrepresentations. In Sect. 3.4 we describe the eigenbasis of the commutative family A(glN ; β) in the (β,N ) } mentioned in the Sect. 1.1 of this introduction. space of states. This is the basis {Xr,λ Sections 4.1–4.5 are concerned with formulation of the Spin Calogero-Sutherland Model in the language of symmetric polynomials. In Sect. 4.1 we summarize notations concerning partitions. In Sect. 4.2 we define the isomorphism which was discussed in Sect. 1.1. Actually we define an infinite family of isomorphisms {K | K ∈ Z} between the space of states FN,n and the space of symmetric Laurent polynomials in variables x1 , . . . , xn . The isomorphisms K are related to each other by the trivial shift: K = (x1 . . . xn )K+1 ,

(1.11)

and the isomorphism whose existence is claimed in Sect. 1.1 is K with an arbitrary integer K. The main part of this section is the proof that each of the K is an isomorphism of Hilbert spaces i.e. that it respects scalar products. In the brief Sect. 4.3 we describe the basis in the space of symmetric Laurent polynomials obtained from the A(glN ; β)-eigenbasis by the map with the isomorphism K for any fixed K. In Sect. 4.4 we define the glN -Jack Polynomials and discuss some of their properties. In Sect. 4.5 we establish the main result of this paper which was described in Sect. 1.1. Finally, in Sects. 5.1 and 5.2 we compute the Correlation Functions. The Appendix contains proofs of some of the statements in the main text.

2. Gauge-Transformed Hamiltonian of the Spin Calogero-Sutherland Model In this part of the paper we summarize some properties of the gauge-transformed Hamiltonian Hβ,N and give the definition of the Hilbert space of states on which it acts. This part mainly follows the paper [1] and our primary objective here is to introduce our notations and to formulate the definition of the model in a way suitable for our subsequent considerations. 2.1. Preliminary remarks and notations. Let N be a positive integer. In this paper N has the meaning of the number of spin degrees of freedom of each particle in the Spin Calogero-Sutherland Model. For any integer k define the unique k ∈ {1, . . . , N } and the unique k ∈ Z by setting k = k − N k. And for a k = (k1 , k2 , . . . , kn ) ∈ Zn set k = (k1 , k2 , . . . , kn ), k = (k1 , k2 , . . . , kn ). For any sequence k = (k1 , k2 , . . . , kn ) ∈ Zn let |k| be the weight: |k| = k1 +k2 +· · ·+kn , and define the partial ordering ( the natural or the dominance ordering [17] ) on Zn by setting for any two distinct k, l ∈ Zn : iff

|k| = |l|,

k>l and k1 + · · · + ki ≥ l1 + · · · + li

for all

i = 1, 2, . . . , n.

(2.1)

n For r ∈ N let L(r) n be a subset of Z defined as n L(r) n = {k = (k1 , k2 , . . . , kn ) ∈ Z | ki ≥ ki+1

and

∀s ∈ Z #{ki | ki = s} ≤ r}. (2.2)


669

(1) In particular the L(n) n is the set of non-increasing sequences of n integers and the Ln is the set of strictly decreasing sequences, i.e. such k = (k1 , k2 , . . . , kn ) ∈ Zn that ki > ki+1 . Let V = CN with the basis {v1 , v2 , . . . , vN } and let V (z) = C[z ±1 ]⊗V with the basis {uk | k ∈ Z}, where uk = z k ⊗ vk . For monomials in vector spaces C[z1±1 , . . . , zn±1 ], ⊗n V and ⊗n V (z) = C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) we will use the convention of multiindices:

z t = z1t1 z2t2 · · · zntn , v(a) = va1 ⊗ va2 ⊗ · · · ⊗ van , u k = uk 1 ⊗ u k 2 ⊗ · · · ⊗ uk n ,

t = (t1 , t2 , . . . , tn ) ∈ Zn ; a = (a1 , a2 , . . . , an ) ∈ {1, . . . , N }n ; k = (k1 , k2 , . . . , kn ) ∈ Zn .

(2.3) (2.4) (2.5)

Let Kij be the permutation operator for variables zi and zj in C[z1±1 , . . . , zn±1 ] ( operator of coordinate permutation ), and let Pij be the operator exchanging ith and j th factors in the tensor product ⊗n V (operator of spin permutation ). Let An be the antisymmetrization operator in ⊗n V (z): X An (uk1 ⊗ uk2 ⊗ · · · ⊗ ukn ) = sign(w)ukw(1) ⊗ ukw(2) ⊗ · · · ⊗ ukw(n) , (2.6) w∈Sn

where Sn is the symmetric group of order n. We will use the notation uˆ k = uk1 ∧ uk2 ∧ · · · ∧ ukn for a vector of the form (2.6), and will call such a vector a wedge. A wedge uˆ k is normally ordered if k ∈ L(1) n , that is k1 > k2 > . . . > kn . Let FN,n be the image of the operator An in ⊗n V (z). Then FN,n is spanned by wedges and the normally ordered wedges form a basis in FN,n . Equivalently the vector space FN,n is defined as the linear span of all vectors f ∈ C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) such that for all 1 ≤ i 6= j ≤ n: Kij f = −Pij f.

(2.7)

This is the definition of FN,n adopted in [1]. 2.2. Scalar product. Here we define a scalar product on the space of states FN,n . Our definition has three steps. First we define scalar products on the vector spaces ⊗n V and C[z1±1 , . . . , zn±1 ] separately. Then we define a scalar product on the tensor product ⊗n V (z) = C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ). Finally we define a scalar product on FN,n considered as a subspace of ⊗n V (z). On ⊗n V define a sesquilinear, i.e., anti-linear in the first argument and linear in the second argument, scalar product ( · , · )N by requiring that pure tensors in ⊗n V be orthonormal: ( v(a) , v(b) )N = δab , a, b ∈ {1, . . . , N }n . (2.8) For w = (w1 , w2 , . . . , wn ) ∈ Cn , |w1 | = |w2 | = . . . = |wn | = 1 and a non-negative real number δ let: Y 1(w; δ) = (1 − wi wj−1 )δ , (2.9) 1≤i6=j≤n

and define for all f (z), g(z) ∈

C[z1±1 , . . . , zn±1 ]

( f (z) , g(z) )0δ =

1 Y n! n

j=1

Z

a scalar product ( · , · )0δ by setting :

dwj 1(w; δ)f (w)g(w), 2πiwj

(2.10)

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D. Uglov

where the integration over each of the variables wj is taken along the unit circle in the complex plane, and the bar over f (w) means complex conjugation. On the linear space ⊗n V (z) = C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) we define a scalar product ( · , · )0δ,N as the composition of the scalar products (2.10) and (2.8), i.e. for f (z), g(z) ∈ C[z1±1 , . . . , zn±1 ]; u, v ∈ ⊗n V we set: ( f (z) ⊗ u , g(z) ⊗ v )0δ,N = ( f (z) , g(z) )0δ ( u , v )N ,

(2.11)

and extend the definition on all vectors by requiring that ( · , · )0δ,N be sesquilinear. Finally, on the subspace FN,n ⊂ ⊗n V (z) a scalar product ( · , · )δ,N is defined as the restriction of the scalar product ( · , · )0δ,N . Note that the normally ordered wedges are orthonormal relative to this scalar product when δ = 0: ( uˆ k , uˆ l )0,N = δkl ,

k, l ∈ L(1) n .

(2.12)

2.3. The gauge-transformed Hamiltonian. Here we briefly recall the relationship between the gauge-transformed Hamiltonian and the Cherednik-Dunkl operators [3, 4, 1]. We will use the following convention: if B is an operator acting on C[z1±1 , . . . , zn±1 ] ( or ⊗n V ) then we will denote by the same letter B the operator B ⊗ id ( or id ⊗ B ) acting on the space C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ). Q π (yi − yj ) and set zj = exp( 2πi Let W = 1≤i<j≤n sin L L yj ). Then the gaugetransformed Hamiltonian Hβ,N is defined as follows [1]: L2 −β W H β,N W β 2π 2 n n X X 2 = Di + β (2i − n − 1)Di

Hβ,N =

i=1

+ 2β

X

(2.13)

i=1

1≤i<j≤n

β 2 n(n2 − 1) , θij Di − Dj + θji (Pij + 1) + 12

where we set: Di = zi ∂/∂zi and θij = zi /(zi − zj ). The Hamiltonian Hβ,N acts in the space FN,n [1]. To see this let us recall, that the Cherednik-Dunkl operators [3, 7]: X X θji (Kij − 1) − θij (Kij − 1), (i = 1, 2, . . . , n), (2.14) di (β) = β −1 Di − i + i<j

i>j

act on C[z1±1 , . . . , zn±1 ] and together with permutations Kij satisfy the defining relations of the degenerate affine Hecke algebra: Kii+1 di (β) − di+1 (β)Kii+1 = 1, Kii+1 dj (β) = dj (β)Kii+1 , (j 6= i, i + 1), di (β)dj (β) = dj (β)di (β).

(2.15) (2.16) (2.17)

These relations imply, in particular, that symmetric polynomials in Cherednik-Dunkl operators commute with permutations Kij and therefore are well-defined operators on the space FN,n . The action of the operator


β2

n X

di (β) +

i=1

+ 2β

X

n+1 2 θij

1≤i<j≤n

2 =

n X i=1

Di2 + β

n X

671

(2i − n − 1)Di

i=1

β 2 n(n2 − 1) Di − Dj − θji (Kij − 1) + 12

(2.18)

on any vector f ∈ FN,n is seen to coincide with the action of the Hamiltonian Hβ,N since Kij f = −Pij f . Thus we may write: 2 n X n+1 2 Hβ,N = β (2.19) di (β) + 2 i=1

as operators on FN,n . Note that from the last equation it follows that the Hβ,N is self-adjoint relative to the scalar product ( · , · )β,N since the Cherednik-Dunkl operators are self-adjoint relative to the scalar product ( · , · )0β as one can easily verify. 2.4. An eigenbasis of the Hamiltonian Hβ,N . In this section we construct a certain eigenbasis of the Hamiltonian Hβ,N . This eigenbasis is not orthogonal with respect to the scalar product ( · , · )β,N . However it has a triangular expansion in the basis of wedges. This is an important property to be used later in Sect. 3. Let k = (k1 , k2 , . . . , kn ) ∈ L(1) n . Note that k1 > k2 > . . . > kn implies, in particular, that k1 ≤ k2 ≤ . . . ≤ kn . And let us act with the Hamiltonian Hβ,N on the (normally ordered) wedge uˆ k = uk1 ∧ uk2 ∧ . . . ∧ ukn . Using the expression (2.18) we find: X hij uˆ k , (2.20) Hβ,N uˆ k = E(k; β)uˆ k + 2β 1≤i<j≤n

where

Pn Pn 2 2 2 −1) E(k; β) = i=1 ki + β i=1 (2i − n − 1)ki + β n(n , 12 and hij (uk1 ∧ . . . ∧ uki ∧ . . . ∧ ukj ∧ . . . ∧ ukn ) =

=

Pkj −ki −1 r=1

(2.21)

(kj − ki − r)(uk1 ∧. . .∧uki −N r ∧. . .∧ukj +N r ∧. . .∧ ukn ). (2.22)

Normally ordering the wedges in the right-hand side of (2.20) we find from (2.22) that: X h(β) ˆ l, (2.23) Hβ,N uˆ k = E(k; β)uˆ k + kl u l∈L(1) n , l>k, l k. Then (2.23) leads to: Proposition 1. (β) For any k ∈ L(1) n there is a unique eigenvector 9k of Hβ,N such that: (2.24) X (β) (β) ˆk + ψkl uˆ l (ψkl ∈ R). 9(β) k =u l∈L(1) n , l>k

Eigenvalue of Hβ,N for this eigenvector is E(k; β). The coefficient

(β) ψkl

vanishes unless l < k.

(2.25) (2.26)

672

D. Uglov

Note that for any integer M the wedge: vac(M ) = uM ∧ uM −1 ∧ · · · ∧ uM −n+1

(2.27)

is an eigenvector of the Hamiltonian Hβ,N as implied by either (2.24) or (2.20, 2.22). We will call any vector of the form (2.27) a vacuum vector. As we have mentioned already, the eigenbasis {9(β) k } is not orthogonal. To construct an orthogonal eigenbasis we need to utilize the Yangian symmetry of the model as discussed in the next part of the paper. 3. Yangian Gelfand-Zetlin Bases and an Orthogonal Eigenbasis of the Spin Calogero-Sutherland Model Our objective in this part of the paper is to construct an orthogonal eigenbasis of the Spin Calogero-Sutherland Model. As was shown in the work [24] to do this it is natural to use the Yangian symmetry of the model. The orthogonal eigenbasis then is defined uniquely up to normalization as the eigenbasis of the commutative family of operators which give the action of the maximal commutative subalgebra in the Yangian action on the space of states FN,n . 3.1. The Yangian of glN and its maximal commutative subalgebra. The Yangian Y (glN ) (s) , where a, b ∈ [6] is an associative unital algebra with generators: the unit 1 and Tab {1, . . . , N } and s = 1, 2, . . . . In terms of the formal power series in variable u−1 : (1) (2) Tab (u) = δab 1 + u−1 Tab + u−2 Tab + ··· ,

(3.1)

the relations of Y (glN ) are written as follows (u − v)[Tab (u), Tcd (v)] = Tcb (v)Tad (u) − Tcb (u)Tad (v),

(3.2)

and the coproduct 1 : Y (glN ) → Y (glN ) ⊗ Y (glN ) is given by: 1(Tab (u)) = PN c=1 Tac (u) ⊗ Tcb (u). The center of Y (glN ) is generated by coefficients of the series X sign(w)T1w(1) (u)T2w(2) (u − 1) · · · TN w(N ) (u − N + 1) (3.3) AN (u) = w∈SN

called the Quantum Determinant of Y (glN ). The Yangian has a distinguished maximal commutative subalgebra A(glN ) [2, 19, 20]. This subalgebra is generated by coefficients of the series A1 (u), A2 (u), . . . , AN (u) defined as follows: X sign(w)T1w(1) (u)T2w(2) (u − 1) · · · Tmw(m) (u − m + 1), Am (u) = (3.4) w∈Sm m = 1, 2, . . . , N. That is to say by the centers of all algebras in the chain: Y (gl1 ) ⊂ Y (gl2 ) ⊂ . . . ⊂ Y (glN ),

(3.5)

where for m = 1, 2, . . . , N − 1 the Y (glm ) is realized inside Y (glN ) as the subalgebra generated by coefficients of the series Tab (u) with a, b = 1, 2, . . . , m.


673

Finite or infinite dimensional Y (glN )-modules with a semisimple action of the subalgebra A(glN ) are called tame, and eigenbases of A(glN ) in irreducible tame modules are called (Yangian) Gelfand-Zetlin bases [20]. Let us now consider certain tame Yangian modules which appear in the context of the Spin Calogero-Sutherland Model. Let f ∈ C and π(f ) : Y (glN ) → End(V ) be the Y (glN )-homomorphism defined by: π(f )(Tab (u)) = δab 1 +

Eba , u+f

(3.6)

where Eba ∈ End(V ) is the matrix unit: Eba vc = δac vb . Also for f = (f1 , f2 , . . . , fn ) ∈ Cn we will denote by π(f ) the tensor product of the homomorphisms (3.6): π(f ) = π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn ) : ⊗n Y (glN ) → End(⊗n V ), so that π(f ) 1(n) (Tab (u)) , where 1(n) is the coproduct iterated n − 1-times, defines a Y (glN )-module structure on ⊗n V . Let now M be an integer ( unrelated to the M in (2.27)) and let p = (p1 , p2 , . . . , pM ) ∈ {1, . . . , N }M be a sequence of positive integers such that: n = p1 + p2 + · · · + pM .

(3.7)

With these (p1 , p2 , . . . , pM ) define integers q0 , q1 , . . . , qM by ps = qs − qs−1 ,

(q0 := 0),

(s = 1, 2, . . . , M ).

(3.8)

For 1 ≤ i < j ≤ n define the partial anti-symmetrization operator A(i,j) ∈ End(⊗n V ) by setting for a = (a1 , . . . , an ) ∈ {1, . . . , N }n : A(i,j) (va1 ⊗ va2 ⊗ · · · ⊗ van ) := X sign(w)va1 ⊗ va2 ⊗ · · · ⊗ vai ⊗ vai+w(1) ⊗ vai+w(2) ⊗

(3.9)

w∈Sj−i

· · · ⊗ vai+w(j−i) ⊗ vaj+1 ⊗ · · · ⊗ van . And let (⊗n V )p be the image in ⊗n V of the operator: Ap :=

M Y

A(qs−1 ,qs ) .

(3.10)

s=1

If we define the set Tp labeled by the sequence p by

then the set

Tp := {a = (a1 , . . . , an ) ∈ {1, . . . , N }n | ai < ai+1 when qs−1 < i < qs for each s = 1, . . . , M },

(3.11)

{ϕ(a) := Ap v(a) | a ∈ Tp }

(3.12)

n

is a basis of (⊗ V )p . Further, let f = (f1 , f2 , . . . , fn ) be a sequence of real numbers satisfying the following two conditions: fi = fi+1 + 1 when qs−1 < i < qs for each s = 1, 2, . . . , M ; and fqs > fqs +1 + 1 for s = 1, 2, . . . , M − 1. Then we have:

(C1) (C2)

674

D. Uglov

Proposition 2. The coefficients of π(f )1(n) (Tab (u)) ∈ End(⊗n V )[[u−1 ]] leave the subspace (⊗n V )p ⊂ ⊗n V invariant. In (⊗n V )p there is a unique up to normalization of eigenvectors π(f )A(glN ) − eigenbasis: {χ(a) | a ∈ Tp }. X c(a, b)ϕ(b), c(a, b) ∈ R. χ(a) = ϕ(a) +

(3.13) (3.14) (3.15)

b>a

π(f )1(n) (Am (u))χ(a) = Am (u; f ; a)χ(a), m = 1, 2, . . . , N ; n Y u + fi + δ(ai ≤ m) where Am (u; f ; a) := . u + fi

(3.16)

i=1

The N -tuples of rational functions in u: A1 (u; f ; a), . . . , AN (u; f ; a) (3.17) are distinct for distinct a ∈ Tp . In other words: the spectrum of the π(f )A(glN ) on (⊗n V )p is simple. In the expression for the eigenvalue Am (u; f ; a) above we have used the convention that for a statement P, δ(P ) = 1 if P is true, and δ(P ) = 0 otherwise. We will often use this convention in this paper. We give a proof of this proposition in the Appendix, Sect. A. 3.2. Yangian in the Spin Calogero-Sutherland Model. The space FN,n admits a Y (glN )i−1 n−i (i) = 1⊗ ⊗ Eab ⊗ 1⊗ ∈ End(⊗n V ), and let action defined as follows [1]: let Eab L(i) ab (u; β) = δab 1 +

(i) Eab . u + di (β)

(3.18)

(n) (2) Set Tab (u; β) = L(1) ac1 (u; β)Lc1 c2 (u; β) · · · Lcn−1 b (u; β), where summation over the indices ci is assumed. The degenerate affine Hecke algebra relations satisfied by the Cherednik-Dunkl operators imply that coefficients of the series Tab (u; β) act on FN,n and satisfy the defining relations of the Yangian [1]. Denote by Y (glN ; β) the Y (glN )action on FN,n defined by the Tab (u; β) and denote by A(glN ; β) the corresponding action of the subalgebra A(glN ). In particular the Quantum Determinant of Tab (u; β) has the form [1]: n Y u + 1 + di (β) , (3.19) AN (u; β) = u + di (β) i=1

and therefore the Hamiltonian Hβ,N (2.3) is an element in the center of the action Y (glN ; β) and hence an element in A(glN ; β). For an operator B acting on FN,n let B ∗ be its adjoint relative to the scalar product ( · , · )β,N and for a series B(u) with operator-valued coefficients we will use B(u)∗ to denote the series whose coefficients are adjoints of coefficients of the series B(u). In [24] it is shown, that generators of the action Y (glN ; β) satisfy the following relations: (3.20) Tab (u; β)∗ = Tba (u; β). These relations imply, in particular, that the action of the subalgebra A(glN ; β) is selfadjoint. That is we have [24]: Am (u; β)∗ = Am (u; β),

m = 1, 2, . . . , N.

(3.21)


675

3.3. Decomposition of the space of states with respect to the Yangian action Y (glN ; β). Here we recall the decomposition of the space of states FN,n into irreducible Yangian representations with respect to the Yangian action Y (glN ; β). For details one may consult the work [24] which contains proofs of the statements made in this section. The Yangian decomposition of the space of states FN,n is constructed by using the Non-symmetric Jack Polynomials (see [21, 5, 16]). Non-symmetric Jack polynomials Et(β) (z) are labelled by t = (t1 , . . . , tn ) ∈ Zn and form a basis of C[z1±1 , . . . , zn±1 ]. For any t ∈ Zn let t+ denote the element of the set L(n) n (2.2) obtained by arranging parts of t in non-increasing order. The polynomials Et(β) (z) have the triangular expansion in the monomial basis: X (β) etr z r , (3.22) Et(β) (z) = z t + r≺t

where e(β) tr are real coefficients and + t > r+ or r≺t ⇔ t+ = r+ and the last non-zero difference ti − ri is negative. Moreover, polynomials Et(β) (z) are eigenvectors of the Cherednik-Dunkl operators: di (β)Et(β) (z) = fi (t; β)Et(β) (z),

(i = 1, 2, . . . , n),

−1

where fi (t; β) = β ti − ρi (t), and ρi (t) = #{j ≤ i | tj ≥ ti } + #{j > i | tj > ti }.

(3.23) (3.24) (3.25)

In [24] it is shown that the space FN,n splits into an infinite number of irreducible ) Yangian submodules Fs labelled by elements of the set L(N n (2.2): ) Fs . FN,n = ⊕s∈L(N n

(3.26)

) To describe the component Fs which corresponds to a given s ∈ L(N let M be the n number of distinct elements in the sequence s = (s1 , s2 , . . . , sn ). And let q0 , q1 , . . . , qM be defined by: q0 = 0, qM = n; sqj > sqj +1 , j = 1, 2, . . . , M − 1. As in (3.8) a sequence p(s) = (p1 , p2 , . . . , pM ) is defined by: pj = qj − qj−1 , j = 1, 2, . . . , M . Clearly we have p1 + p2 + · · · + pM = n. As a Y (glN )-module the space Fs is isomorphic to (⊗n V )p(s) (3.1) where the Yangian action is given by π(f (s))1(n) (Tab (u)) and f (s) = (f1 , f2 , . . . , fn ) with fi = fi (s; β). This isomorphism is explicitly given by the operator U (s; β) : (⊗n V )p(s) → Fs which is defined for any v ∈ (⊗n V )p(s) as follows: X (β) U (s; β)v = Et (z) ⊗ Rt(β) v, (3.27) t∼s

where the sum is taken over all distinct rearrangements t of s, and Rt(β) ∈ End(⊗n V ) is defined by the recursive realtions: Rs(β) = 1, (β) Rt(i,i+1)

= −Rˇ ii+1 (fi (t; β) − fi+1 (t; β))Rt(β)

(3.28) for

ti > ti+1 .

(3.29)

676

D. Uglov

Here Rˇ ii+1 (u) = u−1 + Pii+1 and t(i, i + 1) denotes the element of Zn obtained by interchanging ti and ti+1 in t = (t1 , . . . , tn ). Observe now that, since β > 0, the conditions (C1,C2) are satisfied by the sequence f (s). Therefore setting for all a ∈ Tp(s) : 8(β) (s, a) := U (s; β)ϕ(a), X (β) (s, a) := U (s; β)χ(a);

(3.30) (3.31)

from (3.14 - 3.17) and the fact that Fs is isomorphic to (⊗n V )p(s) we obtain the following: Proposition 3. {X (β) (s, a) | a ∈ Tp(s) } is the unique up to normalization of eigenvectors (3.32) A(glN ; β) − eigenbasis of Fs . X X (β) (s, a) = 8(β) (s, a) + c(a, b)8(β) (s, b), c(a, b) ∈ R. (3.33) b>a

(s, a) = Am (u; f (s); a)X (β) (s, a), m = 1, 2, . . . , N ; n Y u + fi (s; β) + δ(ai ≤ m) Am (u; f (s); a) := . u + fi (s; β)

Am (u; β)X where

(β)

(3.34)

i=1

The N -tuples of rational functions in u:A1 (u; f (s); a), . . . , AN (u; f (s); a) (3.35) are distinct for distinct a ∈ Tp(s) . Thus the set {X (β) (s, a) | a ∈ Tp(s) } is a Yangian Gelfand-Zetlin basis of the irreducible Yangian representation Fs . 3.4. A(glN ; β)-eigenbasis of the space of states FN,n . Here we define the orthogonal eigenbasis of the commutative family A(glN ; β) in the entire space of states FN,n . This eigenbasis is just the union of bases {X (β) (s, a) | a ∈ Tp(s) } taken over all elements s ) from the set L(N n . We prefer, however, to choose a different parameterization of elements in this basis, such that a triangularity of these elements expanded in the basis of normally ordered wedges becomes manifest. For any k = (k1 , k2 , . . . , kn ) ∈ L(1) n the sequence s = (kn , kn−1 , . . . , k1 ) is an element ) of the set L(N . And the sequence a = (kn , kn−1 , . . . , k1 ) belongs to the set Tp(s) defined n by the sequence s as in the previous section. With these k, s and a we have: (−1)

n(n−1) 2

8(β) (s, a) = 9(β) k ,

(3.36)

where the 9(β) k is the eigenvector of the Hamiltonian Hβ,N defined in (2.24). A proof of the equality (3.36) is contained in the Appendix, Sect. B. Now with the same k, s and a as above let us define: Xk(β,N ) := (−1)

n(n−1) 2

X (β) (s, a).

Then we have the following statements about the vectors Xk(β,N )

(3.37)


677

Proposition 4. {Xk(β,N ) | k ∈ L(1) n } is the unique up to normalization of eigenvectors (3.38) A(glN ; β) − eigenbasis of FN,n . X x(β) ˆ l , x(β) (3.39) Xk(β,N ) = uˆ k + kl u kl ∈ R. l∈L(1) n , la

and hence: (π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))ϕ(a) = X = c(u; f ; a, a)v(a) + d(u; f ; a, b)v(b), (d(u; f ; a, b) ∈ R[[u−1 ]]).

(A.11)

b>a

On the other hand the subspace (⊗n V )p is left invariant by the Yangian action, so that

(π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))ϕ(a) = X = g(u; f ; a, b)ϕ(b), (g(u; f ; a, b) ∈ R[[u−1 ]]). b

Comparing this equation with (A.11) and using (A.10) we get

(A.12)

694

D. Uglov

(π(f1 ) ⊗ π(f2 ) ⊗ · · · ⊗ π(fn )) 1(n) (Am (u))ϕ(a) = X = Am (u; f ; a)ϕ(a) + c(u; f ; a, b)ϕ(b), (c(u; f ; a, b) ∈ R[[u−1 ]]).

(A.13)

b>a

Now define the sequence of real numbers h(1) , h(2) , . . . , h(M ) by h(s) := fqs−1 +1 ,

(s = 1, 2, . . . , M ).

(A.14)

And define for any a ∈ Tp : qs X

(s) lm =

δ(ai ≤ m),

(m = 1, 2, . . . , N ; s = 1, 2, . . . , N ).

(A.15)

i=qs−1 +1

Then for the eigenvalue Am (u; f ; a) we have Am (u; f ; a) =

M Y

u + 1 + h(s)

s=1

(s) u + 1 + h(s) − lm

.

(A.16)

The correspondence (s) | s = 1, 2, . . . , M ; Tp → {lm

m = 1, 2, . . . , N }

(A.17)

given by the relation (A.15) is bijective. The conditions (C1, C2) in Sect. 3.1 and the (s) ≤ qs − qs−1 imply that inequalities 0 ≤ lm (s) (s+1) h(s) − lm − h(s+1) + lm > 0,

(s = 1, 2, . . . , M − 1;

m = 1, 2, . . . , N ), (A.18)

so that the N -tuple of rational functions in the variable u: A1 (u; f ; a), A2 (u; f ; a), . . . , (s) | s = 1, 2, . . . , M, m = 1, 2, . . . , N } uniquely. AN (u; f ; a) determines the set {lm This proves the statement (3.17). Now the statements (3.14),(3.15) and (3.16) follow from (A.13) and (3.17). Thus the proof is finished. B. Proof of Equality (3.36) Using the expression (3.27) for the operator U (s; β) we have X (β) Et (z) ⊗ Rt(β) ϕ(a), 8(β) (s, a) =

(B.1)

t∼s

where t ∼ s means that t belongs to the set of all distinct rearrangements of the sequence ) s ∈ L(N n . In view of the triangularity (3.22) of the Non-symmetric Jack Polynomial (β) Et (z) we may split the Et(β) (z) with t ∼ s as follows: Et(β) (z) = Et(β) (z)0 + Et(β) (z)00 , where Et(β) (z)0 =

X rt, r∼s

Both of the vectors

r e(β) tr z ,

and

Et(β) (z)00 =

(B.2) X

) m∈L(N n ,

m<s, r∼m

r e(β) tr z .

(B.3)


X

Et(β) (z)0 ⊗ Rt(β) ϕ(a)

X

and

t∼s

695

Et(β) (z)00 ⊗ Rt(β) ϕ(a)

(B.4)

t∼s

belong to the subspace FN,n since 8(β) (s, a) belongs to FN,n and monomials z r which appear in the decomposition of Et(β) (z)0 as a vector in C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) are distinct from monomials z r which appear in the decomposition of Et(β) (z)00 . Taking into account that Rs(β) = 1 we may write: X

X

Et(β) (z)0 ⊗ Rt(β) ϕ(a) = z s ⊗ ϕ(a) +

t∼s

z t ⊗ ϕ¯ t (a),

(ϕ¯ t (a) ∈ ⊗n V ). (B.5)

t∼s, t≺s

Since the expression above is a vector in FN,n and by using the definition of the vector ϕ(a) in (3.12) we obtain X

Et(β) (z)0 ⊗ Rt(β) ϕ(a) = An (z s ⊗ v(a)) = (−1)

n(n−1) 2

uˆ k ,

(B.6)

t∼s

where An (2.6) is the operator of total antisymmetrization in the space C[z1±1 , . . . , zn±1 ]⊗ (⊗n V ), and as in the paragraph preceding Eq. (3.36) we have k = (k1 , k2 , . . . , kn ) ∈ L(1) n such that s = (kn , kn−1 , . . . , k1 ),

a = (kn , kn−1 , . . . , k1 ).

(B.7)

Furthermore, the expansion of the vector X

Et(β) (z)00 ⊗ Rt(β) ϕ(a)

(B.8)

t∼s

in C[z1±1 , . . . , zn±1 ] ⊗ (⊗n V ) contains only monomials z r such that r+ < s. Therefore expansion of (B.8) in the basis {uˆ l | l ∈ L(1) n } contains only normally ordered wedges uˆ l such that l > k. Taking this, and (B.6) into account we have (−1)

n(n−1) 2

8(β) (s, a) = uˆ k +

X l∈L(1) n,

ϕ(β) ˆl kl u

(ϕ(β) kl ∈ R).

(B.9)

l>k

However according to (2.24) an eigenvector of the Hamiltonian Hβ,N with the above (β) expansion in the basis {uˆ l | l ∈ L(1) n } is unique and equals to 9k . This proves (3.36). Acknowledgement. I am grateful to Kouichi Takemura with whom we collaborated on the paper [24] from which the present work has evolved and to Professor P.J. Forrester for encouragement and discussions during his stay at RIMS, Kyoto University in Summer 1996. I am also grateful to Professors T. Miwa and M. Kashiwara for discussions and support. After this work had been completed I have learned about the unpublished typescript of Y. Kato [12] in which he computes the Green Function of the Spin Calogero-Sutherland Model for N = 2 and β = 1 and conjectures expressions for the Green Function at all integer values of β. The method of Kato is completely different from the method of our paper and is in the spirit of the papers [13, 8]. I am grateful to T. Yamamoto for bringing the paper [12] to my attention.

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References 1. Bernard, D., Gaudin, M., Haldane, F.D.M. and Pasquier, V.: Yang-Baxter equation in long-range interacting systems. J. Phys., A26, 5219–5236 (1993) 2. Cherednik, I.V.: A new interpretation of Gelfand-Zetlin bases. Duke Math. J. 54, 563–577 (1987) 3. Cherednik, I.V.: A unification of the Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras. Inv. Math., 106, 411–432 (1991) 4. Cherednik, I.V.: Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations. Preprint RIMS-776 (1991); Advances in Math. 106, 65–95 (1994) 5. Cherednik, I.V.: Double Affine Hecke algebras and Macdonald’s conjectures. Annals Math. 141, 191– 216 (1995); Non-symmetric Macdonald Polynomials IMRN 10, 483–515 (1995) 6. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212– 216 (1988); “Quantum Groups.” In: Proceedings of the International Congress of Mathematicians, Am. Math. Soc., Providence, RI. 1987 pp. 798–820 7. Dunkl, C.F.: Trans. Am. Math. Soc. 311, 167 (1989) 8. Forrester, P.J.: Int. J. Mod. Phys. B9, 1234 (1995) 9. Ha, Z.N.C.: Phys. Rev. Lett. 73, 1574 (1994); Nucl. Phys., B435[FS], 604 (1995) 10. Jimbo, M.: it Topics from representation theory of Uq (g) – an introductory guide to physicists. Nankai Lectures on Mathematical Physics, Singapore: World Scientific, 1992 11. Kac, V.G. and Raina, A.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. Singapore: World Scientific, 1987 12. Kato, Y.: Green Function of the Sutherland Model with SU (2) internal symmetry. Preprint 13. Kato, Y. and Kuramoto, Y.: Phys. Rev. Lett. 74, 1222 (1995) 14. Konno, H.: Nucl. Phys. B473, 579 (1996) 15. Lesage, F., Pasquier, V. and Serban, D.: Nucl. Phys. 435[FS], 585 (1995) 16. Macdonald, I.G.: Affine Hecke algebra and Orthogonal Polynomials. Séminaire Bourbaki, 47 No. 797, 1–18 (1995) 17. Macdonald, I.G.: Symmetric functions and Hall polynomials 2-nd ed., London: Clarendon Press, 1995 18. Minahan, J. and Polychronakos, A.P.: Phys. Lett. B326, 288 (1994); Phys. Rev., B50, 4236 (1994) 19. Molev, A.I.: Gelfand-Tsetlin Bases for Representations of Yangians. Lett. Math. Phys., 30, 53–60 (1994) 20. Nazarov, M. and Tarasov, V.: Representations of Yangians with Gelfand-Zetlin bases. Preprint UWSMRRS-94-148 (1994). to be published in J. für Reine und Angew. Math. 21. Opdam, E.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995) 22. Stanley, R.P.: Adv. Math. 77, 76 (1989) 23. Sutherland, B.: J. Math. Phys. 12, 246, 251 (1971); Phys. Rev. A4, 2019 (1971); ibid. A5, 1372 (1972) 24. Takemura K. and Uglov D.: The Orthogonal Eigenbasis and Norms of Eigenvectors in the Spin CalogeroSutherland Model. Preprint RIMS-1114 (solv-int/9611006) 25. Uglov, D.: Semi-infinite wedges and the conformal limit of the fermionic spin Calogero-Sutherland model of spin 21 . Nucl. Phys. B478, 401–430 (1996) Communicated by G. Felder

Commun. Math. Phys. 191, 697 – 721 (1998)

Communications in


2-Cocycles and Twisting of Kac Algebras? Leonid Vainerman?? Lab. de Math. Fondamentales, Université Pierre et Marie Curie, Case 191, Aile 46-0, 3-éme e´ tage, 4 place Jussieu, F-75252 Paris Cedex 05, France. E-mail: [email protected] Received: 21 March 1997 / Accepted: 2 June 1997

Abstract: We describe the twisting construction with the help of 2-cocycles on Hopf– von Neumann and George Kac algebras; we show that twisted Kac algebras are again Kac algebras. Using this construction, we give a wide class of new quantizations of the Heisenberg group and describe several series of non-trivial finite- dimensional C∗ -Hopf algebras (Kac algebras) of dimensions 4n and 2n2 (n ∈ N) as twisting of finite groups. 1. Introduction 1.1. This paper can be considered as the development and generalization of [EVa]. There one can find a motivation and some results, notions and notations which we use here. We recall only the most important of them. 1.2. Let us recall that in ([L], §1 and 2), M. Landstad gave a construction of quantum groups, taking the von Neumann algebra L(G) generated by the left regular representation λG of a locally compact group G, and deforming the canonical coproduct 0G s of L(G) defined, for all g in G, by: 0G s (λG (g)) = λG (g) ⊗ λG (g), in order to obtain a new coproduct which can be written, for all x in L(G): ∗ 0 (x) = 0G s (x) ,

where is a “dual” 2-cocycle lifted from an abelian subgroup H of G. The idea of such a construction which is called “twisting” is due to V.G. Drinfeld [D]. ? This research was supported in part by Ukrainian Foundation for Fundamental Studies and by International Science Foundation ?? On leave of absence from: International Solomon University, Zabolotny Street, 38, apt. 61, 252187 Kiev, Ukraine. E-mail: [email protected]

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This procedure, deforming the coproduct of L(G), gives, on the predual A(G) of L(G), a deformation of the product; it is the point of view developed by M. Rieffel in ([R1], §1) or ([R2], §2). This leads to the dual situation, where one takes an algebra of functions on G, deforms the product in this algebra, and leaves untouched the canonical coproduct on this algebra. 1.3. In [EVa] it was shown that, under certain conditions, Landstad’s deformations of locally compact groups are Kac algebras (on the theory of Kac algebras we refer to [ES]), and the analogue of this construction for general Kac algebras was done. It is important to stress that the co-involution of the initial Kac algebra in that framework remained undeformed. The aim of the present paper is to give such a generalization of the twisting construction which includes a deformation of a co-involution, and to show that the result of such a deformation is again a Kac algebra. In this way we also get a much wider class of non-trivial Kac algebras. Notice that another construction of a deformation of a coproduct (or a multiplicative unitary), using a 2-cocycle, can be found in ([BS], 8.24, 8.26). 1.4. The paper is organized as follows: in the second section we give foundations of the theory of 2-cocycles, 2-pseudo-cocycles and the twisting of Hopf–von Neumann algebras. This section is inspired by ([M], 2.3). In the third section we deal with Woronowicz algebras; this is a class of objects introduced in [MN], which contains Kac algebras and also compact and discrete quantum groups. Then, the constructions of Sect. 2 allow us to consider the twisting of the multiplicative unitaries associated with Woronowicz algebras. In Sect. 4 we consider the twisting of Kac algebras and Woronowicz algebras by 2-cocycles lifted from 2-cocycles on their locally compact subgroups. Notice that in [EVa] we worked only with 2-cocycles lifted from special 2-cocycles associated with very special abelian subgroups. Then the main result of this paper is given: in the case of Kac algebras, when a 2-cocycle is of the above mentioned type, and the image of a subgroup belongs to the centralizer of the Haar weight, then the coinvolutive Hopf–von Neumann algebra obtained in Sect. 2, with undeformed Haar weight, is again the Kac algebra. We also describe the deformation of the corresponding multiplicative unitary. In Sect. 5 we give a class of new quantizations of the Heisenberg group, much wider than in ([EVa], 6.3). Some of them have a deformed antipode, and some have an undeformed one; however, all of them are unimodular Kac algebras. In Sect. 6 we describe several series of non-trivial finite-dimensional C∗ -Hopf algebras (Kac algebras) of dimensions 4n and 2n2 (n ∈ N) as twisting of finite groups. It should be noted, that finite-dimensional Kac algebras in the past few years found the important application in studying the irreducible inclusions of factors (see [EN, HS, IK] and the literature cited there).

2. 2-Cocycles and Twisting of Hopf–von Neumann Algebras Definition 2.1. Let (M, 0) be a Hopf–von Neumann algebra ([ES], 1.2.1). An element x ∈ M is said to be group-like if it is unitary: xx∗ = x∗ x = 1 and 0(x) = x ⊗ x. Obviously all group-like elements form a group - an intrinsic group of Hopf–von Neumann algebra (cf. [ES], 1.2.2).

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Definition 2.2. For all unitaries u in M (resp., in M ⊗ M ), where (M, 0) is a Hopf–von Neumann algebra, a notion of a coboundary ∂1 u in M ⊗ M (resp., ∂2 in M ⊗ M ⊗ M ), was defined in ([EVa], 2.2) by: ∂1 u = (u∗ ⊗ u∗ )0(u) (resp., ∂2 = (i ⊗ 0)(∗ )(1 ⊗ ∗ )( ⊗ 1)(0 ⊗ i)() ). A 1-cocycle of (M, 0) is a unitary u ∈ M such that ∂1 u = 1, or, equivalently: 0(u) = (u ⊗ u); A 2-cocycle of (M, 0) is a unitary in M ⊗ M such that ∂2 = 1, or, equivalently: ( ⊗ 1)(0 ⊗ i)() = (1 ⊗ )(i ⊗ 0)(). A 2-pseudo-cocycle of (M, 0) is a unitary in M ⊗ M such that ∂2 belongs to (0 ⊗ i)0(M )0 ([EVa], 2.3). Proposition 2.3. Let (M, 0) be a Hopf–von Neumann algebra, and a unitary in M ⊗ M ; then, let us put, for all x in M : 0 (x) = 0(x)∗ . Then, (M, 0 ) is a Hopf–von Neumann algebra, iff is a 2-pseudo-cocycle of (M, 0); and we shall say that (M, 0 ) (resp., 0 ) is twisted from (M, 0) (resp., from 0) by . Proof. The equality (0 ⊗ i)0 (x) = (i ⊗ 0 )(0 (x)) can be rewritten as ( ⊗ 1)(0 ⊗ i)()(0 ⊗ i)0(x)(0 ⊗ i)(∗ )(∗ ⊗ 1) = (1 ⊗ )(i ⊗ 0)()(i ⊗ 0)0(x)(i ⊗ 0)(∗ )(1 ⊗ ∗ ), or, using 2.2, as ∂2 (0 ⊗ i)0(x) = (i ⊗ 0)(0(x))∂2 , which gives the result.

Let us remark that in ([EVa], 2.5) it was shown that the condition above is sufficient. Proposition 2.4. Let (M, 0) be a Hopf–von Neumann algebra, a unitary in M ⊗ M , u a unitary in M . Let us put: u := (u∗ ⊗ u∗ )0(u). Then, u is a 2-pseudo-cocycle (resp., a 2-cocycle) for (M, 0), iff is. If both and u are 2-pseudo-cocycles, then ∂2 u = ∂2 .

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Proof. ∂2 u = (i ⊗ 0)(0(u∗ )∗ (u ⊗ u))× (1 ⊗ 0(u∗ )(u ⊗ u))((u∗ ⊗ u∗ )0(u) ⊗ 1)(0 ⊗ i)((u∗ ⊗ u∗ )0(u)) = (i ⊗ 0)0(u∗ )(i ⊗ 0)(∗ )(u ⊗ 0(u))(1 ⊗ 0(u∗ )(u ⊗ u))× ((u∗ ⊗ u∗ )0(u) ⊗ 1)(0(u∗ ) ⊗ u∗ )(0 ⊗ i)()(0 ⊗ i)0(u) = (i ⊗ 0)0(u∗ )(i ⊗ 0)(∗ )(1 ⊗ ∗ )( ⊗ 1)(0 ⊗ i)()(0 ⊗ i)0(u) = [(0 ⊗ i)0(u)]∗ ∂2 (0 ⊗ i)0(u). Now one can see that ∂2 u ∈ (0 ⊗ i)0(M )0 iff ∂2 ∈ (0 ⊗ i)0(M )0 , and that ∂2 u = 1 iff ∂2 = 1. Obviously, if both and u are 2-pseudo-cocycles, then ∂2 u = ∂2 . Definition 2.5. 2-pseudo-cocycles (resp., 2-cocycles) 1 and 2 are said to be pseudocohomologous (resp., cohomologous) if ∗1 (2 )u ∈ 0(M )0 (resp., 1 = (2 )u ) for some unitary u ∈ M . Obviously, the condition ∗1 (2 )u ∈ 0(M )0 is equivalent to any from the following conditions: [(2 )u ]∗ 1 ∈ 0(M )0 , [(1 )u ]∗ 2 ∈ 0(M )0 or ∗2 (1 )u ∈ 0(M )0 . Proposition 2.6. Let (M, 0) be a Hopf–von Neumann algebra, 1 a 2-pseudo- cocycle and (M, 01 ) the twisting of (M, 0) as in 2.3. Let 2 be a unitary in M ⊗ M and 02 (x) := 2 0(x)∗2 be a monomorphism from M to M ⊗ M . Then (M, 02 ) is the twisting of (M, 0) isomorphic to (M, 01 ) by means of an inner automorphism π(x) := uxu∗ of M (this means that 02 ◦ π = (π ⊗ π) ◦ 01 , u ∈ M is a unitary) iff 2 is a 2-pseudo-cocycle pseudo-cohomologous to 1 by means of u. Proof. Rewrite the relation 02 ◦ π = (π ⊗ π) ◦ 01 as 2 0(u)0(x)0(u∗ )(2 )∗ = (u ⊗ u)1 0(x)∗1 (u∗ ⊗ u∗ ) or as ∗1 (2 )u 0(x) = 0(x)∗1 (2 )u (∀x ∈ M ). After that the first statement is obvious. If the mentioned condition is satisfied, then (M, 02 ) is a Hopf–von Neumann algebra, from which we have, using 2.3, that 2 is a 2- pseudo-cocycle, pseudo-cohomologous to 1 . Obviously, in this case the twistings (M, 01 ) and (M, 02 ) are isomorphic. Corollary 2.7. Let 1 be a 2-pseudo-cocycle of a Hopf–von Neumann algebra (M, 0) and 2 be a unitary in M ⊗ M such that ∗1 (2 )u ∈ 0(M )0 for some unitary u ∈ M . Then 2 is a 2-pseudo-cocycle, pseudo-cohomologous to 1 . Proposition 2.8. Let (M, 0, κ) be a co-involutive Hopf–von Neumann algebra ([ES], 1.2.5), be a 2-pseudo-cocycle of (M, 0). Then a Hopf-von Neumann algebra (M, 0 ) posesses a co-involution of the form κ (x) = uκ(x)u∗ (u ∈ M is a unitary) iff ς(κ ⊗ κ)()()u ∈ 0(M )0 , where ς(x ⊗ y) = y ⊗ x, x, y ∈ M.


701

Proof. In fact, the relation 0 (κ (x)) = ς(κ ⊗ κ )0 (x) (∀x ∈ M ) is equivalent to the relation 0(uκ(x)u∗ )∗ = ς(u ⊗ u)(κ ⊗ κ)(0(x)∗ )(u∗ ⊗ u∗ ), or 0(u)0(κ(x))0(u∗ )∗ = (u ⊗ u)ς(κ ⊗ κ)(∗ )0(κ(x))ς(κ ⊗ κ)()(u∗ ⊗ u∗ ), or

ς(κ ⊗ κ)()u 0(κ(x)) = 0(κ(x))ς(κ ⊗ κ)()u ,

from which the result is obvious.

Remark 2.9. Let (M, 0, κ) be a co-involutive Hopf–von Neumann algebra, and be a 2-(pseudo-)cocycle for (M, 0); then it is straightforward to check that (κ ⊗ κ)(∗ ) is a 2-(pseudo-)cocycle for (M, ς0), and, therefore, that ς(κ ⊗ κ)(∗ ) is a 2-(pseudo)cocycle for (M, 0). Due to Corollary 2.7, Proposition 2.8 can be reformulated: a twisting (M, 0 ) of a co-involutive Hopf–von Neumann algebra (M, 0, κ) by means of a 2pseudo-cocycle posesses a co-involution of the form κ (x) = uκ(x)u∗ iff a 2-pseudococycle ς(κ ⊗ κ)(∗ ) is pseudo-cohomologous to . Definition 2.10. Let (M, 0, κ) be a co-involutive Hopf–von Neumann algebra, be a 2(-pseudo)-cocycle for (M, 0). We shall say that is pseudo-co-involutive (resp., co-involutive) if a 2(-pseudo)-cocycle ς(κ ⊗ κ)(∗ ) is pseudo-cohomologous to (resp.,ς(κ ⊗ κ)(∗ ) = u ). We shall say that is strongly co-involutive, if ς(κ ⊗ κ)(∗ ) = . Notice that the paper [EVa] was devoted to studying strongly co-involutive 2(-pseudo)cocycles of Hopf–von Neumann and Kac algebras (see, in particular, ([EVa], 2.8, 2.10). Examples 2.11. (i) Let G be a locally compact group, and let us define, for all f in L∞ (G), κa (f ) by the equality, for all s in G: −1 κG a (f )(s) = f (s ). G Then, (L∞ (G), 0G a , κa ) is a co-involutive Hopf–von Neumann algebra ([ES], 1.2.9); let ω be a 2(-pseudo)-cocycle on G, as defined in 2.2. If ω(t, s) satisfies, for almost all s, t in G the equality: ω(s−1 , t−1 ) = ω(t, s),

then it is strongly co-involutive (see also ([EVa], 2.9)). On the other hand, let us suppose that the 2-cocycle ω(t, s) is continuous on G × G and ω(e, s) = ω(s, e) = 1 (∀s ∈ G). Put in the main 2-cocycle equality s1 = s3 = (s2 )−1 = s−1 , then we have: u(s) = ω(s−1 , s) = ω(s, s−1 ) = (κG a u)(s). G ∗ −1 −1 Let us verify that a 2-cocycle ς(κG a ⊗ κa )(ω )(t, s) = ω(s , t ) is always cohomologous to ω(t, s) by means of the mentioned u(s), i.e., that

ω(s−1 , t−1 ) = ω(t−1 , t)ω(s−1 , s)ω(t, s)ω(s−1 t−1 , ts).

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In fact, using twice the main 2-cocycle equality, one has: ω(s−1 , t−1 )ω(s−1 t−1 , ts)ω(t, s) = ω(s−1 , t−1 )ω(s−1 t−1 , t)ω(s−1 , s) = ω(t−1 , t)ω(s−1 , s). So that, any mentioned 2-cocycle on a locally compact group is co-involutive. (ii) In the special case G = Rn the main 2-cocycle equality has the following form: ω(x, y)ω(x + y, z) = ω(y, z)ω(x, y + z) (∀x, y, z ∈ Rn ), and one can easily verify that this equality is satisfied for ω(x, y) := exp(iB(x, y)), where B(x, y) is any real bilinear form on Rn . Obviously, the above 2-cocycle is always co-involutive, and it is strongly co-involutive iff the bilinear form B(x, y) is skewsymmetric.

3. 2-Cocycles of Kac Algebras and Woronowicz Algebras 3.1. One can find the notions of a Kac algebra K = (M, 0, κ, ϕ), its fundamental unitary ˆ = (M ˆ the dual Kac algebra K ˆ , 0, ˆ κ, W , the canonical involutive isometries J and J, ˆ ϕ) ˆ and the duality theory for Kac algebras in [ES]. Similar notions and facts concerning Woronowicz algebras W = (M, 0, κ, τ, ϕ) can be found in [MN]. There one can also find the examples of Kac and Woronowicz algebras. The following statements 3.2, 3.3, 3.4, 3.5 are the generalizations of the corresponding statements ([EVa], 3.6, 3.7, 3.8, 3.9) for the case when the co-involution is also deformed (i.e., u is not necessarily 1). Because of that, the deformation of the fundamental unitary of a Kac algebra is more complicated. Proposition 3.2. Let K = (M, 0, κ, ϕ) be a Kac algebra (or W = (M, 0, κ, τ, ϕ) a Woronowicz algebra), W its fundamental unitary. Let R be unitary from M ⊗ M , and a 2-pseudo- cocycle of (M, 0). With the notations of 3.1, let us put ˜ R˜ = (Jˆ ⊗ J)R∗ (Jˆ ⊗ J) and W,R = W R. Then, for all x in M , we have ∗ 0 (x) = W,R (1 ⊗ x)W,R .

Proof. By the definition of J, we get that R˜ belongs to M ⊗ M 0 ; therefore, the result is trivial. Lemma 3.3. With the hypothesis of 3.2, let us suppose that is a co-involutive 2cocycle for (M, 0, κ), u is a corresponding unitary in M , and R = u V (where V is a unitary from M ⊗ M such that (0 ⊗ i)(V ) = (i ⊗ 0)(V )). We have then: ∗ ˜ ˜ ˜ u )(σ ⊗ i), (0 ⊗ i)(R)( ⊗ 1) = (σ ⊗ i)(1 ⊗ W ∗ )(σ ⊗ i)(1 ⊗ R)(σ ⊗ i)(1 ⊗ W )(1 ⊗

˜ u = (Jˆ ⊗ J)(u )∗ (Jˆ ⊗ J). where σ means the flip on Hϕ ⊗ Hϕ ,


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Proof. Let us write j(x) = Jx∗ J; this way we define an anti-isomorphism from M to M 0 , or from M 0 to M . We have then R˜ = (κ ⊗ j)(R), and, therefore ˜ = (0 ⊗ i)(κ ⊗ j)(R) = (ς(κ ⊗ κ) ⊗ j)(0 ⊗ i)(R). (0 ⊗ i)(R) Using 2.10, we then get: ∗ ˜ ⊗ 1) = (ς(κ ⊗ κ) ⊗ j)[(u ⊗ 1)(0 ⊗ i)(u )(0 ⊗ i)(V )] = (0 ⊗ i)(R)( (ς(κ ⊗ κ) ⊗ j)[(1 ⊗ u )(i ⊗ 0)(u )(i ⊗ 0)(V )] = [(ς(κ ⊗ κ) ⊗ j)(i ⊗ 0)(R)](ς ⊗ i)(1 ⊗ ˜u ), which is equal, using the implementations of j and κ, to ˜ u )(σ ⊗ i), (σ ⊗ i)(Jˆ ⊗ Jˆ ⊗ J)(1 ⊗ W )(σ ⊗ i)(1 ⊗ R∗ )(σ ⊗ i)(1 ⊗ W ∗ )(Jˆ ⊗ Jˆ ⊗ J)(1 ⊗ ˆ is equal to and, using the property linking W , W ∗ and J, J, ˜ u )(σ ⊗ i), (σ ⊗ i)(1 ⊗ W ∗ )(Jˆ ⊗ Jˆ ⊗ J)(σ ⊗ i)(1 ⊗ R∗ )(σ ⊗ i)(Jˆ ⊗ Jˆ ⊗ J)(1 ⊗ W )(1 ⊗

which gives the result.

Proposition 3.4. With the hypothesis and notations of 3.3, we have: (0 ⊗ i)(W,R ) = (W,R )23 (W )13 , where W := W,R for R = u . Proof. Using the definition, we get ∗ ˜ ⊗ 1) = (0 ⊗ i)(W,R ) = ( ⊗ 1)(0 ⊗ i)()(0 ⊗ i)(W )(0 ⊗ i)(R)( ∗ ˜ ⊗ 1). = (1 ⊗ )(i ⊗ 0)()(0 ⊗ i)(W )(0 ⊗ i)(R)( Using then the fact that (0 ⊗ i)(W ) = (1 ⊗ W )(σ ⊗ i)(1 ⊗ W )(σ ⊗ i), we get, using ∗ ˜ ⊗ 1) is equal to 3.3, that (0 ⊗ i)(W )(0 ⊗ i)(R)( ˜ ˜ u )(σ ⊗ i). (1 ⊗ W )(1 ⊗ R)(σ ⊗ i)(1 ⊗ W )(1 ⊗

As (i ⊗ 0)() = (1 ⊗ W )(σ ⊗ i)(1 ⊗ )(σ ⊗ i)(1 ⊗ W ∗ ), we then get that ∗ ˜ ⊗ 1)(1 ⊗ W ∗ ) (i ⊗ 0)()(0 ⊗ i)(W )(0 ⊗ i)(R)( is equal to u

˜ ˜ )(σ ⊗ i) (1 ⊗ W )(σ ⊗ i)(1 ⊗ )(σ ⊗ i)(1 ⊗ R)(σ ⊗ i)(1 ⊗ W )(1 ⊗ ˜ commute, we finally get: and, as (σ ⊗ i)(1 ⊗ )(σ ⊗ i) and (1 ⊗ R) (0 ⊗ i)(W,R ) = (1 ⊗ W,R )(σ ⊗ i)(1 ⊗ W )(σ ⊗ i), which is the result.

Theorem 3.5. Let K = (M, 0, κ, ϕ) be a Kac algebra (or W = (M, 0, κ, τ, ϕ) a Woronowicz algebra), W its fundamental unitary, a co-involutive 2-cocycle of ˜ = (Jˆ ⊗ J)∗ (Jˆ ⊗ J), where J is the canonical involutive isom(M, 0, κ). Let us put etry Jϕ constructed by the Tomita- Takesaki theory, Jˆ is the canonical implementation on Hϕ of the anti-automorphism κ, and W = W ˜u ; then we get: (i) W∗ is a multiplicative unitary; ˜ Jˆ ⊗ J). (ii) W∗ = (Jˆ ⊗ J)u W ( Proof. By 3.2 and 3.4, we get (i); by the formula W ∗ = (Jˆ ⊗ J)W (Jˆ ⊗ J) and the ˜ (ii) is trivial. definitions of ˜u and ,

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4. Twisting of Kac Algebras The results of this section are the generalizations of the results of Sects. 4 and 5 of [EVa] for the case when the co-involution is also deformed (i.e, u is not necessarily 1) and for more general subgroups and 2-cocycles. Because of that, the corresponding calculations and arguments are more complicated. Lemma 4.1. Let (M, 0, κ, ϕ) be a Kac algebra (or W = (M, 0, κ, τ, ϕ) a Woronowicz algebra), W its fundamental unitary. Let K be a locally compact group, α a co-involutive K Hopf–von Neumann morphism from (L∞ (K), 0K a , κa ) to (M, 0, κ), ω(t, s) a continuous counital 2-cocycle from K ⊗ K to T and u(s) = ω(s−1 , s) a corresponding unitary in L∞ (K) as in 2.11, and let = (α⊗α)(ω). If ai , bi ∈ L∞ (K) are such that Σi (ai ⊗bi ) →i ω strongly, and of norm less or equal to 1, then: K ∗ ∗ ∗ u ∗ (i) Σi 0K a (κa (ai ))(bi ⊗ 1) is weakly converging to (u ⊗ 1)(ω ) , where −1 −1 −1 −1 ω u (t, s) := (u∗ ⊗ u∗ )ω0K a (u) = ω(t , t)ω(s , s)ω(t, s)ω(s t , ts); u ∗ ˆ ˜u ˆ (ii) Σi (α(bi )∗ ⊗ Jα(κK a (ai ))J is weakly converging to = (J ⊗ J)( ) (J ⊗ J). K ∗ ∗ Proof. (i) The finite sums Σi 0K a (κa (ai ))(bi ⊗ 1) are converging to the function G ω(s−1 t−1 , t), which equals (u∗ ⊗ 1)ς(κG a ⊗ κa )(ω)(t, s) (to see that, put in the main −1 −1 2-cocycle equality s1 = s , s2 = s3 = t). Since the 2-cocycle ω is co-involutive, the above function equals (u∗ ⊗ 1)(ω u )∗ . (ii) The application x → Jα(x∗ )J from L∞ (K) to M 0 is a homomorphism, and, therefore, all the finite sums Σi (α(bi )∗ ⊗ Jα(κK a (ai ))J are of norm less or equal to 1. Let ξ, ξ 0 , η, η 0 in Hϕ ; we then get 0 0 ((Σi α(bi )∗ ⊗ Jα(κK a (ai ))J)(ξ ⊗ ξ )|η ⊗ η ) = K 0 0 ˆ ((Jˆ ⊗ J)ς(κK a ⊗ κa )Σi (α(ai ) ⊗ α(bi ))(J ⊗ J)(ξ ⊗ ξ )|η ⊗ η ) → → ((Jˆ ⊗ J)ς(κ ⊗ κ)()(Jˆ ⊗ J)(ξ ⊗ ξ 0 )|η ⊗ η 0 ) =

((Jˆ ⊗ J)(u )∗ (Jˆ ⊗ J)(ξ ⊗ ξ 0 )|η ⊗ η 0 ) = ˜ u (ξ ⊗ ξ 0 )|η ⊗ η 0 ). ( Definition 4.2. With the hypothesis of 4.1, we shall say that is an abelian coinvolutive 2-cocycle of (M, 0, κ), constructed by (K, ω, α). As in ([EVa], 4.6), where the definition of an abelian cocycle was done for a more special situation, we can always suppose that α is injective. Remarks 4.3. (i) If W = (M, 0, κ, τ, ϕ) a Woronowicz algebra with ϕ finite (it comes then, using ([BS], §4), from a compact quantum group), or if K = (M, 0, κ, ϕ) is a compact type Kac algebra ([ES], 6.2), and is an abelian co-involutive 2-cocycle of (M, 0, κ) constructed by (K, ω, α); then, as ϕ ◦ α is a bounded left Haar measure on K, so K is compact. (ii) If W = (M, 0, κ, τ, ϕ) is a Woronowicz algebra such that the predual M∗ has a unit (it comes then from a discrete quantum group), or if K = (M, 0, κ, ϕ) is a discrete type Kac algebra ([ES], 6.3), and is an abelian co-involutive 2-cocycle of (M, 0, κ) constructed by (K, ω, α); then, as ◦ α is a unit of L1 (K), so K is discrete.


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Proposition 4.4. Let K = (M, 0, κ, ϕ) be a Kac algebra, W its fundamental unitary, K a locally compact group, α a co-involutive Hopf–von Neumann morphism from K (L∞ (K), 0K a , κa ) to (M, 0, κ), an abelian co-involutive 2-cocycle of (M, 0, κ) constructed by (K, ω, α) as in 4.2, u the corresponding unitary in (L∞ (K)). Let α(L∞ (K)) be included in M ϕ = {x ∈ M, σtϕ (x) = x, ∀t ∈ R}. Then ϕ is left-invariant with respect to 0 , i.e., we have, for all z in M + : (i ⊗ ϕ)0 (z) = ϕ(z)1. Proof. Let ai , bi in L∞ (K) such that Σi (ai ⊗ bi ) →i ω strongly, and all finite sums are of norm less or equal to 1. Let x, y in Nϕ ∩ N∗ϕ , and ξ 0 , η 0 be right bounded vectors with respect to ϕ; using 4.1(ii), we get that Σi (α(bi )∗ ⊗Jα(κK a (ai ))J) is weakly converging to u ∞ ϕ ˜ ˜ u (3ϕ (x) ⊗ 3ϕ (y)) , and, therefore, using that α(L (K)) lies in M , we get that W is the weak limit of W Σi (α(bi )∗ ⊗ Jα(κK a (ai ))J)(3ϕ (x) ⊗ 3ϕ (y)) = ∗ W Σi (3ϕ (α(bi )∗ x) ⊗ 3ϕ (yα(κK a (ai )) )) K ∗ ∗ = 3ϕ⊗ϕ (0(y)(α ⊗ α)(Σi 0K a (κa (ai ) )(bi ⊗ 1))(x ⊗ 1)),

˜ u (3ϕ (x) ⊗ 3ϕ (y)) is the weak limit of and therefore we get that (π 0 (ξ 0 ) ⊗ π 0 (η 0 ))W K ∗ ∗ ((π 0 (ξ 0 ) ⊗ π 0 (η 0 ))3ϕ⊗ϕ (0(y)(α ⊗ α)(Σi 0K a (κa (ai ) )(bi ⊗ 1))(x ⊗ 1)) = K ∗ ∗ 0 0 0(y)(α ⊗ α)(Σi 0K a (κa (ai ) )(bi ⊗ 1))(x ⊗ 1)(ξ ⊗ η )

and is, using 4.1(i), equal to 0(y)(u )∗ (α(u∗ )x ⊗ 1)(ξ 0 ⊗ η 0 ) = 0(yα(u∗ ))∗ (x ⊗ α(u))(ξ 0 ⊗ η 0 ). We then deduce that 0(yα(u∗ ))∗ (x ⊗ α(u)) belongs to Nϕ⊗ϕ and that u

˜ (3ϕ (x) ⊗ 3ϕ (y)) = 3ϕ⊗ϕ (0(yα(u∗ ))∗ (x ⊗ α(u))), W from which we get W (3ϕ (x) ⊗ 3ϕ (y)) = 3ϕ⊗ϕ (0 (yα(u∗ ))(x ⊗ α(u))), and then, taking into account that α(u) ∈ M ϕ , we have (ω3ϕ (x),3ϕ (x) ⊗ ϕ)[(1 ⊗ α(u∗ ))0 (α(u)y ∗ yα(u∗ ))(1 ⊗ α(u))] = (ϕ ⊗ ϕ)[(x∗ ⊗ α(u∗ ))0 (α(u)y ∗ yα(u∗ ))(x ⊗ α(u))] = (ϕ ⊗ ϕ)((x∗ ⊗ 1)0 (α(u)y ∗ yα(u∗ ))(x ⊗ 1)) = ϕ(x∗ x)ϕ(y ∗ y). Now the proof can be finished exactly as in ([EVa], 5.1).

4.5. In the situation described in 4.4 we had the formula: W (3ϕ (x) ⊗ 3ϕ (y)) = 3ϕ⊗ϕ (0(yα(u∗ ))∗ (x ⊗ α(u))) (∀x, y ∈ Nϕ ∩ N∗ϕ ). Let us introduce the following operator: Wu := (Jα(u)J ⊗ Jα(u)J)W (Jα(u∗ )J ⊗ Jα(u∗ )J). Obviously, Wu is equal to

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(1 ⊗ Jα(u)J)W (1 ⊗ Jα(u∗ )J) = (1 ⊗ Jα(u)J)W (Jˆ ⊗ J)0(α(u∗ ))∗ (α(u) ⊗ α(u))(Jˆ ⊗ J)(1 ⊗ Jα(u∗ )J) = (1 ⊗ Jα(u)J)W (Jˆ ⊗ J)0(α(u∗ ))(Jˆ ⊗ J)× ˆ Jˆ ⊗ Jα(u)J)(1 ⊗ Jα(u∗ )J) = (Jˆ ⊗ J)∗ (Jˆ ⊗ J)(Jα(u) ˆ ˆ ˜ Jα(u) ˜ Jα(u) Jˆ ⊗ 1) = W ( Jˆ ⊗ 1). (1 ⊗ Jα(u)J)(1 ⊗ Jα(u∗ )J)W ( We used here the inclusion of the commutative W∗ -algebra Jα(L∞ (K))J into M 0 and the relation W (Jˆ ⊗ J)0(a)(Jˆ ⊗ J) = (Jˆ ⊗ J)(1 ⊗ a)(Jˆ ⊗ J)W, which follows from the definition of W . We also have, using that α(L∞ (K)) ⊂ M ϕ and the above formula for W , that Wu (3ϕ (x) ⊗ 3ϕ (y)) = (1 ⊗ Jα(u)J)W (3ϕ (x) ⊗ 3ϕ (yα(u)) = (1 ⊗ Jα(u)J)3ϕ⊗ϕ (0 (y)(x ⊗ α(u))) = 3ϕ⊗ϕ (0 (x)(y ⊗ 1)). ∗ ˆ ) = α(u)Jˆ is the Lemma 4.6. (Wu )∗ = (Jû ⊗ J)Wu (Jû ⊗ J), where Jû := Jα(u ∗ implementation of κ (·) := uκ(·)u .

Proof. Using 4.5, we have: ˆ ˜ Jα(u) Jˆ ⊗ 1)∗ = (Wu )∗ = (W ( ∗ ˆ ˆ (Jα(u )J ⊗ 1)(Jˆ ⊗ J)(Jˆ ⊗ J)W ∗ (Jˆ ⊗ J)(Jˆ ⊗ J)∗ = ∗ ˆ ˆ ˆ Jˆ ⊗ 1)(Jα(u )J ⊗ 1)(Jˆ ⊗ J) = (Jû ⊗ J)W (Jˆ ⊗ J)∗ (Jˆ ⊗ J)(Jα(u) ∗ ˆ ˆ )J ⊗ 1)(Jû ⊗ J) = (Jû ⊗ J)Wu (Jû ⊗ J). (Jû ⊗ J)Wu (Jα(u

Corollary 4.7. (Wu )∗ is multiplicative unitary. Proof. The statement follows from 3.5 (i) and ([BS], 1.2).

Theorem 4.8. Let K = (M, 0, κ, ϕ) be a Kac algebra, W its fundamental unitary, K a locally compact group, α a co-involutive Hopf–von Neumann morphism from K ∞ ϕ (L∞ (K), 0K a , κa ) to (M, 0, κ) such that α(L (K)) ⊂ M , ω a co-involutive cocycle on K as in 4.1, and = (α ⊗ α)(ω). Then K = (M, 0 , κ , ϕ) is again a Kac algebra whose fundamental unitary is Wu . Proof. Using 4.4, 4.5 instead of ([EVa], 5.1) and 4.6 instead of ([EVa], 3.9 (ii)), one can repeat the proof of ([EVa], 5.2). Remarks 4.9. (i) If a 2-cocycle ω from K × K to T is strongly co-involutive (see 2.10), then u = 1, Theorem 4.8 is valid, even if ω is measurable, κ = κ, and we are in the situation, slightly more general than in ([EV], 5.2). (ii) If in the situation described in 4.8 a Kac algebra K is unimodular, i.e., ϕ = ϕ ◦ κ is the trace on M (see [ES], 6.1.3), then so is K . (iii) It is possible to get in the context of Woronowicz algebras a result similar to Theorem 4.8, if, in addition, α(L∞ (K)) ⊂ M τ = {x ∈ M, τt (x) = x, ∀t ∈ R} (see [MN] for the definition of τt ).


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5. Quantizations of the Heisenberg Group 5.1. The general way of getting examples of non-trivial Kac algebras is the deformation of a locally compact group G with the help of 2-(pseudo)-cocycles lifted from its abelian closed subgroup H such that 1G (s) = 1, for all s in H, where 1G is the modulus of G (see ([EVa], 6.1, 6.2), but now we can deal with more general subgroups and cocycles. This gives a possibility to get a much wider class of deformations 5.2. The Heisenberg group Hn (R) can be considered as the semi-direct product Rn+1 ×α Rn , where the action α of Rn on Rn+1 is given, for a, b in Rn , t in R, by αa (b, t) = (b, t + (a|b)). Therefore, the product rule in Hn (R) is given by (a, a0 , b, b0 in Rn , t, t0 in R): (b, t, a)(b0 , t0 , a0 ) = (b + b0 , t + t0 + (a|b0 ), a + a0 ) and, as the action α leaves the Lebesgue measure of Rn invariant, we are in both situations described in ([EVa], 6.2). The group Hn (R) is unimodular, and the Hilbert space L2 (Hn (R)) can be identified with L2 (Rn ) ⊗ L2 (R) ⊗ L2 (Rn ). So, we get that the left regular representation λ(b, t, a) of Hn (R) is defined by λ(b, t, a)f (v, u, w) = f (v − b, u − t − (a|v − b), w − a), where u belongs to R, v, w to Rn , f to L2 (Rn ) ⊗ L2 (R) ⊗ L2 (Rn ). Let us define a unitary U on that Hilbert space by (see ([EVa], 6.3)): Z 1/2 ˆ uˆ iuuˆ f (v, u, vˆ − w)ei(v|v) e dudv, U f (v, ˆ u, ˆ w) = (|u|/2) ˆ Rn ×R

where u, uˆ belong to R, v, w, vˆ to Rn , f to L2 (Rn ) ⊗ L2 (R) ⊗ L2 (Rn ) and we get that the left regular representation λ(b, t, a) of Hn (R) verifies: U λ(b, t, a)U ∗ f (u, v, w) = ei(b|v)u eitu f (u, v + a, w). Therefore, this representation is equivalent to the representation π on L2 (Rn+1 ) defined, for any φ in L2 (Rn+1 ), by: π(b, t, a)φ(u, v) = ei(b|v)u eitu φ(u, v + a). This representation generates the von Neumann algebra Ł∞ (R) ⊗ L(L2 (Rn )), which is therefore isomorphic to L(Hn (R)); by this isomorphism, there exits a symmetric Kac algebra (Ł∞ (R) ⊗ L(L2 (Rn )), 0s , κs , ϕs ), such that 0s (π(b, t, a)) = π(b, t, a) ⊗ π(b, t, a), κs (π(b, t, a)) = π(−b, −t + (a|b), −a). Since we have

π(b, t, 0) = ((u, v) → ei(b|v)u eitu ),

[ n+1 ) into Ł∞ (R)⊗L(L2 (Rn )) we get from the inclusion then the morphism β1 from L∞ (R Rn+1 ⊂ Hn (R), sends the function (u, v) → eitu ei(b|v) on π(b, t, 0), and, therefore, for [ n+1 ), β (f ) is the function (u, v) → f (u, uv). any f in L∞ (R 1

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As π(0, 0, a) = 1 ⊗ ρ(a), where ρ is the right regular representation of Rn , the cn ) into Ł∞ (R) ⊗ L(L2 (Rn )) we get from the inclusion Rn ⊂ morphism β2 from L∞ (R Hn (R), is given by: β2 (f ) = 1 ⊗ F f F ∗ , cn ) to L2 (Rn ). where F is the Fourier–Plancherel unitary from Ł2 (R [ n+1 by the formula: Using 2.11 (ii), we can construct a 2-cocycle on R ˆ v; ˆ uˆ 0 ,vˆ 0 ) ˆ v; ˆ uˆ0 , vˆ0 ) = eiB(u, , ωB (u,

[ n+1 . b v, cn , B(·, ·; ·, ·) is a real bilinear form on R where u, ˆ uˆ0 belong to R, ˆ vˆ0 belong to R So, B = (β1 ⊗ β1 )(ωB ) is a 2-cocycle for (Ł∞ (R) ⊗ L(L2 (Rn )), 0s , κs ), and B is the operator of multiplication on the fucnction defined on Rn+1 × Rn+1 by 0

0

0

B (u, v, u0 , v 0 ) = eiuu B(u,v;u ,v ) . A deformed coproduct 0 is defined by 0B (π(b, t, a)) = B (π(b, t, a) ⊗ π(b, t, a))∗B , and we obtain 0

0

0

0

0

0

0B (π(b, t, a))f (u, v; u0 , v 0 ) = eiuu [B(u,v;u ,v )−B(u,v+a;u ,v +a)] eit(u+u ) × 0

0

×ei[(v|b)u+(v |b)u ] f (u, v + a; u0 , v 0 + a), where u, u0 belong to R, v, v 0 to Rn , and f belongs to L2 (Rn+1 × Rn+1 ). The operator uB acting in L2 (Rn+1 ) is the operator of multiplication on the function defined on Rn+1 by 2 uB (u, v) = e−iu B(u,v;u,v) . A deformed co-involution κB is defined by κB (π(b, t, a)) = uB κs (π(b, t, a))u∗ B , and we obtain 0B (π(b, t, a))φ(u, v) = eiu

2

[B(u,v−a;u,v−a)−B(u,v;u,v)] iu((a|b)−t) i(v|b)u

e

e

φ(u, v − a),

cn ). for any φ in L2 (R × R According 4.8, this construction gives a series of new unimodular Kac algebras (Ł∞ (R) ⊗ L(L2 (Rn )), 0B , κB , ϕs ). We can consider the Lie algebra L of the Heisenberg group Hn (R), and define, where u belongs to R, v to Rn , vk is the k th component of v, and φ is in the Schwartz algebra S(Rn+1 ), its generators Pk , R, Qk (k ∈ {1, ..., n}) by Pk φ(u, v) =

∂ ∂φ [π(b, t, a)φ(u, v)]|a=0,b=0,t=0 = (u, v), ∂ak ∂vk

∂ [π(b, t, a)φ(u, v)]|a=0,b=0,t=0 = iuφ(u, v), ∂t ∂ [π(b, t, a)φ(u, v)]|a=0,b=0,t=0 = iuvk φ(u, v), Qk φ(u, v) = ∂bk Rφ(u, v) =


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which are the infinitesimal operators of the representation π. These operators are linked by the commutation relations [Pk , R] = [Qk , R] = 0 and [Pk , Qj ] = δjk R. The symmetric coproduct 0s of the enveloping algebra U (L) satisfies: 0s (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk , 0s (R) = R ⊗ 1 + 1 ⊗ R, 0s (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk ; the antipode κs and the involution ∗ verify κs (Pk ) = Pk∗ = −Pk , κs (R) = R∗ = −R, κs (Qk ) = Q∗k = −Qk . Then the above quantization leads to the following formulas on the level of quantized universal enveloping algebra: 0B (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk + R ⊗

∂B 2 ∂B (R , iQ) + 0 (R2 , iQ) ⊗ R, ∂vk ∂v k

0B (R) = R ⊗ 1 + 1 ⊗ R, 0B (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk , and also κB (Pk ) = −Pk + (

∂B ∂B + )(R2 , iQ), P ∗ k = −Pk , ∂vk ∂v 0 k

κB (R) = R∗ = −R, κB (Qk ) = Q∗k = −Qk , where k ∈ {1, ..., n}, Q is the vector with the components Qk . One can see from these formulae that the above deformation is symmetric iff the bilinear form B is symmetric. In particular, for B(u, ˆ v; ˆ uˆ0 , vˆ0 ) = qjk vˆ j vˆ0 k , (where n is more than 1, j, k ∈ ˆ {1, ..., n}, j 6= k, vˆ j (resp., v 0 j ) is the j th component of vˆ (resp., vˆ0 ), qjk ∈ R)), we have on the level of quantized universal enveloping algebra, for l, m ∈ {1, .., n}, l 6= k, l 6= j: 0jk (Pl ) = Pl ⊗ 1 + 1 ⊗ Pl , 0jk (Pj ) = Pj ⊗ 1 + 1 ⊗ Pj + iqjk (R ⊗ Qk ), 0jk (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk + iqjk (Qj ⊗ R), 0jk (R) = R ⊗ 1 + 1 ⊗ R, 0jk (Qm ) = Qm ⊗ 1 + 1 ⊗ Qm , κjk (Pl ) = −Pl , P ∗ l = −Pl , κjk (Pj ) = −Pj + iqjk RQk , P ∗ j = −Pj , κjk (Pk ) = −Pk + iqjk RQj , P ∗ k = −Pk , κij (R) = R∗ = −R, κij (Qm ) = Q∗m = −Qm . cn by the formula: Using 2.11 (ii) again, we can construct a 2-cocycle ω B on R 0

ˆ vˆ ) ω B (v, ˆ vˆ 0 ) = eiB(v, ,

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cn , B(·, ·) is a real bilinear form on R cn . So, B = (β2 ⊗ β2 )(ω B ) is a where v, ˆ vˆ 0 ∈ R ∞ B 2 n 2-cocycle for (Ł (R) ⊗ L(L (R )), 0s , κs ), and as the operator can be written as follows: ˆ vˆ 0 ) (1 ⊗ F ∗ ⊗ 1 ⊗ F ∗ ). B = (1 ⊗ F ⊗ 1 ⊗ F )eiB(v, Let B˜ be an n by n matrix over R corresponding to the bilinear form B : B(v, ˆ vˆ 0 ) = (B˜ v| ˆ vˆ 0 ), and let B˜ t be the corresponding transposed matrix. Then one can calculate: ˆ ˆ = eitu ei(a|v) φ(u, vˆ − ub) (1 ⊗ F ∗ )π(b, t, a)(1 ⊗ F )φ(u, v)

cn ), from which we get: for any φ in L2 (R × R 0

0

ˆ vˆ )) (1 ⊗ F ∗ ⊗ 1 ⊗ F ∗ )0B (π(b, t, a))φ(u, v; ˆ u0 , vˆ 0 ) = eit(u+u ) ei(a|(v+ × 0

0

0

ˆ vˆ ) −iB(v−ub, ˆ vˆ −u b) ×eiB(v, e φ(u, vˆ − ub; u0 , vˆ 0 − u0 b)

cn × R × R cn ), and, using the Fourier transform again, we get for any φ in L2 (R × R 0

0

0

0

0B (π(b, t, a))f (u, v; u0 , v 0 ) = eit(u+u ) e−iuu B(b,b) e[u(v|b)+u (v |b)] × ˜ + a), f (u, v + u0 B˜ t b + a; u0 , v 0 + uBb where u, u0 ∈ R, v, v 0 ∈ Rn . In a similar way one can calculate a deformed co-involution κB : κB (π(b, t, a))f (u, v) = ˆ v) ˆ ˆ v) ˆ (1 ⊗ F )e−iB(v, (1 ⊗ F ∗ )(π(−b, (a|b) − t, −a))(1 ⊗ F )eiB(v, φ(u, v) ˆ = ˆ v) ˆ iu((a|b)−t) −i(a|v) ˆ iB(v+ub, v+ub) ˆ (1 ⊗ F )e−iB(v, e e e ˆ φ(u, vˆ + ub) =

eiu((a|b)−t) eiu

2

B(b,b)

ˆ iu[B(v,b)+B(b, ˆ v)] ˆ (1 ⊗ F )e−i(a|v) e φ(u, vˆ + ub) =

eiu((a|b)−t) e−iu

2

B(b,b) −iu(v|b)

e

f (u, v + u(B˜ + B˜ t )b − a).

This construction gives a series of new unimodular Kac algebras (Ł∞ (R) ⊗ L (L (Rn )), 0B , κB , ϕs ). On the level of a quantized universal enveloping algebra we have 0B (Pk ) = Pk ⊗ 1 + 1 ⊗ Pk , 2

0B (R) = R ⊗ 1 + 1 ⊗ R, 0B (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk − iR ⊗

∂B ∂B (P ) − (P ) ⊗ iR, ∂v 0 k ∂vk

κB (Pk ) = P ∗ k = −Pk , κB (R) = R∗ = −R, κB (Qk ) = −Qk − R(

∂B ∂B + )(iP ); Q∗k = −Qk , ∂v 0 k ∂vk

where k ∈ {1, ..., n}, P is the vector with the components Pk . One can see from these formulae that the above deformation is symmetric iff the bilinear form B is symmetric.


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In particular, for B(v, ˆ vˆ0 ) = qjk vˆ j vˆ0 k , (where n more than 1 j, k ∈ {1, ..., n}, j 6= k, th 0 ˆ vˆ j (resp.,v j ) is the j component of vˆ (resp., vˆ0 ), qjk ∈ R)), we have on the level of a quantized universal enveloping algebra, for l, m ∈ {1, .., n}, l 6= k, l 6= j: 0jk (Pm ) = Pm ⊗ 1 + 1 ⊗ Pm , 0jk (R) = R ⊗ 1 + 1 ⊗ R, 0jk (Qk ) = Qk ⊗ 1 + 1 ⊗ Qk − iqjk (Pj ⊗ R), 0jk (Qj ) = Qj ⊗ 1 + 1 ⊗ Qj − iqjk (R ⊗ Pk ), 0jk (Ql ) = Ql ⊗ 1 + 1 ⊗ Ql , κjk (Pm ) = P ∗ m = −Pm , κij (R) = R∗ = −R, κjk (Qj ) = −Qj − iqjk RPk , Q∗ j = −Qj , κjk (Qk ) = −Qk − iqjk RPj , Q∗ k = −Qk , κjk (Ql ) = Ql ∗ = −Ql . Notice that the quantizations of the Heisenberg group of the above mentioned type corresponding to the simplest skew-symmetric forms B, were considered in ([EVa], 6.3). On the other hand, representing any real bilinear form B as the sum of its symmetric and skew symmetric components, one can show that, up to an isomorphism, the twisting is given essentially by the skew-symmetric part of B.

6. Finite Dimensional Kac Algebras 6.1. First of all, recall that a finite dimensional Kac algebra is nothing but a finite dimensional C∗ -Hopf algebra ([ES], 6.6). This means that M is a semisimple *-algebra over C such that xx∗ 6= o for any x ∈ M , and that 0 and κ commute with ∗. This means also that there exist a linear mapping m : M ⊗M → M which defines the multiplication in M , an ε ◦ κ = κ, (ε ⊗ i)0(x) = (i ⊗ ε)0(x) = x, m(κ ⊗ i)0(x) = ε(x)1, m(i ⊗ κ)0(x) = ε(x)1 for all x ∈ M . Remark that a compact quantum group in the sense of [W1], or a discrete quantum group in the sense of [ER], if it is finite dimensional, is nothing but a finite dimensional Kac algebra (see ([W1, A.2], [ER, Va]). Lemma 6.2. Any counital 2-cocycle of a finite dimensional Kac algebra (i.e., such that (ε ⊗ i) = (i ⊗ ε) = 1), satisfies the condition ς(κ ⊗ κ)(∗ ) = u with a unitary u = m(i⊗κ) = m◦ς(κ⊗i) = κ(u). This means that is co-involutive in the sense of 2.10. Also, ε(u) = 1.

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Proof. a) Rewrite the main 2-cocycle equality with the usual tensor notation = (1) ⊗ (2) ([Abe]): 0

0

0

0

((1) ⊗ (2) )0( (1) ) ⊗ (2) = (1) ⊗ ((1) ⊗ (2) )0( (2) ), 0

0

where (1) ⊗ (2) stands for another copy of . Apply i ⊗ κ ⊗ i to both sides of this equality: 0

0

((1) ⊗ 1 ⊗ 1)[(i ⊗ κ)0( (1) ) ⊗ (2) ](1 ⊗ κ((2) ) ⊗ 1) = 0

0

(1 ⊗ 1 ⊗ (2) )[ (1) ⊗ (κ ⊗ i)0( (2) )](1 ⊗ κ((1) ) ⊗ 1). Apply m ◦ (i ⊗ m) = m ◦ (m ⊗ i) to both sides of this equality, using that m[(a ⊗ 1)(b ⊗ c)(1 ⊗ d)] = a · m(b ⊗ c) · d ∀a, b, c, d ∈ M : 0

0

m(((1) ⊗ 1)[m(i ⊗ κ)0( (1) ) ⊗ (2) ](κ((2) ) ⊗ 1)) = 0

0

m((1 ⊗ (2) )[ (1) ⊗ m(κ ⊗ i)0( (2) )](1 ⊗ κ((1) ))), from where one can get, using the relations of 6.1 and the counitality of : u = (1) κ((2) ) = (2) κ((1) ), or u = m(i ⊗ κ) = m ◦ ς(κ ⊗ i) = κ(u). The property ε(u) = 1 follows from the counitality of . b) Let us show that uu∗ = 1. Indeed, we have 0

0

uu∗ = (1) κ((2) )κ( (1)∗ ) (2)∗ = 0

0

00

(1) κ( (1)∗ (2) ) (2)∗ = 00

00

(1)∗

(1)∗

ε(

00

(2)∗

0

0

0

)(1) κ( (1)∗ (2) ) (2)∗ =

(1) κ( (1)∗ (2) )ε(

00

(2)∗

0

) (2)∗ ,

00

where (1) ⊗ (2) stands for the third copy of . Let us show that the right-hand side of this equality is equal to 1. Rewrite the main 2-cocycle equality as follows: (0 ⊗ i)(∗ ) = (i ⊗ 0)(∗ )(1 ⊗ ∗ )( ⊗ 1) or 00

0((1)∗ ) ⊗ (2)∗ =

(1)∗

00

(1) ⊗ [0(

(2)∗

0

0

)( (1)∗ (2) ⊗ (2)∗ )].

From this we have 00

(i ⊗ κ)0((1)∗ )(1 ⊗ (2)∗ ) =

00

(1)∗

(1)∗

00

(1) ⊗ m(κ ⊗ i)[0( 0

(1) ⊗ κ( (1)∗ (2) )ε(

00

(2)∗

(2)∗

0

0

)( (1)∗ (2) ⊗ (2)∗ )] =

0

) (2)∗ .

Applying the map m to both sides of this equality and using the counitality of ∗ , one has: 00 0 00 0 1 = (1)∗ (1) κ( (1)∗ (2) )ε( (2)∗ ) (2)∗ , which gives uu∗ = 1. Let us show that u∗ u = 1. Indeed, we have 0

0

u∗ u = κ( (1)∗ ) (2)∗ (1) κ((2) ) = κ(

00

(1)

)ε(

00

(2)

0

0

)κ( (1)∗ ) (2)∗ (1) κ((2) ) =

2-Cocycles and Twisting of Kac Algebras 0

00

κ( (1)∗

713 (1)

0

) (2)∗ (1) ε(

00

(2)

)κ((2) ).

Let us show that the right-hand side of this equality is equal to 1. Rewrite the main 2-cocycle equality as follows: (0 ⊗ i)() = (∗ ⊗ 1)(1 ⊗ )(i ⊗ 0)() or 0

0

00

0((1) ) ⊗ (2) = ( (1)∗ ⊗ (2)∗ (1) ⊗ (2) )(

(1)

00

⊗ 0(

(2)

)).

From this we have 0

00

(κ ⊗ i)0((1) ) ⊗ (2) = κ( (1)∗

(1)

0

00

) ⊗ ( (2)∗ (1) ⊗ (2) )0(

(2)

),

and from this: 0

00

1 ⊗ 1 = (κ( (1)∗

(1)

0

) (2)∗ (1) ⊗ (2) )0(

00

(2)

).

Applying the map m ◦ (i ⊗ κ) to both sides of this equality we finally have 0

00

1 = κ( (1)∗

(1)

0

) (2)∗ (1) ε(

00

(2)

)κ((2) ),

which gives uu∗ = 1. Thus, u is unitary. c) Let us show that 0(u) = ∗ (u ⊗ u)ς(κ ⊗ κ)(∗ ), which is obviously equivalent to the main statement of the lemma. Using the main 2-cocycle equality in various forms and some of the above relations, we have 0(u) = 0(m ◦ (i ⊗ κ)()) = (m ⊗ m)(i ⊗ ς ⊗ i)(0 ⊗ 0)((i ⊗ κ)()) = (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))(i ⊗ i ⊗ 0)(0 ⊗ i)() = (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))(i ⊗ i ⊗ 0)(∗ ⊗ 1)(1 ⊗ )(i ⊗ 0)() = (m ⊗ m)(i ⊗ ς ⊗ i)(∗ ⊗ 1 ⊗ 1)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ (i ⊗ 0)())(i ⊗ (i ⊗ 0)0)()] = ∗ (m⊗m)(i⊗ς ⊗i)(i⊗i⊗ς(κ⊗κ))[(1⊗(1⊗∗ )(⊗1)(0⊗i)())(i⊗(0⊗i)0)()] = ∗ (m ⊗ m)(i ⊗ ς ⊗ i)[(i ⊗ i ⊗ ς(κ ⊗ κ))((1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) ))) (1 ⊗ 1 ⊗ ς(κ ⊗ κ)(∗ ))] = ∗ (m⊗m)(i⊗ς⊗i)(i⊗i⊗ς(κ⊗κ))[(1⊗⊗1)((1) ⊗(0⊗i)(0((2) )))]ς(κ⊗κ)(∗ ). Now we have to show only that (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) )))] = u ⊗ u. Indeed, (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) )))] = 0

(m ⊗ m)(i ⊗ ς ⊗ i)[(1 ⊗ (1) ⊗ 1 ⊗ 1)× 0

((1) ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) )))(1 ⊗ 1 ⊗ 1 ⊗ κ( (2) ))] = 0

0

(1 ⊗ (1) )[(m ⊗ m)(i ⊗ ς ⊗ i)((1) ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) )))](1 ⊗ κ( (2) )) = 0

0

((1) ⊗ (1) )[(m ⊗ m)(i ⊗ ς ⊗ i)(1 ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) )))](1 ⊗ κ( (2) )). One can get the following equality for any A ∈ M ⊗ M : (m ⊗ m)[1 ⊗ (ς ⊗ i)(i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(A)] =

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(m ⊗ m)[1 ⊗ (ς ⊗ i)(i ⊗ ς)((i ⊗ κ)0(A(1) ) ⊗ κ(A(2) ))] = (m ⊗ m)[1 ⊗ κ(A(2) ) ⊗ (i ⊗ κ)0(A(1) )] = κ(A(2) ) ⊗ ε(A(1) )1 = ς(1ε ⊗ κ)(A). In particular, we have (m ⊗ m)(i ⊗ ς ⊗ i)(1 ⊗ (i ⊗ ς(κ ⊗ κ))(0 ⊗ i)(0((2) ))) = ς(1ε ⊗ κ)(0((2) )) = κ((2) ) ⊗ 1. That is why we finally get: (m ⊗ m)(i ⊗ ς ⊗ i)(i ⊗ i ⊗ ς(κ ⊗ κ))[(1 ⊗ ⊗ 1)((1) ⊗ (0 ⊗ i)(0((2) )))] = 0

0

0

0

((1) ⊗ (1) )(κ((2) ) ⊗ 1)(1 ⊗ κ( (2) )) = (1) κ((2) ) ⊗ (1) κ( (2) ) = u ⊗ u. Remark 6.3. The similar statement was proved in ([M], 2.3.4) for quasitriangular Hopf algebras, but the quasitriangularity was used there in an essential way. 6.4. Commutative and symmetric (i.e., co-commutative) finite dimensional Kac algebras are associated with finite groups in the way explained in ([ES], 4.2.4, 4.2.5, 6.6). That is why it is interesting to study non-trivial (i.e., non commutative and non symmetric) finite dimensional Kac algebras. Let us specialize our construction for the case of finite group G and its abelian ˆ H > be the duality between subgroup H. Let Hˆ be the dual group for H and < H, them. Then there exists a family of selfadjoint idempotents Phˆ generating the abelian subalgebra C(H) of the group algebra C(G) such that: X X ˆ h > Pˆ , Pˆ = 1 ˆ h > λ(h), < h, < h, Phˆ Pgˆ = δh, ˆ gˆ Ph ˆ , λ(h) = h h ||H|| ˆ H ˆ h∈

0s (Phˆ ) =

X

h∈H

Pgˆ ⊗ Pgˆ −1 hˆ , κs (Phˆ ) = Phˆ −1 ,

ˆ g∈ ˆ H

ˆ ||H|| is the order of H. where g, h ∈ H, g, ˆ hˆ ∈ H, Using these idempotents, one can write the following formulae for , u and ∂2 : X X ω(x, y)(Px ⊗ Py ), u = m(i ⊗ κ) = ω(x, x−1 )Px , = ˆ x,y∈H

∂2 =

X

ˆ x∈H

ω(y, z)ω(x, yz)ω(x, y)ω(xy, z)(Px ⊗ Py ⊗ Pz ).

ˆ x,y,z∈H

Thus, in order to construct a concrete example of a non-trivial finite dimensional Kac algebra, one can choose a finite non commutative group G, its abelian subgroup H and a 2-(pseudo)-cocycle ω : Hˆ × Hˆ → T. Then, using the above formulae, one gets and ˆ u. In the case when ω is a 2-cocycle, also is, but if ω is only a 2-pseudo-cocycle on H, one should verify if is a 2-pseudo-cocycle on (L(G), 0s , κs ). If it is, and it is at least pseudo-coinvolutive (2.10), then we have already new finite dimensional coinvolutive Hopf–von Neumann algebra. If it has a counit, then it is a Kac algebra ([ES], 6.3.5). For this one should choose ω to be counital: ω(e, x) = ω(x, e) = 1 ∀x ∈ H which gives the counitality of : (ε ⊗ i) = (i ⊗ ε) = 1. Finally, if the new coproduct is not symmetric (i.e., non co-commutative), we have a non-trivial Kac algebra. The necessary and sufficient condition for this is, obviously, that the unitary ∗ ς() does not belong to 0(M )0 .


715

Remark 6.5. The construction above and the examples below are valid in the framework of general semisimple Hopf algebras over any algebraically closed field of characteristic zero (to show this one needs some results from [Abe, LR, M], see also [N]). 6.6. We shall describe now two series of non-trivial Kac algebras by twisting two classical series of finite groups: dihedral and quasiquaternion. Let us start with the groups G1 = D2n = Z2n ×α Z2 (2 ≤ n ∈ N), with the following action α of Z2 = {1, b} on Z2n = {ak (k = 0, 1, ..., 2n − 1)}: αb (ak ) = a2n−k (k = 0, 1, ..., 2n − 1); and G2 = Qn having two generators: a of order 2n, (n ∈ N), and b of order 4 such that b2 = an and bab−1 = a−1 . So, any group G1 , G2 has 4n elements: {ak , bak (k = 0, 1, ..., 2n − 1)}, and the group algebra L(G) is in both cases isomorphic to C ⊕ C ⊕ C ⊕ C ⊕ M2 (C) ⊕ . . . ⊕ M2 (C) | {z } n−1

(see [HR], 27.61). Let e1 , e2 , e3 , e4 ,ej11 , ej12 , ej21 , ej22 (j = 1, ..., n−1), be the matrix units of this algebra; we can now write the left regular representation λ of G1 ([HR], 27.61): X j jk(2n−1) j (εjk e22 ), λ(ak ) = e1 + e2 + (−1)k (e3 + e4 ) + n e11 + εn j

λ(bak ) = e1 − e2 − (−1)k (e3 − e4 ) +

X

j (εjk(2n−1) ej12 + εjk n n e21 ),

j

where εn = eiπ/n . Consider the subgroup H = {e, an , b, ban } which is isomorphic to Z2 × Z2 . Since the dual group Hˆ is isomorphic to H, then, following 6.4, one can ˆ compute the orthoprojectors Phˆ , (hˆ ∈ H): Peˆ = e1 +

1 + (−1)n e 4 + q1 , 2

1 + (−1)n e 3 + q2 , 2 1 − (−1)n e 4 + p1 , Paˆ n = 2 1 − (−1)n e 3 + p2 , Pbˆ aˆ n = 2 where p1 , p2 , q1 , q2 are the orthogonal projectors defined by: X0 j (e11 + ej12 + ej21 + ej22 ), p1 = 1/2 Pbˆ = e2 +

j

p2 = 1/2

X0

q1 = 1/2

j

(ej11 − ej12 − ej21 + ej22 ),

X00 j

(ej11 + ej12 + ej21 + ej22 ),

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q2 = 1/2

X00 j

(ej11 − ej12 − ej21 + ej22 ),

P0 P00 where (resp., ) means that the corresponding index of summation takes only odd (resp., even) values. Respectively, one can write the left regular representation λ of G2 ([HR], 27.61): X j −jk j λ(ak ) = e1 + e2 + (−1)k (e3 + e4 ) + (εjk n e11 + εn e22 ), j

λ(ak b) = e1 − e2 + (−1)k (e3 − e4 ) +

X

j (εj(k−n) ej12 + ε−jk n n e21 ),

j

for even n, and λ(ak b) = e1 − e2 + i(−1)k (e3 − e4 ) +

X

j (εj(k−n) ej12 + ε−jk n n e21 ),

j

for odd n, where εn = eiπ/n . Consider the subgroup H = Z4 = {e, b, b2 , b3 }. As the dual group Hˆ is isomorphic ˆ to H, then, following 6.4, one can compute the orthoprojectors Phˆ , (hˆ ∈ H): Peˆ = e1 + e3 + q1 , Pbˆ = p1 , Pbˆ 2 = e2 + e4 + q2 , Pbˆ 3 = p2 , f or even n, Peˆ = e1 + q1 , Pbˆ = e4 + p1 , Pbˆ 2 = e2 + q2 , Pbˆ 3 = e3 + p2 , f or odd n, where p1 , p2 , q1 , q2 are the orthogonal projectors defined by: X0 j (e11 − iej12 + iej21 + ej22 ), p1 = 1/2 j

p2 = 1/2

X0 j

q1 = 1/2 q2 = 1/2

(ej11 + iej12 − iej21 + ej22 ),

X00 j

X00 j

(ej11 + ej12 + ej21 + ej22 ),

(ej11 − ej12 − ej21 + ej22 ),

P0 P00 where (resp., ) means that the corresponding index of summation takes only odd (resp., even) values. There exists a standard structure of a symmetric Kac algebra on L(G) with a symmetric coproduct 0s , involution ∗ and a co-involution κs defined, for all g in G by 0s (λ(g)) = λ(g) ⊗ λ(g), κs (λ(g)) = λ(g), (λs (g))∗ = λ(g −1 ). One can see that e1 is the projection given by the co-unit of L(G) ([ES], 6.3.5). The abelian subalgebra generated by λ(e), λ(an ), λ(b), λ(ban ) (resp., by λ(e), λ(b), λ(b2 ), λ(b3 )), is generated also by the mutually orthogonal orthoprojectors Peˆ , Paˆ n , Pbˆ , Pbˆ aˆ n (resp., Peˆ , Pbˆ , Pbˆ 2 , Pbˆ 3 ). The orthogonal projectors Peˆ + Pbˆ and Paˆ n + Pbˆ aˆ n (resp., Peˆ + Pbˆ 2 and Pbˆ + Pbˆ 3 ) are central.


717

[ [2 2 c c Let us consider now the function ω on (Z 2 ) × (Z2 ) (resp., on Z4 × Z4 ) such that, [ 2 c for all u, h in (Z 2 ) (resp., in Z4 ): ω(e, ˆ u) = ω(u, u) = 1, ω(h, u) = ω(u, h) ˆ = ω(b, ˆ bˆ ˆ an ) = ω(bˆ ˆ an , aˆ n ) = i (resp., ω(b, ˆ bˆ 2 ) = ω(bˆ 2 , bˆ 3 ) = ω(bˆ 3 , b) ˆ = i). and ω(ân , b) [ 2 One can verify that this function is the 2-cocycle on (Z 2 ) (resp., the 2-pseudoc c cocycle on Z4 ; remark that all 2-cocycles on Z4 are symmetric: ω(h, u) = ω(u, h) and so give only symmetric twisting; that is why for getting a non-trivial twisting in this case we are forced to use 2-pseudo-cocycles). Let be the unitary element of L(G) ⊗ L(G) obtained by the lifting construction. For the group G1 we get = Peˆ ⊗ I + Paˆ n ⊗ (Peˆ + Paˆ n + i(Pbˆ − Pbˆ aˆ n )) + Pbˆ ⊗ (Peˆ + Pbˆ + i(Pbˆ aˆ n − Paˆ n )) + Pbˆ aˆ n ⊗ (Peˆ + Pbˆ aˆ n + i(Paˆ n − Pbˆ )). For the group G2 we respectively get = Peˆ ⊗ I + Pbˆ ⊗ (Peˆ + Pbˆ + i(Pbˆ 2 − Pbˆ 3 )) + Pbˆ 2 ⊗ (Peˆ + Pbˆ 2 + i(Pbˆ 3 − Pbˆ )) + Pbˆ 3 ⊗ (Peˆ + Pbˆ 3 + +i(Pbˆ − Pbˆ 2 )). It is useful to note that in both cases we can write = 1 + i2 . [ 2 Since ω is a 2-cocycle on (Z 2 ) , then is a 2-cocycle for (L(G1 ), 0s ). If we consider c c c on Z4 × Z4 × Z4 the function (s1 , s2 , s3 ) → ω(s2 , s3 )ω(s1 , s2 s3 )ω(s1 , s2 )ω(s1 s2 , s3 ), it takes value 1 except if s1 , s2 or s3 equals bˆ or bˆ 3 , for which it takes value -1; therefore, we get: ∂2 = I ⊗ I ⊗ I − 2(Pbˆ ⊗ Pbˆ ⊗ Pbˆ + Pbˆ ⊗ Pbˆ ⊗ Pbˆ 3 + Pbˆ ⊗ Pbˆ 3 ⊗ Pbˆ + Pbˆ 3 ⊗ Pbˆ ⊗ Pbˆ + Pbˆ ⊗ Pbˆ 3 ⊗ Pbˆ 3 + Pbˆ 3 ⊗ Pbˆ ⊗ Pbˆ 3 + Pbˆ 3 ⊗ Pbˆ 3 ⊗ Pbˆ + Pbˆ 3 ⊗ Pbˆ 3 ⊗ Pbˆ 3 ) = I ⊗ I ⊗ I− 2((Pbˆ + Pbˆ 3 ) ⊗ (Pbˆ + Pbˆ 3 ) ⊗ (Pbˆ + Pbˆ 3 )). Thus, ∂2 belongs to Z(L(G2 )) ⊗ Z(L(G2 )) ⊗ Z(L(G2 )) and is therefore a 2pseudo-cocycle for (L(G2 ), 0s ). So, in both cases (L(G), (0s ) ) is a Hopf–von Neumann algebra. It is clear that the above 2-cocycle is strongly co-involutive on (L(G1 ), 0s , κs ); let us show that the 2-pseudo-cocycle is pseudo-co-involutive on (L(G2 ), 0s , κs ) in the sense of 2.10, i.e., that ς(κs ⊗ κs )()(u∗ ⊗ u∗ )0s (u) belongs to 0s (L(G2 ))0 , where u = m(i ⊗ κs )() = Peˆ + Pbˆ 2 + i(Pbˆ 3 − Pbˆ ) (remark that κs (u) = u∗ !). Let us remark now that ς1 = (κs ⊗ κs )(1 ) = 1 ∗ = 1 , ς2 = (κs ⊗ κs )(2 ) = −2 ∗ = −2 , 1 2 = 2 1 , where 1 , 2 were introduced above. Using these relations, one can calculate that ς(κs ⊗κs )()(u∗ ⊗u∗ )0s (u) = (Peˆ +Pbˆ 2 −Pbˆ −Pbˆ 3 )⊗I which belongs to Z(L(G2 )) ⊗ Z(L(G2 )) ⊂ 0s (L(G2 ))0 . Thus, according to 2.8, 2.10 we get that (L(G1 ), (0s ) , κs ) (resp., (L(G2 ), (0s ) , uκs (·)u∗ )) is the co-involutive Hopf–von Neumann algebra. Moreover, in both cases we get that (e1 ⊗ x) = (e1 ⊗ x) = e1 ⊗ x, for all x in L(G); from which we conclude that e1 still gives a co-unit for the new structure; we also get that (0s ) (e1 ) = 0s (e1 ), and, by ([ES], 6.3.5), we get that this new structure is the Kac algebra.

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To prove that this new Kac algebra is non symmetric, it is enough to prove that ς(0s ) (λ(a)) 6= (0s ) (λ(a)), which is: ς(1 + i2 )(λ(a) ⊗ λ(a))(1 + i2 )∗ 6= (1 + i2 )(λ(a) ⊗ λ(a))(1 + i2 )∗ : Using the above relations for 1 , 2 , we see that this inequality is equivalent to: 1 (λ(a) ⊗ λ(a))2 6= 2 (λ(a) ⊗ λ(a))1 , and it is even enought to prove that (p ⊗ q)1 (λ(a) ⊗ λ(a))2 6= (p ⊗ q)2 (λ(a) ⊗ λ(a))1 , where p = p1 + p2 , q = q1 + q2 are in the center of L(G); one can obtain this last inequality, using the above expressions for 1 , 2 and λ(a), by direct calculations (for the case of the group G1 - only if 3 ≤ n, for n = 2 the deformation is co-commutative). Therefore, the Kac algebras in consideration are non symmetric and we obtain two series of non-trivial Kac algebras. Since, by construction, the coproducts 0s and (0s ) are equal on the subalgebra generated by λ(H), it is clear that the corresponding elements belong to the intrinsic group of this Kac algebra (i.e., they are group-like elements) (see 2.1). Therefore, the dual Kac algebra has at least 4 one-dimensional representations. The intrinsic group of the dual Kac algebra is equal to (Z2 )2 for the case of dihedral groups and to (Z2 )2 (if n is even) or to Z4 (if n is odd) for the case of quasiquaternion groups. That is why, at least for odd n, the above Kac algebras are non-isomorphic. Remark that some other description of Kac algebras with the above block-structure of an algebra is also given in ([IK], 5.5, 5.6). Remarks 6.7. (i) The above deformation for the case of dihedral groups was done in ([EVa], 6.5) for n = 3; it was observed in ([N], 5.3) that the construction is valid for general n. (ii) The above deformation for the case of quasiquaternion groups for n = 2 is nothing but the historical Kac-Paljutkin example of a non-trivial Kac algebra (see [KP3], 8.19, [KP2]), its description by means of twisting was done in ([EVa], 6.4); for n = 3, this deformation was done in ([Va1], 6.6); then it was observed in ([N], 5.4) that the construction is valid for general n. (iii) From [Wi, Mas2] one can deduce that the unique (up to an isomorphism) non-trivial semisimple Hopf algebra of dimension less or equal to 11 is the above example by Kac and Paljutkin. Since any finite dimensional semisimple Hopf algebra has no more than one Kac algebra structure (up to isomorphism) (see [And, Vays]), the same is true for Kac algebras of dimension less or equal to 11. On the other hand, using the theory of extensions of groups to Kac algebras (see [K1]) or its modification for finite dimensional semisimple Hopf algebras (see [Mas1] and references there), one can show (see [F]) that there exist exactly 2 (up to an isomorphism) non-trivial Kac algebras of dimension 12, both of them are selfdual and can be presented as extensions of the form 1 → CZ2 → K → CS3 → 1 ∗

and have the same C -algebra structure C ⊕ C ⊕ C ⊕ C ⊕ M2 (C) ⊕ M2 (C). c4 . Now it is clear that these Kac Their intrinsic groups are respectively (Z2 )2 and Z algebras are exactly isomorphic to the above mentioned ones, described by twisting.


719

It was shown in [K2] that all Kac algebras of prime dimension are commutative and symmetric, all of them are associated with the corresponding cyclic group in the sense of 3.2 (i), (ii). Finally, one can conclude from [Mas1, LR] that all Kac algebras of dimensions 14, 15 are commutative or cocommutative. So, the three above described Kac algebras are all the non-trivial Kac algebras of dimension less or equal to 15. (iv) On the other hand, it was shown in ([N], 4.5, 4.6) that there exist non-trivial Kac algebras of dimensions 16 and 2m2 ( m ≥ 3 is prime) which cannot be obtained by twisting of any finite group. Indeed, in [S] a series of non-trivial Kac algebras 2 Km , m ≥ 3 with C∗ -algebra structure Cm ⊕ Mm (C) was constructed. Let now m be prime. We shall show that there is no group algebra with such a C∗ -algebra structure. Let G be any group of order 2m2 , where m ≥ 3 is prime. Then G should contain a subgroup H of order m2 (first Sylow theorem). The index of H is 2, so H is normal ([Kur, 9]). So, there exists a homomorphism α : G → G/H ' Z2 . Let χ1 be a non-trivial character of Z2 , then χ = χ1 ◦ α is a character of G and χ2 is a trivial character of G. The group of characters of G has an even order because it contains an element χ of order 2 (Lagrange’s theorem). Therefore, the number of one-dimensional representations of G is even and is not equal to m2 . So, Km cannot appear as an algebra of a deformation of a symmetric Kac algebra, when m ≥ 3 is a prime. However, below we shall see that the dual Kac algebras for Km can be obtained by twisting of certain finite groups. 6.8. We shall describe now a series of non-trivial Kac algebras by twisting the finite groups G = Zm 2 ×α Z2 (3 ≤ m ∈ N), with the following action α of Z2 = {1, s} on H = Zm 2 = {(a, b) (a, b = 0, 1, ..., m − 1)} : αs (a, b) = (b, a) (we use here the additive notation). So, G has 2m2 elements and any character of G is necessarily symmetric on H; since such a character takes the value 1 or -1 in s, G has exactly 2m characters. This group has an invariant abelian subgroup H of index 2, that is why all other irreducible representations of G are of dimension 2 ([CR], 53.18). Now it is clear that the group algebra L(G) is isomorphic to C ⊕ . . . ⊕ C ⊕ M2 (C) ⊕ . . . ⊕ M2 (C) . {z } | | {z } m(m−1) 2

2m

\2 \ 2 Let us consider now the following function ω on (Z m ) × (Zm ) : ˆ c)). ˆ cˆ, d) ˆ = exp( 2πi (âdˆ − bˆ ω(â, b; m \ 2 One can verify that this function is the 2-cocycle on (Z m ) . Let be the unitary element of L(G) ⊗ L(G) obtained by the lifting construction: =

X ˆ H ˆ H×

exp(

2πi ˆ ˆ (âd − bˆc))P(a, ˆ ⊗ P(c, ˆ, ˆ b) ˆ d) m

where P(·,·) are orthoprojectors generating L(H) obtained as in 6.4. Since ω is a 2∗ \ 2 cocycle on (Z m ) , then is a 2-cocycle for (L(G), 0s ). One can also note that = ς,

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from where one can conclude that 0 is non-symmetric if and only if 2 (λ(s) ⊗ λ(s)) 6= (λ(s) ⊗ λ(s))2 . This inequality can be obtained by straightforward calculation. Thus, we have a series of non-trivial Kac algebras of dimension 2m2 . If m is odd, 2 then the dual Kac algebras have the above C∗ -algebra structure ([S], 3.1): Cm ⊕Mm (C). 2 Remark that we could use above for twisting the finite groups G = Zm ×α Z2 (3 ≤ \2 \ 2 m ∈ N) some other 2-cocycles ω on (Z m ) × (Zm ) : ˆ cˆ, d) ˆ = exp( ω(â, b;

2πi ˆ cˆ, d)), ˆ B(â, b; m

\ 2 where B(·; ·) is general bilinear form on (Z m) . Remark also that some other description of Kac algebras with the above blockstructure of algebra is also given in ([IK], 5.1). Remark 6.9. In the paper [N] the ring of representations (K0 -ring) of a finite-dimensional semisimple Hopf algebra (or Kac algebra) was considered and it was shown there that this ring is the twisting invariant. Using this fact, it was shown that the twisting of a simple finite group is a simple Hopf (or Kac) algebra (i.e., it does not have proper normal Hopf subalgebras). In this case the dual Hopf algebra does not have proper Hopf subalgebras. Using the lifting construction explained in 6.4, a series of nontrivial deformations of the symmetric groups Sn (n > 3) and the alternating groups An (n > 4) was described. The latter Hopf algebras are simple. Acknowledgement. I am deeply grateful to M. Enock for many valuable discussions without which this work could not appear. I am also grateful to V.G. Kac and H.-J. Schneider for important information on finitedimensional Hopf algebras. Finally, I am grateful to D. Nikshych for stimulating discussions.

References [Abe] Abe, E.: Hopf algebras. Cambridge: Cambridge University Press, 1977 [And] Andruskiewitsch, N.: Compact involutions of semisimple quantum groups. Czech. J. Phys. 44, 963– 972 (1994) [BS] Baaj, S. Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés de C∗ -algèbres. Ann. Sci. ENS 26, 425–488 (1993) [CR] Curtis, C.W., Reiner, I.: Representation theory of finite groups and associative algebras New York. Interscience Publishers, 1962 [D] Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad. Math. J. 1, 1419–1457 (1990) [ER] Effros, E.G. and Ruan, Z.-J.: Discrete quantum groups, I; the Haar measure. Int. J. Math. 5, 681–723 (1994) [ES] Enock, M. and Schwartz, J.-M.: Kac algebras and duality of locally compact groups, Berlin: Springer, 1992 [EN] Enock, M. and Nest, R.: Irreducible inclusions of Factors, Multiplicative Unitaries and Kac Algebras J. Funct. Anal. 137, 466–543 (1996) [EVa] Enock, M. and Vainerman, L.: Deformation of a Kac algebra by an Abelian subgroup Commun. Math. Phys. 178, 571–596 (1996) [F] Fukuda, N.: Semisimple Hopf algebras of dimension 12. Preprint [HR] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, v.II. Berlin: Springer-Verlag, 1970 [HS] Hong, J.H. and Szymanski, W.: Composition of Subfactors and Twisted Bicross Products. Preprint (1995) [IK] Izumi, M., Kosaki, H.: Finite dimensional Kac algebras arizing from certain group actions on factors. Preprint (1995) [K1] Kac, G.I.: Extensions of groups to ring-groups. Math. USSR Sbornik 5, 451–474 (1968)


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[K2] Kac, G.I.: Certain arithmetic properties of ring-groups. Funct. Anal. Appl. 6, 158–160 (1972) [KP1] Kac, G.I., Paljutkin: V.G.: Finite ring groups. Trans. Moscow Math. Soc. 15, 251–294 (1966) [KP2] Kac, G.I., Paljutkin, V.G.: An example of a ring group of order eight. (Russian), Soviet Math. Surveys 20 n.5, 268–269 (1965) [Kur] Kurosh, A.G.: it Group theory. Moscow: 1966 [L] Landstad, M.B.: Quantization arising from abelian subgroups. Int. J. Math. 5, 897–936 (1994) [LR] Larson, R.G. and Radford, D.E.: Semisimple Hopf algebras. J. Algebra 171, 5–35 (1995) [M] Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 [MN] Masuda, T. and Nakagami, Y.: A von Neumann Algebra Framework for the Duality of the Quantum Groups. Publ. RIMS, Kyoto Univ. 30 n.5, 799–850 (1994) [Mas1] Masuoka, A.: Semisimple Hopf algebras of dimension 2p. Preprint (1995) [Mas2] Masuoka, A.: Semisimple Hopf algebras of dimension 6 and 8. Preprint (1994) [N] Nikshych, D.: K0 -rings and twisting of finite dimensional Hopf Algebras. To appear in Commun. in Algebra (1997) [R1] Rieffel, M.A.: Compact Quantum groups associated with Toral subgroups. Contemp. Math. 145, 465–491 (1993) [R2] Rieffel, M.A.: Deformation Quantization for Actions of Rd . Memoirs A.M.S. 506 (1993) [S] Sekine, Y.: An example of finite dimensional Kac algebras of Kac-Paljutkin type. To appear in Proc. Amer. Math. Soc. [Va] Vainerman, L.I.: Relations between Compact Quantum Groups and Kac Algebras. Adv. in Soviet Math. 9, 151–160 (1992) [Va1] Vainerman, L.: 2-Cocycles and Twisting of Kac Algebras, Preprint-Nr. gk-mp-9610/41, Mathematisches Institut Ludwig-Maximilians-Universität München, (Oktober 1996) 36pp [Vays] Vaysleb, E.: Cosemisimple Bialgebras and Discrete Quantum Semigroups. Preprint, UCLA (1996) [Wi] Williams, R.: Finite dimensional Hopf algebras, Thesis, Florida State University, 1988 [W1] Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 11, 613–665 (1987) [W2] Woronowicz, S.L.: Compact quantum groups. Preprint (1993) Communicated by A. Connes

Commun. Math. Phys. 191, 723 – 734 (1998)

Communications in


The Kowalevski Top: its r-Matrix Interpretation and Bihamiltonian Formulation I.D. Marshall? Centre des mathématiques et leurs applications, Ecole normale supérieure de Cachan, 61 avenue du président Wilson, 94235 Cachan Cedex, France Received: 18 December 1995 / Accepted: 2 June 1997

Abstract: The Lax pair representation for the Kowalevski top and for the Kowalevski gyrostat are obtained directly via the r-matrix method. The construction of Reyman and Semenov–Tian-Shansky is then used to give a bihamiltonian description of the Kowalevski gyrostat 1. Introduction The Kowalevski top is the third integrable case of a heavy rigid body with a fixed point, rotating in a constant gravitational field. The other two cases are due respectively to Euler and to Lagrange. This third case was published in 1889 [6]. It has historical importance for the field of integrable systems and has been much studied since. It is remarkable that for so long this system resisted any attempts to give it a Lax representation. A significant milestone in the modern subject of integrable systems was the publication in 1987 of three different papers by three different groups of authors. These papers finally established the existence of a Lax representation for the Kowalevski top. In fact in each of the three papers a different Lax pair was found. The results of [4] and of [1] are important as they demonstrate interesting connections between the Kowalevski top and other well-known systems with Lax pairs. In both papers methods of algebraic geometry are used. Haine and Horozov in [4] showed that on any level surface of the invariants, the equations of motion for the Kowalevski top are equivalent to the Neumann system. Adler and van Moerbeke in [1] found connections between the Kowalevski top and the systems of Manakov and Henon-Heiles. A disadvantage with both of these results is that the entries of the matrices of the Lax pair depend on the physical variables via rational transformations. It should be underlined that both sets of results are distinguished by the fact that the Lax pairs can be said to ? Present address: Mathematics Department, University of Glasgow, Glasgow, G12 8QW, UK. E-mail: [email protected]

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have been constructed rather than guessed. This last fact represents a major achievement for each of the approaches of the respective pairs of authors of [7] and [1]. The result of interest for the present paper is that of Reyman and Semenov–TianShansky [8], in which they gave a Lax pair in the form of 4 × 4 matrices, and in which the variables of the system appear explicitly. The Lax pair given in [8] was arrived at by means of a process of symmetry reduction applied to a system of two interacting tops. From this point of view the Kowalevski top is a particular member of a whole family of integrable systems, which could therefore be considered to be a family of multi-dimensional generalisations of the Kowalevski top. The result presented here is supplementary to the result of Reyman and Semenov– Tian-Shansky in that it gives a direct description of the Lax pair in [8] by means of the r-matrix construction. The best reference for the background to the result in the present paper is the article [2] by Reyman and Semenov–Tian-Shansky with Bobenko. There it is explained additionally how with the help of their Lax representation one can study the geometry of the spectral curve and that the form of the solutions one obtains by using the machinery of finite-gap integration theory are simpler than those found by Kowalevski. This should not be understated, for in the work of Kowalevski the explicit solution of the system in terms of physical variables was extremely complicated and difficult to compute, and this part of the result was completed by Kötter [5]. Another important reference is the chapter contributed by Reyman and Semenov–Tian-Shansky to Volume 16 of the Encyclopaedia of Mathematics series [9]. This is a good place to read about the r-matrix construction in the general context and it contains several other relevant references, as well as the Kowalevski top result. The book [3] is highly recommended: it describes the work of Kowalevski (concerning other results as well as the famous top result) in the context of nineteenth century mathematics and helps to makes it clear why this result was so highly thought of and how it represented such a significant advance at the time of its discovery.

Reyman and Semenov–Tian-Shansky defined a family of Hamiltonian systems on T ∗ SO(p, q) by considering the space ` so(p, q), τ of twisted loops based on so(p, q) for q ≤ p, where the involution τ is given by τ (X) = −X T , and using a construction originating in [7]. This construction gives, for any simple Lie group G with Cartan involution σ, a family of Poisson maps from T ∗ G with its canonical symplectic structure into `(g∗ , σ) with its r-matrix Poisson structure arising from the (standard) splitting corresponding to C [λ, λ−1 ] = C [λ] + λ−1 C [λ−1 ]. Restriction to a subfamily followed by the consequent Hamiltonian reduction by the action of SO(q) leads to a Poisson map from T ∗ SO(p) into ` so(p, q), τ . Let us denote this map 8. The r-matrix construction says that the family F of invariant functions on ` so(p, q), τ are in involution and hence the family 8∗ F are in involution on T ∗ SO(p). Due to the fact that sufficiently many of the elements in 8∗ F are independent, this procedure gives integrable systems on T ∗ SO(p). The image under d8 of the equations of motion takes the form of a Lax pair. For p = 3, q = 2 one of these systems turns out to be the Kowalevski top in the stationary reference frame. In physical variables the transformation from the stationary ∗ frame T ∗ SO(p) to the moving frame, where the phase space is so(p) n (Rp )q with its Lie Poisson bracket, is implemented in the loop space by a gauge transformation and this gauge transformation is not a Poisson map. What is important however is that it does preserve the property of the equations of motion that theyare prsented in Lax formg.

The Kowalevski Top

725

∗ Let us denote by 9 the map thus obtained from so(p) n (Rp )q to ` so(p, q), τ . 9 is invertible whilst 8 in general is not invertible. Thus, there are two maps given in [2]: 8 : T ∗ SO(p) → ` so(p, q), τ , which ∗ is Poisson but noninvertible, and 9 : so(p) n (Rp )q → ` so(p, q), τ , which is invertible but not Poisson. The Poisson (non-Poisson) property of 8(9) is with respect to the standard r-matrix Poisson bracket on ` so(p, q), τ . For H(L) = 21 tr L2 |λ0 , 8∗ H gives the Hamiltonian for the generalised Kowalevski top in the stationary reference frame and 9∗ H gives the Hamiltonian for the generalised Kowalevski top in the moving reference frame. The map 8 gives an elegant proof of the integrability of the Kowalevski top as well as its generalisations, but as it is noninvertible it cannot be said to provide a true Lax representation. However the map 9 which, other than by long-hand computation in the moving frame phase space, cannot be used to prove the integrability of the Kowalevski top, is invertible and so does provide a true Lax representation. Indeed it can be used, as was demonstrated for the Kowalevski top case (as opposed to for all of its generalisations), in [2] via the finite-gap method, to solve the equations of motion explicitly. It would be nice to have a map from the physical phase space to a loop algebra which is both Poisson and invertible. In fact it is sufficient just to make 9 be a Poisson map and that is achieved in the present paper. Speaking in general terms, where g is any Lie algebra, if g can be split as a vector space direct sum into two subalgebras A and B, then a (nonstandard) Poisson structure can be defined on `(g∗ ) by splitting the ring `g into λg[λ] + A and λ−1 g[λ−1 ] + B and then specifying an r-matrix in the obvious way corresponding to this splitting of `g. The Lax pair representation of the Kowalevski top in the Euler-Poisson picture is shown to be a Poisson map with respect to a Poisson structure on `(so(p, q), τ ) of this kind. An unforseen advantage of using the new Poisson structure on the loop space is that it permits the use of a method of Reyman and Semenov–Tian-Shansky to find a bihamiltonian version of the integrability of the Kowalevski gyrostat. The most attractive example to treat from the point of view of this paper is the example of the Kowalevski top, or more particularly the Kowalevski gyrostat, for which p = 3 and q = 2. This is because for this case, when we adopt the procedure to be described in Sect. 3, we are led unavoidably to the example of the classical Kowalevski top. The more general (p, q) cases are found by the observation that one can restrict to certain Poisson subspaces. In the present context we can view the form of these subspaces as being suggested by the p = 3, q = 2 example, although of course we know what they are already following [8]. For this reason we devote Sect. 2 and Sect. 3 to the p = 3, q = 2 case for which we use the identification so(3, 2) = sp(4). Then we describe the general case in Sect. 4. In Sect. 5 when we discuss the bihamiltonian formalism for the Kowalevski top and its generalisations, we shall continue to artificially distinguish between the cases g = sp(4) and g = so(p, q) because of the simple form that the results take with the g = sp(4) notation, which then suggests the form that the results should take for the general case. The possibility to generalise the result of Sect. 3 to the general (p, q) case was pointed out to the author by A.G.Reyman [private communication 1996]. 1.1 Notation. Much use will be made of the Kronecker product of matrices. This has a standard notation, but in order to avoid confusion it is defined below. If A is a p × q matrix and B is an m × n matrix A ⊗ B is the pm × qn matrix,

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

a11 B A ⊗ B =  ...

a12 B .. .

...

ap1 B

ap2 B

...

 a1q B ..  . .

(1.1)

apq B

Use will also be made of the standard Pauli matrices σ1 , σ2 , σ3 . In fact we will use σ1 =

0 1

1 0

,

iσ2 =

0 1 −1 0

,

σ3 =

1 0

0 −1

,

(1.2)

as well as the 2 × 2 identity matrix which will be denoted by the symbol 1. The notation mat(p × q, R) means “the set of real p × q matrices”.

2. The Kowalevski Top: Some Known Results The content of this section can be found in [2] (and in [8] or [9]). In the moving reference frame the Kowalevski top is a hamiltonian system on the dual space e(3)∗ of the Lie algebra of the Euclidean group E(3) of rigid motions of three-dimensional Euclidean space. The phase space is six-dimensional and the standard dynamical variables are `1 , `2 , `3 , g1 , g2 and g3 , with Poisson bracket given by {ì , `j } = ijk `k ,

{ì , gj } = ijk gk ,

{gj , gj } = 0 .

(2.1)

In other words the phase space is the dual space of the semidirect product Lie algebra so(3) n R3 , with the Lie–Poisson bracket. There are two Casimir functions, I1 = g 2 and I2 = ` · g, and equating these to constants defines the generic symplectic leaves which are four dimensional. A hamiltonian system on this space will be integrable therefore if it has one non-trivial constant of motion. The Hamiltonian for the Kowalevski top is given by (2.2) H(`, g) = 21 (`21 + `22 + 2`23 ) − g1 . Reyman and Semenov–Tian-Shansky introduced a more general kind of system. They called their system the Kowalevski gyrostat in two fields. Its phase space is ninedimensional and can be thought of as the dual to the semidirect product Lie algebra so(3) n (R3 ⊕ R3 ). That is, its standard coordinates are `1 , `2 , `3 , g1 , g2 g3 , h1 , h2 and h3 and the Poisson bracket is given by {ì , `j } =ijk `k , {ì , gj } = ijk gk , {ì , hj } = ijk hk , {gi , gj } = {gi , hj } = {hi , hj } = 0 .

(2.3)

There are three Casimir functions, C1 = g 2 , C2 = h2 and C3 = g·h and generic symplectic leaves are 6 dimensional. The Hamiltonian function for the Kowalevski gyrostat is given by (2.4) H(`, g, h) = 21 (`21 + `22 + 2`23 ) + γ`3 − (g1 + h2 ) , where γ is a constant. The Kowalevski top is obtained from the Kowalevski gyrostat by imposing the invariant condition h = 0, i.e. C2 = 0, together with γ = 0. The following result is due to Reyman and Semenov–Tian-Shansky:

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727

Theorem [8]. The equations of motion for the Kowalevski gyrostat are equivalent to the Lax equation L˙ = [L, M ] where −`2 1 − `1 iσ2 −γiσ2 0 0 + 21 L(λ) = −λ `2 1 − `1 iσ2 −(2`3 + γ)iσ2 0 σ3 (2.5a) g 3 σ3 + h3 σ1 1 −1 (g1 − h2 )σ3 + (g2 + h1 )σ1 +2λ g 3 σ3 + h3 σ1 −(g1 + h2 )σ3 + (g2 − h1 )σ1 and

M (λ) = λ

0 0

0 σ3

+

1 2

−(2`3 + γ)iσ2 −`2 1 + `1 iσ2

`2 1 + `1 iσ2 (2`3 + γ)iσ2

.

(2.5b)

Two independent commuting integrals for the Kowalevski gyrostat are given by the functions and K2 = tr L4 |λ−2 , (2.6) K1 = tr L4 |λ0 where the subscripts mean respectively “take the λ0 component” and “take the λ−2 component” in the expansion in powers of λ. The fact that the Hamiltonian function of (2.4) (up to addition of a constant) is equal to H = − 21 tr L2 |λ0

(2.7)

is an indication that a straightforward r-matrix description might be possible. Note that the matrix L defined in (2.5a) differs from that in [8] by a factor of a half. This is convenient for present purposes, and of course makes no difference to the validity of the Lax pair.

3. The r-Matrix Construction Applied to the Kowalevski Gyrostat In this section it is shown that the Lax pair given by (2.5) can be obtained as a direct consequence of the r-matrix construction. Let g be sp(4). That is g = sp(4) = {X ∈ gl(4, R) | JXJ = X T } ,

(3.1)

where J = 1 ⊗ iσ2 , so that g = {X = A ⊗ 1 + S1 ⊗ σ1 + S2 ⊗ iσ2 + S3 ⊗ σ3 | AT = −A, SiT = Si } .

(3.2)

Define an involutive automorphism τ : g → g (i.e. τ satisfies τ 2 = id and [τ X, τ Y ] = τ [XY ] ∀X, Y ∈ g) by (3.3) τ (X) = −X T , then X = τ (X) ⇒ X = A ⊗ 1 + S ⊗ iσ2

(3.4)

X = −τ (X) ⇒ X = S1 ⊗ σ1 + S3 ⊗ σ3 .

(3.5)

and The twisted loop algebra associated with τ is given by

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I.D. Marshall

`(g, τ ) = {X ∈ `g | X(−λ) = τ (X(λ))} n X = X= Xk λk ∈ `g | X2n is of the form A ⊗ 1 + S ⊗ iσ2 , k

o X2n+1 is of the form S1 ⊗ σ1 + S3 ⊗ σ3 .

(3.6)

In (3.4), (3.5) and in (3.6) the symbols S, S1 and S3 denote arbitrary symmetric 2 × 2 matrices, whilst A denotes an arbitrary antisymmetric 2 × 2 matrix. Defining the pairing h , i : `g × `g → R by hX, Y i = tr X(λ)Y (λ)|λ0 ,

(3.7)

we have Proposition 3.1. The restriction to `(g, τ ) × `(g, τ ) of the pairing given by (3.7) is nondegenerate and hence it can be used to identify `(g, τ )∗ with `(g, τ ). From now on we will identify `(g, τ )∗ = `(g, τ ) by means of h , i. For any element X ∈ `(g, τ ) we can write X = X+ + X0 + X−

(3.8)

with X+ (λ) a series in λ, X0 independent of λ and X− (λ) a series in λ−1 . Let P± and P0 be the projection operators on `(g, τ ) with respect to the decomposition of (3.8). Let `(g, τ )0 be the λ0 –component of `(g, τ ), i.e. `(g, τ )0 = P0 `(g, τ ) = {A⊗1+S ⊗ iσ2 }. Consider the splitting of `(g, τ )0 as a vector space direct sum of two subalgebras, `(g, τ )0 = A + B ,

(3.9)

with A = {A ⊗ 1 + S ⊗ iσ2 | tr S = 0},

B = {x(1 + σ3 ) ⊗ iσ2 | x ∈ R} .

(3.10)

Note that `(g, τ )0 is u(2) and A is su(2), while B is abelian. Defining ρ ∈ End (`(g, τ )0 ) by

ρ = PA − PB ,

(3.11)

where PA and PB are projection operators on `(g, τ )0 with respect to the splitting given by (3.9), we have the following Proposition 3.2. R ∈ End(`(g, τ )) given by R = P+ + ρ ◦ P0 − P−

(3.12)

Σ = 21 (σ3 − 1) ⊗ σ3 .

(3.13)

is a classical r-matrix. Let Then we have Proposition 3.3. The space K ⊂ `(g, τ ) (∼ `(g, τ )∗ ) given by K = {L(λ) = λΣ + α + λ−1 β} ,

(3.14) ∗

is a Poisson subspace with respect to the R-Lie-Poisson bracket on `(g, τ ) .

The Kowalevski Top

729

Proof. The Proposition follows by a direct calculation and depends crucially on the fact that Σ commutes with B. In fact, if we suppose that C = S1 ⊗ σ1 + S3 ⊗ σ3 satisfies [C, (1 + σ3 ) ⊗ iσ2 ] = 0 (the condition that C be in the centraliser of B), then without loss of generality we can assume that S1 = 0 (otherwise we could make a similarity transformation which changes σ1 and σ3 and leaves 1 and iσ2 unchanged) and then this condition reads S3 (1 + σ3 ) + (1 + σ3 )S3 = 0 ,

(3.15)

i.e. S3 has the form x(1 − σ3 ), for some x ∈ R. If we choose as real coordinates on K the functions ì , gi , hi for i = 1, 2, 3 and γ, given by writing α = − 21 `1 σ1 ⊗ iσ2 − 21 `2 iσ2 ⊗ 1 − 21 `3 (1 − σ3 ) ⊗ iσ2 − 21 γ1 ⊗ iσ2

(3.16)

and β = 21 h1 σ3 ⊗σ1 − 21 h2 1⊗σ3 + 21 h3 σ1 ⊗σ1 + 21 g1 σ3 ⊗σ3 + 21 g2 1⊗σ1 + 21 g3 σ1 ⊗σ3 , (3.17) then we get Proposition 3.4. With respect to { , }R on the Poisson subspace K of `(g, τ )∗ , we have {γ, f } = 0, {ì , `j } = ijk `k ,

for any f ∈ C ∞ (K),

{ì , hj } = ijk hk ,

{ì , gj } = ijk gk ,

{hi , hj } = {gi , gj } = 0 .

(3.18) (3.19) (3.20)

Denoting by Kγ the subset of K given by fixing the value of γ arbitrarily, it follows that ∗ Kγ is isomorphic as a Poisson space to so(3) n (R3 ⊕ R3 ) , which is the phase space for the Kowalevski gyrostat. The r-matrix method guarantees that all coadjoint invariant functions on Kγ commute with respect to { , }R . It follows that the subset of C ∞ (K) generated by the functions Hk,j given by −2j 1 2k tr L(λ) λ (3.21) Hk,j (L) = 0 2k λ

commute amongst themselves. H = −H1,0 is the Hamiltonian function for the Kowalevski gyrostat. H2,0 and H2,−1 are two commuting second integrals. Remarks. (i) It has been shown that the Lax representation for the Kowalevski gyrostat can be obtained by direct application of the r-matrix approach for a particular choice of splitting of the twisted loop algebra based on sp(4). As such, the result is a minor but necessary comment on the result of Reyman and Semenov–Tian-Shansky. (ii) The use of the r-matrix of (3.12), with the operator ρ acting on the λ0 –component, amounts to the identification via the trace form of the dual space to su(2) not with su(2) itself (which is an obvious identification to make), but with the space span{iσ1 , iσ2 , i(1 − σ3 )} ∼ span{σ1 ⊗ iσ2 , iσ2 ⊗ 1, (1 − σ3 ) ⊗ iσ2 } .

(3.22)

730

I.D. Marshall

In effect, the choice of A and B in (3.10) was made in order to force this identification. However many other choices for B, such as span{1 ⊗ iσ2 + σ1 ⊗ iσ2 } with Σ = (1 − σ1 ) ⊗ σ3 , or

(3.23) span{1 ⊗ iσ2 + iσ2 ⊗ 1} with Σ = σ1 ⊗ σ1 + σ3 ⊗ σ3 ,

and so on, would eventually lead to precisely the same system, namely to the Kowalevski gyrostat. 4. The General Case In this section it is shown how to use the r-matrix method to obtain the generalisations of the Kowalevski top given in [8]. The author thanks A.G.Reyman for pointing out that the previous section can be extended in this way. The idea is to do just the same as was done in [8] by starting with the twisted loop algebra `(so(p, q), τ ), where τ : so(p, q) → so(p, q) is given by τ (X) = −X T , but to define on `(so(p, q), τ ) a nonstandard r-matrix analogous to the one used in Sect. 3. Indeed in the case p = 3, q = 2, using the identification so(3, 2) = sp(4), the two r-matrices are just the same. We assume that q ≤ p and we define the p × q matrix E by I (4.1) E= q , 0 where Iq is the q × q identity matrix. In the rest of this section we shall use the blockdecomposition of (p + q) × (p + q) matrices according to which the (1, 1) block is a p × p matrix and the (1, 2) block is a p × q matrix. Let g be any fixed element of the group SO(p). In an analogous fashion to that of Eq. (3.9), let us split `(so(p, q), τ )0 as a vector space direct sum of two subalgebras `(so(p, q), τ )0 = A + B, with

n A=

a 0

o 0 a ∈ so(p) , 0

n B=

gEbE T g T 0

(4.2) o 0 b ∈ so(q) . b

(4.3)

The freedom to choose g arbitrarily corresponds to the freedom in the choice of B mentioned at the end of the last section. We will suppose from now on with no loss of generality that g in (4.3) is the identity. We may note that `(so(p, q), τ )0 is so(p) ⊕ so(q), A is the same as so(p) whilst B is the same as so(q). As in the last section, we now define ρ ∈ End(`(so(p, q), τ )0 ) by ρ = PA − PB ,

(4.4)

where PA and PB are projection operators on `(so(p, q), τ )0 with respect to the splitting given by (4.2), and we go on as in the last section to define the r-matrix R ∈ End(`(so(p, q), τ )) by (4.5) R = P+ + ρ ◦ P0 − P− . We use the same formula as that in (3.7) to identify `(so(p, q), τ )∗ with `(so(p, q), τ ). We let

The Kowalevski Top

731

0 E (4.6) ET 0 and we prove precisely the same proposition in this new context as was proved in the last section. Let us define the Poisson subspace K ⊂ `(so(p, q), τ ) by the same formula as that in (3.14) and consider a generic hamiltonian vector-field on K. If we write O 0 ω 0 0 E −1 + , (4.7) + λ K 3 L(λ) = λ 0T 0 0 ET 0 Σ=

with ω ∈ so(p), ∈ so(q), 0 ∈ mat(p × q, R), then for any function φ ∈ C ∞ (K) the hamiltonian vector-field generated by φ is given by ω˙ = [X − EY E T , ω] + 21 (50T − 05T ), ˙ = −E T [X, ω]E − [Y, ] + 1 E T (05T − 50T )E, 2

(4.8)

0˙ = (X − EY E T )0, where the components X ∈ so(p), Y ∈ so(q), 5 ∈ mat(p × q, R), of dφ are given by d φ(ω + tδω, 0 + tδ0, + tδ) = tr X δω + tr 5T δ0 + tr Y δ. (4.9) dt t=0

The constraint + E T ωE = γ, where γ is an arbitrary constant for q = 2 and γ = 0 for q 6= 2, is invariant under the flow generated by all hamiltonian vector-fields on K and hence the space P of all L ∈ K of the form P 3 L(λ) = L(λ)(ω, 0) = λΣ + α + λ−1 β ω 0 E + =λ T 0 0 E

0 γ − E T ωE

+λ

−1

O 0T

0 , 0

(4.10)

is itself a Poisson subspace of `(so(p, q), τ ). On P the R-Lie-Poisson bracket gives (4.11) {φ, ψ}(ω, 0) = tr ω[δω φ, δω ψ] + tr 0T δω φ δ0 ψ − δω ψ δ0 φ , or equivalently, the hamiltonian vector-field XH for H ∈ C ∞ (P) is given by ω˙ = [δω H, ω] + 21 δ0 H 0T − 0 δ0 H T , 0˙ = δω H 0.

(4.12)

In other words P is the dual of semidirect product Lie algebra so(p) n (Rp )q with its Lie-Poisson bracket. The functions on P of the form 1 tr L(λ)2k λ−2j (4.13) Hk,j = 2k λ0 are in involution and the generalised Kowalevski top on P has the Hamiltonian H1,0 = 21 tr ω 2 + 21 tr ωEE T ωEE T − tr ωEγE T + 2tr E T 0 + 21 γ 2 .

(4.14)

Thus the equations of motion for the generalised Kowalevski top are ω˙ = [EE T ωEE T − EγE T , ω] + (E0T − 0E T ), 0˙ = ω + EE T ωEE T − EγE T 0, where, as we recall, γ ∈ so(q) is zero unless q = 2.

(4.15)

732

I.D. Marshall

5. Bihamiltonian Formulation In this section it is shown how to extend the results of Sects. 3 and 4 in a very simple way, so as to give a bihamiltonian formulation of the Kowalevski gyrostat. We recall a construction of Reyman and Semenov–Tian-Shansky (see §4.2 in [9]): Let `(g) = g ⊗ C[λ, λ−1 ] and let `(g)+ = g ⊗ C[λ] and `(g)− = g ⊗ λ−1 C[λ−1 ]. Let R ∈ End(`(g)) be given by R|`(g)± = ±id and let Rk = R ◦ 3k , where 3 ∈ End(`(g)) is given by (3X)(λ) = λX(λ). Then all the Rk are classical r-matrices and the corresponding R-Lie-Poisson brackets on `(g)∗ are all compatible. If we wish to apply this construction to a twisted loop algebra `(g, τ ) then we must modify it to take into account the order of the involution τ . Thus we must let Rk = R ◦ 3ks , where s is the order of τ , i.e. s is the smallest positive integer for which τ s = id. Indeed for the r-matrices defined in (3.12) and in (4.5) it is easy to check that Rk = R ◦ 32k are all classical r-matrices and, by the construction of Reyman and Semenov–Tian-Shansky, that they give rise to compatible Poisson structures on `(sp(4), τ ) or on `(so(p, q), τ ). For the reader’s convenience the formula is now given for the two Poisson brackets on the whole space `(g, τ ) where, as is the case in thepresent examples, τ is an order 2 involution. For L ∈ `(g, τ ); for φ, ψ ∈ C ∞ `(g, τ ) let ξ = δL φ and η = δL ψ, i.e. φ(L+t1) = φ(L)+h1, ξi+O(t2 ) for all 1 ∈ `(g, τ ) and ψ(L+t1) = ψ(L)+h1, ηi+O(t2 ) for all 1 ∈ `(g, τ ). Then {φ, ψ}k (L) = h3−2k L, [(32k ξ)+ + (32k ξ)0A , (32k η)+ + (32k η)0A ] − [(32k ξ)0B + (32k ξ)− , (32k η)0B + (32k η)− ]i. (5.1) The next problem in applying the construction of Reyman and Semenov–TianShansky is to find convenient Poisson subspaces of the respective loop algebras. As is easily seen, whilst K in Sect. 3 and P ⊂ K in Sect. 4 are Poisson subspaces with respect to the R0 -Lie-Poisson bracket on `(g, τ ) for g = sp(4) or so(p, q) respectively, they are not Poisson subspaces with respect to the other Rk -Lie-Poisson brackets for k 6= 0. However one can check that a simple modification of these spaces gives what we want. The modification consists in adding a λ−2 -term which is to represent a Casimir with respect to the R0 -Lie-Poisson bracket encountered already, so that the extended space is invariant under all hamiltonian flows with respect to the R−1 -Lie-Poisson bracket. Thus for the g = sp(4) case we extend K ⊂ `(sp(4), τ ) and define the space ˆ (5.2) Kˆ = L(λ) = λΣ + α + λ−1 β + λ−2 x1 ⊗ iσ2 , where the matrices Σ, α, β are exactly the same as those in (3.14) and x ∈ R. For g = so(p, q) we extend P ⊂ K ⊂ `(so(p, q), τ ) and define n o 0 0 ˆ Pˆ = L(λ) = λΣ + α + λ−1 β + λ−2 , 0 B

(5.3)

where Σ, α, β are as they were in (4.10) and B ∈ so(q). It can be checked that Kˆ and Pˆ are Poisson subspaces of `(sp(4), τ ) and `(so(p, q), τ ) respectively with respect to the R0 -Lie-Poisson bracket and that the variables x and B respectively are Casimirs. It can also be checked that with respect to the R−1 -LiePoisson bracket both spaces are still Poisson subspaces. It follows that these spaces have compatible Poisson structures given by the R0 – and the R−1 –Lie-Poisson brackets.

The Kowalevski Top

733

It is traditional for discussing bihamiltonian structures to write the compatible Poisson structures as hamiltonian vector-fields. Thus on Kˆ for the sp(4) case, we have  ˙ `=ρ∧`+ξ∧g+η∧h     g˙ = ρ ∧ g − xη X(0) H :  h˙ = ρ ∧ h + xξ    x˙ = 0; (5.4a)  ˙1 = η3 + `2 k `       `˙2 = −ξ3 − `1 k      `˙3 = ξ2 − η1      g˙1 = −(`3 + γ)η1 − `2 ξ3 + `3 ξ2 + (h1 + g2 )k     g˙ = −ρ − (` + γ)η − ` ξ + ` ξ + (h − g )k 2 3 3 2 3 1 1 3 2 1 X(−1) : H g˙3 = ρ2 − (`3 + γ)η3 − `1 ξ2 + `2 ξ1 + h3 k      ˙  h1 = ρ3 + (`3 + γ)ξ1 − `2 η3 + `3 η2 + (h2 − g1 )k      h˙ 2 = (`3 + γ)ξ2 − `3 η1 + `1 η3 − (h1 + g2 )k       h˙ 3 = −ρ1 + (`3 + γ)ξ3 − `1 η2 + `2 η1 − g3 k    x˙ = (g · η − h · ξ) + h1 η2 − h2 η1 + g1 ξ2 − g2 ξ1 − ρ1 `2 + ρ2 `1 , (5.4b) where the components, ρ, ξ, η ∈ R3 and k ∈ R, of dH are given by d H(` + tδ`, g + tδg, h + tδh, x + tδx) = ρ · δ` + ξ · δg + η · δh + kδx. (5.4c) dt t=0 On Pˆ for the so(p, q) case, we have  T 1   ω˙ = [X, ω] + 2 50 − 05 X(0) 0˙ = X0 + 21 5B H :   ˙ B = 0;  ω˙ = 21 E5T − 5E T + [ω, EY E T ]       0˙ = −XE + 21 ω5 + 5E T ωE + 0Y − EY E T 0 X(1) H : B˙ = E T [ω, X]E + [B, Y ]       + 21 0T 5 − 5T 0 + E T (05T − 50T )E ,

(5.5a)

(5.5b)

where the components, X ∈ so(p), Y ∈ so(q), 5 ∈ mat(p × q, R), of dH are given by d H(ω + tδω, 0 + tδ0, B + tδB) = tr X δω + tr 5T δ0 + tr Y δB. (5.5c) dt t=0 It is now a simple exercise to compute, firstly for the sp(4) case: C = 21 x2 ⇒ X(0) = 0 = X(0) and ⇒ X(−1) C K ⇒

X(−1) K

=

X(0) H

for K = −`3 x − 21 (g 2 + h2 ) − γx for H =

1 2 2 (`1

+

`22

+

2`23 )

+ γ`3 − g1 − h2 ;

(5.6)

734

I.D. Marshall

and for the so(p, q) case C = 21 tr B 2 ⇒ X(0) = 0 = X(0) and ⇒ X(−1) C K ⇒

X(−1) K

=

X(0) H

for K = −tr EBE T ω + tr 0T 0 for H =

1 2 tr ω

T

ω + EE ωEE

T

(5.7) + 2tr 0E.

(−1) In both cases, X(0) gives the equations of motion of the Kowalevski top after H = XK setting x = 0 or B = 0. Thus we have a bihamiltonian version of the Kowalevski top.

Note. The results of this section were motivated by a talk of F.Magri. He had mentioned that it was possible to give a bihamiltonian version of the Kowalevski top and this gave the impetus to try and find an appropriate extension of the results of Sects. 3 and 4. It seems that the bihamiltonian formulation presented here differs in some way from that of which Magri spoke, which is due to G.Magnano. An account of the other version can be found in [Magnano 1996]. Acknowledgement. I would like to thank Franco Magri for lengthy discussions on the bihamiltonian formalism, especially concerning the Kowalevski top. I would also like to thank Alexei Reyman, who actually gave a material contribution to this paper, as is clear from §4. This paper was finally completed during a stay at the Institut Henri Poincaré in Paris and the kind generosity of my hosts there is very much appreciated.

References 1. Adler, M., van Moerbeke, P.: The Kowalewski and Hénon-Heiles motions as Manakov Geodesic flows on SO(4) – a two-dimensional family of Lax pairs. Commun. Math. Phys. 113, 659–700 (1988) 2. Bobenko, A.I., Reyman, A.G., Semenov–Tian-Shansky, M.A.: The Kowalevski top 99 years later: A Lax pair, generalisations and explicit solutions. Commun. Math. Phys. 122, 321–354 (1989) 3. Cooke, R.: The Mathematics of Sonya Kowalewskaya. Berlin Heidelberg: Springer-Verlag, 1984 4. Haine, L., Horozov, E.: A Lax pair for Kowalewski’s top. Physica D 29, 173–180 (1987) 5. Kötter, F.: Sur le cas traité par Mme Kowalewskie de rotation d’un solide autour d’un point fixe. Acta Math. 17, 209–264 (1893) 6. Kowalevski, S.: Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12, 177–232 (1889) 7. Reyman, A.G.: Integrable Hamiltonian systems connected with graded Lie algebras. Differential Geometry, Lie groups and mechanics II. Zap. Nauchn. Semin. LOMI 95, 3–54 (1980) [J.Soviet Math. 19, 1507–1545 (1982)] 8. Reyman, A.G., Semenov–Tian-Shansky, M.A.: Lax representation with a spectral parameter for the Kowalevski top and its generalisations. Lett. Math. Phys. 14, 55–62 (1987) 9. Reyman, A.G., Semenov–Tian-Shansky, M.A.: Group-theoretical methods in the theory of finite dimensional integrable systems. Part 2, Chapter 2. In: Encyclopaedia of Mathematical Sciences vol.16, eds. V.I.Arnold & S.P.Novikov, Berlin Heidelberg: Springer-Verlag 1994 Communicated by M.Jimbo This article was processed by the author using the TEX style file pjour1 from Springer-Verlag.

Commun. Math. Phys.191, 735 (1998)

Communications in


Erratum Phase Space Bounds for Quantum Mechanics on a Compact Lie Group Brian C. Hall Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada Received: 30 September 1997 / Accepted: 30 September 1997

Commun. Math. Phys. 184, no 1, 233–250 (1997) ˙ indicating the Three times on p. 238 the symbol γ¯ appears. This symbol should be γ, time derivative of γ. Communicated by D. Brydges

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