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Sine-Gordon theory at rational values of the coupling constant and minimal conformal models
Volume 235, number 3,4
PHYSICS LETTERS B
1 Februa~ 1990
S I N E - G O R D O N T H E O R Y AT R A T I O N A L VALUES OF T H E C O U P L I N G C O N S T A N T AND M I N I M A L C O N F O R M A L M O D E L S Tohru E G U C H l Department of Physics, Universityof Tokyo, Tokyo 113, Japan and Sung-Kil YANG Research lnstituteJor Fundamental Physics, Kyoto University, Kyoto 606, Japan Received 2 November 1989
We discuss sine-Gordon theory at rational values of the coupling constant fl'-/2=p/p'. We point out that at these values of/~2 the theory,possesses an underlying BRST symmetry and the higher soliton sectors are dccouplcd from the physical Hilbert space.
In this article we discuss sine-Gordon theory at rational values o f the coupling constant fl2/2 = p / p ' with p, p' coprime integers and p < p' ~. These are the special values of the coupling constant onto which Virasoro minimal models with the central charge c = I - 6 ( p ' - p ) / p p ' are mapped under perturbation by the ( 1, 3) primary operator [ 1-5 ]. We point out that at these values of the coupling constant sine-Gordon theor3' possesses an underlying BRST invariance and the symmetry of the quantum group SUq(2) with q=exp(i2zt/fl 2). Solitons and anti-solitons form a spin ~ representation of SUq(2 ) and the soliton sectors with spinj>~ ½( p - 1 ) arc decoupied from the physical Hilbert space. We also discuss the absence of conserved currents at grades ( 2 n + l ) X (odd integer) in sine-Gordon theory a t f 1 2 / 2 = 2 / ( 2 n + 3 ) ( n = 0 , 1, 2, ...). Let us consider the sine-Gordon theory in the light-cone quantization O~azO= ( m Z / f l 2)[ H, a:0] ,
(1)
where the hamiltonian is given by
{2)
H = ~ cosfl~(z) d z ,
with fl2/2 = p / p ' . We regard z as the space and -~as the time variable, respectively. 0 (z) in (2) is the sinc-Gordon field in the "interaction rcpresentation" and is a masslcss frec scalar field. Conscrved chargcs of the hamiltonian (2) can be constructed using the idea of conformal field theor3' [6,1 ]. Wc introduce an operator T(z)=-½[0z~(z)]2+io~o0Z,~(z),
o~o=fl/2-1/fl,
(3, 4)
which is the e n e r g y - m o m e n t u m tensor of the minimal conformal model M~,.p, with c= 1 - 12~xg = 1 - 6 (p' - p ) 2 / pp' in the Feigin-Fuchs formalism. It is well known that under the condition (4) .~ exp[iflC,(z) ]dz commutes with all Virasoro operators L n = 4 ( I/2zti) d z T ( z ) z "+' and is the screening operator of the them2¢ Mp,p,. We next construct differential polynomials P~,(z) of T(z) at various grades N which are invariant under ~-, -,e) up to ,t Our normalization offl 2 differs from the conventional one of continuum theory by a factor 4n. 282
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PHYSICS LETTERS B
1 February. 1990
total derivatives (T(z) carries grade 2 and the derivative O~ carries grade I), Then the line integrals I,~,_ l = ~ dz 1~, ( z ) co m mute with both ~ d w exp [ ifl~ (w) ] and ,~d w exp [ - ifl~ ( w ) ] and become con served charges of the sine-Gordon theory. IN is known tO exist at each odd grade N. Note that the vertex operator exp [ - i[3~(w) ] has a dimension h = [ (p' - 3p) 2 - (p' _p)2 ]/4pp' and is identified as the ( 1, 3) operator of the Mp,p, theory. Thus the sine-Gordon system at fl2/2 =p/p' can be interpreted as the minimal theory Mp,p, deformed by the ( 1, 3) operator. In conformal models Virasoro operators are identified as the commutants of the screening operators. In the case ofeqs. (3), (4) screening operators are given by exp[i[3O(w)],
exp[-i(2/fl)~(w)].
(5)
In the sine-Gordon system (2) one of these exp[iflq~(w) ] becomes a part of the Hamillonian, however, the other piece e x p [ - i ( 2 / f l ) ~(w) ] still serves as a screening operator. This is the origin of BRST symmet~' in sine-Gordon theory. In fact it is easy to see that the integral Q = ~ e x p [ - i ( 2 / f l ) ~(w) ] dw
(6)
commutes with the hamiltonian tl and also with the infinity of integrals of motion {1,v} and can be used as the BRST operator in defining the physical sectors of the theory. In order to set up the machinery of the BRST formalism [7 ], we first introduce a lattice in the possible values o f t h c m o m e n t u m P o f t h c scalar field 0 ( z ) as
l'=pr=(r-l)/[3,
reT/
(7)
The subspace of the Fock space of the scalar field with the m o m e n t u m P=Pr is denoted as Ft. Since ~[exp(2zri) z] = O ( z ) + 2 ~ z ( r - 1 )/[3 when ~ acts on F~, the hamiltonian density cosfl@ is well-defined (singlevalued) on Fr. Subspaces F~ with r:~ 1 are interpreted as soliton sectors of the theory. On the module F, the rfold integral @ = ~ . . . ~ e x p [ - i ( 2 / [ 3 ) (o(w~)]...exp[-i(2/[3) 0(w~) ] dwt ... dw~
(8)
is well-defined and maps F~ onto F_, (in (8) the wi are ordered as I w~l > I w21 > ...> I w, I and are integrated from w~ to exp (2~ri) w~ in the counterclockwise direction). When [32/2 is a rational number [32/2 =p/p', @ can also act on F,p+,. ( n e T ) and forms a BRST complex . . . . F2p+~ ~
F2p_~ -9":z, F ~ - Q F_~ Q"v-~ F~_2p . . . . .
Q~Qt,_~=Qv_~Q~=O.
(9, 10)
Physical states arc defined as the BRST cohomology classes Ker Q J i m Qt,-r.
( 11 )
As in the case of unperturbed conformal theory [7], non-trivial cohomology classes occur in the modules F~, r = 1, ..., p - 1. These are the physical sectors of the sine-Gordon theory at [32~2=pip'. All the other sectors F2,w_+~( 1 ~
P=p~,~ = ( r - I )/[3+ ( 1--s)[3/2
(12)
and conformal blocks are labelled by r and s with 1 <~r<~p- 1, 1 <~s<~p' - 1. Under perturbation by the ( r = 1, s = 3 ) operator there occurs an arbitrary number of insertions of the m o m e n t u m P 1,3= - [ 3 and s is no longer a good quantum number. Thus we are left with only the sector quantum number r in sine-Gordon theory,. 283
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PHYSICS LETTERS B
1 February 1990
In order to elucidate the physical significance of the label r it is useful to look at the structure of the sineGordon S-matrix [ 8 ]. The soliton-soliton S-matrix is given by
s ( 0 ) = - (i/g) sh((8g/;,)(i~z-0)) U(O),
~3)
where U(0) = [ ' ( 1 + i80//) I ' ( 1 - 8 g / ~ , - i 8 0 / ' / )
H R,,(O) R , ( i g - 0 ) ,
14)
n=l
F( 2nSg/7+ iSO/7) F( 1 + 2nSn/7+ iSO/;')
15)
R , ( 0 ) = / - ' ( ( 2 n + 1 ) 8rc/7+i80/y) F(1 + ( 2 n - 1 ) 8g/7+i80/7) '
y= 4nil2~ ( 1-fl2/2) and 0 is the rapidity difference of solitons. The anti-soliton-anti-soliton S-matrix is also given by ( 13 ). Transmission and reflection amplitudes of the soliton-anti-soliton scattering are given by Sr(O)=-(i/g)sh((8nO/7)U(O)),
SR(0)=(l/g)sin((8g2/y)
U(O)
(16, 17)
It is well known that the S-matrices (13), (16), (17) assemble into the structure of the R-matrix of the sixvertex model in statistical physics when wc regard (anti-) soliton as up (down) or right (left) arrows coming into the vertex. In particular the sine-Gordon S-matrix commutes with the action of the quantum group S U q ( 2 ) with q = - e x p ( - i 8 g 2 / y ) and the soliton, anti-soliton form a spin ½ representation of SUq(2). In the present case fl2/2 =p/p', q= exp ( - i g p ' / p ) and thus qp= + 1. This is the degenerate case of the quantum group [9] and the allowed values of the spin j are restricted to j = 0 , ~, 1, ..., ½ p - 1. Thus the sectors with spin greater than ~ p - 1 are truncated out of the Hilbert space. Therefore the sector quantum number r is identified as r = 2j+ 1 and corresponds to the height variable of the SOS formulation of the six-vertex model. Solitons and anti-solitons intertwine neighboring sectors r, r' with [r-r'[=l. In the special case o f p = 2, p' = 2n + 3 and [32/2 = 1/ (n + 3), which corresponds to the deformation of minimal non-unitary series M2,2,+ s with c = 1 - 3 (2n + 1 )2/(2n + 3 ), the soliton sectors are completely eliminated from the physical Hilbert space. This result has been obtained by Smirnov [2] based on the analysis o f l b r m factors of sine-Gordon theory. When q is a root of unity, the six-vertex modcl with SUq(2) symmctr2,' can be reformulated as the SOS model with restrictions on the values of its local height variables [ 10]. The relevance of the restricted SOS model to the sine-Gordon S matrix at rational values of the coupling constant has also been noted by LeClair [ 3 ]. The fact that (anti-) sol irons tbrm a representation of the quantum group SUq(2) implies that they obey an unusual (fractional statistics. It is well known [ 11 ] that a q-deformed version of the symmetric group (Hecke algebra) operates in the decomposition of the tensor product of representations of the quantum group into its irreducible components. In the process of symmetrization or anti-symmetrization one multiplies q (or q - l ) when one permutes factors of the tensor product. This means that the solitons obey a fractional statistics with the characteristic phase factor q = - e x p ( - i 8 g 2 / ~ . ' ) . We note that q = + 1 at [32/2= l/n and at these values oft3 the solitons are ordinary fermions as usually considered ( n = 1 is the Kosterlitz-Thouless transition point and n = 2 is the point of free fermion theory). Let us now go back to the models M2.2,,+s and discuss some special features of the sine-Gordon theory, at f12/ 2 = l / ( n + 3) ( n = 1, 2, ...). At these points there exists an additional BRST operator exp[i(2/[3) ~(z) ]. This operator commutes with the hamiltonian and the conserved currents, however, at generic values oft32 it is mutually non-local with the original BRST operator exp [ - i (2/,8) 0 (z) ]. When [32/2 = 1/ ( n + ~), however, exp [i ( 2 / [3) ~(z) ] and exp [ - i (2///) 0 ( z ) ]generate a pole of order 2n + 3 under OPE and have a local anti-commutation rclation. Then 284
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I2,+~={~exp[i(2/13) O(z)]dz,
~exp[-i(2/fl) O(w)]dw}
I Februar),' 1990
(18)
gives an integral of motion at grade 2 n + 1 which is BRST trivial. This means that the conserved charge of grade 2n + 1 disappears from the theory at f12/8~z= 1/ (n + 3). A more general construction I(2,,+ ~,.,a = { ~ ... ~ e x p [ i ( 2 / f l ) ~ ( z , )] ... c x p [ i ( 2 / f l ) O ( z M ) ] dz~ ... dzM,
~... ~ e x p [ - i ( 2 / f l ) ¢ ( w ~ ) ]
... e x p [ - i ( 2 / f l ) 0 ( w M ) ]
dw, ... dwa,}
(19)
gives "missing" currents at grades ( 2 n + I ) M ( M = o d d ) . Thus out of the conserved charges ~lu, N = a l l odd integers} at generic f12, we loose charges {IN, N--- (2n + 1 ) × (odd integer)} at 13z/2 = 1/(n + 3 ). These are exactly the system of conserved currents which were assumed in the computation of the S-matrix in the perturbed Mz,2n+3 theories [12,13 ]. In fact the proposed S-matrices of refs. [12,13] agree exactly with the brother Smatrices in sine-Gordon theory at 132/2= 1 / ( n +3). Note that if we put n =0,132/2 =2 which corresponds to the trivial conformal theory M2,3 with c = 0 . At this point all the conserved charges {l.,v} altogether disappear and in fact the sine-Gordon model becomes a totally trivial theory. Values offl z in the allowed range 0 ~
( 1 ) 132/2=p/p', p, p" coprime, p
(3)132/2=1/n. Cases (2), ( 3 ) are interpreted as the deformed c = 1 conformal field theory at various values of the radius of thc scalar field 0. At generic values of the radius, the conserved charges of the deformed theory occur in the cntire Fock space of the scalar field. At special values of the radius corresponding to the c a s c ( 1 ), however, conserved charges occur in the subspace of the Fock space, i.e. the representation space of the Virasoro algebra of Mp,p, theory. Then the sine-Gordon theory is reinterpreted as the deformation of minimal theories with c < 1. There is neither BRST symmetry not" truncation of soliton sectors in the cases (2), (3). Next we c o m m e n t on the extra conserved charge at grade n - 1 (mod 2 ( n - 1 ) ) in thc sine-Gordon theory of case (3) notcd in ref. [5 ]. In this casc thc BRST operators exp [ + i(2/13) 0 ( z ) ] arc interpreted as chiral current operators with dimension h=n. Thus their integrals givc conserved charges at grade n - 1. The antisymmetric combination ~{exp [ i ( 2/13) 0 (z) ] - exp [ - i ( 2/fl) 0 ( z ) ] } dz is odd under charge conjugation and prohibits the rctlection amplitudes in soliton-anti-soliton scattering. The absence of the reflection at these values of the coupling constant flz/2 = 1/n is well known in the classical soliton theory. Extra conserved charges at gradcs ( n - 1 ) M ( M = 3 , 5 .... ) are similarly constructed by M-fold integrals o f c x p [ q_-i(2/fl) 0 ( z ) ]. Now we would like to discuss the relationship between the sine-Gordon descriptions of deformed conformal models and those by thc G i n z b u r g - L a n d a u cffective lagrangian [ 14]. In the Ginzburg-Landau description of unitary conformal theory Mp=m,p,=m+lone introduces an effective potential V c ( ~ ) = ~ 12(m- 1) which represents the ( r n - 1 )th fold degenerate nature of its critical point. When (Z2-invariant) relevant perturbations are applied to the theory, V(0) is turned into a polynomial of degree 2 ( m - 1 ), V ( O ) = ~ i - ~ m - ~ ) a , O z~. V(O) genericaIly has m - 1 distinct minima. On the other hand it is well known that the conformal theory M,,.,,,+ ~ is reproduced at a critical point of the m-state restricted SOS (RSOS) model [ 10] on a border of its parameter space [ 15 ]. From the specific heat exponent one sees that the departure from the criticality in the RSOS model corresponds to applying the ( 1, 3) perturbation to the M .... + t theory. One of the crucial features of RSOS models is the existence of vacuum 285
Volume 235, number 3,4
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degeneracy at off-criticality. In the case o f the m-state RSOS m o d e l there exist ( m - 1 ) degenerate vacua (in the m o r e general class o f m o d e l s c o r r e s p o n d i n g to m i n i m a l theories Mp,p, there exist p - 1 degenerate vacua [ 16 ] ). T h u s the coefficients o f the effective potential m u s t be suitably a d j u s t e d so that its m i n i m a have exactly the same energy. O n the other h a n d in the s i n e - G o r d o n d e s c r i p t i o n the v a c u u m degeneracy is m a n i f e s t due to the periodicity o f the cosine f u n c t i o n . While in the " i r r a t i o n a l " cases o f ( 2 ) a n d ( 3 ) the v a c u u m is infinitely degenerate, in the rational case ( 1 ) with fl2/2 =p/p' the theory c a n n o t s u p p o r t higher soliton sectors a n d there exist only ( p - 1 ) degenerate vacua c o r r e s p o n d i n g to thc presence o f ( p - 1 ) B R S T sectors. T h u s d u e to the t r u n c a t i o n o f the Hilbert space the cosine f u n c t i o n gets effectively t u r n e d into a p o l y n o m i a l o f degree 2 ( p - 1 ). T h e s i n e - G o r d o n d e s c r i p t i o n o f the p e r t u r b e d c o n f o r m a l theories thus appears consistent with the characteristic features o f RSOS m o d e l s a n d the ideas o f the effective lagrangian. Finally, we note that the c o m m e n s u r a t e - i n c o m m e n s u r a t e effect in the s i n e - G o r d o n system has been also observed in some other contexts [ 17,18 ]. Recently Schroer [ 19 ] has e m p h a s i z e d its possible significance in c o n n e c t i o n with the theory" o f fractional statistics a n d a n t i - f e r r o m a g n e t i c q u a n t u m spin systems.
References [ 1] T. Eguchi and S.-K. Yang, Phys. Left. B 224 ( 1989 ) 373. [2] F.A. Smirnov, Intern. J. Mod. Phys. A 4 (1989) 4231; Reduction of quantum sine-Gordon model as perturbations of minimal models ofconformal field theory., LOMI preprint E-4-89 (1989). [ 3] A. LeClare, Restricted sine-Gordon theory and the minimal conformal series, Princeton preprint PUPT-1124 (1989). [4] T.J. Hollowood and P. Mansfield, Phys. Lett. B 226 (1989) 73. [ 5 ] V.A. Fateev and A.B. Zamolodchikov, Conformal field theory and purely elastic S-matrix, Landau Institute preprint ( 1989). [6] R. Sasaki and I. Yamanaka, Adv. Stud. Pure Math. 16 (1988) 271. [7] G. Fclder, Nucl. Phys. B 317 (1989) 215; G. Felder, J. Frohlich and G. Keller, Commun. Math. Phys. 124 (1989) 647. [8] A.B. Zamolodchikov, Sov. Phys. JETP Lett. 25 (1977) 499; M. Karowski and H.J. Thun, Nucl. Phys. B 130 (1977) 295. [ 9 ] G. Lustig, Modular representations and quantum groups, MIT preprint ( 1988 ); H. Saleur, Representations of Uqsl(2) for q a root of unity, Saclay preprint ( 1989 ). [ 10] G.E. Andrews, R.J. Baxter and P.J. Forrester, J. Star. Phys. 35 (1984) 93. [ 11 ] M. Jimbo, Lecture Notes in Physics, Vol. 246, (Springer, Berlin, 1986)p. 335. [ 12] J.L. Cardy and G. Mussardo, Phys. Lett. B 225 (1989) 275. [ 13 ] P.G.O. Freund, T.R. Klasser and E. Melzer, S-matrices for perturbations of ccrtain conformal field theories, Chicago preprint EFI 89-29 (1989). [ 14] A.B. Zamolodchikov, Sov. J. Nucl. Phys, 44 (1986) 529. [ 15 ] D.A. Huse, Phys. Rev. B 30 (1984) 3908. [ 16] P.J. Forrester and R.J. Baxter, J. Stat. Phys. 38 (1985) 435. [ 17 ] A.G. Izergin and V.E. Korepin, Len. Math. Phys. 5 ( 1981 ) 199. [ 18] J. Frohlich, in: Renormalization theo~, cd. G. Velo ct al. (Reidel, Dordrecht, 1976) p. 371. [ 19] B. Schroer, High T¢ superconductivity and a new symmetry' concept in low dimensional quantum field theories, Berlin preprint (1989).
286
PHYSICS LETTERS B
1 Februa~ 1990
S I N E - G O R D O N T H E O R Y AT R A T I O N A L VALUES OF T H E C O U P L I N G C O N S T A N T AND M I N I M A L C O N F O R M A L M O D E L S Tohru E G U C H l Department of Physics, Universityof Tokyo, Tokyo 113, Japan and Sung-Kil YANG Research lnstituteJor Fundamental Physics, Kyoto University, Kyoto 606, Japan Received 2 November 1989
We discuss sine-Gordon theory at rational values of the coupling constant fl'-/2=p/p'. We point out that at these values of/~2 the theory,possesses an underlying BRST symmetry and the higher soliton sectors are dccouplcd from the physical Hilbert space.
In this article we discuss sine-Gordon theory at rational values o f the coupling constant fl2/2 = p / p ' with p, p' coprime integers and p < p' ~. These are the special values of the coupling constant onto which Virasoro minimal models with the central charge c = I - 6 ( p ' - p ) / p p ' are mapped under perturbation by the ( 1, 3) primary operator [ 1-5 ]. We point out that at these values of the coupling constant sine-Gordon theor3' possesses an underlying BRST invariance and the symmetry of the quantum group SUq(2) with q=exp(i2zt/fl 2). Solitons and anti-solitons form a spin ~ representation of SUq(2 ) and the soliton sectors with spinj>~ ½( p - 1 ) arc decoupied from the physical Hilbert space. We also discuss the absence of conserved currents at grades ( 2 n + l ) X (odd integer) in sine-Gordon theory a t f 1 2 / 2 = 2 / ( 2 n + 3 ) ( n = 0 , 1, 2, ...). Let us consider the sine-Gordon theory in the light-cone quantization O~azO= ( m Z / f l 2)[ H, a:0] ,
(1)
where the hamiltonian is given by
{2)
H = ~ cosfl~(z) d z ,
with fl2/2 = p / p ' . We regard z as the space and -~as the time variable, respectively. 0 (z) in (2) is the sinc-Gordon field in the "interaction rcpresentation" and is a masslcss frec scalar field. Conscrved chargcs of the hamiltonian (2) can be constructed using the idea of conformal field theor3' [6,1 ]. Wc introduce an operator T(z)=-½[0z~(z)]2+io~o0Z,~(z),
o~o=fl/2-1/fl,
(3, 4)
which is the e n e r g y - m o m e n t u m tensor of the minimal conformal model M~,.p, with c= 1 - 12~xg = 1 - 6 (p' - p ) 2 / pp' in the Feigin-Fuchs formalism. It is well known that under the condition (4) .~ exp[iflC,(z) ]dz commutes with all Virasoro operators L n = 4 ( I/2zti) d z T ( z ) z "+' and is the screening operator of the them2¢ Mp,p,. We next construct differential polynomials P~,(z) of T(z) at various grades N which are invariant under ~-, -,e) up to ,t Our normalization offl 2 differs from the conventional one of continuum theory by a factor 4n. 282
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
Volume 235, number 3,4
PHYSICS LETTERS B
1 February. 1990
total derivatives (T(z) carries grade 2 and the derivative O~ carries grade I), Then the line integrals I,~,_ l = ~ dz 1~, ( z ) co m mute with both ~ d w exp [ ifl~ (w) ] and ,~d w exp [ - ifl~ ( w ) ] and become con served charges of the sine-Gordon theory. IN is known tO exist at each odd grade N. Note that the vertex operator exp [ - i[3~(w) ] has a dimension h = [ (p' - 3p) 2 - (p' _p)2 ]/4pp' and is identified as the ( 1, 3) operator of the Mp,p, theory. Thus the sine-Gordon system at fl2/2 =p/p' can be interpreted as the minimal theory Mp,p, deformed by the ( 1, 3) operator. In conformal models Virasoro operators are identified as the commutants of the screening operators. In the case ofeqs. (3), (4) screening operators are given by exp[i[3O(w)],
exp[-i(2/fl)~(w)].
(5)
In the sine-Gordon system (2) one of these exp[iflq~(w) ] becomes a part of the Hamillonian, however, the other piece e x p [ - i ( 2 / f l ) ~(w) ] still serves as a screening operator. This is the origin of BRST symmet~' in sine-Gordon theory. In fact it is easy to see that the integral Q = ~ e x p [ - i ( 2 / f l ) ~(w) ] dw
(6)
commutes with the hamiltonian tl and also with the infinity of integrals of motion {1,v} and can be used as the BRST operator in defining the physical sectors of the theory. In order to set up the machinery of the BRST formalism [7 ], we first introduce a lattice in the possible values o f t h c m o m e n t u m P o f t h c scalar field 0 ( z ) as
l'=pr=(r-l)/[3,
reT/
(7)
The subspace of the Fock space of the scalar field with the m o m e n t u m P=Pr is denoted as Ft. Since ~[exp(2zri) z] = O ( z ) + 2 ~ z ( r - 1 )/[3 when ~ acts on F~, the hamiltonian density cosfl@ is well-defined (singlevalued) on Fr. Subspaces F~ with r:~ 1 are interpreted as soliton sectors of the theory. On the module F, the rfold integral @ = ~ . . . ~ e x p [ - i ( 2 / [ 3 ) (o(w~)]...exp[-i(2/[3) 0(w~) ] dwt ... dw~
(8)
is well-defined and maps F~ onto F_, (in (8) the wi are ordered as I w~l > I w21 > ...> I w, I and are integrated from w~ to exp (2~ri) w~ in the counterclockwise direction). When [32/2 is a rational number [32/2 =p/p', @ can also act on F,p+,. ( n e T ) and forms a BRST complex . . . . F2p+~ ~
F2p_~ -9":z, F ~ - Q F_~ Q"v-~ F~_2p . . . . .
Q~Qt,_~=Qv_~Q~=O.
(9, 10)
Physical states arc defined as the BRST cohomology classes Ker Q J i m Qt,-r.
( 11 )
As in the case of unperturbed conformal theory [7], non-trivial cohomology classes occur in the modules F~, r = 1, ..., p - 1. These are the physical sectors of the sine-Gordon theory at [32~2=pip'. All the other sectors F2,w_+~( 1 ~
P=p~,~ = ( r - I )/[3+ ( 1--s)[3/2
(12)
and conformal blocks are labelled by r and s with 1 <~r<~p- 1, 1 <~s<~p' - 1. Under perturbation by the ( r = 1, s = 3 ) operator there occurs an arbitrary number of insertions of the m o m e n t u m P 1,3= - [ 3 and s is no longer a good quantum number. Thus we are left with only the sector quantum number r in sine-Gordon theory,. 283
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In order to elucidate the physical significance of the label r it is useful to look at the structure of the sineGordon S-matrix [ 8 ]. The soliton-soliton S-matrix is given by
s ( 0 ) = - (i/g) sh((8g/;,)(i~z-0)) U(O),
~3)
where U(0) = [ ' ( 1 + i80//) I ' ( 1 - 8 g / ~ , - i 8 0 / ' / )
H R,,(O) R , ( i g - 0 ) ,
14)
n=l
F( 2nSg/7+ iSO/7) F( 1 + 2nSn/7+ iSO/;')
15)
R , ( 0 ) = / - ' ( ( 2 n + 1 ) 8rc/7+i80/y) F(1 + ( 2 n - 1 ) 8g/7+i80/7) '
y= 4nil2~ ( 1-fl2/2) and 0 is the rapidity difference of solitons. The anti-soliton-anti-soliton S-matrix is also given by ( 13 ). Transmission and reflection amplitudes of the soliton-anti-soliton scattering are given by Sr(O)=-(i/g)sh((8nO/7)U(O)),
SR(0)=(l/g)sin((8g2/y)
U(O)
(16, 17)
It is well known that the S-matrices (13), (16), (17) assemble into the structure of the R-matrix of the sixvertex model in statistical physics when wc regard (anti-) soliton as up (down) or right (left) arrows coming into the vertex. In particular the sine-Gordon S-matrix commutes with the action of the quantum group S U q ( 2 ) with q = - e x p ( - i 8 g 2 / y ) and the soliton, anti-soliton form a spin ½ representation of SUq(2). In the present case fl2/2 =p/p', q= exp ( - i g p ' / p ) and thus qp= + 1. This is the degenerate case of the quantum group [9] and the allowed values of the spin j are restricted to j = 0 , ~, 1, ..., ½ p - 1. Thus the sectors with spin greater than ~ p - 1 are truncated out of the Hilbert space. Therefore the sector quantum number r is identified as r = 2j+ 1 and corresponds to the height variable of the SOS formulation of the six-vertex model. Solitons and anti-solitons intertwine neighboring sectors r, r' with [r-r'[=l. In the special case o f p = 2, p' = 2n + 3 and [32/2 = 1/ (n + 3), which corresponds to the deformation of minimal non-unitary series M2,2,+ s with c = 1 - 3 (2n + 1 )2/(2n + 3 ), the soliton sectors are completely eliminated from the physical Hilbert space. This result has been obtained by Smirnov [2] based on the analysis o f l b r m factors of sine-Gordon theory. When q is a root of unity, the six-vertex modcl with SUq(2) symmctr2,' can be reformulated as the SOS model with restrictions on the values of its local height variables [ 10]. The relevance of the restricted SOS model to the sine-Gordon S matrix at rational values of the coupling constant has also been noted by LeClair [ 3 ]. The fact that (anti-) sol irons tbrm a representation of the quantum group SUq(2) implies that they obey an unusual (fractional statistics. It is well known [ 11 ] that a q-deformed version of the symmetric group (Hecke algebra) operates in the decomposition of the tensor product of representations of the quantum group into its irreducible components. In the process of symmetrization or anti-symmetrization one multiplies q (or q - l ) when one permutes factors of the tensor product. This means that the solitons obey a fractional statistics with the characteristic phase factor q = - e x p ( - i 8 g 2 / ~ . ' ) . We note that q = + 1 at [32/2= l/n and at these values oft3 the solitons are ordinary fermions as usually considered ( n = 1 is the Kosterlitz-Thouless transition point and n = 2 is the point of free fermion theory). Let us now go back to the models M2.2,,+s and discuss some special features of the sine-Gordon theory, at f12/ 2 = l / ( n + 3) ( n = 1, 2, ...). At these points there exists an additional BRST operator exp[i(2/[3) ~(z) ]. This operator commutes with the hamiltonian and the conserved currents, however, at generic values oft32 it is mutually non-local with the original BRST operator exp [ - i (2/,8) 0 (z) ]. When [32/2 = 1/ ( n + ~), however, exp [i ( 2 / [3) ~(z) ] and exp [ - i (2///) 0 ( z ) ]generate a pole of order 2n + 3 under OPE and have a local anti-commutation rclation. Then 284
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I2,+~={~exp[i(2/13) O(z)]dz,
~exp[-i(2/fl) O(w)]dw}
I Februar),' 1990
(18)
gives an integral of motion at grade 2 n + 1 which is BRST trivial. This means that the conserved charge of grade 2n + 1 disappears from the theory at f12/8~z= 1/ (n + 3). A more general construction I(2,,+ ~,.,a = { ~ ... ~ e x p [ i ( 2 / f l ) ~ ( z , )] ... c x p [ i ( 2 / f l ) O ( z M ) ] dz~ ... dzM,
~... ~ e x p [ - i ( 2 / f l ) ¢ ( w ~ ) ]
... e x p [ - i ( 2 / f l ) 0 ( w M ) ]
dw, ... dwa,}
(19)
gives "missing" currents at grades ( 2 n + I ) M ( M = o d d ) . Thus out of the conserved charges ~lu, N = a l l odd integers} at generic f12, we loose charges {IN, N--- (2n + 1 ) × (odd integer)} at 13z/2 = 1/(n + 3 ). These are exactly the system of conserved currents which were assumed in the computation of the S-matrix in the perturbed Mz,2n+3 theories [12,13 ]. In fact the proposed S-matrices of refs. [12,13] agree exactly with the brother Smatrices in sine-Gordon theory at 132/2= 1 / ( n +3). Note that if we put n =0,132/2 =2 which corresponds to the trivial conformal theory M2,3 with c = 0 . At this point all the conserved charges {l.,v} altogether disappear and in fact the sine-Gordon model becomes a totally trivial theory. Values offl z in the allowed range 0 ~
( 1 ) 132/2=p/p', p, p" coprime, p
(3)132/2=1/n. Cases (2), ( 3 ) are interpreted as the deformed c = 1 conformal field theory at various values of the radius of thc scalar field 0. At generic values of the radius, the conserved charges of the deformed theory occur in the cntire Fock space of the scalar field. At special values of the radius corresponding to the c a s c ( 1 ), however, conserved charges occur in the subspace of the Fock space, i.e. the representation space of the Virasoro algebra of Mp,p, theory. Then the sine-Gordon theory is reinterpreted as the deformation of minimal theories with c < 1. There is neither BRST symmetry not" truncation of soliton sectors in the cases (2), (3). Next we c o m m e n t on the extra conserved charge at grade n - 1 (mod 2 ( n - 1 ) ) in thc sine-Gordon theory of case (3) notcd in ref. [5 ]. In this casc thc BRST operators exp [ + i(2/13) 0 ( z ) ] arc interpreted as chiral current operators with dimension h=n. Thus their integrals givc conserved charges at grade n - 1. The antisymmetric combination ~{exp [ i ( 2/13) 0 (z) ] - exp [ - i ( 2/fl) 0 ( z ) ] } dz is odd under charge conjugation and prohibits the rctlection amplitudes in soliton-anti-soliton scattering. The absence of the reflection at these values of the coupling constant flz/2 = 1/n is well known in the classical soliton theory. Extra conserved charges at gradcs ( n - 1 ) M ( M = 3 , 5 .... ) are similarly constructed by M-fold integrals o f c x p [ q_-i(2/fl) 0 ( z ) ]. Now we would like to discuss the relationship between the sine-Gordon descriptions of deformed conformal models and those by thc G i n z b u r g - L a n d a u cffective lagrangian [ 14]. In the Ginzburg-Landau description of unitary conformal theory Mp=m,p,=m+lone introduces an effective potential V c ( ~ ) = ~ 12(m- 1) which represents the ( r n - 1 )th fold degenerate nature of its critical point. When (Z2-invariant) relevant perturbations are applied to the theory, V(0) is turned into a polynomial of degree 2 ( m - 1 ), V ( O ) = ~ i - ~ m - ~ ) a , O z~. V(O) genericaIly has m - 1 distinct minima. On the other hand it is well known that the conformal theory M,,.,,,+ ~ is reproduced at a critical point of the m-state restricted SOS (RSOS) model [ 10] on a border of its parameter space [ 15 ]. From the specific heat exponent one sees that the departure from the criticality in the RSOS model corresponds to applying the ( 1, 3) perturbation to the M .... + t theory. One of the crucial features of RSOS models is the existence of vacuum 285
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degeneracy at off-criticality. In the case o f the m-state RSOS m o d e l there exist ( m - 1 ) degenerate vacua (in the m o r e general class o f m o d e l s c o r r e s p o n d i n g to m i n i m a l theories Mp,p, there exist p - 1 degenerate vacua [ 16 ] ). T h u s the coefficients o f the effective potential m u s t be suitably a d j u s t e d so that its m i n i m a have exactly the same energy. O n the other h a n d in the s i n e - G o r d o n d e s c r i p t i o n the v a c u u m degeneracy is m a n i f e s t due to the periodicity o f the cosine f u n c t i o n . While in the " i r r a t i o n a l " cases o f ( 2 ) a n d ( 3 ) the v a c u u m is infinitely degenerate, in the rational case ( 1 ) with fl2/2 =p/p' the theory c a n n o t s u p p o r t higher soliton sectors a n d there exist only ( p - 1 ) degenerate vacua c o r r e s p o n d i n g to thc presence o f ( p - 1 ) B R S T sectors. T h u s d u e to the t r u n c a t i o n o f the Hilbert space the cosine f u n c t i o n gets effectively t u r n e d into a p o l y n o m i a l o f degree 2 ( p - 1 ). T h e s i n e - G o r d o n d e s c r i p t i o n o f the p e r t u r b e d c o n f o r m a l theories thus appears consistent with the characteristic features o f RSOS m o d e l s a n d the ideas o f the effective lagrangian. Finally, we note that the c o m m e n s u r a t e - i n c o m m e n s u r a t e effect in the s i n e - G o r d o n system has been also observed in some other contexts [ 17,18 ]. Recently Schroer [ 19 ] has e m p h a s i z e d its possible significance in c o n n e c t i o n with the theory" o f fractional statistics a n d a n t i - f e r r o m a g n e t i c q u a n t u m spin systems.
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