Integrable Quantum Field Theories and Their Applications

INTEGRABLE QUANTUM FIELD THEORIES AND THEIR APPLICATIONS Proceedings of the APCTP Winter School

ed b,

tiangrim Ahn haiho Rim r u Sasaki

INTEGRABLE QUANTUM FIELD THEORIES AND THEIR APPLICATIONS

INTEGRABLE QUANTUM FIELD THEORIES AND THEIR APPLICATIONS Proceedings of the APCTP Winter School Cheju Island, Korea 28 February - 4 March 2000

Edited by

Changrim Ahn Ewha Womans University, Korea

Chaiho Rim Chonbuk National University, Korea

Ryu Sasaki Kyoto University, Japan

V f e World Scientific V I

New Jersey 'London •Singapore Singapore ••Hong Kong

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V

Preface The APCTP Winter School on Integrable Quantum Field Theories and Applications was held in Cheju Island, KOREA, 28 February - 4 March 2000. The School was to promote the quantum field theories of integrable models and to propagate the recent developments of the non-perturbative results and new techniques to various background scholars. The research of integrable models had been limited to a few groups of scholars around the world and been mostly considered as a branch of mathematical physics of their own, not much related to the real physics world. This view, however, turns out to be too narrow. The non-perturbative approach has recently opened a new horizon to strongly correlated systems in low dimensional physics and provided invaluable tools otherwise nowhere to analyze analytically. This trend will continue as non-perturbative quantum phenomena are enhanced in low dimensional physics and the importance of the research will be emphasized in physics community. During the School, lectures were made on the fundamentals and elementary techniques of the integrable field theories and also on their applications to low dimensional physics system. New results of recent years are also presented in seminars. Through the lectures and seminars the School was arranged to increase the opportunity of communication among participant scholars ranging from researchers of many years to graduate students, and from high energy physicists to condensed matter physicists. In addition, the natural atmosphere of the Cheju Island was an invaluable asset to the informal discussion, being known for her beautiful landscape and mild temperature during winter season. Organizers of this winter school thank for lecturers and participants from all over the world and especially for Asia Pacific Center for Theoretical Physics (APCTP) for her generous financial support as well as for the time consuming secretarial works. Please notice that the contributions to this proceedings are arranged in alphabetic order of the first authors.

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Contents Preface

v

Applications of Reflection Amplitudes in Toda-Type Theories C. Ahn, C. Kim and C. Rim

1

Lax Pairs and Involutive Hamiltonians for C# and BCJV Ruijsenaars-Schneider Models Kai Chen, B.-Y. Hou and W.-L. Yang

35

Fateev's Models and Their Applications D. Controzzi and A. M. Tsvelik

55

The ODE/IM Correspondence P. Dorey, C. Dunning and R. Tateo

74

Integrable Sigma Models P. Fendley

108

Lorentz Lattice Gases and Spin Chains M. J. Martins

179

Quantum Calogero-Moser Models for Any Root System R. Sasaki

195

Quasi-Particles in Conformal Field Theories for Fractional Quantum Hall Systems K. Schoutens and R. A. J. van Elburg Towards Form Factors in Finite Volume F. A. Smirnov Static and Dynamic Properties of Trapped Bose-Einstein Condensates T. Tsurumi, H. Morise and M. Wadati

241

243

269

viii

Integrability of the Calogero Model: Conserved Quantities, the Classical General Solution and the Quantum Orthogonal Basis H. Ujino, A. Nishino and M. Wadati Conformal Boundary Conditions J.-B. Zuber

289

318

1

APPLICATIONS OF R E F L E C T I O N A M P L I T U D E S IN T O D A - T Y P E THEORIES CHANGRIM AHN Department

of Physics, Ewha Seoul 120-750,

Womans Korea

Department

of Physics, Seoul National Seoul 151-742

University,

CHANJU KIM University,

CHAIHO RIM Department

of Physics, Chonbuk Chonju 561-756,

National Korea

University,

Reflection amplitudes are defined as two-point functions of certain class of conformal field theories where primary fields are given by vertex operators with real couplings. Among these, we consider (Super-)Liouville theory and simply and nonsimply laced Toda theories. In this review we explain how to compute the scaling functions of effective central charge for the models perturbed by some primary fields which maintains integrability. This new derivation of the scaling functions are compared with the results from conventional TBA approach and confirms our approach along with other non-perturbative results such as exact expressions of the on-shell masses in terms of the parameters in the action, exact free energies. Another important application of the reflection amplitudes is a computation of onepoint functions for the integrable models. Introducing functional relations between the one-point functions in terms of the reflection amplitudes, we obtain explicit expressions for simply-laced and non-simply-laced affine Toda theories. These nonperturbative results are confirmed numerically by comparing the free energies from the scaling functions with exact expressions we obtain from the one-point functions.

1

Introduction

There is a large class of 2D quantum field theories (QFTs) which can be considered as perturbed conformal field theories (CFTs) 1. These theories are completely defined if one specifies its CFT data and the relevant operator which plays the role of perturbation. The CFT data contain explicit information about ultraviolet (UV) asymptotics of the field theory while its long distance property is the subject of analysis. If a perturbed CFT contains only massive particles, it is equivalent to the relativistic scattering theory and is completely defined by specifying the S-matrix. Contrary to CFT data the S-matrix data exibit some information about long distance properties of the theory in an explicit way, while the UV asymptotics have to be derived.

2

A link between these two kinds of data would provide a good view point for understanding the general structure of 2D QFTs. In general this problem does not look tractable. Whereas the CFT data can be specified in a relatively simple way, the general S-matrix is very complicated object even in 2D. However, there exists an important class of 2D QFTs (integrable theories) where scattering theory is factorized and S-matrix can be described in great details. In this case one can apply the nonperturbative methods based on the Smatrix data. One of these methods is thermodynamic Bethe ansatz (TBA) 2 3 ' . It gives the possibility to calculate the ground state energy E(R) (or effective central charge ceff(i?)) for the system on the circle of size R. At small R the UV asymptotics of cesf(R) can be compared with that following from the CFT data. Usually the UV asymptotics for the effective central charge can be derived from the conformal perturbation theory. In this case the corrections to ceff(0) = CCFT have a form of series in R1 where 7 is defined by the dimension of perturbing operator. However, there is an important class of QFTs where the UV asymptotics of ceg(R) is mainly determined by the zero-mode dynamics (see for example 4'?-?>?). In this case the UV corrections to CCFT have the form of series in inverse powers of log(l/i?). This UV expansion is also encoded in CFT data 6 . The simplest integrable QFT with the logarithmic expansion for the effective central charge is the sinh-Gordon (ShG) model, which is an integrable deformation of Liouville conformal field theory (LFT). It was shown in paper 6 that the crucial role in the description of the zero-mode dynamics in the ShG model is played by the "reflection amplitude" of the LFT, which determine the asymptotics of the ground state wave function in this theory. (The reflection amplitudes in CFT define the linear transformations between different exponential fields, corresponding to the same primary field of chiral algebra.) In this paper, we want to show that these zero-modes dynamics are quite general features of 2D QFTs with exponential interactions. Imposing the integrability, we will consider the Bullough-Dodd model which is another integrable perturbation of the LFT, supersymmetric ShG models 7 , and affine Toda field theories (ATFTs) associated with both simply laced and non-simply laced Lie algebras 8 ' ? . Each model shows quite unique properties in the Smatrix data; some has non-diagonal S-matrix, some has many particles with different masses, and so on. In this review of our recent works 7'8>9>10; we want to describe UV asymptotic behaviours of these 'Toda-type' models, i.e. a class of the integrable QFTs with exponential interactions, in unifying way in terms of the zero-mode dynamics. The perturbing term in the model re-

3

stricts the zero-mode dynamics to a box of size / ~ log(l/i?) in the auxiliary space with dimension equal to the number of independent zero-modes. This leads to the quantization condition for the momentum P conjugated to the zero-modes and the solution P{R) determines all logarithmic terms in the UV asymptotics of the effective central charge ceff(.R). Although it may seem quite model-dependent, it turns out that the basic dynamics in common is the reflection of a zero-mode off the interacting potential of the LFT. In all cases the results agree perfectly with TBA results based on the 5-matrix data. The remarkable feature is that effective central charge calculated from the CFT data with subtracted bulk free energy term (like in TBA approach) gives a good agreement with the TBA results even outside the UV region (at R ~ 0(1)). This "empirical" fact still needs the explanation. Finally the reflection amplitudes can be used n ' 1 2 > 1 3 to find the exact onepoint functions of this class of integrable models. One needs only symmetric properties of the given Lagrangian and analytic properties of the one-point function. Explicit calculation is given for various models 6 '->-' 13 '---'-'-. In the section 2, we introduce the reflection amplitudes for the CFTs we are interested in. These amplitudes are interpreted as quantum mechanical reflections off the potential wall of the zero-modes dynamics in sect.3. In sect.4 we analyze the off-critical integrable models. Due to the integrable perturbations, the zero-modes are confined in the potential well and the conjugate momenta are quantized. Using this, we calculate the UV asymptotics for the effective central charges. In sect.5 we compare this asymptotics with numerical solutions of TBA equations. We derive the exact one-point functions and free energies in sect.6. 2

Normalization Factors and Reflection Amplitudes

In this section, we introduce the reflection amplitudes for the Toda-type CFTs with background charges. 2.1

Toda-type CFTs

We start with non-affine Toda theories (NATTs) whose actions are given by

A = jd2x i - ( ^ ) 2 + ^ Mi e" e -^

(1)

where e^, i = 1 , . . . , r are the simple roots of the Lie algebra G of rank r. For simply laced algebras, the /Vs are all the same as \i. Non-simply laced ATFTs have standard simple roots with e? = 2 and nonstandard simple roots with

4

e 2 = £ 2 (^ 2). We choose the corresponding parameters /i; as \i (for standard roots) and \J! (for nonstandard ones) respectively". The simplest case is the A\ NATT, or the LFT with an action A=fd2x

l.(d^)2

+

fie^f

(2)

With appropriate background charges, these Toda-type QFTs are the CFTs. To describe the generator of conformal symmetry we introduce the complex coordinates z = X\ + 1x2 and ~z = X\ — 1x2 and vector:

ct = bp + \p\

PV = ^ E « V '

P= ^ £ ° ' 0

(3)

a>0 v

where the sum in definition of Weyl vector p (p ) runs over all positive roots a (co-roots a v ) of G. The holomorphic stress-energy tensor T{z) = -\{dz8^) + inb2ipipeb* + ^ A e 2 6 * .

(5)

2

The super-LFT with the background charge Q = b+l/b has the central charge cSL = 3/2(1 + 2Q 2 ). 2.2

Reflection

Amplitudes

NATTs Besides the conformal invariance the NATT possesses extended symmetry generated by W(G)-algebra. The full chiral W/(G)-algebra contains r holomorphic fields Wj(z) (W2(z) = T(z)) with spins j which follows the exponents of Lie algebra G. The primary fields $„, of W(G) algebra are classified by r a

W e choose the convention that the length squared of the long roots are four for C). two for the other untwisted algebras.

and

5

eigenvalues Wj, j = 1 , . . . , r of the operator Wj j0 (the zeroth Fourier component of the current Wj(z)): WJi0$w = Wj$w,

Wjtn$w

= 0,

n > 0.

(6)

The exponential fields Va(x) = e(Q+«)-v(*)

(7)

are spinless conformal primary fields with dimensions A(a) = W2(a) = (Q 2 — a 2 ) / 2 . The fields V^ are also primary with respect to all chiral algebra W(G) with the eigenvalues Wj depending on a. The functions Wj(a), which define the representation of H / (G)-algebra possess the symmetry with respect to the Weyl group W of Lie algebra G 1 6 '', i.e. Wj(sa) = Wj(a); for any s € W. It means that the fields Vsa for different s G W are reflection images of each other and are related by the linear transformation: Va(x) = Rs(a)VBa(x)

(8)

where R§{a) is the "reflection amplitude". This function is an important object in CFT and plays a crucial role in the calculation of the one-point functions in perturbed CFT 19 . To calculate the function Rg(a) for simply laced NATTs, we introduce the fields $w: $w(x)=N-1(a)Va(x)

(9)

where normalization factor N(a) is chosen in the way that field $w satisfies the conformal normalization condition = i — ^ (io) 1^ — 2/1 The normalized fields $w are invariant under reflection transformations and hence;

«• = I S , -

0 and oscillators satisfy V

=

% 0 ~ o ^ ~ '

Tft [am,0.n] = -W&m+m [am,an]

l acpo

2

TTX = —Sm+n.

(29)

2

T h e Virasoro generators can be written in terms of these modes. T h e space of states is now represented as A0 = £ 2 ( - o o Q < o o ) ® T

(30)

where £2 is the two-dimensional phase space spanned by 0 —> —00 asymptotic limit. In particular, the wave functional for the primary state vp corresponds t o * » P [ 0 ( S I ) ] = ( e i P 0 ° + S(P)e-iP*°)

|0)

as (h0 -»• - 0 0

(31)

where S(P) is the reflection coefficient of the asymptotic wave functional. One can check t h a t the wave functional of asymptotic form Eq.(31) has correct conformal dimension by acting LQ. T h e coefficient S(P) should be the reflection amplitude SL(P) introduced earlier since the wave functional ^y_P for the primary state V-p is SL(—P)^VP along with SL(P)SL(-P) = 1.

(32)

In this framework, one can check the validity of the reflection amplitude by taking semiclassical limit b —> 0 and using duality. Since P is of the order of 0(b), one can neglect the oscillators and keep only the zero-mode • 0. Since P is small of order of 0(b), one can neglect the oscillator part in Eq.(44) and study only the dynamics of zero-modes. In the (NS) sector, only bosonic zero-mode appears so that the Hamiltonian becomes ff

™=-!-(&)'+"v*v"''

which is essentially the same as that of the LFT, hence the reflection amplitude becomes SNS(P)

(^VfPT(l

= - (TJ

+ iP/b)

T{1_iP/by

On the other hand, in the (R) sector, additional fermionic zero-mode is introduced in the hamiltonian by 22 H* = - i ^ - \

+ T r V V e 2 ^ + 27ri A i&Vo^e i *°.

Since the fermionic zero-mode satisfies {^o,^o} = 0,

^ 0 = ^ 0 = 2'

13

we can represent it by 1

-

1

-

i

TT/X&V^S

= (H+ £ V

and the Hamiltonian becomes

H* = V2+ TrVVe26^ The solution of H+ can be obtained as

g>+(0o) = fy^[Ki/2-iP/b{xHK1/2+iP/b(x)]

\

x =

w

^

0

where Kv(x) is the modified Bessel function. By taking the asymptotic limit 0o - • - c o , one can find the non-vanishing component is given by $ ~ eiP0° +

SR{P)e~iP4'0

with

c (p,(^Y^pn\

+ iPlb)

- \T)

Vi - iP/b) ra-iP/bY

SR{P)

These are consistent with the exact result Eq.(25) in the b —> 0 limit. 4

Quantization Conditions and Scaling Functions

In this section we derive the scaling functions for the various Toda-type models on a cylinder with circumference R. In the deep UV region R -» 0, the wave functional interpretation introduced in the previous section is used to obtain the quantization condition for the zero-mode momenta and the vacuum energies. 4-1

ShG model

We start by reviewing the analysis of 6 for the ShG model or A\ ATFT defined first on a circle of circumference R with periodic boundary condition. By rescaling the size to 2ir, one can write the action as /•27T

/

i

/ p\

2+26

(45) 2

2

dx2j dx! ^ ( ^ ) + / W | H (e^+e" ^) where fi ~ [mass]2"1"26 is the dimensional coupling constant with b the coupling constant.

14

We are interested in the ground state energy E(R) or, more conveniently, the finite-size effective central charge Ceft{R) = -—E(R)

(46)

7T

in the ultraviolet limit R —• 0. Since we are interested in the ground-state energy, only the zero-mode contribution counts. So the corresponding effective central charge at R —> 0 is determined mainly by P ceS(R) = 1 - 24P 2 + 0(R)

(47)

up to power corrections in R. For the ground state energy, one can consider only the zero-mode dynamics where the wave functional of o is confined in the potential well due to the ShG interaction term. The ShG potential introduces a quantization condition for the momentum P which depends on the finite size R. As R —>• 0, in particular, the wave functional is confined in the potential well where the potential vanishes in the most of the region and becomes nontrivial only at 2b(f>0 ~ ±ln/u(i?/27r) 2+26 near the left and right edges. Near these edges of the potential well, the potential becomes that of the LFT and the wave functional will be reflected with the reflection amplitude of the LFT introduced earlier. Therefore, the quantization condition is given by (R/2n)-4y/*iPQ

Sl(P)

= 1.

(48)

In terms of the reflection phase 6L (P) defined by SL(P) = -ei5^p\

(49)

the ground state momentum is qunatized as «5i(P) = 7r + 2 ^ P Q l n ^ .

(50)

Z7T

Thus determined quantized momentum will give the scaling function ceg(R) in the UV region by Eq.(47). To see this explicitly, one can expand the reflection phase in the odd powers of P, SL(P)

= S1(b)P + S3(b)P3 + S5(b)P5 + ...

(51)

where the coefficients can be obtained from the reflection amplitude Eq. (23) as follows: SJb) = hnb2 o

r ( 2 w ) r ( 1 + 2HW) -2Qln — —r^

+7E

(52)

15

s3(b)^-am3 h(b)

+ b'

(53)

:((5)(b5+b-5)

(54)

with Euler constant 7E- Now solving Eq.(51) iteratively, we get CeS(R)

= l+C^ + ^

C +

± + ..

(55)

where I = Si{b) d

-2V2Qln(R/2n) 2

= -6TT ,

C2 = 12TT%(&),

6

c3 = 12ir 65(b).

(56) (57)

The Gamma functions appear in 0

with / = L - 2jE(bhv

+ h/b) = L - L0

(72)

The above equation can be solved iteratively in powers of \/l. Inserting the solution into Eq.(69), we find: Ceff

=

r

- r(h + l)/i v (?fj

+ ^C(3)[C 4 (G V )6 3 + C4(G)/fo3]

- ^ C ( 5 ) [ C 6 ( G V ) 6 5 + C 6 (G)/6 5 ] ( y )

+ 0(r»),

(-^ (73)

19 where the coefficients C(G) are defined as: C 4 (G v ) = ^ p l ,

C4(G) = £ p Q / £ v ,

C6(G^=J2p6a-

C6(G) = Y,P~Pl*, a>0

(74)

a>0

For simply laced algebras, these coefficients have the values:

™ ^nV-l)(2n2-3), -i-n2(n2-l)(n2-2)(3n2-5), 168 {l

3 ">\ = —(16n - 45n 2 + 27n + 8)n(n - l)(2ra - 1), 30 v 1 C6(£»{11)) = — (48n5 - 213n4 + 262n3 + 6n 2 - lOln - 32)n(n - l)(2n - 1 42

j(D For the non-simply laced algebras £?„ and C«( i ) , we can express the results through these values. Namely, we find:

CiicP)

= CiiD^),

For exceptional algebras G2

Ci(W) and F\

- jI C Ci{G22 ( l )) ^— ^ 44(D. ^ 4Vh

= Ci(D™n),

(i = 4,6). (75)

, we obtain:

392,

C6(G 6/2 for SL(P) given in Eq.(23) and Q = b + l/b,

Q' = 6/2 + 2/6.

(79)

Using the phase shifts defined as -ei5^p\

SL(P) =

S'L(P) ==

-ei6 AP)

the quantization condition becomes 6(P) = TT + 4QP\n^-

(80)

where 6(P) = \(SL(P)

+ 8'L(P)),

Q = \{Q + Q').

The relation between P and R in Eq.(80) gives the scaling function ceff as a continuous function of R, Eq.(73), with Q replaced by Q and J's with S's

21

defined by power series expansion of the phase shift in P 5(P) = 61P + 63P3 + 55P5 + ... S1

^lnj-2{Q

=

x


(82)

where l = d

(83)

51(b)-4Qla(R/2w)

= -24TT

2

c2 = 48TT 4 4(6)

c 3 = 4 8 * % (6). 4-4

the SShG model

Now we consider an integrable model obtained as a perturbation of the superLFT, the SShG model. By rescaling the size to 2ir, one can express the action of the SShG model by A. SShG

/ d%2 I

dx\

- ^ ( 3 « 0 ) 2 - - ^ W + lW)

R \ 1+b2 ( B \ 2+2b2 2 2 + 2inb ( — J i>i> cosh(&) + -Kfi b I — \ 2

[cosh(26(A) - 1] (84)

In the UV limit, the exponential potential becomes negligible except in the region where o goes to ±oo. This means that the SShG model is effectively described by the super-LFT as R —> 0. From the ground state energy for the

22

primary state labelled by P , the effective central charge can be obtained by CeffCR) = ^-12P2+0{R) 2

= -12P

(NS)

+ 0{R)

(R).

(85)

For the (NS) sector, P corresponding to the ground state is determined again by the quantization condition coming from the super-LFT reflection amplitudes: SNS(P)=n

+ 2QP\n^,

(86)

where (5NS (-P) is the phase factor of (NS) reflection amplitudes. This quantization condition can be solved iteratively by expanding 0 so that the quantum number n should be 1 as in Eq.(86), the wave functional for the (R) sector becomes constant corresponding to n = 0. Therefore, the quantization condition becomes SR(P) = 2QPIn ~ .

(88)

Z7T

Obvious solution is P = 0 so that ceS(R) = 0 + O(R).

(89)

In the & —»• 0 limit, one can verify this from the (R) sector zero-mode dynamics of the SShG model which is governed by the Hamiltonain H^ = - ( -— I +

4TT 2 /X 2 & 2

s i n h 2 b0 cosh b0.

(90)

23

This is a typical supersymmetric quantum mechanics problem and in general there exists a zero-energy ground-state 23 if the supersymmetry is not broken. Explicitly, the wavefunction of the state is found to be / „ — 2TTH cosh b<j>o \

*o(o)=(^

J-

0

(91)

This state is normalizable and its energy is exactly zero. Thus at least in 6 —» 0 limit, ceff is exactly zero regardless of r without any power correction. 5

Comparison with the T B A results

A standard approach to study the scaling behaviour of integrable QFTs is to solve the TBA equations. In this section we compute the scaling functions in the UV region from the TBA equations and compare them with the results in the previous section based on the reflection amplitudes. 5.1

TBAs of ATFTs and BD model

The TBA equations for the ATFTs are given by (i = 1, • • •, r) C

(TBA) eff

(R) = J 2 ^

1

/ c o s h 9 l o g ( l + e - ^ ' f i > ) dQ.

(92)

where m^'s are particles masses and functions Ci(6,R) (i = l , - - - , r ) satisfy the system of r coupled integral equations: TTHRcosh9 = 6i(8, R) + Y,

f V>ij{0 ~ e')

lo

S 0 + e- £ ' ( e '' f l ) ) ^ ,

(93)

with the kernels ifiij, equal to the logarithmic derivatives of the 5-matrices Sij(6) Of ATFTS 24,25,26,27^

Equation (93) becomes the TBA equation of BD model when r = 1 and the kernel is given by b2 f{d) = *2/ 3 (0) + *-B/ 3 (0) + *(B-2)/ 3 (0).

With

B =

1 +

.

where ,

...

4sin7rxcosh# cos 2TTX - cosh 29

,_,

24

The function E(TBA"> (R) defined from the TBA equations differs from the ground state energy E(R) of the system on the circle of size R by the bulk term: E^TBA\R) = E(R) - fR, where / is a specific bulk free energy 3 . To compare the same functions we should subtract this term from the function E(R) defined by Eq.(73) i.e.

&BA,(*)=c£A)(*) + —/(G).

05)

7T

The specific bulk free energy f(G) can be calculated by Bethe Ansatz method. For ATFTs 28,29,9,10^ m2sin(7i7/i)

-/^ /(G)

m S[n{i /H) f

l

^

flr) J { )

^

= 8Sin(,B/h)Sin(!(l-B)/h)

:

,^^

° =A D E

'

„

G = B?\C?\ J

T

8sin(7rS/H) sin(7r(l - B)/H) '

m 2 cos(7r(l/3-l/g))

_

S6neS

T

'

_

16cos(jr/6)sin(7rB/ff)sin(7r(l-B)/H)'

(1) u

'

(1)

2 > ^4 , l»°;

where

^-rr^'

ff

-TT62-

(97)

For the BD model 19 , f = —r • 08) 16-v/3sin(7rB/6)sin(7r(2-B)/6 The contribution of bulk term / ( G ) becomes quite essential at R ~ 0(1). The TBA equations (93) are solved numerically for various algebras. The effective central charge Cgff (R) is then computed from Eq.(92) for many different values of parameter TnR. After taking into account the bulk term, the numerical solution for Cgff (R) is fitted with the expansions (73) in l/l considering the coeffiecients as fitting parameters. To compare the numerical TBA results with analytical ones from reflection amplitudes, we need to know the exact relations between parameters Hi of the action and masses of particles m,. This is because TBA equations are derived from S-matrix data while the method of reflection amplitudes deals with the paramters of the action directly. These "mass-//" relations 18>29>9 are given in the Appendix. With the help of them, we can express ceff(R) obtained in the previous section purely in terms of particles masses m^. For

25

example, the function L(R) of ATFTs defined in Eq.(67) becomes -T(h + b

b2hv)\n ? M G ) r

(i_£)r(1 + |

-ln(6 2 "(C 2 /2) z ).

(99) Similarly we can rewrite fi and / / in (81) in terms of the particle mass m as, 6 b2 S1 = ^ln--2(Q

+ Q') In

-

r-/,^

x

2V5r(|)

- + 7E

(100)

Comparing the results, we found that, up to the order l// 7 , the numerical TBA results are in excellent agreement with the analytic results given in previous sections. (For the details of the comparison, see 8 and 9 .) To see the agreement more concretely, we plot the functions ceff (R) and ce({ (R) for non-simply laced ATFTs setting B — 0.5. The first function is computed numerically from TBA equations. The second one is calculated using Eqs.(66) and (69), based on the reflection amplitudes, with taking into account the bulk free energy term according to Eq.(95). Figure 2 shows that, for all models, the two curves are almost identical without essential difference in the graphs even at R ~ (9(1). This good agreement outside the UV region looks not to be accidental. However, at present, we have no satisfactory explanation of this interesting phenomena in ATFTs. 5.2

TBA of SShG model

Finally we briefly describe the TBA analysis of SShG model (see 7 for details). There are two sectors in SShG model and the corresponding TBA equations are different from each other. They can be written as 7 . ceff(r) = ^

fcosh9\n(l±e-ei^de

(101)

where the pseudo-energies are the solution of the equations, —-(0-6l')ln[l±e-£2(o'>], rift'

/

|%(0-0')m[l±e-^'>].

(102)

In the above equations, the plus (minus) sign corresponds to the NS (R) sector and the kernel is given by tp(B) = - * B ( 0 )

with B = ^ 5 1 ,

26

0.01

0.001

0.1

1

R

o

i

0.001

• —

• —

0.01

0.1

•

•

' • • '

1

R

Figure 2. (a) Plot of c eff for A 2 , A 3 , A 4 , D4, A 6 , E6 ATFTs and BD model at B = 0.5. (b) Plot of c0

One may simply prove that [Rl] and [R2] are satisfied. Requirement [R3] is checked by confirming the result for the Weyl reflection Si with respect to any simple root e^ : G(Q + iPi) = SL{eil,P)

G{Q + isiP)

where Pi = P • e^. Here are two ingredients to be checked, a2 and which under the reflection, result in (Q + i P ) 2 - (Q + isiP)2 = 2i(b + i ) P i = 2iQPi o

(115) I(a,t),

(116)

30

and I(a,t)

= / ( Q + iP) - 7(Q + iSiP)

(117)

2

2

= - (cosh[2biPit + h(l + b )t] - cosh[-26iP^ + h{\ + b )i\) = smh{2biPit) sinh/i(l + b2)t. Combining this two reflection property, G(a) satisfies the Eq. (115). The symmetry operation r in the the requirement [R4] is the Dinkyn diagram symmetry of ADE series. Obviously a2 is invariant under r. One needs to prove that I{a,t) is invariant. Let us rearrange I(a,t) in Eq. (113) as I(a,t)

= -[J{a,t)-J(ai=0,t)]

(118)

where J(a, t) = ^2

cosh

i2b(a -a-Q-a)t

+ h(l + b2)t)\,.

(119)

a>0

The problem reduces to prove J(a, t) invariant under r. Under r, if a positive root goes to another positive root, this only reshuffles the terms in J(a,t). However, there are cases where a postive root goes to a negative root, —/3 = TOL. Then apart from the reshuffling, J(a,t) contains the term, cosh (2ba • (-/3)f - 26Q • at + h{\ + b2)t)

(120)

This is the case when r changes a simple root aT to zeroth root eo : raT = eo. Noting that in ADE series, any postive root can contain at most one such a root a r , and the root satisfies an identity, p- {ex - ret) = p • (aT - e 0 ) = h.

(121)

From this one can find a unique positive root a satisfying 2foQ • a - h(l + b2) = - 2 6 Q • 0 + h{l + b2) and therefore, J(a,t) 6.3

(122)

is invariant under r.

One-point function in nonsimply-laced

ATFTs

One-point function of non-simply laced ATFTs can be obtained in the same way as follows 9 '': G{a) = K(a) x exp f — ( a 2 e " 2 t - ^ ( a , *))

(123)

31

^here ,9i

,2

(a.*)(

9(P-x)\, J

(24) Here, the condition A 9 ft = p. — v means that the summation is over roots ft such that for 3u £ A p-v

= ft £ A.

So does for A 9 ft = —/i + v. 3.1

Cn model

The set of Cn roots consists of two parts, long roots and short roots: AC„=ALUAS,

(25)

in which the roots are conveniently expressed in terms of an orthonormal basis of Rn: AL = {±2ej: As = {±ej±ek,:

j = l,...,n}, j , k = 1,... ,n}.

In the vector representation, vector weights A are

(26)

42 A

c„ ={ej,-ej

:

j = l,...,n}.

(27)

The Hamiltonian of Cn model is given by Hc

"

=

2 *52 (exP(M-p) x

H

II

f(P-x)+exp{-(j,-p)

gW-x)).

Ac„9/3=-M+"

(28)

/

From the above data, we notice that either for A^-i or Cn Lie algebra, any root a € A can be constructed in terms with vector weights as a = (i — v where p., v £ A. By simple comparison of representation between AN-\ or Cn, one can found that if replacing ej+n with — ej in the vector weights of A2n-i algebra, we can obtain the vector weights of Cn one. Also does for the corresponding roots. This hints us it is possible to get the Cn model by this kind of reduction. For A2n-i model let us set restrictions on the vector weights with ej+n + ej = 0,

for j = 1 , . . . , n,

which correspond to the following constraints on the phase space of type RS model with (_rj = ^6j_)_n -f- Ci) ' X — Xi + Xi-\-n

(29) A2n-i-

— U,

Gi+n = (ei+n + ei) -p = Pi +Pi+n = 0 , i-l,...,n,

(30)

31

Following Dirac's method , we can show {GUHA^-O,

for

Vte{l,...,2n},

(31)

i.e. HA2„-! is the first class Hamiltonian corresponding to the above constraints Eq. (30). Here the symbol ~ represents that, only after calculating the result of left side of the identity, could we use the conditions of constraints. It should be pointed out that the most necessary condition ensuring the Eq. (31) is the symmetry property Eq. (17) for the Hamiltonian Eq. (2). So that for arbitrary dynamical variable A, we have A = {A^HA^D

~{A,HA2n_,},

= {A^HA^}

-

{A^^Ar^Gj^A^}

i,j = l,...,2n,

(32)

43

where AlJ = {Gl,GJ}=2^_0Id™y

(33)

and the {,}D denote the Dirac bracket. By straightforward calculation, we have the nonzero Dirac brackets of {Xi,Pj}D

{xi,pj+n}D

= {xi+n,Pj+n}D

= {xi+n,pj}D

=^j.j,

(34)

= - 2&,:>'•

Using the above data together with the fact that HAN_1 is the first class Hamiltonian (see Eq. (31), we can directly obtain Lax representation of Cn RS model by imposing constraints Gk on Eq. (18)

{-k/tsn-i'-ft^n-iJ-D = {^yl 2n _ 1 ,i?>l2„_i}|Gfc,*=l,...,2n,

{LA2n_1,HA2_1}D

= [MA2„_„LA^_A\Gh,k=l,...,2n = {LCn,HCn},

= [MCn,LCJ,

(35) (36)

where Hcn

-

Lc„

—

2^2n-i|Gfc,fc=l,...,2n, LA2n-i\Gk,k=l,...,2n,

MCn =Mj42„_1|Gfc,fc=l,...,2n,

(37)

so that Lcn = {LCn ,HCJ = [MCn ,LCJ.

(38)

Nevertheless, the (H^~)AN_1 is not the first class Hamiltonian, so the Lax pair given by many authors previously can't reduce to Cn case directly by this way. 3.2

BCn model

The BCn root system consists of three parts, long, middle and short roots: ABC„ = A x , U A u A s ,

(39)

44

in which the roots are conveniently expressed in terms of an orthonormal basis of Rn: AL = {±2ej: A = {±ej±ek: As = {±ej:

j = l,...,n}, j,k = 1,... ,n}, j = l,...,n}.

(40)

In the vector representation, vector weights A can be ABCn={ej,-ej,0:

j = l,...,n}.

(41)

The Hamiltonian of BCn model is given by H

BC„

=2 ]C

x

[expiVP)

J]

II

f(0-x)+exp(-n-p)

9(fi-x)\.

ABc„9/3=-M+'/

(42)

/

By similar comparison of representations between AN_I or BCn, one can found that if replacing e J + n with — e,- and e2ra+iwith 0 in the vector weights of A-in Lie algebra, we can obtain the vector weights of BCn one. Also does for the corresponding roots. So by the same procedure as Cn model, it is expected to get the Lax representation of BCn model. For Ain model, we set restrictions on the vector weights with ej+n + ej = 0 ,

for j = 1 , . . . , n,

e2n+i = 0,

(43)

which correspond to the following constraints on the phase space of ^42n"type RS model with CTJ =z y€i-\~n "r 6jJ • X — Xj -r Xi±n

Gi+n = (ei+n + ei) -p = pi+ pi+n (*2n+l = e 2 n + l ' X = X2n+1

= 0,

G2n+2

= 0.

= C2n+1 • P = Pln+1

— U,

= 0, i =

l,...,n,

(44)

Similarly, we can show {Gi,HAaJ~0,

for

V » e { l , . . . , 2 n + l,2n + 2}.

(45)

45

i.e. HA2n is the first class Hamiltonian corresponding to the above constraints Eq. (44). So LBC„ and MBC„

LBCn

can

be constructed as follows

= LA2n\G'

fc=l,...)2n+2' k'

'

'

•

MBCn = M A a J G i i f c = l i . . 2 n + 2 ,

(46)

#BC„ = 2^2»|G*,*=l,-..,2n+2,

(47)

while HBC„ is

due to the similar derivation of Eq. (32-38). 4 4-1

Lax r e p r e s e n t a t i o n s a n d involutive H a m i l t o n i a n s for Cn a n d BC„ RS models Cn model

The Hamiltonian of Cn RS system is Eq.(28), so the canonical equations of motion are = ePibi-e-pibi,

xi = {xi,H} p. PJ

=

r„. # )

=

(48)

^TsePih.(L^2lL

_ /j[£l±fil\

U.,,//) 2. { e 6 , ( / ( ^ } 9{Xji) bi{1

f{2Xl)

& (

' %(2^)

/(x.+ X | ) )

g(xj+Xi) +

^f{xlj) +

f^{9(xij)

+

+

f^+xj)" g(xt+Xj))h

where

/ w = = ™ .•. fc = / ( 2 H ) U(/(ar< - a;*)/(n + xk)),

(49)

46

b

i = 9{2xi) ~[[(9(xi - xk)g{xi + xk)).

(50)

The Lax matrix for Cn RS model can be written in the following form for the rational case (LcJ^-e

*bv

^ - ^

,

(51)

which is a 2n x 2n matrix whose indices are labelled by the vector weights, denoted by \i, v G Ac„, Mc„ can be written as MCn =D + Y,

(52)

where D denotes the diagonal part and Y denotes the off-diagonal part Y^=e"-pbv*(xltl„\) +e~>ipbl/$(x^,\ D^

+ N1),

(53)

= (C(A) + C ( 7 ) ) e " ^ - (C(A + 7) + E

C W ^ " ^

( ( C ( ^ + 7) - ax»»))e"vK

+ i % ^ ^ $ ( ^ > A + Ny)e-"-"b'v)

(54) (55)

and

/^-^

n

K=

A c „ 3/3=/*-"

*V =

II

9(13-x),

(56)

A c „9/3=M-i /

z,n/

:

={n-v)-x.

(57)

The Lc„, MQ„ satisfies the Lax equation LCn = {LCn ,HCn}

= [MCn, LCJ,

(58)

which equivalent to the equations of motion Eq.(48) and Eq.(49). The Hamiltonian Hcn can be rewritten as trace of Lc„

HCn = trLCn = \

Y. M6A C „

^'PK

+ e""-^).

(59)

47

The characteristic polynomial of the Lax matrix Lcn generates the involutive Hamiltonians

det(L(A) -

v

( C T ( A

• Id) = £

^1^

where (H0)cn = (#271)0, = 1 and (Hi)Cn

J 7 )

(-^)2"-J(^)c.

= (H2n-i)cn

{(Hi)c„,(Hj)cJ=0,

(60)

Poisson commute

i,j = l,...,n.

(61)

This can be deduced by verbose but straightforward calculation to verify that the (Hi)A2n_1,i = l,...2n is the first class Hamiltonian with respect to the constraints Eq.(30), using Eq.(15), (32) and the first formula of Eq.(37). The explicit form of (Hi)cn are {Hi)cn=

ex

Yl

P(Pej)^J;^cf/^,/-|J|.

/ = l , . . . , n , (62)

JC{1....,»},|J|P=

i,j'€J

i£J

i '

where

h = f(xi)f(2xi)

J J ( / ( x * - xk)f(xi

+ xk)),

n

b'i = g(xi)g(2xi)

Y[(g(xi - xk)g{xi + xk)),

n

bo = Y[f(xi)g(xi).

(67)

i=l

The Lax pair for BCn RS model can be constructed as the form of Eq.(51)-(55) where one should replace the matrices labels with [i,v G A-BC„, and roots with /? € &BCn •

49

The same as for the C„ model, the Hamiltonian HBC^ can also be rewritten as trace of the Lax matrix LBC„

HBcn=trLBCn=\

Yl

(e^b^+e-^).

(68)

M6ABC„

The characteristic polynomial of the Lax matrix LBcn Hamiltonians 2

det(L(A) -vld)= where (H0)Bcn commute

-

{ a {

£

f ' ^

J 1

\ - v r ^ ( H

= 1 and (Hi)Bcn

(H2„)BC„

{(Hi)BcT,,(Hj)BCn}

= 0,

generates the involutive

j ) B

c ,

= (H2n+i-i)BCn

Poisson

i,j = l,...,n.

(69)

This can be deduced similarly to C„ case to verify that the (Hi)A2n ,i — 1, ...2n is the first class Hamiltonian with respect to the constraints Eq.(44). The explicit form of {H{)Bcn are (Hi)BCn=

^

^viPeriFej-^cUjc^^,

I = l,...,n,

(70)

JC{l,...,n},|J|
Ei=±i,jeJ

ith PeJ = 5 Z £iPi> Fej-K = J J f{ejXj 3<j'

+erxr)

2£ x

J J f(ejXj +xk)f(ejxj

- xk)

keK

x n f( i i) n /(e^i). Ui,P=

Yl

II

fixik)f{xjJrxk)g{xjk)g{xj+xk)

\i'\=[P/2}kei\r x

f Ui€i\r f(xi)g(xi), lYli'ei' f(xi')9(xi>),

(p odd), (peven).

(7i)

50

5

Degenerate cases

Let us now consider its various special degenerate cases. As is well known, if one or both the periods of Weierstrass sigma function a(x) become infinite, there will occur three degenerate cases associated with trigonometric, hyperbolic and rational systems respectively. The degerate limits of the functions $(x, A) , a(x) and ((x) will give corresponding Lax pairs which include spectral parameter. Moreover, when the spectral parameter takes on certain limit, the Lax pairs without spectral parameter will be derived. 5.1

Trigonometric limit

The limit can be obtained by sending w3 to ioo with Wi = | , so that a\x) —> e 6

sinar,

((x) —> cot a; + -x, o

and the function $(x, A) = //jfwA reduce to $(x, A) -> (cot A - cot x)e*xu. By replacing the corresponding functions (x, A), a{x) and C,(x) to the form given above for the Lax pairs, we will get the corresponding spectral parameter dependent Lax pairs. For the simplicity, we notice that the exponential part of the above functions can be removed by applying suitable "gauge" transformation of the Lax matrix on which condition the functions can be valued as follows: cr{x) —>• sin a;, Q(x) —> cotx, $(ar, A) -> (cot A - cotx). As for the spectral parameter independent Lax pair, furthermore, we can take the limit A —> ioo, so the function $(i,A) ->

, sinx while the corresponding Lax matrix become to L,v = evnv

Sin 7 . . sm((/i — v) • x + 7)

(73)

51 which are exactly the same as the spectral parameter independent Lax matrix given in . 5.2

Hyperbolic limit

In this case, the periods can be choosen as by sending by sending Wi to ioc with u>3 = f, so following all the procedure in achieving the result of trigonometric case, we can find the hyperbolic Lax pairs by simple replacement of the functions appeared in trigonometric Lax pair as follows: sin a; —> sinhi, cos a; —> cosh a;, cot a: —> cotha;. The same as for the trigonometric case, we can get the Lax pairs with and without spectral parameter. 5.3

Rational limit

As far as the form of the Lax pair for the rational-type system is concerned, we can achieve it by making the following substitutions a(x) —> x,

C(x) -> I *(*,A)-± + ± for the spectral parameter dependent Lax pair, while furthermore, taking the limit A —> ioo, we can obtain the spectral parameter independent Lax pair. The explicit form of Lax matrix without spectral parameter is

£„, = e"-*b„-

7——-

(74)

(fj, - v) • x + 7

which completely coincide with the spectral parameter independent Lax matrix given in 32 . 6

Nonrelativistic limit to the Calogero-Moser s y s t e m

The Nonrelativistic limit can be achieved by rescaling pM i—> /3p^, 7 i—• /?7 while letting /? i—> 0, and making a canonical transformation

52

Pn^—>Pn+l

Y^

C(»7 -a;),

(75)

here pM = fi • p, such that L^Id

+ pLCM

+ 0(f32),

(76)

and + 2P2HCM + 0(P2).

H^N

(77)

where N = 2n for C„ model and N = 2n + 1 for B C n model. L C M can be expressed as LCM

=p-H

+ X,

(78)

where X^

= 7 $ ( x ^ , A)(l - JM„).

The Hamiltonian / ? C M of CM model can be given by

HCM = 2p2 ~ \ ^ 2

p Q

(

' *)

=

4 i r Z / 2 + c o n s *'

( 7Q )

( where const = ^ ' 7 p(A). All of the above results are identified with the results of Refs. 6.9>io,11,12 up to a suitable choice of coupling parameters. As for the various degenerate cases, one can follow the same procedure as for the RS model(please refer to Eq.(73)-(74)).

7

Summary and discussions

In this paper, we have proposed the Lax pairs for elliptic Cn and BCn RS models. The spectral parameter dependent and independent Lax pairs for the trigonometric, hyperbolic and rational systems are derived as the degenerate limits of the elliptic potential case. Involutive Hamiltonians are showed to be generated by the characteristic polynomial of the corresponding Lax matrix. In the nonrelativistic limit, the system leads to CM systems associated

53

with the root systems of Cn and BCn which are known previously. There are still many open problems, for example, it seems to be a challenging subject to carry out the Lax pairs with as many independent coupling constants as independent Weyl orbits in the set of roots, as done for the Calogero-Moser systems 6 ' 8 ' 9 ' 10 ' 11,12 . What is also interesting may generalize the results obtained in this paper to the systems associated with all of other Lie Algebras even to those associated with all the finite reflection groups 11 which including models based on the non-crystallographic root systems and those based on crystallographic root systems. Acknowledgement One of the authors K. Chen is grateful to professors K. J. Shi and L. Zhao for their encouragement. This work has been supported financially by the National Natural Science Foundation of China.

References 1. K. Hasegawa, Commun. Math. Phys. 187, 289 (1997). 2. F.W. Nijhoff, V.B. Kuznetsov, E.K. Sklyanin and O. Ragnisco, J. Phys. A: Math. Gen. 29, L333 (1996). 3. A. Gorsky, A. Marshakov, Phys. Lett. B 375, 127 (1996). 4. N. Nekrasov, Nucl. Phys. B 531, 323 (1998). 5. H.W. Braden, A. Marshakov, A. Mironov and A. Morozov, Nucl. Phys. B 558, 371 (1999). 6. M.A. Olshanetsky and A.M. Perelomov, Phys. Rep. 7 1 , 314 (1981). 7. VI. Inozemtsev, Lett. Math. Phys. 17 (1989) 11 8. A.J. Bordner, E. Corrigan and R. Sasaki, Prog. Theor. Phys. 100, 1107 (1998). 9. A.J. Bordner, R. Sasaki and K. Takasaki, Prog. Theor. Phys. 101, 487 (1999). 10. A.J. Bordner and R. Sasaki, Prog. Theor. Phys. 101, 799 (1999). 11. A.J. Bordner, E. Corrigan and R. Sasaki, Prog. Theor. Phys. 102, 499 (1999). 12. E. D'Hoker and D.H. Phong, Nucl. Phys. B 530, 537 (1998). 13. J.C. Hurtubise and E. Markman, e-print, "Calogero-Moser systems and Hitchin systems", e-print math/9912161. 14. Y. Komori, K. Hikami, J. Math. Phys. 39, 6175 (1998).

54

15. Y. Komori, "Theta Functions Associated with the Affine Root Systems and the Elliptic Ruijsenaars Operators," e-print math.QA/9910003. 16. J.F. van Diejen, J. Math. Phys. 35, 2983 (1994). 17. J.F. van Diejen, Compositio. Math. 95, 183 (1995). 18. K. Hasegawa, T. Ikeda, T. Kikuchi, J. Math. Phys. 40, 4549 (1999). 19. K. Chen, B.Y. Hou, W.-L. Yang and Y. Zhen, Chin. Phys. Lett. 16, 1 (1999); High energy physics and nuclear physics. Vol.23, No. 9, 854 (1999). 20. B.Y. Hou and W.-L. Yang, Commun. Theor. Phys. 33, 371 (2000); J. Math. Phys. 4 1 , 357 (2000). 21. B.Y. Hou and W.-L. Yang, Phys. Lett. A 261, 252 (1999); K. Chen, H. Fan, B.Y. Hou, K.J. Shi, W.-L. Yang and R. H. Yue, Prog. Theor. Phys. Suppl. 135, 149 (1999). 22. T. Kikuchi, "Diagonalization of the elliptic Macdonald-Ruijsenaars difference system of type C2," e-print math/9912114. 23. S.N.M. Ruijsenaars, Commun. Math. Phys. 110, 191 (1987). 24. M. Bruschi and F. Calogero, Commun. Math. Phys. 109, 481 (1987). 25. I. Krichever and A. Zabrodin, Usp. Math. Nauk, 50:6, 3 (1995). 26. Y.B. Suris, "Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-operators as the Calogero-Moser ones?" e-print hep-th/9602160. 27. Y.B. Suris, Phys. Lett. A225, 253 (1997). 28. K. Chen, B.Y. Hou and W.-L. Yang, "The Lax pair for C 2 -type Ruijsenaars-Schneider model," e-print hep-th/0004006. 29. S.N.M. Ruijsenaars, Commun. Math. Phys. 115, 127 (1988). 30. Y. Ohta, "Instanton Correction of Prepotential in Ruijsenaars Model Associated with N=2 SU(2) Seiberg-Witten Theory", e-print hep-th/9909196. 31. Paul A.M. Dirac, Lectures on Quantum Physics(Yeshiva University, New York, 1964) 32. K. Chen, B.Y. Hou, W.-L. Yang, "Integrability of the Cn and BCn Ruijsenaars-Schneider models" e-print hep-th/0006004.

55

FATEEV'S MODELS A N D THEIR APPLICATIONS D.CONTROZZI AND A.M.TSVELIK Department of Physics, Theoretical Physics, 1 Keble Road, 0X1 3NP Oxford, UK. We present two families of integrable models recently introduced by Fateev and suggest some possible application to physical systems.

1 1.1

Fateev's models Introduction

Recently Fateev introduced various families of models x'2,3 that, beside being amusing examples of integrable field theories, may also have some interesting physical applications. In this paper we revise some aspects of two of these families and discuss some possible physical applications. The complete list of models can be found in 1'2. These theories exist in two representations -bosonic and fermionic - and the weak coupling limit of one representation corresponds to the strong coupling of the other one. The two families are characterized by the presence of one or two fermionic/bosonic fields respectively, coupled to Toda chains. We will refer to them as type I and type II models. After introducing the models, we report some of the mathematical results obtained in 4 . We will also present two possible applications to physical systems. A common ingredient in both these interpretations is the presence of a phononic background that interacts with some additional bosonic or fermionic degrees of freedom. We will discuss in some details one of this applications. According to this interpretation the Euclidean version of Fateev models is seen as Ginzburg-Landau free energy functionals describing thermal fluctuations in superconducting films interacting with an insulating substrate. The major result of this analysis is that the effective critical temperature increases with the thickness on the insulating stratum. Some of this results were presented before in 10 . The paper is organized as follow. In the first section we introduce Fateev's models and briefly describe some possible physical applications. In Sec.II we recall few essential results from the exact solution, and in the last section we will study the physical consequences of our interpretation of the models.

56

1.2

Type I models

The models of type I have a Lagrangian formulation in terms of a complex fermion field {ip,ip) and n scalar fields ip = (