Sine-Gordon theory with higher-spin N = 2 supersymmetry and the massless limit

NUCLEAR PHYSICS B [FS] ELSEVIER

Nuclear Physics B 419 [FS] (1994) 632—646

Sine-Gordon theory with higher-spin N = 2 supersymmetry and the massless limit * H. Itoyama, T. Oota Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan

Received 22 September 1993; revised 21 December 1993; accepted 25 January 1994

Abstract The sine—Gordon theory at fl2/8m = 2/(2n + 3), n = 1,2,3,..., has a higher-spin generalization of the N = 2 supersymmetry with the central terms arising from the affine quantum group Uq (sI (2)). We observe that the algebraic determination of S-matrices (~quantum integrability) requires the saturation of the generalized Bogomolny bound. The S-matrix theory considered is a variant of the sine—Gordon theory at this value of the coupling whose spectrum Consists of a doublet of fractionally charged solitons as well as that of anti-solitons in addition to the ordinary breathers. This is in contrast to the theory considered by Smirnov which is obtained by the truncation to the breathers. The allowed values for the fractional part of the fermion number are also determined. The central charge in the massless limit is found to be c = 1 from the TBA calculation for nondiagonal S-matrices. The attendant c = 1 conformal field theory is the gaussian model with Z 2 graded chiral algebra at the radius parameter r = V2n + 3. In the course of the calculation, we find 4n + 2 zero modes from the (anti-)soliton distributions.

1. introduction The sine—Gordon theory has been a prototype of relativistic continuum field theories that are integrable in 1 + 1 dimensions. The mass spectra and Smatrices as well as the relationship with the massive Thirring model have been well-studied for a long time [1,21. In recent years, attention has been paid to This work is supported in part by a Grant-in-Aid for Scientific Research (05640347) from the Ministry of Education, Japan.

*

0550-3213/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDIO55O-32l3(94)00050-0

H. Itoyama, T. Oota /Nuclear Physics B 419 IFS] (1994) 632—646

633

identify the universality class which the model belongs to, for a variety of values of the coupling constant. This is not a trivial question as the model at special values of the coupling constant permits a self-consistent truncation, which can be revealed by the consistency of the bootstrap framework or by the structure of the vacuum. This point becomes evident in view of the perturbation of the conformal field theory at the fixed point (see, for example, ref. [3]). Formulating thermodynamics from the physical S-matrices [4], often referred to as the thermodynamic Bethe ansatz (TBA) [5], provides an efficient method to reveal ground-state properties of the model. In some cases, it is sufficient to identify the attendant conformal field theory. One series of special points considered so far is 2 p2 8ir

=

2n + 3~

n

=

1,2,3

(1.1)

This lies in the regime in which the Thirring coupling is attractive. It was noted [6], on the basis of the structure of the form factors, that keeping breathers alone is a self-consistent truncation of the model. In ref. [6], the massless limit of this series of models with truncation has been identified with the nonunitary minimal conformal field theory indexed by (2, 2n + 3). The TBA calculation has confirmed this [7]. Another series of points where the TBA calculation has been done [8,91 includes

p2

1

(1.2)

lying in the repulsive regime (see ref. [10] for TBA at other series of points). Both the calculation [81 based on the structure of the vacuum considered from the microscopic Bethe ansatz diagonalization [111 and the one based on the “RSOS” scattering matrices proposed [9] have led to the same set of the algebraic equations. The central charge in the massless limit is found to be that of the unitary minimal model c = 1 6/I? (I? + 1) [8,9 1. The neutral excitations (or distributions) play a prominent role here 1 From the representation-theoretical point of view, it is well-known that the soliton—(anti-)soliton S-matrices of the sine—Gordon are naturally regarded as the trigonometric ~(v) matrices in the vector representation which have the quantized universal enveloping algebra [13,14] (or the affine quantum group) Uq (sl(2)) as a commutant (or symmetry): —

[~(v),it®~(A(çb(X)))]0,

XEUq(Sl(2)).

(1.3)

Here it denotes the representation and 4 is the co-product. In order to find a finite-dimensional representation, e 0 and fo have been represented through the restriction q’ [14] of Uq (sl(2)) onto the nonaffine part Uq (sl(2)). In what Similar features have been seen in the microscopic calculation of ref. 121 for the critical RSOS

spin chain.

634

H. Itoyama, T. Oota /Nuclear Physics B 419 [FS] (1994) 632—646

follows, we will employ Eq. (1.3) as a definition of the word “sine—Gordon”. The theory, therefore, has Uq (sI(2)) at 8m2i

(1.4)

as genetic on-shell symmetry of S-matrices. It has been noted that, at q2 = —1, some of the defining relations of Uq (si (2)) can be converted into anticommutators [15]. In this vein, Smirnov’s truncation at Eq. (1.1), which yields q2 = —1, is the case in which the symmetry Uq (sl(2)) is nonlocally realized. In more physical terms, it is the one in which “charge carriers” are confined by a dynamical reason. An interesting variant has been considered by Fendley and Intrilligator [161 at fl2/8it = They propose an S-matrix theory where the spectrum consists of a doublet of the solitons and that of the anti-solitons. This is distinct from the original sine—Gordon theory. Each of the two doublets provides a representation of the N = 2 supersymmetry algebra with the central terms [17], which is possible only when the Bogomolny bound is saturated. The solitons and the anti-solitons carry noninteger fermion numbers [18]. This theory is interpreted as the one in which the symmetry Uq (si (2)) is locally realized by the doublet of solitons and that of anti-solitons. The central charge c in the massless limit is found to be c = 1 [16]. Note that this point is the intersection (n = 0) of the two series listed in Eqs. (1.1) and (1.2), where the theory considered in refs. [8,9] predicts c = 0. The existence of the c = 1 S-matrix theory of ref. [16] at fl2/8ir = suggests that there be two distinct (S-matrix) theories for every point of the series Eq. (1.1): the one is that of ref. [6] where Uq(sl(2)) is nonlocally realized and the other is the possibility in which Uq(~l(2))is locally realized and which has not been considered before. In what follows, we consider a variant of the sine—Gordon theory at fl2/8ir = 2/(2n + 3) as S-matrix theory, filling up this missing possibility. The Uq (sl(2)) is locally realized by a doublet of solitons as well as that of anti-solitons whereas the breathers (neutral particles) are considered as a nonlocal realization of

4.

4

Uq(sl(2)). The argument from perturbed conformal field theory tells us that the Lorentz spin of the anticommuting charges is ±I /y = + (8it/fl2 I) = ±(n + (the second reference of ref. [15]). The algebra is, therefore, regarded as a higher-spin generalization of the N = 2 supersymmetry algebra —

with central terms. (Examples of conserved charges which form a higher-spin analog of supersymmetry have been given in ref. [19].) The two doublets again form representations where the Bogomolny bound is saturated. In the next section, we first derive the higher-spin generalization of N = 2 supersymmetry with the central charge from Uq(sl(2)) at q = ±i.We note that the homomorphism q5 : Uq(sl(2)) Uq(sl(2)) is equivalent to the statement that the Bogomolny bound is saturated. We present the soliton—antisoliton 5-

H. Itoyama, T. Oota/Nuclear Physics B 419 IFS] (1994) 632—646

635

matrices by demanding the algebra to be on-shell symmetry. We also determine the allowed values of the fractional part of the fermion number. In sect. 3, we outline the diagonalization of the TBA for the nondiagonal soliton—antisoliton S-matrices. (see ref. [20]). We find 4n + 2 fermionic zero modes around the (anti-)soliton distributions after the diagonalization. In sect. 4, we present our calculation of the the central charge in the massless limit. We find c = 1 for n = 1,2,3... . In sect. 5, the c = 1 conformal field theory attendant with our model is identified to be the gaussian model having the radius parameter r = ‘/2n + 3 and a Z~graded chiral algebra. 2. Construction of S-matrices from Uq(sl(2)) at q

=

Let us first derive the higher-spin generalization of N = 2 supersymmetry algebra with the central terms from Uq (sl(2)) at q = ±i.From the generators of Uq(sl(2)) we define .F ~/(h 2c/(ei)qF, Q~ 1), Q+ M ht M’~/2c/~e 1!2çb(f 2~(fo)qF. Here we 0)q, M M and 1)q~, Mh/ have introduced a Q+ mass= scale q? is Q the homomorphism mentioned in the introduction. Then

~

[F,Q~]= +Q~,

{ Q~~Q~}z~,

=

{Q~~}=

=

{Q—~}=o~

(2.1)

where Z(±) ~M2’~’ (1 — q±4F)commutes with F, Q±,Q+ and in fact a center of the algebra generated by these. In the second line of Eq. (2.1), we have used q = exp(—iti(n + ~)). The expression for the center tells us that it can be represented as a nonvanishing number only when the fermion number of the state is fractional.

—

+

—+—--

Define P {Q Q }, P {Q , Q }. The q-analog of the Serre relations reads [2, (Q~)2]= 0, [~, (~~)~] = 0 for q = exp(—iti(n + ~)). It is known that one can set ,

(Q±)2= 0,

(~±)2= 0

(2.2)

consistently [21]. We also find, for q [PQ~]

=

exp(—iri(n +

=

—

[(Q±)2Q+]

=

0,

[~~]

=

0,

=

—

[(~±)2~Rz]

=

0,

[~,Q~]

=

0,

(2.3)

from a simple calculation. The algebra defined by Eqs. (2.1), (2.2) and (2.3) is the higher-spin N

=

2 supersymmetry algebra with the central terms.

636

H. Itoyama, T. Oota /Nuclear Physics B 419 IFS] (1994) 632—646

The co-product is translated into 4(Q~)= Q~® 1 + q+2.F ® Q±4(e) = Q ®l+q±2F®Q We consider a finite-dimensional representation of the algebra in which P and ~ are diagonal. Writing the eigenvalue ofP and that of ~ as m2~+1 e(2n + I )v and m2t1+l e~2u1+1)v, respectively, we label a generic state by m and v, which are considered as the mass and the rapidity of the particle, and by the fermion number f. As in the case of the ordinary N = 2 supersymmetry, the Bogomolny bound can be derived [17] by considering the positive semi-definiteness of {Af, A} with A Q+ (~ZL)) 1/2 (~z(+))1/2: this gives M4’~2sin2 ( (2n + 3)irf) ~ m4’~2.The two-dimensional representation is possible only when ~-

the bound gets saturated and the central terms are nonvanishing. The latter condition leads to the noninteger fermion number. We note that the homomorphism q~is equivalent to A = 0 and, therefore, implies the saturation of the Bogomolny bound. In fact ~(e = 2~~4~)v translates into 0) A == 2f1,~(ei) 0 evaluated e1, the q?~(fo) = ~ 9f(f1) Quantum = fi,2 = integrability, ie( at one-particle states. which permits us to construct S-matrices algebraically, demands that the bound be saturated. The old wisdom is now found to be a reality. We are thus led to consider the two-dimensional representation with noninteger fermion number. Following ref. [16], we postulate that the soliton and the anti-soliton individually form doublets in contrast to the original sine— Gordon case. The rationale for the two doublets comes from the semi-classical description of the corresponding Landau—Ginzburg lagrangian [16,221 and is closely related to the chiral algebraic structure of the theory as we will see. The soliton doublet and the anti-soliton doublet are denoted respectively by (~)and (f). The fermion number of the soliton doublet is denoted by (ff 1) and that of the anti-soliton doublet by In the case in which more than one doublet are present in the theory, the charge conjugation symmetry alone does not force the fermion number to be + ~ [231. The action of the charges on the one-soliton states must be consistent with Eq. (2.1). Determining the phases allowed, we find 2 Q+ Id(v)) = (me~/ Q~ u(v)) = (me +1/2 d(v)),

(‘j).

u(v))

=

e~~(m c_v )fl+1/2

d(v))

=

e_1a(me~v)~~/2 u(v)).

(2.4)

Here e’~= iexp(—(2n + 3)irif). The other action of the charges on the one-soliton states vanishes. The equations for the nonvanishing action of the charges on the one anti-soliton states are the same as Eq. (2.4). We write a generic two-soliton state as A(vi)B(v 2)) =1 A(v1))® I B(v2)). The action of Q~as well as that of on this tensor product state is in accordance with

H. Itoyama, T. Oota/Nuclear Physics B 419 IFS] (1994) 632—646

637

the co-product. We now turn to consider the elastic scattering among the solitons, the antisolitons and the breathers. As is discussed in the introduction, we will construct a variant of the sine—Gordon S-matrices at fl2/8n = 2/(2n + 3). The breather—breather scattering and the breather—(anti-)soliton scattering are taken from those of the original sine—Gordon S-matrices. We determine the

soliton—antisoliton S-matrices by demanding that the algebra (Eqs. (2.1) and (2.3)) be on-shell symmetry i.e. the symmetry of the S-matrices. Let the S-matrix of the soliton—antisoliton scattering be U(V2)U(V1)

d(v

u(v1)ü(v2) ( c(v1 —v2) d(v1)ci(v2) ~ b(v1 ~2)

2)d(v1) b(vi t)2) c(v1 —v2)

u(v2)d(v1) u(v1)d(v2) (a(vi_v2) d(v1)ü(v2) ~ 0

d(v2)ü(v1) 0 a(v1 —v2) )•

(25)

Let us demand that the= charges with S-matrix. Wevalues find that f 2(2f~~) 1. Thiscommute determines thethe allowed set of {{J~}} mustthesatisfy q~ part of the fermion number: for fractional

2(n+1—fl+l

3

4n+6

£eZ,

3

—(n+~)
(2.6)

We also find that a(v), b(v) and c(v) must be of the form a(v) = Z(v)cosh(n+~)v,b(v) =i(—1)~~’~Z(v)sinh(n+~)v,c(v) =Z(v).The Z-factor is determined by the requirements of the crossing symmetry and the

unitarity and the mass spectra of the breathers; mt =2msin(~iryI?), Z(v)=

I?

=

1,2,... < 1/yE 8it/fl2— 1.

(2.7)

(—1 “i ‘ / cosh(n +

I

~ xexp I —i/ — ,J

t

+ ~)vt/m)sinh((n -

sinht/2cosh((n +

)t/2)

,j

(2.8)

.

The choice of the sign factor conforms to the case of the ordinary sine—Gordon

model at fl2/8ir = 1 / (n + ~). The soliton—breatherS-matrices Sas (v) as well as the breather—breather S-matrices are, with Fv(v)

sinhv + isinc~it tanh~(v+ iUm) . . = sinhv — zsincElr tanh~(v ic~2r) .

—

(ref. [24]),

=

Sat (v)

638

H. Itoyama, T. Oota /Nuclear Physics B 419 IFS] (1994) 632—646 5as

=

HFU_y(a_2r))/2,

5aa’

=

Fy(a+a~)/2

Fy(a+a~_2r)/2

Fy(a+a~_2r)/2.

(2.9)

3. TBA with reflections and the diagonalization In what follows, we generically denote by ~(ic)

PK(v)

=

lim

~(K)

1+1

L-.ooL(V,+l

— v)

the density of states of species K per unit rapidity and unit length and by PK(V) =

L=~L(v 1~1 -vi)

the density of particles of species K. Here, L is the size of the box and the number of particles in the system is denoted by .iV. The integer n~labels the ith level belonging to the species i~. We introduce e,~(v) — ~ / T by PK(V) = exp(~uK/T—K(v)) PK(v)

—

1 + exp(~K/T—K(v))

We first need to diagonalize the quantity (3.1)

~

under the periodic boundary condition exp(imL sinh v, ) Tab (v1) ~Pb = ~ in the infinite-volume limit Al, L cc. From the unitarity, we see that S~’(0) = ±ô~ô~ holds. The above equation can then be written as exp ( imL sinh Vk) [± ~.IbTbb (v = vk) ] ~ = ~, where the quantity >b Tbb (v) is the transfer matrix. The S-matrix seen in Eq. (3.1) is taken to be our S-matrix of Eq. (2.5) describing the scattering of u- and d-solitons —*

du ud ud(b du~c b)’

fl

uu dd uu (a 0 ddk~0

(32) ,

with a = a, b = b and c = c~.For notational simplicity, we have identified d = u and ü = d. The diagonalization in this section does not lose generality by this identification. This S-matrix satisfies the condition a(v)â(v) + b(v)b(v) — c(v)~(v) = 0 which is nothing but a free-fermion condition for the six-vertex model.

H. Itoyama, T. Oota /Nuclear Physics B 419 IFS] (1994) 632—646

We omit the procedure of diagonalizing the transfer matrix

I~ The

639 eigenval-

ues of the transfer matrix are ‘)(v;{{yr}})=CflZ(v_vj)H(_1)(~_t)icosh((n+~)(v_yr))

fi

x

sinh((n+~)(v-yr))~

(3.3)

r=n’+l

where {{Yr}} are the roots of the following equation:

fl

1=1

i(—l)(~~)sinh(n+ ~)(Yr — v1) cosh(n + ~)(YrVi)

=

(—l)~’~’.

(3.4)

Here C is a function which has no pole in v for {{Yr}} satisfying Eq. (3.4) and is bounded at infinity. It must therefore be independent of v. In the physical strip, the roots of Eq. (3.4) are ~~k)

=

Zr

+ k +

(k)

Zr

k

~

=

0,l,2,...2n, (3.5)

.

1_ni,

Zr2 11t1, _.~i, , ,... fl. are real numbers. Let us denote these 4n + 2 modes in Eq. (3.5) by

Yr

Here

2

2n+l k+~

—

—

{{O,1,2,...,2n}} + ~

LEB~.

(3.6)

The density of states for these 4n + 2 modes is denoted by p(l) (z~’~) for £

e

~

From Eq. (3.4), we find 1(2n + 1) [dv Pm(V) j 2m cosh(2n + l)(Z( ) —v) P~(z~) = (—l) forI? e ~

in the Al

—f

Pm(Z)

cc,

L

—p

cc

(3.7)

limit. Here,

Ellm(k)(k)

has no dependence on the superscript (k). We refer to this as the density of particles (occupancy) for the massive mode m. On the other hand, comparing Eq. (3.7) with the periodic boundary condition of TBA, we see that the 4n + 2 modes lying along the purely imaginary direction are interpreted as massless

modes appearing from the soliton distributions. The semi-classical treatment The procedure is well-known as the quantum inverse method and is at the free-fermion point in our case. See ref. [16,20].

640

H. Itoyama, T. Oota INuclear Physics B 419 IFS] (1994) 632—646

of these modes should correspond to a generalization of the discussion of the fractional fermion number in refs. [18,23]. We now turn to the (anti-)soliton—breather S-matrices Sas ( Sat) which are diagonal. Using the monodromy operator T(v) = Tab (v), the periodic boundary condition for the soliton with mass m reads exp ( imL sinh v) X UjSas(V — v~)T(v)~I’ = W. From this equation, m . 1 d 1~(v;{{yr}}) ~—coshv + L—~oc lim --Im-~~-—log). LIV i~ uV n

(3.8) where cbam(v)

=

—i(logSa

5)’, Pa(V) is the density of particles for the ath breather, and Pm (v) is the density of states for m. We would like to express

1Im~log~(v)~’~(v;{{yr}}) dv lim in terms of the densities of the (4n + 2) massless modes in Eq. (3.6). For that purpose, we note that the 2~eigenvalues of the transfer matrix i~P1’)(V,{{yr}}) fl’ = 1,...A~in Eq. (3.3) are determined by choosing between sinh((n + ~)(v_Yr)) and cosh((n + .L)(vyr)) for each Yr. We L—.cxjL

define the density p~ (Z) £ = k (or k) E £ (or ~) by the occupancy of the mode k (or k) at y(/C) = Z + [(k+ ~.)/(2n + l)]mi (or Y~ = Z— [(k+ ~)/(2n + 1)]mi ) which contributes a cosh to t(~’)(v,{{y~}})for £ = k (or a sinh for £ = k). Similarly, we define the density p~(Z) by the occupancy of the mode k = £ (or I?) which contributes a sinh (or a cosh). By construction, P(t)(v) = p~(v) + p~2(v)which permits us to think of + modes as “particles” and — modes as “holes”. We need to derive an expression for .1 d lim —Im—log2(v) dv

L-~ooL

in terms of the above quantities. This calculation is a direct but nontrivial generalization of the one carried out in ref. [16] (see ref. [20]). Space permits us to write only the final expression derived from Eq. (3.8), ~-coshv

+

~

v’)~(v-v’)+p~P(v’)~2(v-v’)}

tELec

n

+~f~_~am(v_v’)Pa(v’)Pm(V)~

(3.9)

H. Itoyama, T. Oota /Nuclear Physics B 419 IFS] (1994) 632—646

641

where (t) ~

(~l)1~~~

-

1

+

~

+

miti/(2n + 1))

+(-1)’ (n + / cosh(2n + l)v So far, we have obtained a system of nonlinear equations from soliton— antisoliton scattering (Eq. (3.7)) and from breather-soliton scattering (Eq. (3.9)). The breather—breather scattering which is diagonal, and the corresponding set of nonlinear equations are well-known [4] with the S-matrices taken from Eq. (2.9). In order to derive a closed set of nonlinear equations, we have to minimize a free energy with chemical potentials under the above conditions. Having done this, we find a system of nonlinear equations which determine ~(v) as functions of temperature and chemical potentials: mA [dv’ / eA(v)=—~-coshv— ~ -~--—ç~(v —v)

j

K =m,f,a

xlog(l +expCuK/T—K(v’))),

(~ a—i

~ab(v)

b—i

=

—4coshv

=

sin(a+b—2r)m/(2n + 1) cos(a + b — 2r)27r/(2n + 1) cbam (v) = —4 cosh v cosh2v

q?ma (v)

(3.10)

+

(311)

—

sin(~ (a + 2r)/(2n + l))it r=0C05h2’L~—cos(~— (a + 2r)/(2n + l))2it

(3.12)

—

~im(v)= q~mm(v)=

~mi

(v)

~

=

(2n + 1)~U

l)v’

(3.13)

f~_cb~(v_v’)cbtm(v’).

(3.14)

£ Ef

4. The massless limit We now proceed to take the massless limit of the system and compute the central charge ~ in the limit. Let the chemical potential for K be /1K = ic~KT and let the entropy per unit length be S = ~itc(m/T)T. Then cEllmc(m/T)

=

(L[f(e~(0) - ~~)]

-

L[f

(EK(cc)

-

642

H. Itoyama, T. Oota /Nuclear Physics B 419 fF5] (1994) 632—646 EC

(4.1)

0—C~.

Ref. [25] tells us

C—

=

12 (hmin +

~min)

with c the central charge of the

corresponding conformal field theory in the massless limit and hmin + hmin the lowest scaling dimension. The function L (x) is the Rogers dilogarithm. How to determine K(0) and K(oo) is well-known by now. The K(0) satisfies the algebraic equations exp(EK(0))

fJ(l +exp(/1K’/T—eK’(0)))’~’,

=

(4.2)

where Nab

=

—~J

dv~ab(v).

As for the other limit r

—~

J]~(1

=

—~cc,

only the massless modes survive and

K=a,m,

exp(CK(cc))=+oc,

exp(e,(cc))

0, v

+ exp(/1K’/T—fK’(cc)))’~’,

£ E LEI.

(4.3)

Turning to the evaluation of NKK’, we find fa+b— 1,

ja—bI

I2mln(a,b), Nma = a,

la—bi ~2,

=

0,1,

Nabc

Nam

=

N~+ 2(’)2fl+ Nt_ = ~(—l) 2n + NtmNmt Nmm~

~~1)t ~

IC

1’

N~IV~m

£~C

~(_l12fl~~

~)

x (_~(_l)1)

=

n.

(4.4)

£EL~L

Let us solve Eqs. (4.2) and (4.3) for zero chemical potentials. Define = exp(—EA(O)). Eq. (4.2) reads 2b(1 + a

=

b=i fl(1

+ xb)

Xa)2a_l

ft c=a+1

(1 + X~)2a(l+

Xm)a.

H. Itoyama, T. Oota/Nuclear Physics B 419 IFS] (1994) 632—646

=fl(l

—

Xodd

Xeven

+xb)b(1 +Xm)~(1+XevenY~(l

(1

=

643

+XoddY.

(4.5)

2.

+Xm)”

Here we recall that the indices a, b refer to n kinds of breathers while the subscript m denotes the massive mode. The 4n + 2 massless modes of £ ~ are divided into the 2n + 2 even modes and the 2n odd modes. After some guesswork for n = 1,2,3 cases, we find a solution to Eq. (4.5): exp(ea(0)) =a(a + 2), exp(em(0))

=

CXp(even(0))

=

(n + 1)2 2n + 3 exp(—codd(O))

=

+

~,

(4.6)

which yields ~o = 2n + 2. For EA(cc), the corresponding set of algebraic equations is obtained by removing the massive mode m and the breathers a = 1 n in Eq. (4.5): Cm (cc) = cc, ~a (cc) = cc. The solution is exp (Ceven (cc)) = exp(odd(cc)) = 1, which means = 2n + 1. We obtain ~, from Eq. (4.1), =

5. The c

1

=

for n

=

0,1,2

(4.7)

1 conformal field theory

We now turn to identify the C = 1 conformal field theory that our model belongs to in the massless limit. The known c = 1 conformal field theories (see for example ref. [26] for a review), except the cases with isolated moduli [27], are characterized by the two marginal directions, namely, the gaussian line and the orbifold line. To fix our notation, we write a compactified free massless scalar and the two-point functions as ~ (Z, ~) = ~ (q~( Z) + ~ ( i)), (~(w)~(z))= —ln(w — z), (c7(t~~)~(±)) = —ln(rii — ±).The radius parameter r is introduced through ~(z, ±) q (z, ±)~- 2jtr. In the massless limit, the higher-spin N = 2 supersymmetry algebra derived in Eqs. (2.1) and (2.3) separates into the unbar and the bar subalgebras: the central charges Z(±),which are spinless operators, become zero. This can be done by m —p 0,v v + A, A +cc, keeping me’s finite in Eq. (2.4) for u(v) and d(v) and by m —p 0, v —p v A, A +cc, me’s finite in the corresponding formula for ü (v) and d (v). The resulting algebra is regarded as a centerless subalgebra of a Z 2 graded chiral algebra for the C = 1 conformal —~

—~

—

—~

We have also carried out calculation in the presence of purely imaginary chemical potentials. We have determined the roots of the algebraic equation (4.2) for the case of the fermion number f = ~ [20].

644

H. Itoyama, T. Oota /Nuclear Physics B 419 [ES] (1994) 632—646

field theory. In the case n = 0, this chiral algebra is the well-known N = 2 super-conformal algebra. Focusing on the holomorphic part, we write the limit as Q~—p f dZ G~(z). Let us denote the conformal weights of the chiral operator G±(Z)by (hG+ ,hG±= 0). This operator can be represented by the chiral vertex operator (5.1) G~(z) =exp(±ipç~(Z)), hG±= ~p2. Recall that the Lorentz spin of Q~is n + ~ at off-criticality. Taking the massless limit, we conclude that the spin of G~(z) associated with the euclidian rotation is n + ~. On the other hand, the primary fields of U (1) xU (1) current algebra are a set of vertex operators Vmt(Z, ~) exp(ip~(z) + ij~(~))with the conformal weights (hji)

=

(~p2,~2),

(m,I?)

=

{{m

E Z,I? E

p

Z}},

=

~.

+I?r,

p

=

~—I?r,

(5.2)

or

={{me2Z,I? EZ}}~{{mE2Z+ 1,I? EZ+ ~}}.

(5.3)

The spin is mI?. Eq. (5.3) is derived on the basis of the mutual locality of Vmt [28]. In the latter case of Eq. (5.3) [29], the GSO projection [30] must be given in order to render the spectrum modular invariant. Let us find the value of the radius parameter r. For the chiral vertex_operator of Eq. (5.1), we should set h = 0 in Eq. (5.2). This gives r = ~/m/2I?. From the spin, we find hG~= mI? = n + ~. We conclude that r=~/2n+3,

m=+(2n+3),

£=±~.

(5.4)

The model belongs to the latter possibiity of Eq. (5.3). The algebra generated by the operators Eq. (5.1) may be called ~2 graded chiral algebra. Closure of this algebra tells us to include_the chiral vertex operators exp(iQç~(z)) with Q an integer_multiple of ~/2n + 3. The operators vk(z) = exp(iq~~(Z)) with = k/’../2n + 3, k = 0, 1,.. . 2n + 2 provide representations of this algebra: [1], [v 1],... [v2~~2]. Now we turn to the question of which marginal line the model lies in. As in the case of the original Zamolodchikov S-matrices [2], the poles appearing in our soliton—antisoliton S-matrix are interpreted as the ones in the direct channel of the soliton—antisoliton scattering: they are understood as the bound state poles of a soliton and an antisoliton. The conservation of the U (1) quantum number is thus seen. We conclude that our model lies in the gaussian line. A natural question arises. Can we have a freedom to change our 5matrices while maintaining the calculation and the discussion in the previous sections, so that the model is converted to belong orbifoldByline? This 2/8ir = I/n, to n =the2,3 changing is possible theS-matrix, reflectionless points fl the bound-state poles appear in the the sign ofatthe we can make crossed channel without changing the assumed mass spectrum [7]. The poles

H. Itoyama, T. Oota/Nuclear Physics B 419 IFS] (1994) 632—646

645

are then interpreted as the soliton—soliton bound states in this case. In those points fl2/8it = 2/(2n + 3), n = 1,2,... we consider, however, this is not the case by the following reason. Let the mass of a bound state be mb = 2m sin w 0
=

2msin

(it

—

(it

—

Ut)))

=

2msin

2n + 1

This is not the mass spectrum we assumed originally in Eq. (2.7) and the presence of the crossed channel pole is simply contradictory to the rest of our discussion. Acknowledgement We acknowledge Professor E. Date for valuable remarks on quantum groups. We also thank Katsushi Ito, Yutaka Matsuo and Hisao Suzuki for useful discussions. References [1] R. Dashen, Hasslacher and A. Neveu, Phys. Rev. Dli (1975) 3424; J. Johnson, S. Krinsky and B. McCoy, Phys. Rev. A8 (1973) 526; H. Bergknoff and H.B. Thacker, Phys. Rev. Lett. 42 (1979) 135; Phys. Rev. Dl9 (1979)

3666;

YE. Korepin, Teor. Mat. Fiz. 41 (1979) 169 [Theor. Math. Phys. 41 (1979) 9531 [2] A.B. Zamolodchikov, Commun. Math. Phys. 55 (1977) 183; A.B. Zamolodchikov and Al.B. Zamolodchikov, Ann. Phys. (NY) 120 (1979) 253 [3] A.B. Zamolodchikov, Adv. Stud. Pure Math. 19 (1989) 191 [4] Al.B. Zamolodchikov, NucI. Phys. B342 (1990) 695 [51C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115 [6] F.A. Smirnov, mt. J. Mod. Phys. A4 (1989) 4213; Nucl. Phys. B337 (1990) 156 [7] T.R. Klassen and E. Melzer, NucI. Phys. B338 (1990) 485 [8] H. Itoyama and P. Moxhay, Phys. Rev. Lett. 65 (1990) 2102; in Proc. Argonne Quantum group workshop, April 1990, ed. T. Curtright et al. (World Scientific, Singapore, 1991) [9] AI.B. Zamolodchikov, NucI. Phys. B358 (1991) 497; P. Fendley, H. Saleur and Al.B. Zamolodchikov, BU-HEP 93-6, hep-th 930451 110] P. Christe and M.J. Martins, Mod. Phys. Lett. AS (1990) 2189; C. Ahn and S. Nam, Phys. Lett. B271 (1991) 329; C. Ahn, CLNS 91/1117, hep-th 9111022 [11] YE. Korepin, Commun. Math. Phys. 76 (1980) 165 [121V.Y. Bazhanov and NYu. Reshetikhin, mt. J. Mod. Phys. A4 (1989) 115 [13] V.G. Drinfeld, Soy. Math. Doki. 32 (1985) 254; in Proc. mt. Congress of mathematicians, Berkeley, California, 1987

646

H. Itoyama, T. Qota /Nuc/ear Physics B 419 [ES] (1994) 632—646

[14] M. Jimbo, Lett. Math. Phys. 10 (1985) 63; Commun. Math. Phys. 102 (1986) 537; P. Kulish and N. Reshetikhin, J. Soy. Math. 23 (1983) 2435; E. Sklyanin, Funct. Anal. Appi. 16 (1982) 263 [15] D. Bernard and V. Pasquier, Int. J. Mod. Phys. B4 (1990) 913; D. Bernard and A. LeClair, Commun. Math. Phys. 142 (1991) 99 [16] P. Fendley and K. Intriligator, NucI. Phys. B372 (1992) 533 [17] E. Witten and D. Olive, Phys. Lett. B78 (1978) 97 [18] R. Jackiw and C. Rebbi, Phys. Rev. D13 (1976) 3398 [19] E. Witten, Nuci. Phys. Bl42 (1978) 285; K. Schoutens, NucI. Phys. B344 (1990) 665 [20] H. Itoyama and T. Oota, OU-HET-l82, September 1993, hep-th 9309117 [211 G. Lustig, Comtemp. Math. 82 (1989) 59; Geometriae Dedicata 35 (1990) 89; A. LeClair and C. Vafa, Nucl. Phys. B401 (1993) 413; T. Klassen and E. Melzer, NucI. Phys. B382 (1992) 441 [22] 5. Cecotti and C. Vafa, NucI. Phys. B367 (1991) 359; R. Dijkgraaf, H. Verlinde and E. Verlinde, Nuci. Phys. B352 (1991) 59 [23] W.P. Su, JR. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; Phys. Rev. B22 (1980) 2099 [24] M. Karowski and H.J. Thun, NucI. Phys. B130 (1977) 295 [25] H.W.J. Bläte, J.L. Cardy and M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742; I. Affleck, Phys. Rev. Lett. 56 (1986) 746; C. mtzykson, H. Saleur and J.B. Zuber, Europhys. Lett. 2 (1986) 91 [26] P. Ginsparg, in Fields, strings and critical phenomena, Les Houches 1988, ed. E. Brezin and J. Zinn-Justin (North-Hollland, Amsterdam, 1989) [27] P. Ginsparg, NucI. Phys. B295 (1988) 153 [28] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nuci. Phys. B241 (1984) 333 [29] D. Friedan and S. Shenker, in Conformal invariance and application to statistical mechanics, ed. C. Itzykson, H. Saleur and J.B. Zuber (World Scientific, Singapore, 1988) [30] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253