Physics Reports vol.318

F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83 1 PATTERN FORMATION AND COMPETITION IN NONLINEAR OPTICS F. Tito...

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F.T. Arecchi et al. / Physics Reports 318 (1999) 1}83

1

PATTERN FORMATION AND COMPETITION IN NONLINEAR OPTICS

F. Tito ARECCHI , Stefano BOCCALETTI, PierLuigi RAMAZZA Istituto Nazionale di Ottica, Largo E. Fermi, 6, 150125, Florence, Italy Department of Physics, University of Florence, Florence, Italy Dept. of Physics and Applied Mathematics, Universidad de Navarra, Irunlarrea s/n, Pamplona, Spain

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 318 (1999) 1}83

Pattern formation and competition in nonlinear optics F. Tito Arecchi , Stefano Boccaletti, PierLuigi Ramazza * Istituto Nazionale di Ottica, Largo E. Fermi, 6, I50125, Florence, Italy Department of Physics, University of Florence, Florence, Italy Dept. of Physics and Applied Mathematics, Universidad de Navarra, Irunlarrea s/n, Pamplona, Spain Received January 1999; editor: I. Procaccia Contents 1. Introduction 1.1. Optical patterns 1.2. Aspect ratios 1.3. Classi"cation of laser systems depending on damping rates 1.4. Outline of this review 2. Patterns in active optical systems 2.1. The theory of patterns in lasers 2.2. Experiments with lasers 2.3. Patterns in photorefractive systems 3. Patterns in passive optical systems 3.1. Filamentation in single-pass systems 3.2. Solitons in single-pass systems 3.3. Counterpropagating beams in a nonlinear medium 3.4. Nonlinear medium in an optical cavity 3.5. Nonlinear slice with optical feedback 3.6. Nonlocal interactions 4. Defects and phase singularities in optics 4.1. Phase singularities and topological defects in linear waves 4.2. Phase singularities in nonlinear waves

4 4 4 7 9 11 11 14 17 24 24 25 28 32 38 45 51

5. Open problems and conclusions 5.1. Localized structures in feedback systems 5.2. Control of patterns 5.3. Patterns in atom optics Acknowledgements Appendix A. A reminder of nonlinear optics A.1. Nonlinear susceptibility A.2. The two level approximation A.3. The s and s nonlinear optics A.4. The photorefractive (PR) e!ect Appendix B. Rescaling the Maxwell}Bloch equations to account for detuning and a large aspect ratio Appendix C. Multiple scale analysis of the bifurcation problem for the non lasing solution of the Maxwell}Bloch equations Appendix D. Symmetries and normal form equations References

51 53

* Corresponding author. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 0 7 - 1

57 57 60 62 65 65 65 65 68 70

71

71 73 76


3

Abstract Pattern formation and competition occur in a nonlinear extended medium if dissipation allows for attracting sets, independently of initial and boundary conditions. This intrinsic patterning emerges from a reaction di!usion dynamics (Turing chemical patterns). In optics, the coupling of an electromagnetic "eld to a polarizable medium and the presence of losses induce a more general (di!raction-di!usion) mechanism of pattern formation. The presence of a coherent phase propagation may lead to a large set of unstable bands and hence to a richer variety with respect to the chemical case. A review of di!erent experimental situations is presented, including a discussion on suitable indicators which characterize the di!erent regimes. Vistas on perspective new phenomena and applications include an extension to atom optics. 1999 Elsevier Science B.V. All rights reserved. PACS: 05.45.#b Keywords: Nonlinear optics; Nonlinear dynamics; Pattern formation

4


1. Introduction 1.1. Optical patterns Pattern formation in extended media is the result of the interaction between a local nonlinear dynamics and space gradient terms which couple neighboring spatial regions. Furthermore, nonlocal terms may bring interactions from far away either in space or time. Some previous review papers are available on the matter [1,2]. A recent comprehensive review [3] is mainly devoted to #uid dynamic or chemical patterns, where the gradient terms result either from momentum transport of from di!usion processes. Optical patterns, on the other hand, are characterized by a wave transport, mainly pointing in one direction. Furthermore, use of laser sources and resonant media to enhance the nonlinearities restricts the time dependence to a quasi-monochromatic behavior. Therefore, the functional terms we have to deal with in optics are of the type f (x, y, z, t) e IX\SR ,

(1)

where the exponential term accounts for a plane wave moving along a direction z ((x, y) being the plane orthogonal to z) and the residual z and t dependence is slow, i.e.

Rf ;k" f ", Rz

Rf ;u" f " . Rt

(2)

This is currently called SVEA (slowly varying envelope approximation) and it is a sensible approximation even when the slow and fast scales di!er by a factor less than 10, as it occurs e.g. for femto-second pulses. Based upon this wave transport feature, optical patterns present all classes of relevant phenomena reported elsewhere, plus some ones which are speci"c of optics. It seems then appropriate to introduce a general classi"cation of optical patterns, within which it is easy to include also classes of phenomena observed in #uid and chemistry. For convenience, we have collected in Appendix A some introductory facts on nonlinear optics, together with the corresponding jargon. 1.2. Aspect ratios In the classi"cation we distinguish between the longitudinal space direction z and the transverse plane (x, y), since the boundary conditions are usually drastically di!erent for the two cases. We classify patterns as 0, 1, 2, or 3-dimensional depending on the functional space dependence of the envelope f. Notice that, at variance with condensed matter instabilities, a 0-dimensional dynamics (i.e. ruled by an ordinary di!erential equation for f ) still refers to a wave pattern which is mono-directional and mono-chromatic. Crucial parameters to estimate the dimensionality are the aspect ratios, which will be de"ned as follows. Let us con"ne the optical dynamics within a rectangular box of sides ¸ , ¸ , ¸ and take V W X ¸ X *u (8) C" X *u GG> de"nes a longitudinal aspect ratio. If C (1 only one longitudinal resonance can be excited, and X the cavity "eld is uniform along z, whereas if C '1 many excited longitudinal modes give rise to X a short pulse whose spatial length is smaller than the cavity length ¸ . X In Fig. 1a we report the frequency position of the modes and the medium line for 0-, 1- and 2-dimensional cases, in Fig. 1b we report the corresponding wavenumbers. In fact, for the 1- and 2-dimensional cases, we do not consider di!erent wavenumbers, but we use a pseudo-spectral method consisting in Fourier expanding around the central wavenumber k , and considering the residual spread as a slow time-space dependence in terms of evolution equations including the nonlinearities.


7

Table 1 Classi"cation of optical pattern forming systems depending on the longitudinal and transverse aspect ratios Dimension

X

>

Z

Systems

0

F (1 V

F (1 W

C (1 X

䢇

Single longitudinal and transverse mode laser

1

F (1 V

F (1 W

C '1 X

䢇

Single transverse mode, longitudinally multi-mode laser Temporal solitons in "bers

䢇

1

F '1 V

F (1 W

C (1 X

䢇

䢇

Linear arrays of semiconductor lasers Nonlinear optics of slab interferometers Single mode laser with delayed feedback Spatial solitons in planar waveguides

䢇䢇

2

F '1 V

F (1 W

C (1 X

䢇

n.l.o. of planar waveguides

2

F "F '1 V W

F "F '1 V W

C (1 X

䢇

Transverse optical patterns in active media (lasers, PRO) and passive media (liquid crystals, resonant gases)

F "F '1 V W

F "F '1 V W

C (1 X

䢇

3

Co-propagating beam interaction as e.g. four wave mixing

䢇

Break-up and "lamentation of beams in n.l.o.

Note: n.l.o."Nonlinear optics, PRO"Photorefractive oscillator.

In Table 1 we list optical pattern forming systems of di!erent dimensions. It is understood that along a dimension there is no evolution whenever the corresponding aspect ratio is less than 1. 1.3. Classixcation of laser systems depending on damping rates It is well known that a discrete nonlinear dynamical system can undergo a chaotic motion, that is, at least one of its Liapunov exponents can be positive, only when the number of degrees of freedom (phase space dimension) is at least 3. We "nd it convenient to refer to dissipative systems, that is, systems with damping terms for which the phase-space volume is not conserved. In such systems the sum of all Liapunov exponents is negative, and initial conditions tend asymptotically to an attractor [4]. For dimensions N"1, the attractor is a "xed point, for N"2 a "xed point or a limit cycle, for N"3 it can be a "xed point (all the three Liapunov exponents negative), a limit cycle (two Liapunov exponents negative and one zero), or a torus (one Liapunov exponent negative and two zero), or even a chaotic attractor (one Liapunov exponent negative, one zero and one positive). An example of chaotic motion is o!ered by the Lorenz model of hydrodynamic instabilities [5], which corresponds to the following equations, where the parameter values have been chosen so as

8


to yield one positive Liapunov exponent: 8 x "!10x#10y, y "!y#28x!xz, z "! z#xy , 3

(9)

where now x, y, z are suitable variables, and dots denote temporal derivatives. Also the fundamental equations for "eld matter interaction can exhibit chaotic features. Indeed, if we couple Maxwell equations with Schroedinger equations for N atoms con"ned in a cavity, and expand the "eld in cavity modes, keeping only the "rst mode which goes unstable, its amplitude E is coupled with the collective variables P and D describing respectively the atomic polarization and the population inversion. The resulting equations are EQ "!kE#gP,

PQ "!c P#gED , , (10) DQ "!c (D!D )!4gPE . , For simplicity, we consider the cavity frequency at resonance with the atomic one, so that we can take E and P as real variables and we have three real equations. Here k, c , c are the loss rates for , , "eld, polarization and population, respectively, g is a coupling constant and D is the population inversion which would be established by a pump mechanism in the atomic medium, in the absence of the coupling. While the "rst equation comes from the Maxwell equation, the two others imply the reduction of each atom to a two-level atom resonantly coupled with the "eld, that is, a description of each atom is an isospin space of spin 1/2. The last two equations are like Bloch equations which describe the spin precession in the presence of a magnetic "eld. For such a reason, Eqs. (10) are called Maxwell}Bloch equations. The presence of loss rates means that the three relevant degrees of freedom are in contact with a sea of other degrees of freedom. In principle, Eqs. (10) could be deduced from microscopic equations by statistical reduction techniques [6,365]. The similarity of Eqs. (10) with Eqs. (9) would suggest the easy appearance of chaotic instabilities in single mode, homogeneous-line lasers. However, time-scale considerations rule out the full dynamics of Eqs. (10) for most of the available lasers. Eqs. (9) have damping rates which lie within one order of magnitude of each other. On the contrary, in most lasers the three damping rates are widely di!erent from one another. The following classi"cation has been introduced [7]. Class A lasers (e.g. He}Ne, Ar, Kr, dye): c Kc M , (2.17) '()* where the symbol > denotes the equality when acting on e. The space of solutions now depends on the value of k'()*,Z '(Z)* "ZZ. If k"0, Eq. (2.17) implies (Z'()"M and C"M. Since Tr C"0, the 32;32 matrix C has 16 eigenvalues M and 16 eigenvalues !M, and therefore

N.A. Obers, B. Pioline / Physics Reports 318 (1999) 113}225

125

Eq. (2.16) is satis"ed for a dimension-16 space of vectors e. The state with charges Z'( is therefore annihilated by half the supersymmetry generators Q , and has a mass ? M"Z'(Z'( .

(2.18)

The condition ZZ"0 means that the antisymmetric charge matrix Z'( has rank 2, i.e. that only parallel M2-branes are superposed. If on the other hand ZZO0, we may rewrite Eq. (2.17) as Ce"(M!M)e, C"k'()*C , '()*

(2.19)

and we are lead back to an equation similar to Eq. (2.16). Squaring again yields C"(k'()*)#(k ) k)'()*C #(kk)'()*+,./C '()* '()*+,./ >(M!M) ,

(2.20)

where (k ) k)'()*"k'(+,k)*+,. As before, if k ) k"kk"0, this equation implies (k'()*)"(M!M)"C. Since Tr C"0, Eq. (2.19) is satis"ed by half the supersymmetries, but Eq. (2.16) by a quarter only. We therefore get a 1/4-BPS state with mass squared: M"Z'(Z'(#(k'()*k'()* ,

(2.21a)

k'()*"Z '(Z)* .

(2.21b)

This expression reduces to Eq. (2.18) for a 1/2-BPS state, i.e. when k'()*"0. On the other hand, if k ) k or kkO0 do not vanish, the state is at most 1/8-BPS and we have to carry the same analysis one step further. Note that the conditions k ) kO0 (resp. kkO0) can only be satis"ed when d56 (resp. d58), in agreement with the absence of 1/8-BPS states in more than "ve space-time dimensions. 2.3. BPS solutions of 11D SUGRA In want of a microscopic formulation of M-theory (or of non-perturbative type IIA string theory), it is certainly di$cult to determine what representations of the eleven-dimensional PoincareH superalgebra actually occur in the spectrum. However, this is achievable for BPS states, since supersymmetry protects these from quantum e!ects and in particular determines their exact mass formula. They can be studied at arbitrarily low energy, and in particular in the 11D SUGRA limit of M-theory. Instead of describing the equations implied by the BPS condition on the supergravity con"guration, we refer the reader to existing reviews in the literature [102,104,292,306,307], and content ourselves with recalling the four 1/2-BPS standard solutions: the pp-wave and three extended solutions, the membrane (or M2-brane), "vebrane (M5-brane) and the Kaluza}Klein monopole, also known as the KK6-brane. The eleven-dimensional metric describing the extended solutions splits into two parts: the world-volume, denoted by EN, including the time and p world-volume directions, and the

126


transverse Euclidean part E\N. These solutions are given in terms of a harmonic function H on the transverse space, which we choose as a single pole k , H(r)"1# r\N

(2.22)

although any superposition of such poles would do (this is stating the no-force condition between static BPS states; the constant shift in Eq. (2.22) ensures the asymptotic #atness of space-time, required for a soliton interpretation). The constant k depends on Newton's constant i and on the p-brane tension, and is quantized by the requirement that the space-time be smooth (we will henceforth choose the smaller quantum). The pp-wave and KK6-brane solutions only involve the metric, and read [167,304] pp-wave: ds "!dt#do#(H!1)(dt#do)#ds(E) , k H"1# . r

(2.23a) (2.23b)

KK6-brane: ds "ds(E)#ds (y) , 2,

(2.24a)

ds "HdyGdyG#H\(dt #< (y)dyG), i"1, 2, 3 , 2, 2, G

(2.24b)

k

; X" . (2.27) L C(L>) The tensions (or mass per unit world-volume) of these four basic BPS con"gurations can be easily evaluated from ADM boundary integrals and Dirac quantization, or more easily yet by dimensional analysis: R 1 KK6-brane: T " , KK-state: T " , R l N (2.28) 1 1 M5-brane: T " . M2-brane: T " , l l N N The tension (i.e. mass) of the pp-wave with momentum along a compact direction of radius R (occasionally denoted as R ) is the one expected for a massless particle in eleven dimensions; the 2, tension of the KK6-brane is easily obtained from the latter by electric}magnetic duality, after reading o! from Eq. (2.6) the Kaluza}Klein gauge coupling 1/g "R/l: )) N R T . (2.29) T" " g l N )) All these BPS states have been inferred from a classical analysis of 11D supergravity. They should in principle arise from a microscopic de"nition of M-theory, which would allow a full account of their interactions. Nevertheless, it is still possible to formulate their dynamics in terms of their collective coordinates which result from the breaking of global symmetries in the presence of the soliton [127]. Supersymmetry gives an important guideline, since (the unbroken) half of the 32 supercharges has to be realized linearly on the world-volume, while the other half is realized non-linearly. This "xes the dynamics of the M5-brane to be described in terms of the chiral (2, 0) six-dimensional tensor theory [57], while the membrane is described by the 2#1 supermembrane action [40,84]. Unfortunately, the quantization of these two theories remains a challenge. As for the KK6-brane, the description of its dynamics is still an unsettled problem [152]. 2.4. Reduction to type IIA BPS solutions Upon compacti"cation on a circle (with periodic boundary conditions on the fermion "elds), the supersymmetry algebra is una!ected and the generators merely decompose under the reduced Lorentz group. The 32-component Majorana spinor Q decomposes into two 16-component ? Majorana}Weyl spinors of SO(1,9) with opposite chiralities, and the N"1 supersymmetry in 11D

128


Table 2 Type IIA central charges, gauge "elds and extended objects Z

ZGH

ZG

ZGHIJK

ZGHIJ

ZG

Z

ZGHIJ

ZGHI

A D0

C GH D2

B G F1

E GHIJK NS5

R GHIJ D4

none D8

none 9-brane

K K_KNOPQ KK5

R JKLNOP D6

gives rise to non-chiral N"2 supersymmetry in 10D. However, it is convenient not to separate the two chiralities explicitly, and rewrite the supersymmetry algebra as 1 +Q , Q ,"(CCI) P #(CC ) Z# (CC ) ZIJ#(CC C ) ZI IJ ?@ I Q ?@ ? @ ?@ I Q ?@ 2 1 1 # (CC ) ZIJMNO# (CC C ) ZIJMN , IJMNO ?@ IJMN Q ?@ 5! 4!

(2.30)

where the eleventh Gamma matrix C is identi"ed with the 10D chirality operator C C 2C . The Q eleven-dimensional central charges give rise to the charges Z, ZI, ZIJ, ZIJMN, ZIJMNO whose interpretation is summarized in Table 2, where we omitted the momentum charge P . In this table, I K denotes the 6-form dual to g after compacti"cation of the direction m. K_KLNOPQR IK Under Kaluza}Klein reduction, the BPS solutions of 11D SUGRA yield BPS solutions of type IIA supergravity. This reduction can, however, be carried out only if the eleventh dimension is a Killing vector of the con"guration. This is automatically obeyed if the eleventh direction is chosen along the world-volume EN, and reduces the eleven-dimensional p-brane to a tendimensional (p!1)-brane with tension T "RT ; this procedure is called diagonal or double N\ N reduction [103], and we shall call the resulting solutions wrapped or longitudinal branes. One may also want to choose the eleventh direction transverse to the brane, but this is not an isometry, since the dependence of the harmonic function H on the transverse coordinates is non-trivial. However, this can be easily evaded by using the superposition property of BPS states, and constructing a continuous stack of parallel p-branes along the eleventh direction. The harmonic function on E\N turns into an harmonic function on E\N:

1 dxQ & . \N o\N \[(xQ)#o]

(2.31)

We therefore obtain an unwrapped or transverse p-brane in ten dimensions with the same tension T as the one we started with. This procedure is usually called vertical or direct reduction. It has N also been proposed to reduce along the isometry that arises when the sphere S\N in the transverse space E\N is odd-dimensional, hence a ;(1) Hopf "bration [105], but the status of the solutions obtained by this angular reduction is still unclear. Applying this procedure to the four M-theory BPS con"gurations, with tensions given in Eq. (2.28), we "nd, after using the relations (2.11), the set of BPS states of type IIA string theory listed in Table 3. Table 3 shows that we recover the set of all 1/2 BPS solutions of type IIA string theory, which include the KK excitations, the fundamental string and the set of solitonic states


129

Table 3 Relation between M-theory and type IIA BPS states M-theory

Mass/tension

Type IIA

Longitudinal M2-brane

R 1 T " Q" l l N Q 1 1 T " " l g l N QQ R 1 T " Q" l g l N QQ 1 1 T " " l gl N Q Q 1 1 T " " R gl Q QQ 1 1 T " " R R G G R R R T " Q 2," 2, l gl N Q Q R 1 T " Q" l g l N QQ R R T " 2," 2, gl l Q Q N

F-string

Transverse M2-brane Longitudinal M5-brane Transverse M5-brane Longitudinal KK mode Transverse KK mode Longitudinal KK6-brane KK6-brane with R "R 2, Q Transverse KK6-brane

D2-brane D4-brane NS5-brane D0-brane KK mode KK5-brane D6-brane 6-brane

comprised by the NS5-brane, KK5-brane and the Dp-branes with p"0, 2, 4, 6. The NS5-brane is a solitonic solution that is magnetically charged under the Neveu}Schwarz B-"eld [57]. The Dp-branes are solitonic solutions, electrically charged under the RR gauge potentials R (or N> magnetically under R ) [251]. The tension of these BPS states does not receive any quantum \N corrections perturbative or non-perturbative, which is why these objects are useful when considering non-perturbative dualities. States electrically (resp. magnetically) charged under the Neveu} Schwarz gauge "elds have tensions that scale with the string coupling constant as g (resp. 1/g), Q Q whereas states charged under the Ramond "elds have tensions that scale as 1/g . Q The last line in Table 3 is an unconventional solution, which we call a 6-brane, obtained by vertical reduction of the KK6-brane in a direction in the 1 part of the Taub}NUT space [52]. The integration involved in building up the stack is logarithmically divergent, and, if regularized, yields a non-asymptotically #at space. However, as we will see in more detail in Section 4.9, at the algebraic level this solution is required by U-duality symmetry. At that point we will also explain The letter D stands for the Dirichlet boundary conditions in the 9!p directions orthogonal to the world-volume of the Dp-brane, which force the open strings to move on this (p#1)-dimensional hyperplane. There is also an 8-brane coupling to a nine-form, whose expectation value is related to the cosmological constant [36,38,251].

130


our nomenclature for this (and other) non-conventional solutions. It is also interesting to note that all the tensions obtained above are not independent, since they follow from the basic relations (2.11). This already hints at the presence of a larger structure that relates all these states, a fact that we will establish using the conjectured U-duality symmetry of compacti"ed M-theory. The dimensional reduction can also be carried out at the level of the supergravity con"guration itself. For example, using the relation (2.12) between the 11D metric and 10D string metric, one "nds that a solution with 11D metric of the form ds "HGds(EN)#HHds(E\N) yields two 10D solutions with metric and dilaton ds "H?ds(ENY)#H@ds(E\NY), e\("HA , where diagonal: p"p!1, vertical: p"p,

i 3i 3i a" , b"j# , c"! , 2 2 2

3j 3j j a"i# , b" , c"! , 2 2 2

(2.32)

(2.33a)

(2.33b) (2.33c)

for diagonal and vertical reduction respectively. As explained in the beginning of this subsection, in the "rst case the harmonic function is the same as the original one, and in the second case it is a harmonic function on a transverse space with one dimension less. The reduction of the gauge potentials can be worked out similarly. The resulting 10D type IIA con"gurations are then described by the following solutions: F-string: ds "H\ds(E)#ds(E) , k B "H\, e\("H, H"1# r NS5-brane: ds "ds(E)#Hds(E) , k dB"夹 dH, e\("H\, H"1# , r Dp-brane: ds "H\ds(EN)#Hds(E\N) , k e\("HN\, H"1# , r\N

(2.34a) (2.34b) (2.35a) (2.35b) (2.36a) (2.36b)

FN>"Vol(EN)dH\, p"0, 1, 2 , (2.36c) C F\N"夹 dH, p"4, 5, 6 , (2.36d) K \N F"F#F, p"3 , (2.36e) C K where, for completeness, we have included the Dp-brane solutions for all p"0,2, 6, although we note that only even p occurs in type IIA. The subscripts e and m indicate whether the p-branes are electrically or magnetically charged under the indicated "elds. One also "nds the ten-dimensional


131

gravitational solutions, consisting of the pp-waves and KK5-brane, which have a metric analogous to the eleven-dimensional case (see Eqs. (2.23a), (2.23b) and (2.24a)}(2.24c)), with harmonic functions on a transverse space with one dimension less. Of course, one may explicitly verify that all of these solutions are indeed solutions of the tree-level action (2.7). In contrast to the M2-brane and M5-brane, the dynamics of Dp-branes has a nice and tractable description as (p#1)-dimensional hyperplanes on which open strings can end and exchange momentum with [251]. The integration of open string #uctuations around a single D-brane at tree level yields the Born}Infeld action [17,56,205],

1 dN>me\((g( #BK #l F . (2.37) S " Q ' lN> Q Here, the hatted "elds g( , BK stand for the pullbacks of the bulk metric and antisymmetric tensor to the world-volume of the brane, and F is the "eld strength of the ;(1) gauge "eld living on the brane. The coupling to the RR gauge potentials is given by the topological term [95,140]

S "i e K >JQ $R , 00

(2.38)

where R" R denotes the total RR potential. N N In the zero-slope limit, the Born}Infeld action becomes the action of a supersymmetric Maxwell theory with 16 supercharges. In the presence of N coinciding D-branes the world-volume gauge symmetry gets enhanced from ;(1), to ;(N), as a consequence of zero mass strings stretching between di!erent D-branes [315]. The non-Abelian analogue of the Born}Infeld action is not known, although some partial Abelianization is available [309], but its zero-slope limit is still given by ;(N) super-Yang}Mills theory. 2.5. T-duality and type IIA/B string theory So far, we have discussed M-theory and its relation to type IIA string theory. In this subsection, we turn to type IIB string theory and its relation, via T-duality, to type IIA [78,93]. We "rst recall that the massless sector of type IIB consists of the same Neveu}Schwarz "elds (2.8a) as the type IIA string, but the Ramond gauge potentials of type IIB now include a 0-form (scalar), a 2-form and a 4-form with self-dual "eld strength, a, B , D , (2.39) IJ IJMN with 夹D "D . The low-energy e!ective action has a form similar to that in Eq. (2.7), with the appropriate "eld strengths of the even-form RR potentials in Eq. (2.39), as long as the 4-form is not included. The standard 1/2-BPS solutions of type IIB are the fundamental string, NS5-brane, Dp-branes with odd p, pp-waves and KK5-brane.

There is also a gravitational term required for the cancellation of anomalies [136], but it does not contribute on #at backgrounds. A local covariant action for the self-dual four-form can be written with the help of auxiliary "elds [79], but for most purposes the equations of motion are su$cient.

132


In order to describe the precise T-duality mapping, we again write the ten-dimensional metric as a ;(1) "bration ds "R(dx#A dxI)#g dxIdxJ, k, l"0,2, 8 . I IJ

(2.40)

T-duality on the direction 9 relates the "elds in the type IIA and type IIB theories in the Neveu}Schwarz sector as l l ¹ : R Q , g Q g , A B , Q R Q I I R

B B !A B #A B IJ IJ I J J I

(2.41)

leaving g and the string length l invariant. The Ramond gauge potentials are furthermore IJ Q identi"ed on both sides according to ¹ : Rdx ) R#dxR, R" R , N N

(2.42)

where ) and denote the interior and exterior products, respectively. In other words, the 9 index is added to the antisymmetric indices of R when absent, or deleted if it was already present. These identi"cations actually receive corrections when BO0, and the precise mapping is [39,109,140] e RPdx ) (e R)#dx(e R)

(2.43)

in accord with the T-duality covariance of the RR coupling in Eq. (2.38). Whereas one T-duality maps the type IIA string theory to IIB and should be thought of as a change of variables, an even number of dualities corresponds to a global symmetry of either type IIA or type IIB theories. This symmetry will be discussed in Section 3, and its non-perturbative extension in Section 4. The action on the BPS spectrum can again be easily worked out, at the level of tension formulae or of the supergravity solutions themselves. As implied by the exchange of the Kaluza}Klein and Kalb}Ramond gauge "elds A and B , states with momentum along the 9th direction are I I interchanged with fundamental string winding around the same direction. On the other hand, T-duality exchanges Neumann and Dirichlet boundary conditions on the open string world-sheet along the 9th direction, mapping Dp-branes to D(p#1)- or D(p!1)-branes, depending on the orientation of the world-volume with respect to x [35,78]. This of course agrees with the mapping of Ramond gauge potentials in Eq. (2.42). Similarly, NS5-branes are invariant or exchanged with KK5-branes, according to whether they are wrapped or unwrapped, respectively [109,244]. This can also be easily seen by applying the transformation (2.41) to the tension formulae, as summarized in Table 4 for a T-duality ¹ on an arbitrary compact dimension with radius R . G G T-duality can then be used to translate the relation between strongly coupled type IIA theory and M-theory in type IIB terms. In this way, it is found that the type IIB string theory is obtained Whereas the worldvolume dynamics of type IIB NS5- and D5-branes is described by a non-chiral (1,1) vector multiplet, the type IIB KK5-brane is chiral and supports a (2,0) tensor multiplet. Indeed, it is T-dual to the chiral type IIA NS5-brane [10]. On the other hand, the type IIA KK5-brane, dual to the type IIB NS5-brane, is nonchiral.


133

Table 4 T-duality of type II BPS states Type IIA (B)

Tension

¹ -dual tension G

Type IIB (A)

KK mode

1 M" R G

R M" G l Q

Winding mode

Wrapped Dp-brane

R " G N\ g lN> QQ R T " G gl Q Q 1 T " gl Q Q

1 " N\ g lN QQ R T " G gl Q Q R T " G gl Q Q

Unwrapped D(p!1)-brane

Wrapped NS5-brane Unwrapped NS5-brane

T

T

Wrapped NS5-brane Unwrapped KK5-brane

by compactifying M-theory on a two-torus ¹, with vanishing area, and a complex structure q equated to the type IIB complex coupling parameter [269]: i q"a# . g Q

(2.44)

Here, a is the expectation value of the Ramond scalar and g the type IIB string coupling. Q We focus for simplicity on the case where the torus is rectangular, so that q is purely imaginary and hence the RR scalar a vanishes. In this case, the relation between the M-theory parameters and type IIB parameters reads R l l g " Q , l" N , R " N , Q R Q R RR Q Q

(2.45)

where R , R are the radii of the M-theory torus and R the radius of the type IIB 9th direction. Q The uncompacti"ed type IIB theory is obtained in the limit (R , R )PR, keeping R /R "xed. Q Q From Eq. (2.45), we can then identify the type IIB BPS states to those of M-theory compacti"ed on ¹. The results are displayed in Table 5 for states still existing in uncompacti"ed type IIB theory, and in Table 6 for states existing only for "nite values of R . As in Table 3, we see in the last entry of Table 6 a non-standard BPS state with tension scaling as g\, which we have called a 7 -brane. As this brane will turn out to be related to the D7-brane by Q S-duality (see Section 4.5) it may also be referred to as a (1,0) 7-brane. This and other non-standard solutions will be discussed in more detail in Section 4.9.

3. T-duality and toroidal compacti5cation Having discussed how dualities of string theory lead to the idea of a more fundamental eleven-dimensional M-theory, we now turn to the symmetries that this theory should exhibit, with

134


Table 5 Relations between M-theory and type IIB BPS states M-theory

Mass/tension

Type IIB

M2-brane wrapped around xQ

R 1 Q" l l N Q R 1 " l g l N QQ RR 1 Q " l g l N QQ RR 1 Q " l g l N QQ RR 1 Q" l gl N Q Q

Fundamental string

M2-brane wrapped around x M5-brane wrapped on xQ, x KK6-brane wrapped on x, charged under g

IQ

KK6-brane wrapped on xQ, charged under g I

D1-brane (D-string) D3-brane D5-brane NS5-brane

Table 6 More relations between M-theory and type IIB BPS states M-theory

Mass/tension

Type IIB

M2-brane wrapped on xQ, x

R R 1 Q" l R N 1 R " l gl N Q Q 1 R " l g l N QQ R R Q" l g l N QQ R R " gl l Q Q N R R Q" l g l N QQ R R Q" gl l N Q Q

KK mode

Unwrapped M5-brane Unwrapped M2-brane M5-brane wrapped on xQ M5-brane wrapped on x Unwrapped KK6-brane, charged under g IQ Unwrapped KK6-brane, charged under g I

KK5-brane with R "R 2, Q Wrapped D3-brane Wrapped D5-brane Wrapped NS5-brane Wrapped D7-brane Wrapped 7 -brane

the hope of getting more insight into its underlying structure. For this purpose, it is convenient to consider compacti"cations on tori, which have the advantage of preserving a maximal amount of the original super-PoincareH symmetries, while bringing in degrees of freedom from extended states in eleven dimensions in a still manageable way.


135

The approach here is similar to the one that was taken for the perturbative string itself, where the study of T-duality in toroidal compacti"cations revealed the existence of spontaneously broken `stringya gauge symmetries (see [132] for a review). Given the analogy between the two problems, we shall "rst review in this section how T-duality in string theory appears at the level of the low-energy e!ective action and of the spectrum, with a particular emphasis on the brane spectrum. We shall then apply the same techniques in Sections 4 and 5 in order to discuss U-duality in M-theory. 3.1. Continuous symmetry of the ewective action Compacti"cation of string theory on a torus ¹B can be easily worked out at the level of the low-energy e!ective action, by substituting an ansatz similar to Eq. (2.4) ds "g (dxG#AG dxI)(dxH#AH dxJ)#g dxIdxJ , GH I J IJ i, j"1,2, d, k, l"0,2, (9!d)

(3.1a) (3.1b)

in the ten-dimensional action

l 1 (3.2) S " dx(!g e\( R#4(R )! Q (dB) , l 12 Q where we omitted Ramond and fermion terms. We have also split the ten-dimensional two-form B into d(d!1)/2 scalars B , d vectors B and a two-form B . GH GI IJ Concentrating on the scalar sector, and rede"ning the dilaton as <e\("lBe\(B where Q g 1 1 Q , , . (3.11b) ¹ : (g , R , R ) Q R R R R For convenience, we followed the double T-duality on directions 1 and 2 by an exchange of the two radii, included the action on the coupling constant and set the string length l to 1. Altogether, the Q Weyl group of SO(d, d, 9) is the "nite group

W(SO(d, d))"9 ( )S (3.12) B generated by the T-duality transformation ¹ and the permutation group S of the d directions of B the torus. The Weyl group of SO(d, d) can actually be written as the semi-direct product S ( (9 )B\, where the commuting B 9 's are the double inversions of R and, say, R . G

138


On the other hand, Borel generators include the Borel elements of the modular subgroup, acting as c Pc #c on the homology lattice of the lattice, as well as the integer shifts of the expectation G G H value of the two-form in the internal directions B PB #1. Any element in SO(d, d, 9) can be GH GH reached by a sequence of these transformations. Weyl and Borel generators can be given a more precise de"nition as operators on the weight space of the Lie group or algebra under consideration (see for instance Ref. [173] for an introduction to the relevant group theory). Weyl generators correspond to orthogonal re#ections with respect to planes normal to any root and generate a "nite discrete group, while Borel generators act on the weight lattice by translation by a positive root. Any "nite-dimensional irreducible representation (of the complex Lie algebra) can then be obtained by action of the Borel group on a, so-called, highest-weight vector, and splits into orbits of the Weyl group with de"nite lengths. 3.4. Weyl generators and Weyl reyections Weyl generators encode the simplest and most interesting part of T-duality. It is very easy to study the structure of the "nite group they generate, by viewing them as orthogonal re#ections in a vector space (the weight space) generated by the logarithms of the radii. More precisely, let us represent the scalar moduli (ln g , ln R ,2, ln R ) as a form u on a vector space < with basis Q B B> e , e ,2, e , and associate to any weight vector j"xe #xe #2#xBe , its tension B B B (3.13) T"e6PH7"gV RV RV 2RV . Q B The vector j should be seen as labelling a state in the BPS spectrum, with tension T. The generators (3.11a) and (3.11b) are then implemented as linear operators on < with matrices B> 1 1

S" G

1 1

!1

!1

, ¹"

!1

!1

.

(3.14)

( ( B\ B\ These operators S and ¹ in Eq. (3.14) are easily seen to be orthogonal with respect to the G signature (!#2#) metric ds"!(dx)#(dxG)#dx(dx#2#dxB) ,

(3.15)

and correspond to Weyl re#ections a)j jPo (j)"j!2 a ? a)a

(3.16)

From this point of view, Weyl generators are not properly speaking elements of the group, but can be lifted to generators thereof, at the cost of introducing 9 phases in their action on the step operators E . See for instance Appendix ? B in Ref. [207], for a discussion of this issue in the physics literature. One could omit the x coordinate since g can be absorbed by a power of the invariant Planck length R /g, but we Q G Q include it for later convenience.


139

with respect to planes normal to the vectors a "e !e , i"1,2, d!1 (3.17a) G G> G a "e #e . (3.17b) The group generated by S and ¹ is therefore a Coxeter group, familiar from the theory of Lie G algebras (see [173] for an introduction, and [119,174] for a full account). Its structure can be characterized by the matrix of scalar products of these roots: (a )"(a )"2 , (3.18a) G a ) a "a ) a "!1 . (3.18b) G G> This precisely reproduces the Cartan matrix D of the T-duality group SO(d, d, 1), summarized in B the Dynkin diagram: (3.19) The only delicate point is that the signature of the metric (3.15) on < is not positive-de"nite. B> This can be easily evaded by noting that the invariance of Newton's constant R /g implies that G Q all roots are orthogonal to the vector d"e #2#e !2e (3.20) B with negative proper length d"!(d#4), so that the re#ections actually restrict to the hyperplane < normal to d: B d ) x"x"0 . (3.21) The Lorentz metric on < then restricts to a positive-de"nite metric g "d on < . The dualities B> GH GH B S and ¹ therefore generate the Coxeter group D , which is the same as the Weyl group of the Lie G B algebra of SO(d, d, 1). In order to distinguish the various real and discrete forms of D , one needs to B take into account the Borel generators, which we defer to Section 3.7. The Dynkin diagram (3.19) allows a number of simple observations. We may recognize the Dynkin diagram A of the Lorentz group Sl(d, 1) (denoted with #), extended with the root B\ * into the Dynkin diagram of the T-duality symmetry SO(d, d, 1). T-duality between type IIA and type IIB corresponds to the outer automorphism acting as a re#ection along the horizontal axis of the Dynkin diagram. The chain denoted with *'s represents a dual Sl(d, 1) subgroup, which is nothing but the Lorentz group on the type IIB T-dual torus. The full T-duality group is generated by these two non-commuting Lorentz groups of the torus and the dual torus. Decompacti"cation of the torus ¹B into ¹B\ is achieved by dropping the rightmost root, which reduces D to D . When the root a is reached, the diagram disconnects into two pieces, B B\ corresponding to the identity SO(2, 2, 1)"Sl(2, 1);Sl(2, 1), or to the decomposition of the torus moduli space into the ¹ and ; upper half-planes. Finally, for d"1 the T-duality group SO(1, 1, 9) becomes trivial, while the generator of O(1, 1, 9) corresponds to the inversion R1/R, not a symmetry of either type IIA or type IIB theories. The extra 9 exchanging the two Sl(2, 1) factors belongs to O(2, 2, 1) but not to SO(2, 2, 1).

140


3.5. BPS spectrum and highest weights Having proved that the transformations S and ¹ indeed generate the Weyl group of SO(d, d, 9), G we can use the same formalism to investigate the orbit of the various BPS states of string theory. According to Eq. (3.13) the mass or tension can be represented as a weight vector in < , and one B> should let Weyl and Borel generators act on it to obtain the full orbit. Each orbit admits a highest weight from which all other elements can be reached by a sequence of Weyl and Borel generators (Weyl generators alone are not su$cient, because they preserve the length of the weight). All highest weights can be written as linear combinations with positive integer coe$cients of the fundamental weights R j"e !e PM " , " g Q

(3.22a)

R R j"e #e !2e PM " , ,1 g Q

(3.22b)

R 2R B\ , jB\"e #2#e !2e PM 2 " (3.22c) B\ ,1 g Q 1 jB\"e #2#e !2e &!e PM " , (3.22d) B\ B $ R B 1 j"!e PM " (3.22e) " g Q dual to the simple roots, that is jG ) a "!d . We used the symbol & for equality modulo the H GH invariant vector d in Eq. (3.20), and the notation F, D and NS for fundamental, Dirichlet and Neveu}Schwarz states, respectively, depending on the power of the coupling constant involved, and w for each wrapped direction (the notation wF is justi"ed by the fact that the Kaluza}Klein states are in the same multiplet as the string winding states). This is summarized in the Dynkin diagram

(3.23)

which shows the highest weights associated to each node of the Dynkin diagram. In particular, we see from Eq. (3.23) that the type IIA D-particle mass (M"1/g l ) lies in the QQ spinor representation dual to a , just as do the type IIB D-string tension (T "1/g l) and QQ The minus sign shows that we are really considering lowest-weight vectors, but we shall keep this abuse of language.


141

D-instanton action (T "1/g ), whereas the type IIB D-particle mass (M"R /g l) and type IIA \ Q G QQ D-string tension (T "R /g l) and D-instanton action (T"R /g l ) transform in the spinor G QQ G QQ representation dual to a , of opposite chirality. On the other hand, the Kaluza}Klein states lie in a vector representation. All highest-weight representations can be obtained from the tensor product of these `extremea (from the point of view of the Dynkin diagram) representations. T-duality on a single radius exchanges the two spinor representations, as it should. 3.6. Weyl-invariant ewective action In the previous subsections, we have discussed how the Weyl group of SO(d, d) arises as the "nite group generated by the permutations and double T-duality (3.11a) and (3.11b), whereas the low-energy action itself is invariant under the continuous group SO(d, d, 1). This has been checked in the scalar sector in Eq. (3.4), by direct reduction of the 10D e!ective action on ¹B. It is however possible to rewrite the full action in a manifestly Weyl-invariant way, by a step-by-step reduction from 10D, as was originally developed in Ref. [213] in the context of 11D supergravity. This procedure leads to a clear identi"cation of `dilatonica scalars, which appear through exponential factors in the action and include the dilaton g and the radii R of the torus, versus `Peccei}Quinna Q G scalars which have constant shift symmetries and are better thought of as 0-forms with a 1-form "eld strength. , with internal indices Each "eld strength FN gives rise to "eld strengths of lower degree FO G2GO i 2i (given by the exterior derivative of a (q!1)-form up to Chern}Simons corrections), while O the metric gives rise to Kaluza}Klein two-form "eld strengths FG and one-form "eld strengths FG, i(j, of the vielbein components in the upper triangular gauge H g "E. E/ g , (3.24a) +, + , ./

(3.24b) where EJ denotes the vielbein in the uncompacti"ed directions. The action (2.7) in the Neveu} I Schwarz sector then takes the simple form:

< RR R G # GFG R#(R )# S " d\Bx(!g ,1\B gl R R H Q Q G H GH G

l l Q F # (R FG)#(lF)# Q F # , G Q R G R R GH G G H G G GH

(3.25)

142


where the "rst "ve terms come from the reduction of the Einstein}Hilbert term and the last three terms from the kinetic term of the two-form. Putting the forms of the same degree together, we see that their coe$cients form the Weyl orbit U , of the string tension (F), the Weyl orbit U of the Kaluza}Klein and winding states (F), Q H )) H and the set of positive roots U "+e $e , i(j, (F). We can therefore rewrite the action in the > G H ? Weyl-invariant form:

< R#Ru ) Ru# e\6P?7(F) S " d\Bx(!g ? ,1\B gl Q Q ?ZU>

# e\6PH7(F)# e\6PH7(F) , H H HZUQ HZU))

(3.26)

where u"(ln g , ln R ,2, ln R ) is the vector of dilatonic scalars, 1u, j2 the duality bracket in Q B Eq. (3.13) and Ru ) Ru the Weyl-invariant kinetic term obtained from the non-diagonal metric (3.15). A diagonal metric on the dilatonic scalars is recovered upon going to the Einstein frame. The Weyl group acts by permuting the various weights appearing in Eq. (3.26), and the invariance in the gauge sector is therefore manifest. As for the scalars, the set of positive roots U is > not invariant under Weyl re#ections, but the Peccei}Quinn scalars undergo non-linear transformations APe\6P?7A that compensate the sign change [215]. The Peccei}Quinn scalars therefore appear as displacements along the positive (non-compact) roots. Together with the dilatonic (non-compact) scalars u, they generate the solvable Lie subalgebra that forms the tangent space of the moduli space H [6}8,308]. We have so far concentrated on the Neveu}Schwarz sector, but the same reasoning can be applied to the full type II action. The T-duality Weyl symmetry can, however, be exhibited only by dualizing the p-form gauge "elds GN"dRN\ into lower rank (10!d!p)-form gauge "elds when possible, and keeping them together when their dual when the self-duality condition 10!d!p"p is satis"ed. We then obtain, for the action of the Ramond "elds

< e\6PH7(G)# e\6PH7(G) S " d\Bx(!g H H 00 gl U Q Q HZ "' HZU"

# e\6PH7(G)# e\6P H7(G) , H H HZU" HZU"

(3.27)

where U , U , U , U denote the Weyl orbits with highest weight 1/g R , 1/g l , R /g l, 1/g l, "' " " " Q G QQ G QQ QQ respectively, corresponding in turn to the two spinor representations. 3.7. Spectral yow and Borel generators Having discussed the structure of the Weyl group we now want to investigate the full SO(d, d, 9) symmetry. For this purpose, it is instructive to go back to the perturbative multiplet of Kaluza}Klein and winding states. The action of the Weyl group on the highest weight 1/R of the B vector representation generates an orbit of 2d elements, 1/R and R . However, a particle can have G G


143

any number of momentum excitation along each axis, and wind along any cycle of the torus ¹B. It is therefore described by integer momenta m and winding numbers mG, so that its mass on an G arbitrary torus reads M"m gGHm #mGg mH, i, j"1,2, d , (3.28) G H GH when B "0. This mass formula is then invariant under modular transformations cGPcG#*AG cH GH H of the torus, i.e. integer shifts AG PAG #*AG of the o!-diagonal term of the metric (no sum on i) H H H ds"R(dxG#AG dxH)#g dxHdxI , (3.29) B G H HI upon transforming the momenta and winding as m Pm !*AG m , mIPmI#dI*AG mH. (3.30) I I I G G H This transformation generates a spectral yow on the lattice of charges m and mG. G In addition, being charged under the gauge potential B , the momentum of the particle shifts IG according to m Pm "m #B mH, yielding the mass (3.8a) and (3.8b). From this, we see that the G G G GH Borel generator B PB #*B induces a spectral #ow GH GH GH m Pm #*B mH, mIPmI. (3.31) I I HI The two spectral #ows (3.30) and (3.31) can be understood in a uni"ed way as translations on the weight lattice by positive roots. Indeed, the set of all positive roots of SO(d, d) includes the Sl(d) roots e !e , i(j, images of the simple roots a "e !e , 14i4d!1 under the Weyl group H G G G> G S of Sl(d), as well as the roots e #e , which are images of the T-duality simple root a "e #e . B G H The translation by a root e !e generates in"nitesimal rotations in the (i, j) plane: H G *"!e 2"!*AG "!e 2, *"e 2"dI*AG "e 2 (3.32) I I G I G H H equivalent to the spectral #ow in Eq. (3.30), whereas translations by a root e #e generate an G H in"nitesimal B shift: GH *"!e 2"*B "e 2, *"e 2"0 (3.33) I HI H I as in Eq. (3.31). The moduli AG and B can therefore be identi"ed as displacements on the moduli H GH space H along the positive roots e !e and e #e . We note that the two displacements do not G H G H necessarily commute and that only integer shifts are symmetries of the charge lattice. 3.8. D-branes and T-duality invariant mass In order to study the analogous properties of the D-brane states, we may try to write down the moduli matrix M 3SO(d, d, 1)/SO(d);SO(d) in the spinorial representation and look for the 1 transformations of charges that leave the mass mM m invariant, when now m is a spinor of 1 D-brane charges. It is in fact much easier to study the D-brane con"guration itself and compute its Born}Infeld mass [149,250]. The Borel generators E actually either translate the weight vectors j or annihilate them. ?

144


BPS D-brane states are obtained by wrapping Dp-branes on a supersymmetric p-cycle of the compacti"cation manifold. In the case of a torus ¹B, this is simply a straight cycle, and in the static gauge the embedding is speci"ed by a set of integer (winding) numbers NG : ? XG"NG p?, i"1,2, d, a"1,2, p , (3.34) ? where p? and XG are the space-like world-volume and embedding coordinates respectively. The numbers NG can, however, be changed by a world-volume di!eomorphism, and one should instead ? look at the invariant mGHIJ"e?@ABNG NH NINJ , (3.35) ? @ A B where we restricted to p"4 for illustrative purposes. mGHIJ is a four-form integer charge that speci"es the four-cycle in ¹B. In addition, the D-brane supports a ;(1) gauge "eld that can be characterized by the invariants 1 mGH" e?@ABNG NH F , ? @ AB 2

1 m" e?@ABF F , ?@ AB 8

(3.36)

which are again integer-valued, because of the #ux and instanton-number integrality. The charges N"+m, mGH, mGHIJ,2, constitute precisely the right number to make a spinor representation of SO(d, d, 9) when p"d or p"d#1 (depending on the type of theory and dimensionality of the torus); indeed, the spinor representation of SO(d, d) decomposes under Sl(d) as a sum of even or odd forms, depending on the chirality of the spinor. The Chern}Simons coupling (2.38) can be rewritten in terms of these charges (up to corrections when BO0) as

1 1 e K >?Y$R"mR # mGHR # mGHIJR #2 2 GH 4! GHIJ

(3.37)

so that (for p"4) the instanton number m can be identi"ed as the D0-brane charge, the #ux mGH as the D2-brane charge and mGHIJ as the D4-brane charge. Con"gurations with mO0 exist in SYM theory on a torus, even for a ;(1) gauge group, and correspond to torons [147,148,299]. The mass of the wrapped D-brane can be evaluated by using the Born}Infeld action (2.37), and depends only on the parametrization-independent integer charges m, mGH, mGHIJ,2 . Explicitly, we obtain, for p"d, the T-duality invariant mass formula: 1 1 1 m # (m GH)# (m GHIJ)#2 , M" 2gl 4!gl gl Q Q Q Q Q Q m "m#mGHB #mGHIJB B #2 , GH GH IJ m GH"mGH#mIJGHB #2 , IJ m GHIJ"mGHIJ#2 , This expression was originally derived in Ref. [250] by a sequence of T-dualities and covariantizations.

(3.38a) (3.38b) (3.38c) (3.38d)


145

where the dots stand for the obvious extra terms when d54. A similar expression holds for p"d#1 and yields the tension of D-strings: 1 1 1 (m G)# (m GHI)# (m GHIJJ)#2, T" gl 3!gl 5!gl Q Q Q Q Q Q m G"mG#mHIGB #mHIJKGB B #2, HI HI JK m GHI"mGHI#mJKGHIB #2, JK m GHIJK"mGHIJK#2,

(3.39a) (3.39b) (3.39c) (3.39d)

where the integer charges read, e.g. for p"5, mGHIJK"e?@ABCNG NH NINJ NK , (3.40a) ? @ A B C mGHI"e?@ABCNG NH NIF , (3.40b) ? @ A BC NG F F . (3.40c) mG"e ?@ABC ? @A BC The mass formulae (3.38a)}(3.38d) and (3.39a)}(3.39d) hold for 1/2-BPS states only; they are the analogues of Eqs. (3.8a) and (3.8b) for the two spinor representations of SO(d, d). They can be derived by analysing the BPS eigenvalue equation in a similar way as in Section 2.2. This analysis is carried out in Appendix A.3, and yields, in addition, the conditions for the state to be 1/2-BPS, as well as the extra contribution to the mass in the 1/4-BPS case. In the d46 case, we "nd a set of conditions: kGHIJ,m GHmIJ #mmGHIJ"0 ,

(3.41a)

kG_HIJKL,mG HmIJKL #mmGHIJKL"0 ,

(3.41b)

kGH_IJKLNO,nGHnIJKLNO#nGH IJnKLNO "0

(3.41c)

analogous to the level-matching condition ""m"""0 on the perturbative states. In contrast to the latter, they have a very clear geometric origin, since they can be derived by expressing the charges m in terms of the integer numbers NG (Eqs. (3.38a)}(3.38d)). For d"6, they transform in ? a 15#36#15"66 irrep of the T-duality group SO(6, 6, 9). The last line in Eqs. (3.41a)}(3.41c) drops when d"5, giving a 5#5"10 irrep of SO(5, 5, 9). When d"4, only the k"mm#mm,0 component remains, which is a singlet under SO(4, 4, 9). When the conditions n"0 in Eqs. (3.41a)}(3.41c) are not met, the state is at most 1/4-BPS, and its mass receives an extra contribution, e.g. for d"5:

1 1 1 1 1 m # (m GH)# (m GHIJ)# (kI GHIJ)# (kI G_HIJKL) , M" 4!l 2l 4!l 5!l gl Q Q Q Q Q Q where the shifted charges are given by kI GHIJ"kGHIJ#B kK_LGHIJ, KL

kI G_HIJKLN"kG_HIJKLN .

(3.42)

(3.43)

146


For d"6, there are still conditions to be imposed in order for the state to be 1/4-BPS instead of simply 1/8-BPS, which are now cubic in the charges m and transform as a 32 of SO(6, 6, 9) (see Appendix A and Section 5.9).

4. U-duality in toroidal compacti5cations of M-theory T-duality is only a small part of the symmetries of toroidally compacti"ed string theory, namely the part visible in perturbation theory. We shall now extend the techniques of Section 3 in order to study the algebraic structure of the non-perturbative symmetries, which go under the name of U-duality. In this section, we focus on the subgroup of the U-duality symmetry that preserves compacti"cations on rectangular tori with vanishing expectation values of the gauge potentials. The most general case of non-rectangular tori with gauge potentials, for which the full U-duality symmetry can be exhibited, is discussed in the next section. 4.1. Continuous R-symmetries of the superalgebra As in our presentation of uncompacti"ed M-theory in Section 2, the superalgebra o!ers a convenient starting point to discuss the symmetries of M-theory compacti"ed on a torus ¹B. The N"1, 11D supersymmetry algebra is preserved under toroidal compacti"cation: the generators Q merely decompose as bispinor representations of the unbroken group SO(1, 10!d);SO(d), and ? form an N-extended super-PoincareH algebra in dimensions D"11!d. The "rst factor SO(1, 10!d) corresponds to the Lorentz group in the uncompacti"ed dimensions and is actually part of the superalgebra, while the second only acts as an automorphism thereof, and is also known as an R-symmetry. There can be automorphisms beyond the obvious SO(d) symmetry, however, and these are expected to be symmetries of the "eld theory. This symmetry enhancement can be observed at the level of the Cli!ord algebra itself [177,215]. The Gamma matrices C , M"0, d#1,2,10 of eleven-dimensional supersymmetry can be kept + to form a (reducible) Cli!ord algebra of SO(1, 10!d), while the matrices C , I"1,2, d form an ' internal Cli!ord algebra. Note that we have chosen here, in contrast to the notation of the rest of the review, the internal indices running from 1 to d. The generators C generate the SO(d) '( R-symmetry, but they can be supplemented by generators C to form the Lie algebra of a larger ' R-symmetry group SO(d#1). It was the attempt to exhibit the SO(8) symmetry of 11D SUGRA compacti"ed on ¹ that led to the discovery of hidden symmetries [71]. The R-symmetry group is actually larger still. Consider the algebra generated by C , C , C , C , where the subscripts denote the number of antisymmetric internal indices, and the corresponding generators are dropped when the number of internal directions is insu$cient: E For d"2, the only generator C "C generates a ;(1) R-symmetry. '( The R-symmetry is actually part of the local supersymmetry, but we are only interested in its global #at limit. This is the basis for the Twelve-dimensional S-theory proposal [25]. It is important that these generators commute with the momentum charge CC . I


147

Table 7 Classi"cation of the supercharges and central charges w.r.t the Lorentz/R-symmetry group SO(1, 10!d);H. Irreps of H are in bold face. Charges in parenthesis are PoincareH -dualized (moved) into charges in square brackets. Adapted from Ref. [25] d

Q? ?

p"0 Z', Z'( Z'()*+

p"1 ZI, ZI' ZI'()*

p"2 ZI' ZIJ'()

p"3 ZIJM'(

p"4 ZIJMN'

p"5 ZIJMNO

H

1

($, 16)

1#0 #0

1#1 #0

1 #0

0

1

1> #1\

1

2

(2, 16)

2#1 #0 "2#1

1#2 #0 "2#1

1#0 "1

1

[1] #2 "2#1

(1) move

SO(2)

3

(2, 8>) #(2, 8\)

3#3 #0 "3>#3\

1#3 #0 "3#1

1#1 "1#1

3#[1] "3#1

3> #3\ "3>#3\

(1) move

SO(2) ;;(1)

4

(4, 8)

4#6 #0 "10

1#4 #1 "5#1

1#4 #[1] "5#1

6#[4] "10

(4) move

(1) move

SO(5)

5

(4, 4 ) #(4 , 4)

5#10 #1 "(4,4)

1#5 #5#[1] "(5, 1) #(1, 5) #2(1, 1)

1#10 #[5] "(4, 4)

10>#10\ "(10, 1) #(1, 10)

(5) move

(1) move

SO(5) ;SO(5)

6

(8, 4)

6#15 #6 #[1] "27#1

1#6 #15#[6] "27#1

1#20 #[15] "36

(15) move

(6) move

(1) move

;Sp(8)

7

(8>, 2) #(8\, 2 )

7#21 #21 #[7] "28 A

1#7 #35#[21] "63#1

1!#35! "36 A

(21) move

(7) move

0

S;(8)

8

(16, 2)

8#28 #56 #[28] "120

1#8#70 #[1#56] "135#1

(1#56) move

(28) move

0

0

SO(16)

E For d"3, C and C commute, and generate an SO(3);;(1) symmetry. E For d"4, C "C C , where C is the space-time or internal chirality (see Eq. (A.1)) and, > > together with C , generates an SO(5) symmetry. E For d"5, C $C C generate two commuting SO(5) subgroups. >

148


E For d"6, C appears in the commutator [C , C ] and a ;Sp(8) is generated. E For d"7 (resp. d"8) the generator C comes into play and one obtains an S;(8);;(1) (resp. SO(16)) R-symmetry group. The various R-symmetry groups are summarized in the right column of Table 7, which furthermore gives the decomposition of the 528 central charges on the right-hand side of Eq. (2.13a) under the Lorentz group SO(1, 10!d) in the uncompact directions and the R-symmetry group. The various columns correspond to distinct SO(1, 10!d) representations, after dualizing (moving) central charges into charges with less indices when possible. In all these cases, the superalgebra can be recast in a form manifestly invariant under the R-symmetry. Here we collect the cases D"4, 5, 6, including the central charges, which transform linearly under the R-symmetry: E For D"4 (d"7), the 32 supercharges split into 8 complex Weyl spinors transforming as an 8 8 of S;(8): +Q , Q Q M ,"pI Q P d M , ?@ I ? @

(4.1a)

+Q , Q ,"e Z , ? @ ?@ +Q M , Q Q M ,"e Q ZHM M , ? @ ?@

(4.1b) (4.1c)

where k"0, 1, 2, 3 are SO(3, 1) vector indices, a, a"1, 2 are Weyl spinor indices, and A, AM "1, 2, 8 are 8, 8 indices of S;(8). The central charges are incorporated into a complex antisymmetric matrix Z . E For D"5 (d"6), the 32 supercharges split into 8 Dirac spinors of SO(4, 1), transforming in the fundamental representation of ;Sp(8). The N"8 superalgebra in a ;Sp(8) basis is +Q , Q ,"P (CcI) X #C Z , ? @ I ?@ ?@

(4.2)

where k"0, 1, 2, 3, 4 are SO(4, 1) vector indices, a"1, 2, 3, 4 are Dirac spinor indices, A"1,2,8 are indices in the 8 of ;Sp(8), and X is the invariant symplectic form and Z is the central charge matrix. E For D"6 (d"5), the 32 supercharges form 4 complex spinors transforming in the (4, 1)#(1, 4) of SO(5);SO(5) and the superalgebra takes the form +Q?, Q@ ,"u?@cI p , ? @ ?@ I

(4.3)

+Q?, QM @ ,"d Z?@ , ? @ ?@

(4.4)

where a, b"1,2, 4 are SO(5) spinor indices and u?@ is an invariant antisymmetric matrix, from the local isomorphism SO(5)";Sp(4). The 16 central charges are incorporated in a matrix Z?@ transforming as a bispinor under the R-symmetry SO(5);SO(5) and satisfying the reality condition ZH"uZuR. The R-symmetries that we have discussed here will be of use in the next section to determine the scalar manifold of the compacti"ed 11D SUGRA and hence the global symmetries.


149

4.2. Continuous symmetries of the ewective action In our discussion, in Section 3.1, of the continuous symmetry of the e!ective action of the toroidally compacti"ed type IIA theory, we have intentionally focused our attention on the Neveu}Schwarz sector, and have brie#y described how the Ramond "elds would transform under the symmetries of the Neveu}Schwarz scalar manifold. The distinction between Neveu}Schwarz and Ramond sectors is however an artefact of perturbation theory and, as we discussed in Section 2, the two sets of "elds are uni"ed in the 11D SUGRA description. They mix under the eleven-dimensional Lorentz symmetries unbroken by the compacti"cation on ¹B, namely Sl(d, 1). The low-energy e!ective action therefore admits a continuous symmetry group G containing B SO(d!1, d!1, 1) ( ) Sl(d, 1) , (4.5) where the symbol ( ) denotes the group generated by the two non-commuting subgroups. As found by Cremmer and Julia [70,178], the groups G turn out to correspond to the E series, listed in B BB Table 8. The notation E denotes a particular non-compact form of the exceptional group E , namely its BB B normal real form, and from now on this distinction will be omitted. As evident from their Dynkin diagrams shown in Table 9, the groups E form an increasing family, whose members are related by B a process of group disintegration re#ecting the decompacti"cation of one compact direction in ¹B. This is displayed in Table 9, and will be discussed more fully in the next subsection. The occurrence of these groups can be understood by "tting the number of scalar "elds (including the duals of forms of higher degree) to the dimension of a coset space G /H , where H is B B B the R-symmetry of the superalgebra described in the previous section. In order to have a positive metric for the scalars, it is necessary that H be the maximal compact subgroup of G . Together B B with the dimension of the scalar manifold, this su$ces to determine G . B Scalar "elds arise from the internal components of the metric g of the torus ¹B, and from the '( expectation value of the three-form gauge "eld C on ¹B; they also arise from the expectation '() value E on ¹B of the six-form dual to C in eleven dimensions, or equivalently the '()*+, +,. expectation value of the scalar dual to the three-form C in D"5, the axion scalar dual to the IJM two-form C in D"4, or to the one-form C in D"3; similarly, the Kaluza}Klein gauge IJ' I'( potentials g can be dualized in D"3 into scalars K , which can be interpreted as the expectation I' ' value K on ¹B of the magnetic gauge potential dual to g in eleven dimensions. The '_()*+,./0 +, counting is summarized in Table 10. The factor 1> appearing in D"10 and D"9 corresponds to the type IIA dilaton, and generates a scaling symmetry of the e!ective action, called trombonne symmetry in Ref. [76]. Note that a quite di!erent U-duality group would be inferred if one did not dualize the Ramond "elds into "elds with less indices [72,214], or if one would consider Euclidean supergravities [75,171]. An analogous counting has been performed in Tables 11 and 12 for one-form and two-form potentials, inducing particle and string electric charges, respectively. The latter can be put in Note that d has been upgraded by one unit with respect to the previous section. The normal real form has all its Cartan generators and positive roots non-compact, and is the maximal noncompact real form of the complex algebra E (") [129,155]. B

150


Table 8 Cremmer}Julia symmetry groups and their maximal compact subgroups D

d

G "E B BB

H B

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1> Sl(2, 1);1> Sl(3, 1);Sl(2, 1) Sl(5, 1) SO(5, 5, 1) E E E

1 ;(1) SO(3);;(1) SO(5) SO(5);SO(5) ;Sp(8) S;(8) SO(16)

one-to-one correspondence to the central charges of the supersymmetry algebra discussed in the previous section, with two exceptions. Firstly, the Lorentz-invariant central charge Z in "ve dimensions, where 024 denote the "ve space-time dimensions, does not correspond to any one-form potential [25,29]. This truncation of the superalgebra is consistent with U-duality and is of no concern, except for the twelve-dimensional origin of M-theory. Secondly, there are only 120 Lorentz singlet central charges in D"3 for 128 gauge potentials (equivalently, there are only 64 Lorentz vector charges in D"4 for 70 two-form gauge "elds). As we shall see shortly, U-duality implies that there should in fact be 248 electric charges in D"3 (133 string charges in D"4), yielding a linear representation of the duality group E (resp. E ). Of course, the notion of electric charge is ill-de"ned in D"3, where a one-form (or a two-form in D"4) is PoincareH -dual to a zero-form and a particle (or a string) to an instanton. Another manifestation of the pathology of the D"3 case is the non-asymptotic #atness of the point-like solitons (or string-like in D"4), and the logarithmic divergence of the kernel of the Laplacian in the transverse directions. In spite of these di$culties, we shall pursue the algebraic analysis of these cases in the hope that they can be resolved. If the charges m under the gauge "elds can be put in one-to-one correspondence with the central charges Z, they are nevertheless not equal: the gauge charges are integer-quantized, as we will discuss in the next subsection, whereas the central charges are moduli-dependent linear combinations of the latter: Z"V ) m ,

(4.6)

where V is an element in the group G containing the moduli dependence; it is de"ned up to the left B action of the compact subgroup K"H , inducing an R-symmetry transformation on Z. B The local H gauge invariance can be conveniently gauge-"xed thanks to the Iwasawa decompoB sition (see for instance [197,227]) V"k ) a ) n3K ) A ) N

(4.7)

Equivalently, the central charges Z, Z2 transform as a vector in six space-time dimensions. These charges could be attributed to a KK6-brane, if only the KK6-brane did not need six compact directions to yield a string, and seven to yield a particle state.


151

Table 9 Dynkin diagrams of the E series. The group disintegration proceeds by omitting the rightmost node. The integers shown B are the Coxeter labels, that is the coordinates of the highest root on all simple roots

of G into the maximal compact K, Abelian A and nilpotent N. A natural gauge is obtained by B taking K"1, in which case the `vielbeina V becomes a (generalized) upper triangular matrix V"a ) n. The Abelian factor A is parametrized by the `dilatonic scalarsa, namely the radii of the internal torus, whereas the nilpotent factor N incorporates the `gauge scalarsa, namely the expectation values of the gauge "elds (including the o!-diagonal metric, three-form and their duals) on the torus. G acts on the charges m from the left and on V from the right. The transformed B

152


Table 10 Scalar counting and scalar manifolds in compacti"ed M-theory D

d

g

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 3 6 10 15 21 28 36

C

1 4 10 20 35 56

E

1 7 28

K _

8

Total

Scalar manifold

1 3 7 14 25 42 70 128

1> Sl(2, 1)/;(1);1> Sl(3, 1)/SO(3);Sl(2,1)/;(1) Sl(5,1)/SO(5) SO(5,5,1)/SO(5);SO(5) E /;Sp(8) E /S;(8) E /SO(16)

V can then be brought back into an upper triangular form by a moduli-dependent R-symmetry compensating transformation on the left. This implies that the central charges Z transform nonlinearly under the continuous U-duality group G . For the case of T-duality in type II string theory B this decomposition is given in Eq. (3.7). In Section 5, we shall obtain an explicit parametrization of V in terms of the shape of the torus and the various gauge backgrounds. 4.3. Charge quantization and U-duality As in the case of T-duality, the continuous symmetry E (1) of the two-derivative e!ective BB action cannot be a symmetry of the quantum theory: the gauge potentials transform non-trivially under E , and the continuous symmetry is therefore broken by the existence of states charged under B these potentials. At best there can remain a discrete subgroup E (9), which leaves the lattice of BB charges invariant. For one thing, a subset of the charges corresponds to the Kaluza}Klein momentum along the internal torus, and are therefore constrained to lie in the reciprocal lattice of the torus. Another subset of charges corresponds to the wrapping numbers of extended objects around cycles of ¹B, and are then constrained to lie in the homology lattice of ¹B. A way to determine the remaining discrete subgroup is to consider M-theory compacti"ed to D"4 dimensions, in which case PoincareH duality exchanges gauge one-forms with their magnetic duals [172]. In this dimension, Dirac}Zwanziger charge quantization takes the usual form mGn!mGn 39 (4.8) G G for two particles of electric and magnetic charges mG and n respectively, and i runs from 1 to 28, as G read o! from Table 11. This condition is invariant under the electric}magnetic duality Sp(56, 9), under which (mG, n ) transforms as a vector. The exact symmetry group is therefore at most G E (9)LE (1)5Sp(56, 9) . (4.9) This translates into a condition on E (9) for d47 by the embedding E (9)LE (9). A similar BB BB condition can be obtained in D"3, where all one-forms are dual to scalars. The condition (4.9) requires a precise knowledge of the embedding of E (1) in Sp(46, 1). Instead, we shall take another approach, and postulate that the U-duality group of M-theory


153

Table 11 Vectors and particle charge representations in compacti"ed M-theory D

d

g

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

C 1 3 6 10 15 21 28

E

1 6 21 56

K _

7 36

Total

Charge representation

1 3 6 10 16 27 56 128

1 3 of Sl(2) (3, 2) of Sl(3);Sl(2) 10 of Sl(5) 16 of SO(5, 5) 27 of E 56 of E 248 of E

Total

Charge representation

1 2 3 5 10 27 70

1 2 of Sl(2) (3, 1) of Sl(3);Sl(2) 5 of Sl(5) 10 of SO(5,5) 27 of E 133 of E

Table 12 Two-forms and string charge representations in compacti"ed M-theory D

d

10 9 8 7 6 5 4

1 2 3 4 5 6 7

g

C 1 2 3 4 5 6 7

E

1 5 15 35

K _

6 28

compacti"ed on a torus ¹B is generated by the T-duality SO(d!1, d!1, 9) of type IIA string theory compacti"ed on ¹B\, and by the modular group Sl(d, 9) of the torus ¹B: E (9)"SO(d!1, d!1, 9) ( ) Sl(d, 9) . (4.10) BB The former was argued to be a non-perturbative symmetry of type IIA string theory, as discussed in the previous section, while the latter is the remnant of eleven-dimensional general reparametrization invariance, after compacti"cation on a torus ¹B: it is therefore guaranteed to hold, as long as M-theory, whatever its formulation, contains the graviton in its spectrum. The above construct is therefore the minimal U-duality group, and since it preserves the symplectic condition (4.8) also the maximal one. In the d"2 case, the U-duality group (4.10) is the modular group Sl(2, 9) of the M-theory torus, which in particular contains the exchange of R and R ; translated in type IIB variables, this is Q simply the Sl(2, 9) S-duality of type IIB theory (in 9 or 10 dimensions), which contains the strong-weak coupling duality g P1/g , as can be seen from Eq. (2.45). Note that we do not expect Q Q A veri"cation of this statement requires a precise knowledge of the branching functions of Sp(56) into E .

154


Table 13 Discrete subgroups of E B D

d

E (1) BB

E (9) BB

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 Sl(2, 1) Sl(3, 1);Sl(2, 1) Sl(5, 1) SO(5, 5, 1) E (1) E (1) E (1)

1 Sl(2, 9) Sl(3, 9);Sl(2, 9) Sl(5, 9) SO(5, 5, 9) E (9) E (9) E (9)

any quantum symmetry from the trombonne symmetry factor 1>. For d"3, the T-duality group splits into two factors Sl(2, 9);Sl(2, 9), one of which is a subgroup of the modular group Sl(3, 9) of the M-theory torus ¹. The de"nition (4.10) therefore yields E (9)"Sl(3, 9);Sl(2, 9) and is the natural discrete group of E . For d"4, SO(3, 3, 9) is isomorphic to a Sl(4, 9) (in the same way as SO(6)&S;(4)), which does not commute with the modular group Sl(4, 9) of M-theory on a torus ¹. Altogether, they make the Sl(5, 9) subgroup of E (1)"Sl(5, 1). For d"5, we obtain the SO(5, 5, 9) subgroup of E (1)"SO(5, 5, 1). For d56, this provides a dexnition of the discrete subgroups of the exceptional groups E (1). These groups are summarized in the rather BB tautological Table 13. We note that it is crucial that the groups E be non-compact in order BB for an in"nite discrete group to exist. The maximal non-compact form is also required in order that all representations be real (i.e. that the mass of a particle and its anti-particle be equal, see Section 4.8). 4.4. Weyl and Borel generators A set of generators of the U-duality group can easily be obtained by conjugating the T-duality generators under Sl(d, 9). The Weyl generators now include the exchange of the eleven-dimensional radius R with any radius of the string-theory torus ¹B\, in addition to the exchange of Q the string-theory torus directions among themselves and T-duality on two directions thereof. It is interesting to rephrase the latter in M-theory variables, using relations (2.1), (3.11a) and (3.11b): l l l l (4.11) ¹ : R P N , R P N , R P N , lP N . H RR Q RR N RRR GH G RR Q G G H G H Q H Q These relations are symmetric under permutation of i, j, s indices, and using an R R transformaI Q tion, we are free to choose i, j, s along any direction of the M-theory torus ¹B. The M-theory This is particularly interesting in the d59 case, where we obtain discrete versions of a$ne and hyperbolic groups, see Section 4.6.


155

T-duality therefore reads l l l l N ¹ : R P N , R P N , R P N , lP (4.12) '() ' R R ( R R ) RR N RRR ( ) ) ' ' ( ' ( ) and in particular involves three directions, contrary to the naive expectation. We emphasize that the above equation summarizes the non-trivial part of U-duality, and arises as a mixture of T-duality and S-duality transformations. It can in particular be used to derive [322] the duality between the heterotic string compacti"ed on ¹ and type IIA compacti"ed on K in the Horava}Witten picture, and thus unify all vacua with 16 supersymmetries. We however restrict ourselves to the maximally supersymmetric case in this review. The Weyl group can be written in a way, similar to Eq. (3.12): W(E )"9 ( )S (4.13) B B but it should be borne in mind that the algebraic relations between the 9 symmetry ¹ and the permutations S are di!erent from those of the T-duality generators ¹ and S ; in addition '( GH d di!ers by one unit from the one we used there. We also note that the transformations ¹ and '() S preserve Newton's constant '( R < 1 '" 0 , " (4.14) l l i N N B where we have de"ned < to be the volume of the M-theory compacti"cation torus. 0 On the other hand, the Borel generators now include a generator c Pc #c that mixes the G G Q eleven-dimensional direction with the other ones, as well as the T-duality spectral #ow B PB #1, from which, by an R R conjugation, we can reach the more general M-theory GH GH Q G spectral yow C : C PC #1 . (4.15) '() '() '() We should also include a set of generators shifting the other scalars from the dual gauge potentials, as explained in Section 4.2: E : E PE #1 , (4.16a) '()*+, '()*+, '()*+, K : K PK #1 . (4.16b) '_()*+,./0 '_()*+,./0 '_()*+,./0 These scalars and corresponding shifts are needed for d56 and d58 respectively. For d59, as will become clear in Section 4.6, the enlargement of the symmetry group to an a$ne or Kac}Moody symmetry requires an in"nite number of such Borel generators. As we shall see in Section 5.4, the Borel generators (4.16) can be obtained from commutators of C transformations. '() This equation holds for d53 only; when d(3 the 9 symmetry (4.12) collapses and only the permutation group S remains. B As discussed in Section 5.4, the C shift actually has to be accompanied by E and K shifts to be a symmetry of the equations of motion.

156


Table 14 S-dual type IIB BPS states State

Tension

S-dual

Dual state

D1-brane

1 g l QQ 1 g l QQ 1 g l QQ R gl Q Q 1 g l QQ 1 g l QQ

1 l Q 1 g l QQ 1 gl Q Q R gl Q Q 1 gl Q Q 1 gl Q Q

F-string

D3-brane D5-brane KK5-brane D7-brane D9-brane

D3-brane NS5-brane KK5-brane 7 -brane 9 -brane

4.5. Type IIB BPS states and S-duality Before studying the structure of the U-duality group, we shall pause and brie#y discuss the action of the extra Weyl generator R R on the type IIB side. Using the identi"cation (2.45) to Q convert to type IIB variables, this action inverts the coupling constant and rescales the string length as 1 l lg , (4.17) g , Q Q Q Q g Q in such a way that Newton's constant 1/(gl) is invariant. Its action on the BPS spectrum can be Q Q straightforwardly obtained by working out the action on the masses or tensions, and is summarized in Table 14. In this table, we have displayed the action of the 9 Weyl element only. Under more general duality transformations, the fundamental string and the NS5-brane generate orbits of so called (p, q) strings and (p, q) "ve-branes. The former can be seen as a bound state of p fundamental strings and q D1-branes, or (in the Euclidean case) as a coherent superposition of q D1-branes with p instantons [195]. The (p, q) "ve-branes similarly correspond to bound states of p NS5-branes and q D5-branes. On the other hand, the action of S-duality on the D7 and D9-brane yields states with tension 1/g and 1/g, respectively. These exotic states will be discussed in Section 4.9, where our Q Q nomenclature will be explained as well. Again, such states have less than three transverse dimensions, and do not preserve the asymptotic #atness of space-time and the asymptotic constant value of the scalar "elds. In particular, the D7-brane generates a monodromy qPq#1 in the complex scalar q at in"nity. Its images under S-duality then generate a more general Sl(2, 9)


157

monodromy

M"

1!pq

p

!q

1#pq

(4.18)

ascribable to a (p, q) 7-brane. We "nally remark that the relations in Table 14 can also be veri"ed directly using the R R #ip and the M-theory/IIB identi"cations as (un)wrapped M-theory Q branes, given in Tables 5 and 6. 4.6. Weyl generators and Weyl reyections In order to understand the occurrence of the E U-duality group, we shall now apply the same BB technique as in the T-duality case and investigate the group generated by the Weyl generators. We choose as a minimal set of Weyl generators the exchange of the M-theory torus directions S : R R , where I"1,2, d!1, as well as the T-duality ¹"¹ on directions 1, 2, 3 of ' ' '> the M-theory torus. Adapting the construction of Ref. [108] and Section 3.4, we represent with basis the monomials u"(ln l, ln R , ln R ,2, ln R ) as a form on a vector space < N B B> e , e , e ,2, e , and associate to any weight vector j"xe #xe #2#xBe its `tensiona B B T"e6PH7"lVRVRV2RVB . (4.19) N B The generators S and ¹ can then be implemented as linear operators on < , with matrix ' B> 2 1 1 1 1 !1 !1 !1 1 , ¹" !1 !1 . (4.20) !1 S" ' 1 !1 !1 !1 ( B\ ( B\ The operators S and ¹ in Eq. (4.20) are easily seen to be orthogonal with respect to the Lorentz ' metric

ds"!(dx)#(dx') ,

(4.21)

and correspond to Weyl re#ections a)j a jPo (j)"j!2 ? a)a

(4.22)

along planes orthogonal to the vectors a "e !e , I"1,2, d!1, ' '> '

a "e #e #e !e .

(4.23)

It has also been proposed that the IIB 7-branes transform as a triplet of Sl(2, 9) [225]. In Ref. [108], the discussion was carried out from the gauge theory side, and the U-duality invariant (4.14) was used to eliminate the vector e , except when d"9. This vector can, however, be kept for any d, and, as we shall momentarily see, appears as an extra time-like direction. T actually has the dimension of a p-brane tension T , with p"!3x!x!2!xB!1. N

158


It is very striking that l appears on the same footing as the other radii R , but with a minus sign in N ' the metric: it can be interpreted as the radius of an extra time-like direction, much in the spirit of certain proposals about F-theory [25,310]. The only non-vanishing (Lorentzian) scalar products of these roots turn out to be (a )"(a )"2 , a ) a "a ) a "!1 ' ' '>

(4.24)

summarized in the Dynkin diagram: * "

(4.25)

# ! ! ! !2! . B\ This is precisely the Dynkin diagram of E as shown in Table 9, in agreement with the analysis B based on moduli counting. In Eq. (4.25) it is easy to recognize the diagrams of the SO(d!1, d!1, 9) (denoted by *'s) and Sl(d, 9) (denoted by #'s) subgroups. The branching of the Sl(d) diagram on the third root re#ects the action of T-duality on three directions. The full diagram can be built from the M-theory Lorentz group Sl(d, 9) denoted by #'s, and from the type IIB Lorentz group Sl(d!1, 9) generated by the roots a , a ,2, a . Under decompacti"cation, the rightmost root has to be dropped, so B\ that E disintegrates into E . When the root at the intersection is reached, the diagram falls B B\ into two pieces, corresponding to the two Sl(2) and Sl(3) subgroups in D"8. The root a itself disappears for d"2, leaving only the root a of Sl(2, 1). Again, the action of the Weyl group on < is reducible, at least for d48. Indeed, the B> invariance of Newton's constant R /l implies that the roots are all orthogonal to the vector ' N d"e #2#e !3e , B

(4.26)

with proper length d"d!9, so that the re#ections actually restrict to the hyperplane < normal B to d: x#2#xB#3x"0 .

(4.27)

The Lorentz metric on < restricts to a metric g "d !1/9 on < , which is positive-de"nite for B> '( '( B d48, so that S and ¹ indeed generate the Weyl group of the Lie algebra E (1). The order and ' B number of roots of these groups are recalled in Table 15 [174]. When d"9, however, the invariant vector d becomes null, so that < no longer splits into B> d and its orthogonal space; the generators act on the entire Lorentzian vector space < , and the B> From this point of view, the U-duality is a consequence of general coordinate invariance in M and type IIB theories [204]. There is a notable exception for d"8, where E disintegrates into E ;Sl(2). This is because the extended Dynkin diagram of E has an extra root connected to a . Only Sl(2) singlets remain in the spectrum, however. The same happens in d"4, where E "Sl(3);Sl(2) in E "Sl(5) is not a maximal embedding.


159

Table 15 Order and number of roots of E Weyl groups B d

2

3

4

5

6

7

8

9

E B Order Roots

A 2 2

A ;A 6;2 6#2

A 5! 20

D 25! 40

E 23 5 72

E 23 5 7 126

E 2357 240

EK R R

generators S and ¹ no longer span a "nite group. Instead, they correspond to the Weyl group of ' the azne Lie algebra E "EK . This is in agreement with the occurrence of in"nitely many conserved currents in D"2 space-time dimensions. This case requires a speci"c treatment and will be discussed in Section 4.12. For d'9, that is compacti"cation to a line or a point, the situation is even more dramatic, with the occurrence of the hyperbolic Kac}Moody algebras E and E , about which very little is known. The reader should go to [126,179,180,234] for further discussion and references. 4.7. BPS spectrum and highest weights Pursuing the parallel with our presentation on T-duality, we now discuss the representations of the U-duality Weyl group. The fundamental weights dual to the roots a ,2, a , a are easily B\ computed: R j"e !e PT " , l N R R j"e #e !2e PT " , l N R R R j"e #e #e !3e PT " , l N R R R R j"e #2#e !3e PT " , l N 2 R R jB\"e #2#e !3e PT " 2 B\ , B\ \B l N 1 jB\"e #2#e !3e &!e PM" , B\ B R B 1 j"!e PT " , l N

(4.28a) (4.28b) (4.28c) (4.28d)

(4.28e) (4.28f) (4.28g)

160


where the symbol & in Eq. (4.28f) denotes equality modulo d, that is up to a power of the invariant Planck length. In the above equations, we have translated the weight vectors into monomials, and interpreted it as the tension T of a p-brane: N> E The weight jB\ corresponds to the Kaluza}Klein states, with mass 1/R , as well as its ' U-duality descendants. We shall name its orbit the particle multiplet, or yux multiplet, for reasons that will become apparent in Section 6.9. E The weight j on the other hand has dimension 1/¸, and corresponds to the tension of a membrane wrapped on the direction 1: it will go under the name of string multiplet, or momentum multiplet. The latter name will also become clear in Section 6.9. E The weight j is the highest weight of the membrane multiplet containing the fundamental membrane with tension 1/l, together with its descendants. N E The weights j and j both correspond to threebrane tensions T and T . Even though they are inequivalent under the Weyl group, it turns out that j is a descendant of j under the full U-duality group. The U-duality orbit of the state with tension T is therefore a subset of the orbit of the state with tension T , and j is the true highest-weight vector of the threebrane multiplet. E The same holds for j associated to a membrane tension T and descendant of the highest weight j of the membrane multiplet under U-duality, as well as for j and j. E The weight j corresponds to a "vebrane tension T , but is again not the highest weight of the xvebrane multiplet, which is instead a non-fundamental weight: 1 T " Pj"!2e "2j . l N

(4.29)

Similarly, the weight j corresponds to a fourbrane tension T , and is not the highest weight of the fourbrane multiplet, which is instead a non-fundamental weight: R T " Pj"e !2e "j#j . l N

(4.30)

E Finally, the instanton multiplet does not appear in Eqs. (4.28a)}(4.28g). An instanton con"guration can be obtained by wrapping a membrane on a three-cycle, and corresponds to a weight vector R R R T " Pj"a . \ l N

(4.31)

Since this vector is a simple root, it corresponds to a multiplet in the adjoint representation. It is, however, not the highest weight of the U-duality multiplet, which is instead the highest root t whose expansion coe$cients on the base of the simple roots are given by the Coxeter labels We should, however, warn the reader that it is not the representation arising in non-perturbative couplings, as we shall discuss in Section 5.8.


161

in Table 9. An explicit computation gives d"4: t"d!j!j ,

(4.32a)

d"5: t"d!j ,

(4.32b)

d"6: t"d!j ,

(4.32c)

d"7: t"d!j ,

(4.32d)

d"8: t"d!j .

(4.32e)

Since the fundamental weights jG are dual to the simple roots a , it is clear that t ) a "d , ' ' G' where I is the index appearing on j in Eqs. (4.32a)}(4.32e) at a given d, and moreover it can be easily checked that t"2. The highest root can therefore be added as an extra root in the Dynkin diagrams in Table 9, and turns them into extended Dynkin diagrams. The previous considerations are summarized in the diagram 1 l N "

(4.33)

R R R R R R R R R R 1 ! ! ! !2! l l l l R N N N N B where we have indicated the highest weight associated to each node of the Dynkin diagram. For simplicity, we shall henceforth focus our attention on the particle and string multiplets, corresponding to the rightmost node with weight jB\ and leftmost node with weight j, respectively. 4.8. The particle alias yux multiplet The full particle multiplet can be obtained by acting with Weyl and Borel transformations on the Kaluza}Klein state with mass 1/R . Instead of working out the precise transformation of the ' supergravity con"gurations, we can restrict ourselves to considering the masses of the various states in the multiplet. We note that the action of S and ¹ on the dilatonic scalars R is '( '() ' independent of the dimension d of the torus, so that we can work out the maximally compacti"ed case D"3, and obtain the higher-dimensional cases by simply deleting states that require too many di!erent directions on ¹B to exist. The results are displayed in Table 16, where distinct letters stand for distinct indices. The states are organized in representations of the Sl(8, 9) modular group of the torus ¹. These representations arrange themselves in shells with increasing power of l; since l is invariant under Sl(8, 9), this N N corresponds to the grading with respect to the simple root a . Generalized T-duality ¹ may '() See Ref. [216] for the construction of U-duality multiplets of p-brane solutions, and Ref. [113] for a discussion of the continuous U-duality orbits of p-brane solutions.

162


Table 16 Particle/#ux multiplet 248 of E Mass M

Sl(8) irrep

Charge

1 R ' RR ' ( l N RRR R R ' ( ) * + l N RR R R R R R < ' ( ) * + , ., 8 0 l l N N RRR R R R R R ' ( ) * + , . / l N RRR RR R R R ' ( ) * + , . / l N RRR RR R RR ' ( ) * + , . / l N

8

m

28

m

56

m

1#63

m_

56

m_

28

m_

8

m__

move from one shell to the next or previous one, whereas S acts within each shell. Eight states '( with mass < /l have been added in the middle line, corresponding to zero-length weights that 0 N cannot be reached from the length-2 highest state. These states are, however, necessary in order to get a complete representation of the modular group Sl(8, 9), and can be reached by a Borel transformation in Sl(8, 9). They can be thought of as the eight ways to resolve the radius that appears squared in the mass of the other states on the same line into a product of two distinct radii. This is not required for the other lines, since all squares can be absorbed with a power of Newton's constant. In the last column of Table 16, we have indicated the representation of Sl(8, 9) that yields the same dimension. The superscripts denote the number of antisymmetric indices, and no symmetry property is assumed across a semicolon. In other words, m_ correspond to the HZU

(4.40) # e\6PH7(F)# e\6PH7(F)#2 , H H HZU HZU where u"(ln l, ln R ,2, ln R ) is the vector of dilatonic scalars (whose "rst component is N B non-dynamical), 1u, j2"x ln l#x ln R #2 is the duality bracket (4.19) and Ru ) Ru the N Weyl-invariant kinetic term (Rl"0) obtained from the metric (4.21). In addition to the equations N of motion from (4.40), the duality equations FN"*F\B\N should also be imposed. As in the


169

case of T-duality, the set of positive roots U is not invariant under Weyl re#ections, but the > Peccei}Quinn scalars undergo non-linear transformations APe\6P?7A that compensate for the sign change [215]. 4.12. Compactixcation on ¹ and azne EK symmetry As we pointed out in Section 4.6, the compacti"cation on a nine-torus ¹ to two space-time dimensions gives rise to a qualitative change in the U-duality group: the invariant vector d in Eq. (4.26) corresponding to the dimensionless Newton constant becomes light-like w.r.t. the Lorentzian metric !(dx)#(dx)#2#(dx), so that the action of the U-duality group generated by S and ¹ in Eq. (4.20) cannot be restricted to its orthogonal subspace. Instead, it generates the Weyl ' group of the EK a$ne algebra, as was shown in Ref. [108]; we shall recast their construction in the notation of this review, at the same time settling several issues. In order to see the a$ne symmetry EK arise, we simply note that the Dynkin diagram of E (see Table 9) is nothing but the extended Dynkin diagram of E , where the additional root with Coxeter label 1 corresponds to a "e !e . The roots a , a ,2, a generate the E horizontal Lie algebra, whereas a and d" e !3e are the extra dimensions needed to represent the central charge ' ' K and degree D generators of the standard construction of a$ne Lie algebras (see e.g. Ref. [118]). To make the identi"cation precise, we recall that the simple roots of an a$ne Lie algebra GK can be chosen as a( "(a , 0, 0), I"1,2, r, a( "(!t, 0, 1) (4.41) ' ' in the basis (k, k, d) of the Minkovskian weight space < "1P#1 with norm k#2kd. Here, P> t is the highest root of G, r is the rank of G, k is the a$ne level, and d the ¸ eigenvalue. In the case at hand, we have G"E so r"8 and want to "nd the change of basis between the roots a , I"0,2, 8 and null vector d of our formalism and the standard roots a( , I"0,2, 8 and ' ' vectors c"(0, 0, 1), i"(0, 1, 0). From Eqs. (4.32a)}(4.32e) we have, t"e #2#e #2e !3e "d!a , so that, comparing with Eq. (4.41), we can identify d with c"(0, 0, 1) and

(4.42)

a( "a , I"1,2,7 , (4.43a) ' ' a( "a , (4.43b) a( "a . (4.43c) The vector i"(0, 1, 0) can be easily calculated from the requirements that i"i ) a( "0, ' I"1,2, 8 and i ) d"1: 1 d i" (!e !2!e #e #3e )"e ! . 2 2

(4.44)

In order to keep with the standard notation, the simple roots of the Lie algebra are now labelled by subscripts ranging from 1 to r, as opposed to our notation for the simple roots of the U-duality groups E , which carry labels 0 P to r!1.

170


The level k and degree d of any weight vector j3< can now be obtained from the products d ) j and d ) i, respectively, and they both have a simple interpretation: k"d ) j"x#2#x#3x is simply the length dimension of the associated monomial RV'lV, and ' N d"i ) j"x!k/2

(4.45)

(4.46)

counts the power of R appearing in the same monomial, up to a shift k/2. This was expected, since the horizontal subalgebra E LEK does not a!ect R and by de"nition commutes with ¸ . ¸ (n'0) generators, on the other hand, bring additional powers of R and increase the \L degree d. In particular, the ¸ eigenvalues are integer-spaced, as they should. We proceed by considering the particle/#ux and string/momentum multiplets introduced in Sections 4.8 and 4.10, with highest weights jB\"!e and j"e !e , respectively (see Eqs. (4.28a)}(4.28g)). The particle multiplet is therefore a level !1 representation with trivial ground state k"0 (that is, in the chiral block of the identity). A bit of experimentation reveals the "rst Sl(9) representations occurring in the particle multiplet: m__, m__, m__, m__, m__, m___,2

(4.47)

with tensions scaling from 1/l to 1/l, in addition to the representations already present in d"8, N N given in Table 16. However, the full orbit is in"nite. On the other hand, the string multiplet is a level !2 representation with ground state in the 3875 of E . In both cases, the representations are in"nite-dimensional, and need to be supplemented with weights of smaller length as in the E and E cases. The instanton multiplet, on the other hand, is a level-0 representation of EK , with a non-singlet ground state in the adjoint of E . This makes it obvious that the usual unitarity restrictions for compact a$ne Lie algebras do not apply in our case. This concludes our analysis of the d"9 case, and we now restrict ourselves to the better understood d48 case.

5. Mass formulae on skew tori with gauge backgrounds We would now like to generalize the mass formulae of the U-duality multiplets obtained so far for rectangular tori and vanishing gauge potentials to the more general case of skew tori and arbitrary gauge potentials, which will exhibit the full U-duality group. This will also allow a better understanding of the action of Borel generators on the BPS spectrum. We will concentrate on the d"7 #ux multiplet, but the same method applies to the other multiplets. 5.1. Skew tori and Sl(d, 9) invariance We have already argued that BPS states could be labelled by a set of tensors of integer charges describing their various momenta and wrappings. In particular, for the case of the d"7 #ux multiplet, the charges m , m, m, m_

(5.1)


171

describe the Kaluza}Klein momentum, membrane, "vebrane and KK6-brane wrappings. The position of the index has been chosen in such a way that we obtain the correct mass by contracting each of them with the vector of radii R' or inverse radii 1/R . Note that for d"7 the tensor m_ is ' really a tensor m, but the extra seven indices account for an extra factor of the volume in the tension. Of course, a BPS state with generic charges m will not be 1/2-BPS state in general (for d55): some quadratic conditions on m have to be imposed, as already discussed in Sections 2.2 and 3.8. We shall henceforth assume these conditions ful"lled, deferring the study of the latter to Section 5.9. The 1/2-BPS state mass formula for a non-diagonal metric g can be straightforwardly obtained '( by replacing contractions with the vector of radii by contractions with the metric, and inserting the proper symmetry factor and power of the Planck length on dimensional grounds: M"(m )#(m)#(m)#(m_) 1 1 m'()*+g g g g g m,./01#2 . "m g'(m # m'(g g m)*# ') (* ', (. )/ *0 +1 ' ( 2!l 5!l N N

(5.2)

This formula is invariant under Sl(d, 9), but not yet under the T-duality subgroup SO(d!1, d!1, 9) of the U-duality group. It only holds when the expectation value of the various gauge "elds on the torus vanish. To reinstate the dependence on the three-form C , we apply the '() following strategy. E Decompose the #ux multiplet as a sum of T-duality irreps. E Include the correct coupling to the NS two-form "eld B using the T-duality invariant mass GH formulae. E Study the T-duality spectral #ow BPB#*B. E Covariantize this #ow under Sl(d, 9) into a CPC#*C #ow. E Integrate the CPC#*C #ow to obtain the U-duality invariant mass formula. 5.2. T-duality decomposition and invariant mass formula We have already discussed the "rst step in Section 4.9, and we only need to restrict ourselves to the case d"7. Table 18 then truncates to its upper left-hand corner displayed in Table 22, as can be read from the d"7 particle multiplet mass formula (5.2) written with s and i indices:

M"

(m) m (mQ) (m) (mQ_Q) (m_Q) Q #(m ) # (mQ)# # # # # g g g g g g Q Q Q Q Q Q

(5.3)

corresponding to three SO(6, 6) irreps, JQ R. Formula (5.23) reduces to the d"5 result of Ref. [89] for vanishing expectation values of the gauge backgrounds (see also [294]). 5.7. String multiplet and U-duality invariant tension formula Exactly the same analysis can be done for the momentum multiplet. We give here the result for d"6. The contributing charges n, n, n_ decompose into SO(6, 6) T-duality multiplets I"(nQ),

S"(n, nQ, nQ_Q),

and the boosts ¸\G, ¸>\ do not depend on the dynamics, while the generator P\ generates the translations in the x> direction and plays the role of the Hamiltonian. The usual dispersion relation H"(PGP #M in equal-time quantization, is G replaced in the light-cone quantization by PGP #M G , P\" 2P>

(6.1)

exhibiting Galilean invariance on the transverse space. Particles, with positive energy P\'0, necessarily have positive longitudinal momentum P>, while antiparticles will have negative P>. The vacuum of P\ is hence reduced to the Fock-space state "02, and the negative-norm ghost states are decoupled as well. This simpli"cation of the theory is at the expense of instantaneous non-local interactions due to the P>"0 pole in Eq. (6.1). Discrete light-cone quantization proceeds by compactifying the longitudinal direction x\ on a circle of radius R : J x Kx #2nR . (6.2) \ \ J This results into a quantization of the longitudinal momentum of any particle i according to n (6.3) P>" G . G R J Because the total momentum is conserved, the Hilbert space decomposes into "nite-dimensional superselection sectors labelled by N" n . Note that the "nite dimension does not require G imposing any ultraviolet cut-o! on the eigenvalues n , but follows from the condition n '0. G G It is important to note that, because the x\ direction is a light-like direction, the length R of the J radius is not invariant, but can be modi"ed by a Lorentz boost ¸>\,

x

P

x

cosh b

!sinh b

x

!sinh b

cosh b

x

,

(6.4)


183

which amounts to R Pe@R , P\Pe@P\, P>Pe\@P> . (6.5) J J This implies that the Hamiltonian P> depends on the radius R through an over-all factor J P\"R H , (6.6) J , so that the mass M"2P>P\ is independent of R . J 6.2. Why is Matrix theory correct? Following Ref. [274], we will now derive the Hamiltonian H describing the DLCQ of , M-theory, and obtain the BFSS Matrix-theory conjecture. The basic idea is to consider the compacti"cation on the light-like circle as Lorentz-equivalent to a limit of a compacti"cation on a space-like circle. Acting with a boost (6.4) on an ordinary space-like circle, we "nd

cosh b

!sinh b

0

R !1#e\@ 1 !RJ " J P , (6.7) !sinh b cosh b R 1#e@ R (2 (2 Q J where R "R e\@. Sending bPR while keeping R "nite, we see that the light-like circle is Q J J Lorentz-equivalent to a space-like circle of radius R P0. Q In order to keep the energy "nite, which from Eq. (6.6) and on dimensional ground scales as R /l, J N we should also rescale the Planck length (and any other length) as l "e\@l . Altogether, NQ N M-theory with Planck length l on the light-like circle of radius R in the momentum P>"N/R N J J sector is equivalent to M-theory with Planck length l on the space-like circle of radius R in the NQ Q momentum P"N/R sector, with Q R "R e\@, l "e\@l (6.8) Q J NQ N in the limit bPR. Eliminating b, we obtain the following scaling limit: R R R P0, M" Q " J""xed . (6.9) Q l l NQ N Following Ref. [274], we shall denote the latter theory as M I theory. Since the space-like circle R shrinks to zero in l units, this relates the DLCQ of M-theory to Q NQ weakly coupled type IIA string theory in the presence of N D0-branes carrying the momentum along the vanishing compact dimension. Using Eq. (2.1), the scaling limit becomes R g "(R M), a"l" Q , R P0, Q Q Q Q M

M""xed .

(6.10)

In particular, g and a go to zero, so that the bulk degrees of freedom decouple, and only the Q leading-order Yang}Mills interactions between D0-branes remain. This validates the BFSS conjecture, up to the possible ambiguities in the light-like limit bPR [50,156]. Several di$culties have also been shown to arise for compacti"cation on curved manifolds [100,101], but since we are only concerned with toroidal compacti"cations, we will ignore these issues.

184


6.3. Compactixcation and Matrix gauge theory For toroidal compacti"cations of M-theory, we consider the same scaling limit as in Eq. (6.9), and keep the torus size constant in Planck length units, that is

R R Q , r " ' ""xed . (6.11) R "r ' l ' ' M NQ However, comparing Eqs. (6.9) and (6.11), we "nd that the size of the torus goes to zero in the scaling limit. To avoid this it is convenient to consider the theory on the T-dual torus ¹I B, obtained by a maximal T-duality in all d directions. From Eq. (2.5), this has the e!ect that,

IIA with N D0-branesP

IIA with N Dd-branes

d"even ,

IIB with N Dd-branes

d"odd .

(6.12)

Using the maximal T-duality transformation B ¹ , with ¹ given in Eq. (2.5), the type II ' ' ' parameters then become R 1 (R M)\B , a"l" Q , RI " , (6.13a) g" Q Q M ' rM Q r ' ' R R R P0, M" Q ""xed, r " ' ""xed , (6.13b) Q ' l l NQ NQ so that, in particular, the size of the dual torus is "xed in the scaling limit. We will sometimes refer to the type II theory in this T-dual picture as the II I -theory. The behaviour of the string coupling in the scaling limit is now di!erent according to the dimension of the torus:

0,

d(3 ,

g P Finite, Q R,

d"3 ,

(6.14)

d'3 .

In particular for d(3 we still have weakly coupled type IIA or IIB string theory in the presence of N Dd-branes, so that M-theory is described by the SYM theory with 16 supercharges living on the world-volume of the N Dd-branes. The gauge coupling constant of this Matrix gauge theory and the radii s of the torus on which the D-branes are wrapped read ' 1 M\B , < , r , s "RI " (6.15) g "g lB\" P ' ' ' 7+ QQ rM < ' P ' showing, in particular, that g is "nite in the scaling limit. 7+ The special case of Matrix theory on a circle (d"1) yields (after an S-duality transforming the background D1-strings into fundamental strings) Matrix string theory [91,92,229], in which an identi"cation between the large-N limit of two-dimensional N"8 supersymmetric YM theory and type IIA string theory is established. We will not further discuss this topic here, and refer to


185

Refs. [114,287] for the next case d"2 and its relation to type IIB string theory. Moving on to the case d"3, the same conclusion as in the d(3 case continues to hold, since although the string coupling is "nite, the string length goes to zero so that loop corrections are suppressed in the aP0 limit. Consequently, the d"3 Matrix gauge theory is N"4 supersymmetric Yang}Mills theory. For d'3, however, the coupling g blows up, and the weakly coupled string description of the Q D-branes is no longer valid. This coincides with the fact that the Yang}Mills theory becomes non-renormalizable and strongly coupled in the UV. Hence, in order to de"ne a consistent quantum theory, one needs to supplement the theory with additional degrees of freedom. In the following we brie#y review the proposals for d"4 and d"5, and show the complication that arises for d"6. These proposals follow from the above prescription, using further duality symmetries, which will be examined in more detail in Section 7. Other decoupling limits have been considered in [168]. 6.4. Matrix gauge theory on ¹ In the case d"4, it follows from (6.12) that the e!ective theory is 4#1 SYM coming from the type IIA D4-brane world-volume theory. In the scaling limit the type IIA theory becomes strongly coupled and using the correspondence between strongly coupled IIA theory and M-theory a new eleventh dimension is generated, which plays the role of a "fth space dimension in the gauge theory [43,45,260]. Using Eqs. (2.11) and (6.13a), the radius and 11D Planck length are 1 , lI "gl "RM\B> spectrum, as demanded by eleven-dimensional Lorentz invariance. 7.1. Weyl transformations in Matrix gauge theory The discussion of Matrix gauge theory from M-theory in Section 6 has been restricted to rectangular tori with vanishing gauge potentials, so that we "rst focus on the transformations in the Weyl subgroup of the U-duality group W(E (9))"9 ( )S . (7.1) BB B The permutation group S that interchanges the radii R of the M-theory torus obviously still B ' permutes the radii s of the Matrix gauge theory T-dual torus. On the other hand, the generalized ' in Eq. (4.12), using the dictionary (6.22a) and (6.22b), translates into the following T-duality ¹ '() transformation of the Matrix gauge theory parameters:

gB\ g P 7+ , 7+ =B\

S : '(E )

s Ps , ? ? g s P 7+s , ? = ?

=, s , ?$' ( ) ? a"I, J, K ,

(7.2)

aOI, J, K .

For d"3 the transformation (7.2) is precisely the (Weyl subgroup of ) S-duality symmetry of N"4 SYM in 3#1 dimensions [124,295]: g P1/g , (7.3) 7+ 7+ obtained for zero theta angle. The transformation (7.2) generalizes this symmetry to the case d'3, by acting as S-duality in the (3#1)-dimensional theory obtained by reducing the Matrix gauge theory in d#1 dimensions to the directions I, J, K and the time only [108]. Indeed, the coupling constant for the e!ective (3#1)-dimensional gauge theory reads = 1 " , g g 7+ and the transformation (7.2) becomes

(7.4)

(g , s , s )P(1/g , s , g s ) . (7.5) ? ? ? ? To summarize, we see that from the point of view of the Matrix gauge theory the U-dualities are accounted for by the modular group of the torus on which the gauge theory lives (yielding the We restrict to the case d53; the case d"1 has trivial Weyl group, while for the case d"2 there is only the permutation symmetry S .

192


Sl(d, 9) subgroup) as well as by generalized electric}magnetic dualities (implementing the Tdualities of type IIA string). We now discuss in more detail the d"4, 5, 6 cases, in order to give more support to the proposals discussed in Section 6. Explicitly, one obtains d"4: S

d"5: S

'()

'()

g s , 7+ ? s Ps , ? ?

aOI, J, K , a"I, J, K ,

g Pg , 7+ 7+ g a, bOI, J, K , s P 7+, ? s @ s Ps , a"I, J, K , ? ?

(7.6a)

(7.6b)

g g P 7+ , a, b, cOI, J, K , 7+ s s s ?@A d"6: S '() g (7.6c) s P 7+ , ? ss @A s Ps , a"I, J, K . ? ? For d"4 we see that Eq. (7.6a) induces a permutation of the YM coupling constant with the radii, in accordance with the interpretation (6.17) of the YM coupling constant as an extra radius. For d"5, Eq. (7.6b) takes the form of a T-duality symmetry (2.41) of the non-critical string theory living on the type IIB NS5-brane world-volume with the YM coupling related to the string length as in Eq. (6.19). Finally, for d"6, we see by comparing Eq. (7.6c) with the U-duality transformation in Eq. (4.12) that we recover the symmetry transformation ¹ in M-theory with the YM coupling '() constant related to the Planck length by Eq. (6.21). At this point, it is also instructive to recall the full U-duality groups for toroidal compacti"cations of M-theory, as summarized in Table 8, and discuss their interpretation in view of the Matrix gauge theories for d"3, 4, 5 (see Table 23). For d"3, the Sl(3, 9);Sl(2, 9) U-duality symmetry is the product of the (full) S-duality and the reparametrization group of the three-torus. For d"4, the Sl(5, 9) symmetry is the modular group of the "ve-torus, corroborating the interpretation of this case as the (2, 0) theory on the M5-brane [260]. Finally, for d"5 the SO(5, 5, 9) symmetry should be interpreted as the T-duality symmetry of the string theory living on the NS5-brane [45,89]. The E (9) symmetry is by no means obvious in the IMF description discussed in Section 6.6, but this is expected since part of it are Lorentz transformations broken by the IMF quantization. The interpretation of the exceptional groups E (9), d"7, 8 is not obvious either, since a consistent BB quantum description for these cases is lacking as well. In Sections 7.4}7.6, the precise identi"cation of the full U-duality groups for d"3, 4, 5 will be discussed in further detail. Note also that as we are considering M-theory compacti"ed on a torus From the point of view of type IIB theory, it can be shown that the latter also account for the restoration of the transverse Lorentz invariance [287].


193

Table 23 Interpretation of U-duality in matrix gauge theory D

d

U-duality

Origin

8 7 6 D45

3 4 5 d56

Sl(3, 9);Sl(2, 9) Sl(5, 9) SO(5, 5, 9) E (9) BB

S-duality;symmetry of ¹ Symmetry of ¹ of M5-brane T-duality symmetry on NS5-brane Unclear

times a light-like circle, it has been conjectured that the E (9) U-duality symmetry should be BB extended to E (9), as a consequence of Lorentz invariance. This extended U-duality B>B> symmetry will be discussed in Section 7.7. Finally, we can translate the U-duality invariant Newton constant (4.14) in the Matrix gauge theory language. The most convenient form is obtained by writing

< B>

204


Table 33 Flux, momentum and rank multiplets

D

d

U-duality E (9) B

Flux +m,

Mom. +n,

Rank +N,

Total +M,

10 9 8 7 6 5 4 3

1 2 3 4 5 6 7 8

1 Sl(2) Sl(3);Sl(2) Sl(5) SO(5, 5) E E E

1 3 (3, 2) 10 16 27 56 248

1 2 (3, 1) 5 10 27 133 3875

1 1 1 1 1 1#1 56#1#1#1 R

3 6 10 16 27 56 248 R

direction I on ¹B: R R . (7.34) J ' The action of this Weyl transformation leaves the other R 's and l invariant. In particular, ( N Newton's constant in 11!(d#1) dimensions < R B> N'0, and it seems to require a revision both of the interpretation of N as the rank of a gauge theory and of the relation between N and the light-cone momentum P>. Finally, let us comment in some more generality on the occurrence of this extended U-duality group. At least at low energies, the Matrix gauge theory describing the DLCQ of M-theory compacti"ed on ¹B is nothing but the gauge theory on the N Dd-brane wrapped on ¹B. The latter is certainly invariant under the T-duality SO(d, d, 9), and not only SO(d!1, d!1, 9) ( ) Sl(d). Its spectrum of excitations, or equivalently bound states, is therefore invariant under SO(d, d, 9), and very plausibly under the extended duality group E (9). On the other hand, we have B>B> expanded the bound-state mass in the limit where the N Dd-branes are much heavier than their bound partners, whereas T-duality can exchange the Dd-branes with some of their excitations. SO(d, d, 9) is therefore explicitly broken, and E (9) is broken to E (9). The invariance of B>B> BB the mass spectrum can be restored by using the full non-commutative Born}Infeld dynamics instead of its small a Yang}Mills limit [68]. While not relevant for M(atrix) theory anymore, interesting insights can certainly be obtained by studying these stringy gauge theories.

Acknowledgements The material presented in this review grew from lectures given by the authors on various occasions during the 1997}1998 academic year, including CERN-TH Workshop and journal club, Nordita and NBI, Tours meeting of the Working Group on integrable systems and string theory, Ecole Normale SupeH rieure and CEA SPhT seminars, Amsterdam Summer Workshop on String Theory and Black Holes, Hamburg Workshop on Conformal Field Theory of D-branes, Corfu Summer Institute on Elementary Particle Physics. We are very grateful to the organizers for invitation and support. It is our pleasure to thank E. Rabinovici for an enjoyable collaboration and early participation to this work. We are grateful to C. Bachas, I. Bars, D. Bernard, J. de Boer, E. Cremmer, E. Eyras, K. FoK rger, A. Giveon, F. Hacquebord, M. Halpern, C. Hofman, K. Hori, R. Iengo, B. Julia, E. Kiritsis, M. Krogh, D. Olive, S. Ramgoolam, H. Samtleben, C. Schweigert, K. Stelle, W. Taylor, P. di Vecchia, E. Verlinde and G. Zwart, for useful remarks or discussions. This work is supported in part by the EEC under the TMR contract ERBFMRX-CT96-0090 and ERBFMRX-CT96-0045. See Ref. [258] for a discussion of DLCQ with negative light-cone momentum.

206


Appendix A. BPS mass formulae Here, we analyse the BPS eigenvalue equation (2.14) for various choices of non-vanishing central charges. This gives a check on the mass formulae obtained on the basis of duality, and yields the conditions on the charges for a state to preserve a given fraction of supersymmetry. A.1. Gamma matrix theory In order to maintain manifest eleven-dimensional Lorentz invariance, we use the 11D Cli!ord algebra [C , C ]"2g , with signature (!,#,2), even after compacti"cation. The matrices + , +, C are then 32;32 real symmetric except for the charge conjugation matrix C"C , which is real + antisymmetric. All products of Gamma matrices are traceless except for C C 2C C "1 , (A.1) Q where we denote by s the eleventh direction. We de"ne C 2"C C 2 if the p indices M, N,2 + , +, are distinct, zero otherwise, and abbreviate it as C . We have N

(A.2) (C C )"(!1) N\ (C )"(!1) N , , N N where the p indices are non-zero and the square brackets denote the integer part. Furthermore, C C #(!)NOC C " N O O N

I N>O\IYN\O

C C !(!)NOC C " N O O N

C , N>O\I

I N>O\\IYN\O

C

N>O\\I

(A.3a) ,

(A.3b)

with no restrictions on the p#q indices. On the right-hand side of Eq. (A.3a) (resp. Eq. (A.3b)), a contraction between the "rst 2k (resp. 2k#1) indices of C and the "rst 2k (resp. 2k#1) indices N of C is implied. In particular, N [C , C ]"C , (A.4) N N since C generates Lorentz rotations. A.2. A general conxguration of KK-M2-M5 on ¹ Here we consider M-theory compacti"ed on ¹, and allow for non-vanishing central charges Z , ' Z ,Z , where the indices I, J,2 are internal indices on ¹. We therefore look for solutions to '( '()*+ the eigenvalue equation Ce"Me , C,Z C'#Z'(C #Z'()*+C . ' '( '()*+ Squaring this equation, we obtain Z Z +C , C ,!Z'(Z +C , C ,#Z'()*+Z,./01+C ,C , ' ( ' ( )* '( )* '()*+ ,./01 #2Z Z()[C', C ]#2Z Z()*+,+C', C ,!2Z'(Z)*+,.[C , C ]>M , ' () ' ()*+, '( )*+,.

(A.5)

(A.6)


207

where the symbol > denotes the equality when acting on e. Using the identities (A.3a), (A.3b), this reduces to (Z )#(Z'()#(Z'()*+)##Z Z'(C #(Z Z+'()*#Z'(Z)*)C >1 . ' ( ' + '()*

(A.7)

A 1/2-BPS state is obtained under the conditions k',Z Z'("0 , (

(A.8a)

k'()*,Z Z+'()*#Z'(Z)*"0 , +

(A.8b)

which indeed form a string multiplet 10 of E "SO(5, 5), and has a mass given by M"(Z )#(Z'()#(Z'()*+). '

(A.9)

k'()*/4!, C "C and rewrite If the conditions are not satis"ed, we can de"ne k "e ' '()*+ Eq. (A.7) as k'C #k C C' > M!M . ' '

(A.10)

Note that the SO(5, 5) vector (k , k') is null: k k'"0. Squaring again yields the 1/4-BPS state mass ' ' formula M"(Z )#(Z'()#(Z'()*+)#((k')#(k ) . ' '

(A.11)

This result can be straightforwardly made invariant under the full U-duality group by including the couplings to the gauge potentials through the lower charges as found for the particle and string multiplet in Eqs. (5.23) and (5.27). A.3. A general conxguration of D0, D2, D4-branes on ¹ We now consider the D-brane sector of M-theory on ¹, that is a general con"guration of D0, D2, D4-branes. The eigenvalue equation becomes Ce"Me ,

(A.12a)

C,ZC #ZGHC #ZGHIJC , Q GH GHIJQ

(A.12b)

where Z, ZGH, ZGHIJ denote the D0, D2, D4-brane charges, respectively, and i, j,2 run from 1 to 5. Squaring this equation, we obtain 2Z#ZGHZIJ+C , C ,#ZGHIJZKLNO+C C ,#4Z ZGHIJC GH IJ GHIJ KLNO GHIJ #2ZGHZIJKL[C ,C ]C > M . GH IJKL Q

(A.13)

Using identities (A.3a), (A.3b), this becomes Z#(ZGH)#(ZGHIJ)#kGHIJC #(k)GHIJC > M , GHIJ GHIJQ

(A.14)

208


where we de"ned kGHIJ,Z GHZIJ #ZZGHIJ ,

(A.15a)

kYGHIJ,ZK GZHIJ K .

(A.15b)

The second combination can be rewritten on ¹ as a form kG_HIJKL"ZG HZIJKL . Then, k and k_ can be dualized into a 10 null vector (k , kG) of the T-duality group SO(5, 5). A state with k"k"0 is G 1/2-BPS with mass M"(Z)#(ZGH)#(ZGHIJ) .

(A.16)

If these conditions are not met, we can rewrite Eq. (A.14) as k CGC #kCGC C > M!M , G G Q

(A.17)

implying a mass formula M"(Z)#(ZGH)#(ZGHIJ)#2((kG)#(k ) G

(A.18)

or, in terms of the natural undualized charges, M"(Z)#(ZGH)#(ZGHIJ)#2((kGHIJ)#(kG_HIJKL) .

(A.19)

A.4. A general conxguration of KK}w}NS5 on ¹ Finally, we consider the Neveu}Schwarz sector of the theory considered in the Appendix A.3, namely the bound states of NS5-branes, winding and Kaluza}Klein states. The eigenvalue equation then reads (z CG#zGC #zGHIJKC )e"Me . G QG GHIJK

(A.20)

Taking the square gives z#(zG)#(z )#2zzGC #2zzGC !2C zGz > M , G G QG Q G

(A.21)

so the 1/2-BPS conditions appear to be zzG"zz "zGz "0 . G G

(A.22)

This agrees with the vanishing of the entropy zzGz and its "rst derivatives, as obtained in Ref. [113]. G We can go further and "nd the complete 1/8-BPS mass formula: multiply Eq. (A.20) by zC : !zz C !zzGC !ze > zMC G G QG

(A.23)

and combine with Eq. (A.21) to obtain: (!z#(zG)#(z )#2zMC !2zGz C ) > M . G G Q

(A.24)


209

Now C and C commute, are traceless and square to 1, so this is a second-order equation: Q !z#(zG)#(z )$2zM$2zGz > M , (A.25) G G with solutions M"$z$((z $zG) G or, equivalently:

(A.26)

(A.27) M"z#(z )#(zG)#2"z zG"#2"z"((z )#(zG)#2"z zG" . G G G G This reduces to the usual mass formula for perturbative string states (z"0) and for KK}NS5 or w}NS5 bound states. For momentum and winding charges along a single direction, this reduces to M"$z$z $z, in agreement with the identi"cation of central charges in Ref. [113]. The U-duality invariant generalization of this mass formula is however unclear.

Appendix B. The d"8 string / momentum multiplet For completeness, we give in Table 34, the content of the string/momentum multiplet for d"8 in the 3875 of E . It comprises the 2160 states in the Weyl orbit of the highest weight R /l of length G N 4, together with 7 copies of the 240 weights of length 2 with tension < T" 0;(d"8 particle multiplet) , l N as well as 35 zero weights with tension

(B.1)

< 0 . (B.2) l N As in d"7, the resulting multiplet exhibits a mirror symmetry, which relates each state with tension R?\/l?, a"1,2, 6 to another state with tension R\?/l\? through the relation N N T"

< 0 , (B.3) l N where < is the volume of the eight-torus. For this reason, Table 34 only gives the explicit form of 0 the tensions for the lower half a"1,2, 5 and the self-mirror part a"6. The second column gives the Sl(8) irreps at each level graded by 1/l?, while the last column lists the corresponding charges. N Here the notation is as follows: a semicolon denotes an ordinary tensor product as before (so in general contains more than one Sl(8) irrep); two superscripts (p; q) grouped within parentheses and separated by a semicolon denote the irrep, whose Young tableau is formed by juxtaposition of a column with p rows and one with q rows. As an aid to the reader, we give the charges of the dual states at level l\?: N a"1: n___, a"2: n___, a"3: n___ , (B.4) MM"

a"4: n___, n___, a"5: n___, n___ .

210


Table 34 String/momentum multiplet 3875 of E Mass

Sl(8) irrep

Charge

R ' l N RRR R ' ( ) * l N RR R R R R < ' ( ) * + ,, 7 0 l R l N 'N RRR R R R R ' ( ) * + , . l N < R < R R 0 ', 7 0 ' ( l l N N RRR RR R R ' ( ) * + , . l N < RR R R < R R R R R 0 ' ( ) *, 7 0 ' ( ) * + l l N N < RRR R R R < RR R R R R R is enhanced to a Sl(2, 1)/;(1) factor in four dimensions, where the Kalb-Ramond two-form B can be dualized into an axionic "eld b and combined with the IJ four-dimensional coupling into a modular parameter S"b#i< /g l . Accordingly, there is && a non-perturbative symmetry SP!1/S [327,328] which transforms the heterotic string length and coupling (for b"0) as M\? where the extra e M\? term comes from the spherical Bessel functions (e.g., terms such as h /(h ) ). Hence, the "elds (F) can be written as ? F"F(aligned)#F(dipole)#F(quadrupole) , (85)

The analysis here is largely repeating Section 5.2 of Ref. [4], except for a few typos in Ref. [4]: Eq. (16d) should be multiplied by a factor of 2 and the multiplication factors (before and after Eq. (25)) di!er for B and E by a factor of a (cf. our Eq. (88)).

F.C. Michel, H. Li / Physics Reports 318 (1999) 227}297

257

where F(aligned) is given as a B "2B cos m cos h , P r a cos m sin h , B "B F r B "0 , ( a E "E cos m(1!3 cosh) , P r a E "!E cos m sin 2h , F r E "0 , (

(86)

F(dipole) is h /o sin m sin h e ( , B "2B P (h /o) ?

h o h # sin m cos h e ( , B "B F h o ? o h h # i sin m e ( , B " ( h o ? E "0 , P h sin m e ( , E "!E F (h ) ? h i sin m cos h e ( E "!E ( (h ) ? and F(quadrupole) is B "0 , P

o B "B h sin m cos h e ( , F oh #h ? o h cos 2h i sin m e ( , B "B ( oh #h ?

(87)

258


h o sin m sin 2h e ( , E "E 3 P oh #h o ?

o oh #h cos 2h sin m e ( , E "E F oh #h o ? o oh #h i sin m cos h e ( . E "E ( oh #h o ?

(88)

Finally, adding the electrostatic monopole discussed above, we can recast Deutsch's solutions Eqs. (82) and (83) as, after taking real parts, a +cos m cos h#sin m sin h[d cos t#d sin t], , B "2B P r a +cos m sin h!sin m cos h[(q #d )cos t#(q #d )sin t], , B "B F r a sin m+![q cos 2h#d ]cos t#[q cos 2h#d ]sin t, , B "B ( r

(89)

a 3 a 2 cos m# cos m(1!3 cos h)! sin m sin 2h[q cos t#q sin t] , E "E P r 3 r o

a a ! cos m sin 2h#sin m[(q cos 2h!d )cos t#(q cos 2h!d )sin t] , E "E F r r a sin m cos h+(q !d )cos t!(q !d )sin t, , E "E ( r where t" #o!a (again, the extra term o!a comes from the Bessel functions) and ao#1 , d " a#1 o!a d " , a#1 1#ao!o , d " a#1

A subroutine on solving these "elds around a neutron star is available by sending an email to [email protected].

(90)


259

(o!1)a#o d " , a#1 3o(6a!a)#(3!o)(6a!3a) , q " a!3a#36 (3!o)(a!6a)#3o(6a!3a) q " , a!3a#36 (o!6o)(a!6a)#(6!3o)(6a!3a) q " , o(a!3a#36) (6!3o)(a!6a)#(6o!o)(6a!3a) , q " o(a!3a#36)

(91)

where all the d and q terms come from F(dipole) and F(quadrupole), respectively. For those who G G prefer Gaussian units over SI units, change E to E /c in all the above equations. We will now discuss these "elds in near (o&a) and far (oa

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