Quantum Kac-Moody symmetry in integrable field theories

NUCLEAR

P H VS I C S B

Nuclear Physics B 369 (1992) 433—460 North-Holland

_________________

Quantum Kac—Moody symmetry in integrable field theories Samir D. Mathur Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 7 March 1991 (Revised 24 June 1991) Accepted for publication 19 September 1991

We show that just as rational conformal theories are characterised by quantum algebras, integrable deformations of such theories are characterised by quantum affine (i.e. Kac—Moody) algebras. More precisely, we study the perturbation series around the conformal theory and observe that each term can be decomposed by an algebraic trick into a sum of products of 2) holomorphic and antiholomorphic “blocks”. For the perturbation giving integrability, the SUq( describing the conformal theory extends to an SUq(2) for the “blocks” of the deformed theory. The primary fields c6i~(to which we restrict for simplicity) correspond to level-k integrable representations of SU 2÷= 2p/p’, k = p’ —2. Of the two independent null 5(2), with q =one e’~-,a vectors in such a representation is the BRST closure condition of the conformal theory, while the other reflects the finiteness of the primary operator content of the theory.

1. Introduction Two different algebraic structures arise in the study of two-dimensional conformal field theories. The conformal algebra, or an extension of it, links a primary state to an infinite number of secondaries. This is a holomorphic symmetry, because it appears in a direct tensor product with its antiholomorphic counterpart, and it leads to a “holomorphic block structure” of correlation functions [1]. A rational conformal theory contains a finite number of primaries, which are by definition not related by the holomorphic symmetry. The second algebraic structure in the theory is the braiding and fusion relations of the blocks corresponding to different primaries. It is now quite well established that this “non-holomorphic” structure of the theory is described by quantum algebras [2—6]. It was shown by Zamolodchikov [71that rational conformal theories possess deformations that are not conformal but are nevertheless integrable. An example is the deformation of the c < 1 minimal models by the energy operator ~ In this paper we show that these c < 1 models perturbed in this fashion are characterised by a quantum affine (i.e. Kac—Moody) algebra, which reduces in the conformal 0550-3213/92/$05.00 © 1992

—

Elsevier Science Publishers B.V. All rights reserved

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limit to the quantum algebra of the conformal theory. More precisely, we show that the perturbation terms arising from the deformation arise in a “holomorphic block structure”, with the blocks forming a realisation of the quantum affine algebra. The grading of the affine algebra corresponds to the orders of perturbation theory. There are different algebras corresponding to a~and a charges; we concentrate on the a~sector in this paper. The “left” and “right” sectors also give independent algebras, though their representations are paired in obtaining the complete correlator. In all that follows we assume the diagonal A type modular invariant for the theory. In more detail, our approach and results are described by the following steps: (i) We discuss an algebraic trick which converts a two-dimensional integral f d2z~1 f(z1,..., z,~,z) 2 into a sum of products of holomorphic and antiholomorphic contour integrals. Thus real integrals of this type can be expressed in terms of holomorphic “blocks” and their complex conjugates, where the blocks are given by contour integrals of the f~ starting and ending at one of the singularities z3 of the f1. (ii) Before addressing the perturbed theory we apply this algebraic result to the c < 1 rational conformal theories. We recall that correlation functions in these theories can be expressed in terms of Feigin—Fuchs contour integrals attached to the vertices of the correlator [8,9]. We use the notation of Felder [9] rather than that of Dotsenko and Fateev [8] so that no charge is placed at infinity and charge balance is required in the finite complex plane. We show that the integral

f fl d2z~(fl emn÷4(

fl

Z~, z~) N ~

M

I({ Z1, 2~})

z 1, 2k))

(1.1)

with M ~Ji gives on resolution into therational correct blocks for the correlator 4(zi,contours 2~)in the conformal field theory of fields Thus represented by e1~~a+ (RCFT). in particular the integral (1.1) manifests all the fusion rules of the RCFT, including the “truncation from above”. (Blocks violating these rules vanish identically or arise with vanishing coefficient in the decomposition of (1.1) by contours.) The above result can be re-expressed by saying that the correlator of the ~ be computed using the action =

S

=

fd2z(Th~

~4+ e~’~-~’).

(1.2)

Note that in (1.2) there is no coupling to the curvature, and so the boson without the perturbing exponential describes a c 1 theory. It turns out that one may equally well use the action =

S

=

fd2Z

34 34

+

—i 2 e1’~’— a..— 4~r

R4

(1.3)

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(R is the curvature of the Z space) if one changes appropriately the amount of charge corresponding to curvature concentrations and primary fields. It has been suggested before [10] that the Liouville action (1.3) “formally” characterises a conformal theory. The term “formal” here refers to the possibility that the Liouville theory may not incorporate the BRST truncations required to give the minimal model from the bosonic theory. It had also been suggested before [8] that the contour integrals defining correlation functions for a~not a rational number might be obtainable from an integration trick like the one mentioned above. What we find here is that for a~ equalling a rational number the correctly truncated theory automatically appears in the consideration of correlation functions of physical states. In the equivalence of (1.2) and (1.3) we are observing the solution of a conformal Liouville theory having c 1 3(a~—2/a~)2 by means of a Backlund transformation to a free c 1 theory [11,12]. (iii) We consider the c < 1 rational theories perturbed by the energy operator =

—

=

4~L3’which

is expressed in the Dotsenko—Fateev language by ~ seen that the action S=fd2z(a~ 3~+emn±4+A ema÷4~(a+_ ~)R~)

It is readily

(1.4)

considered perturbatively in A generates the perturbed series for the deformed conformal theory. (We may equivalently consider the action without the boundary charge, by what was said above.) Expressing all two-dimensional integrals in terms of contours, we have contour integrals over the locations of both ~ and e’~~’ insertions. These two different kinds of contour integrals satisfy commutation relations appropriate to the two simple roots of an SU(2) affine quantum Kac— Moody algebra. Further, the contours that can “enclose” a charge state corresponding to a primary field (J ~1,2J+I can be placed in correspondence with the states in a level-k representation of the quantum affine algebra. Here the quantum deformation parameter is q e”~= e2~’~”, and k =p’ 2. This representation has null states corresponding to each of the two simple root vectors of the affine algebra. The null state corresponding to elt~*~~, the screening charge, is the statement that K] Q 0, where Q is the BRST charge introduced by Felder [9]. The null state corresponding to the perturbation ~ can be expressed as K] I Q 0, where Q is a charge constructed in a fashion analogous to Q. This null vector expresses the fact that the set of primary fields in the rational conformal theory is finite. We must include a curvature 2ir at each external vertex in a correlator, corresponding to the situation where each such operator is placed at the end of a semi-infinite cylinder emerging from the two-dimensional surface. Correspondingly, there must be two points for every external vertex carrying *

=

—

=

=

*

Our definition of q is the square of the more conventional choice of definition.

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curvature —ir and no charge. These points act as level-i representations of the quantum affine algebra. “Fusion” of two vertices requires the absorption of two of these negative curvature points, giving a level of the fused operator equal to k + k + 2 k since the level is measured only mod k + 2 in the quantum algebra. Thus our level-k representations are closed under fusion. This quantum affine symmetry appears to be a quantum version of the Kac— Moody algebra (in the spectral parameter) which arises in affine Toda theories [131.It is larger than the quantum symmetry seen in the solitonic asymptotic states of the perturbed theory [14,15], as it acts independently on the left and right sectors. It is known of course that Yang—Baxter algebras characterise integrable quantum field theories [16,171and that solutions of the Yang—Baxter equation can be found in correspondence with affine Lie algebras [18,191.An important distinction between these studies and our results is that in our case the R-matrix acts on infinite-dimensional representations of the quantum affine algebra with in general non-vanishing level. As may be expected by analogy with the case of ordinary (i.e. non-quantum) affine algebras, the non-vanishing of the level appears to preclude finite-dimensional representations in our physical context. The level is related to the curvature in the physical two-dimensional space (on which the model lives) at the point of insertion of the quantum field theory operator to which the representation corresponds. While this curvature is a natural object in the continuum theory that we are studying (in fact the curvature summed over the various curvature carrying points must total 4~rfor a finite area sphere) it does not appear in a lattice approach to the sine-Gordon equation [20]. In ref. [21] it is shown how starting from a zero-level finite-dimensional representation with spectral parameter of the affine quantum group one can obtain the abstract affine quantum group relations including the central term. We obtain representations of this full algebra, while R-matrices obtained from the scattering of solitons in integrable theories correspond to the above-mentioned finite-dimensional representation. The plan of this paper is the following. In sect. 2 we discuss the integrals arising in the perturbation theory, and establish the decomposition of two-dimensional integrals with “holomorphic block integrands” into contours. In sect. 3 we show that the blocks for any correlator in a c < 1 RCFT arise from the decomposition into contours of a single real integral. In sect. 4 we investigate the contour structure arising from the perturbed theory, and establish that it corresponds to a quantum affine symmetry. Sect. 5 discusses further directions, some conjectures and possible relations to other works.

2. Perturbation terms as contour integrals In this section we recall how the correction terms to any correlation function in a perturbed rational conformal theory can be expressed in terms of contour

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integrals. The essential algebraic idea is not new. It was suggested by Dotsenko and Fateev [81 that in a conformal theory the contours appropriate to the Feigin—Fuchs integral representations of correlations could be deduced, for irrational values of a~,by adding a term f ~ 2) d2z to the action of a free boson. In ref. [221one- and two-point functions on the torus were computed using such a trick. For the more general integrals arising in perturbation theory about the conformal theory, a generalisation of this contour argument was shown to occur in refs. [23,241where in the former the perturbation theory was examined in the plane, and in the latter it was found more natural to examine it on the cylinder and the torus. One version of this algebraic trick also appears in expressing closed string amplitudes in terms of open string amplitudes [25]. We assume in the following discussion that we are working on the cylinder with its usual flat metric. In sect. 4 we will switch to more general metrics which share the property of being flat everywhere except at isolated points. Consider a rational conformal theory perturbed by a relevant primary field CD having left and right dimensions (h~ h~).This strength replaces the action S ofdimensions the theory 2wCD(w, iii), where the 1~, coupling A has mass by Sh~,1 + A f d he). Correlation functions in the perturbed theory are given by (1 —

—

~ exp[_

(s + Afd2wCD(w,

~))] ~))] flCD~(w

1,~,)

Kf]CD~(w1, ~

=

,

(2.1)

~ exp[_(S+Afd2wCD(w,

where the summation denotes the statistical sum over microscopic configurations. Expanding (2.1) in powers of A, the nth order perturbation term is given by n

K[TCD~(w~,wj))n

=

~ —

ffld2v1KflCD~(w~,~3~)flCD(v1, i3~)~

disconnected terms.

(2.2)

Here the “disconnected terms”, arising from the denominator in (2.1), correspond to some of the CD-fields correlating among themselves; they have the form n—rn

(in d

~ (in In

2v

1,( flCD~(w1,~) flCD(v1~,

2v~~~K flCD(v~, d (2.3)

In the above integrations over the locations of the CD there arise short-distance singularities (ultraviolet divergences). We assume that these are regulated by analytic continuation of all expressions in the dimension h,~,,from a region where

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Moody symmetry

such singularities do not arise. There are also infrared divergences, but as long as we are working on the cylinder these are completely removed by the subtraction of the disconnected terms in (2.2). (This follows because if a field other than the identity propagates between two clusters of fields then the correlator is exponentially suppressed with the distance between the clusters.) The typical integral arising in (2.2) has the form

~

id2w

~

(2.4)

fa(W)Qa~f~(~),

a,13=1

where the fa are singular at some points {wk}, k 0, i,.. m and the finiteness of N in (2.4) follows from the structure of the conformal theory. It is convenient for graphical purposes to map the cylinder coordinate w to a plane coordinate z, by the transformation =

Z

. ,

e~v.

=

(2.5)

Thus we will write (J1CD~(z1,2k))

KI[JCD~(z~(w1),2~(iii~))~KfJCD~(w~, iP,)). =

(2.6)

Note that this is only a coordinate change, not a conformal map; thus there are no factors like (dw/dz)h arising in the conformal transformation of the fields CD1. We adopt the notation that in this section all correlation functions, whether in the w-coordinate or in the z-coordinate, describe the physics on the cylinder. Let f (z) denote the vector {fa~)}, a 1,..., N and define the quadratic form =

N

~

(f,g)Q~ ~

(2.7)

a,131

Since it represents a physical correlator, the integrand (1, f)Q in (2.4) is real; further the f~(Z) are linearly independent functions. These facts imply that Q must be a hermitian quadratic form: (f, g)Q

=

(g, f)~.

(2.8)

Further, since the integrand in (2.4) is monodromy invariant, Q is invariant under the linear transformations froming the monodromy group of the f in (2.4). Let D be any simply-connected region of S without any singular points of f. Let P be any point in D, and ~(z)=fdZf(z)

(2.9)

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439

Cl

P

4

I

I

.

S ~J7~2

z0=ü

-S2

~,

-S Z2

-C2

Fig. 1. The plane cut into annuli avoiding the singularities off.

where the path of integration lies in D. We have

jd2Z(f,

f)Q

=

jd2z(a2~, f)Q = fd2z

~

f)Q

=

~if d2(i, f)Q,

(2.10)

where in the second step we have used the fact that, for holomorphic g, (f, g)Q is antiholomorphic in the argument of g. To evaluate (2.4), we partition the plane into annuli by circles (cf. fig. 1) about z0 0, with m (Z~ are the singular points of the f1). We have 1‘A’radius where Z~ A I, i 0,. L~fl=~j 1 is the annulus between Z11 I and Z1 I, and Zm+i is also a branch point of the fa in general. Further, cut each A1 by a pair of contours linking Z11 to Z1, obtaining a simply-connected region A1 for each annulus A1. Let A1 be the region D in (2.10). It has a boundary C1 + S~ C2 ~2’ linking points P1,..., P4 as shown in fig. 1. Define, for a contour C extending from point P to P’: =

=

. . ,

=

=

—

1c~

f”d2(~, f)Q(Z, 2),

~

f”dZ

f(Z).

~

—

(2.ii),(2.12)

~

The boundary of

A

1

gives the following contributions to the last expression in

(2.10): C1:

~ifP2d2(f

f(Z’) dZ’,

f)=

c,

(2.13)

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S~:~ifP3d2(fP2f(zI) “2

=

—

Moody symmetry

dz’

+

“1

iv’)

dZ’,

f) Q

~i[(I~, Is)Q +J~],

(2.14)

C2: _~if~3d2(f”~f(z’)dZ’+ ff(z’) dZ’, f) —~i[(i~2,1c2)Q+JC},

=

(2.15)

S2: _~if”~d2(ff(Z’) dz’, f)= —~LJ~2.

(2.16)

The term J~,equals Js2~as the integrand (f, f)Q is invariant under the monodromy around the annulus. These terms therefore cancel. Jc, cancels, for the same reason, against a J-term from the integration over the annulus A1~1,and cancels against J-term from the annulus A1_1. (~~20 if C2 shrinks to a point, by our assumption about regularity of the integral for the parameters involved; one may be analytically continuing the result in the parameters later.) Further, we have the relations =

Ic,

+

I~ I~ I~2 0, —

—

=

I~ M1I~, =

(2.17) ,(2.18)

where M1 is the monodromy matrix of the blocks f around the annulus A1. Adding the contributions of all the A1 to cancel the J-terms, and using eqs. (2.17) and (2.18), we get m+1

EIA,

,=

=

~ ([(1 —M~1) ‘—(1

—

M~)’jI~,IE)Q,

(2.19)

where Ii f~ z ~1 dZ f(z), and M1 are the monodromies introduced above. We have repeated the ideas in refs. [23,24] but in a basis of contours more suited to a Hilbert space interpretation of the “screening charges” which arise in the conformal blocks in the conformal theory. This choice of basis is important for manifesting the symmetry algebra described by the contours for an integrable perturbation of a conformal theory. =

=

3. RCFT from real integrals Our final goal is to show that the contours arising from a Feigin—Fuchs representation of the conformal correlators appear on a common footing with the

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441

contours arising from a perturbation in the integrable direction ~ For this purpose we need to understand the contour structure of multiple two-dimensional integrals, which is described in subsect. 3.1. In subsect. 3.2 we show that the Feigin—Fuchs contours describing any RCFT correlation function can be obtained from such multiple two-dimensional integrals. 3.1. MULTIPLE INTEGRALS

We illustrate with an example some of the subtleties and algebraic manipulations involved in converting multiple two-dimensional integrals to contour form. We take the example of the two-point function of ~13 fields in the conformal theory. Vertex operators e~4’ are placed at the origin and at z. Charge balance requires two insertions of the screening charge e”~’.We therefore consider the following real integral:

~=

id2Zl

d2Z 2Ke

fd2Zi

~~(0)

2Z d 2I f(z1,

2)

~

ema~(Z1,2~)e~~(z2,22))

(3.1)

2

Z2)

where f(Z1, Z2) ~f10,21(Z1,

Z2) —Z

Z1~Z2~(Z1

—Z)~~(z2 —Z)~~(Z1 —z2)~.

(3.2) We integrate first over z2 keeping z1 fixed:

1=1

2+f

d2Zlfd2z

d2zlid2z2If(zi,z2)~

O<~z1r
2If(Zl,z2)12. zI
(3.3) Define V~(z)~e1~(z),

V~(z)

~‘HIz1

e1a*~(z~) e1Z, dz” e”~~~(z’) ema*~(zII)e’~(z),

dZ’ I z’ 1,1 z”I

=

zI

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Kac— Moody symmetry

and so on. In the contour integrals we assume, in keeping track of phases, that Iz’I=IzI—e, IZ”l=lz’I—E,etc.,withe--’Ot To avoid certain cancelling singularities, we replace the operators at 0, z by V_(a+±~), Vta~

1~ respectively, and take the limit ~ —~ 0 in the final result.

Let q

=

e’~-,q4

~

=

Using eq. (2.19) we obtain

I

(0<

=

z I < I zII

2z d

2

2If(z1,

Z2)1

~i[(1 -(1 ~q~)~1] x IKV_(a++~)(0)~±(Zl)V_(a~±_~)(Z)) 12 _q2q~~l

x+~i[(1 I KV(a~ +8)(0)~( -(1 Zl)V~(a q2)1] _~~( z)) I 2 (3.5) with a similar expression from the range I Z I Iz 1 I
2~I KV_(a +~)(0)~÷( Zl)V(a

Iz

_~~( z))

2

1 I
~

+J terms,

101<1 z~I
d~z~ 1KV_ta ±6)(0) Va (Z1)V 2q~)~ii KV_(a +a)(0W-(a+

=

~i(1

—

q

~

_~)(Z)) Z))

I

2 +j

2

terms.

The J-terms can be ignored as they will cancel when the integral over

I zI

(3.6)

I

z

1I <~ is added. We note that in the first of (3.6) the phase for V(z1) going around a state of charge 2IT. a isThis the has phase change of from Z~±aZ~~~(Zi —z’Y’~ when all argucontributions Z ments increase by 1 circling the origin, z’ circling the origin, and also from z going around z’ which happens because I Z’ I I z1 I —E. The contour generated by the integral of I’~(Z1) can be expressed as a double contour attached to z, but with the additional factor which gives the modulus squared prefactor in the first of (3.6). =

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Kac — Moody symmetry

Computing similarly the contribution from 5

—~

I zI

I

<

Z

1

443

I


taking the limit

0 we obtain I

=

—

~(q’/~

+ q_1/2)~(q

—

— q_t)~(q3/2

q_3/2)~,

(3.7)

which is manifestly real. 3.2. CONFORMAL THEORY CONTOURS FROM TWO-DIMENSIONAL INTEGRALS

In this subsection we discuss the structure of Feigin—Fuchs contours [8] that arise in the c < 1 correlation functions. Our goal is to show that these holomorphic contour integrals, and their antiholomorphic counterparts, arise from the decomposition of two-dimensional integrals into holomorphic and antiholomorphic contours by (2.19). More precisely, we consider a correlation function of primary fields ~ 1=0, 1,...,N, placed at points (Z1, 2~)(Z0=0). We adopt the language of ref. [9] rather than ref. [8], so that we place no boundary charge at infinity. As mentioned in sect. 1, the action (1.2) leads, by charge balance, to the following two-dimensional integral: M

I((Z1, 2~})

f]~Jd2~1(fl V~(

N

fl V_jia+( z1,

Z1, 2~)

2~)),

(3.8)

where j~ (m1 1)/2, M El1 and Va is defined in (3.4). We wish to show that (upto an overal normalisation constant) I({Z,, 2~))gives the correlation function of the fields {41i~1(Zl, 2~)}. We need to show that the linear span of holomorphic contour integrals arising from (3.8) is the same as the one occurring in ref. [9]. The holomorphic and anti-holomorphic blocks in the RCFT correlator should be paired to yield a real monodromy invariant function; this is manifestly assured by the definition of (3.8). The holomorphic block structure of the c < 1 conformal theories is described by the SU(2) quantum algebra fusion rules [3,6]. To be able to compare with what we obtain from (3.8), we list the rules4~i,n for making blocks by contours in the basis of ref. [91.As in (3.8), primary fields 1 represented by vertex operators V_jiac are placed at points (Z1, 2~),I 0,... N (Z0 0). Screening charges may be attached to points z1, / * 0, giving screened vertex operators V211a(zi), where r1 is the number of contours of f dZ,’ ema±~(Z,I)attached to the point Z1. The correlation function is composed of blocks KU~.0 V2IIa(Zi~2~)) and their complex conjugates. Let a~=2p/p, k =p’ 2. Then 0 ~
—

=

,

=

=

—

jenclosed(Z)

=

—

—f Iz’I=IzI

dZ’

d~fr(Z’).

(3.9)

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Kac— Moody symmetry

The blocks are constrained by the following rules: (i) r1 ~ 2j1enclosed where J1enclosed E~’0(j1 r1~)is the charge of the state acted upon by V2ja(Zi). (ii) r1 ~ 2j~,so that the charge (I~ r1) “carried” at the point Z1 remains within the limits —j~~ (j1 r1) ~j1. (iii) 21enclosed + 2], r1
—

—

—

—

We now sketch the proof that the integral (3.8) reproduces (i)—(iii). Proof of (i). First we note that the multiple contours contained in the block KV_j,,a+(0)Vja±(2i)~) give zero because of phase cancellations if r1 > 2j~.Let z be for simplicity on the positive x-axis, and squeeze the contours to run parallel to the x-axis. The r1 contours in the above expression yield the integral

f

fl d~i(Vjoa*(O) 11 Va±(Yi) V_jia

(Z1)...)

x

=

fl (1 j1

C 1=1

f

fldyjKV_10a±(0)U

yl<...

—

q23(I+)_1)(

j-1 i’=O

(3.10)

~÷(Yj)V_j~a~(Zi)...).


Here the subscript C on the l.h.s. denotes circular contours at radius I I and the on the r.h.s. run along the positive x-axis. For r1 2j~+ 1 the factor 1 q 23o + TI—I occurring in the r.h.s. of eq. (3.10) vanishes, establishing the claim. =

—

—

This establishes (i) for the case when 11enclosed is due to a single vertex operator. To prove the result for the general case we use the fact that screened vertex operators commute with the BRST charge. The BRST charge is defined by [9]

Qmf

1BRST(Z)dZ, Iz’I=IzI rn-I

JBRST(Z)

=

f [1

dZ;Va(Z;),

(3.11)

where m 21enclosed + 1, with Jenclosed the charge enclosed at radii I z I. The contours in run in circles of constant radius from Z’ to Z’, with I Z I I z,’,, -1 I giving the rulemfor const.X avoidingV~t”~, coincident points. Then we have ViI-Ja+ Q Ja± =

JBRST

=

const. x

~

(3.12)

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/ Kac

—

Moody symmetry

445

The relation (3.12) holds with finite nonzero constants for 1 m 1, 0 ~j ~ k/2, 0 r ~ 2]. The desired result (i) follows on noting that the r1~1 2j~0~l(IZ1~1 I —E) contour integrals form a BRST charge, which may be commuted by (3.12) through successive screened vertex operators until it encloses just one of them, when we get zero by the computation following (3.10). Proof of (ii). We first consider the case where 1enclosed is composed of a single vertex operator. Thus we need to show that a block KV_joa+(O)Via+(Zi)~) vanishes, or appears with a vanishing coefficient in the evaluation of (3.8) by contours, for r1 ~ 2j~+ 1. For Jo ~ we already know from our proof of (i) above that the block vanishes. Thus let j~>j1. Again assume for simplicity that Z1 is on the positive real axis. Then the r1 contour integrals can be replaced by Cfyi<... k/2 for I Z1 I < I z I < I Z1÷1I. Then jenclosed(Z) k/2 + 1/2 in this range, and further the vertex operator at Z1~1 carries one screening charge. The contour for this screening charge can be changed to a closed contour at a radius I Zi+i I < I Z I < being I Z1~2I;a the closure the multiple of of p’/2. contour is due to the enclosed charge (k + 2)/2 The vertex operator at Z 1~3 carries a screening charge too, by our assumption ~

—

~

=

=

.,

=

=

. ,

=

. .

=

2

=

p’/

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S.D. Mathur

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Kay — Moody symmetry

about I. “Attaching” the closed contour to z1~3we find the relations

f

2l/ dZ’Va(Z’)V’l/2a(Zl+3)

=

const.X V

2a(zl±s)

{Iz,±21
=

V_l/2a+(Zl+S)f

dz’Va(Z’)f Iz+/
dZ’Va(z”), Iz”l=Iz’~

(3.13) where the z’ contour begins and ends at Z and the z contour is closed because + 1)/2. We continue in this fashion, gaining contours when pass-

jeaclosed(Z)

=

(p’

ing through a vertex operator Vil/2a~ losing contours when passing through V_l/2a*, but never having zero contours in the “charge” because of our assumption about I. This finally gives a charge at I z I > I z,~I of the form ~ fdZ’v(z)fflv(Z;), (3.14)

where the z~integrals begin and end at z, and the z integral is closed. Moving this set of contours to infinity, we find that the block identically vanishes. We have thus established that the integral (3.8) generates all the blocks, correctly paired, for any correlation function of primary fields t~1n in the confor2z~ e~’~’ o describe such a mal theory. We may therefore use insertions of fd correlator, which is what we do in sect. 4.

4. Quantum affine symmetry In this section we consider the decomposition of the terms in the perturbation

series (2.2) into holomorphic and antiholomorphic contour integrals. We show that the rules governing allowed blocks in the perturbed theory correspond to the rules for generating the states in integrable representations of a quantum affine algebra with level k =p’ 2 (a~.=2p/p’). As before, we consider only the a~sector of the theory and ignore the a_ sector. In subsect. 4.1 we discuss the nature of “primary” operators in the deformed theory. In subsect. 4.2 we recall the basic structure of (quantum) affine algebras and their representations. In subsect. 4.3 we find the rules giving allowed blocks of the integrable theory. Subsect. 4.4 contains a discussion of the implications of some of these rules. —

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4.1. BASIC OPERATORS IN THE PERTURBED THEORY We will restrict our attention to correlation functions of operators that represented primary 41,n fields in the critical theory. The structure of the perturbed theory is sensitive to the metric on the two-dimensional space, in particular to the metric in the vicinity of the operator insertions in the correlation functions. We describe our choice of metric, with some motivating arguments. The insertion of an external operator on a Riemann surface occurs at a puncture Z0, with some coordinate system Z around z0. The metric around the puncture is given by 2)1 dZ 12 I z~ dZ 2 which is flat except at the puncture, and2IT describes a semi-infinite cylinder. hasthis a curvature solid angle. The primary field is Such placeda atpuncture infinity on cylinder. equivalent to (One needs to correspondingly rescale the vertex operator generating this primary p(Z,

=

field, to obtain a finite non-vanishing operator insertion at z 0). The metric is chosen to be flat away from such operator insertions, save for a finite number of points carrying a curvature equivalent to IT solid angle. These points are necessary to have the total curvature equal the required amount for a given genus Riemann surface, and their number depends on the number of external insertions. (There are two —IT curvature points for every external insertion, apart from a genus-dependent number *~)Note that the metric chosen here resembles the minimal area metrics used in the construction of string field theory vertices [26]; it is however not essential to have all the restrictions on cycle lengths which appear in the latter case. It turns out that the above choice of metric at the operator insertion is important in manifesting the quantum affine symmetry of the theory. We interpret this circumstance in the following way. Screening charges in the conformal theory, and the more general contour integrals in the perturbed theory, map one charge sector of the free boson Fock space to another charge sector. Thus it is natural to expect that any symmetry of such mapping operators will be manifest only if the Hilbert spaces involved do not change from one “time” to another. Such is the case for a flat metric on the cylinder, but not for example for the flat metric on the plane where equal “time” surfaces are circles of increasing radii. We proceed to examine the perturbed theory. In order to extend the results of sect. 43 to the perturbed case insertion. we need Let to know the phasee’~”~4 changes or encircle an operator the operator be when placede”~~’ at the e~dI* origin in a disc D which contains no other singularities. Because of the curvature at the operator insertion the metric in D is ds2 I z~ dz I 2 which is equivalent to saying that the operator is placed at ~c on the cylinder with coordinate w + —

=

—

*

We consider only genus 0 in what follows.

=

T

448

iu log =1 =

S.D. Mathur Z

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Kac — Moody symmetry

and a flat metric in the w coordinate. Since dZ/dw

fdw eu~+1~(w) =

f

=

z and dim(e1t~4~)

dZ —e~4(Z)z

(4.1)

fdZ ema+4(z),

=

which just reflects the conformal invariance of the screening charges of the conformal theory under a conformal transformation. Thus no change occurs in this case in the phase computations of sect. 3 due to the curvature at the operator insertion. On the other hand dim(e”~4) a~—1 and we find =

fdw

ema+4(w)

=

f~e_ma*4(z)Z~1

=

fdzea+4(Z)a~2.

(4.2)

Thus the effect of the operator insertion e’~~’~4’ and its accompanying curvature is as follows. The operator e~a+4picks up a phase e2~h1x’+ on circling the insertion, while the operator e~a+l~ picks up the phase e2 j+1)a~. on encircling the insertion. With a similar computation one finds that the points with no operator insertion but curvature IT have the following property. The operator eu~~ encounters no phase in encircling such a point, but e Ia~ encounters a phase e These facts will be important in subsect. 4.3. —

~

4.2. QUANTUM AFFINE ALGEBRAS

A Lie algebra G is presented in the Chevally basis by means of positive and negative simple root generators E 1±,and Cartan subalgebra elements H,, i 1,.. . , 1 where 1 is the rank of G. These generators satisfy the relations =

[H,, E1~j [E7, EJ (adE±)’_A,JE±

=

=

±A11Efh, ~-

,

~

i *1,

(4.3)

where A,~is the Cartan matrix of G. The affine extension 6 of G has an extra simple root, so that (4.3) is satisfied with i, j 0, 1,. . . ,I. The relation of this presentation to the more commonly encountered presentation in terms of modes, =

~

J~]

jfabcje

+mk5’~’8m+,,t1,

(4.4)

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449

is given, for the case of ~i~(2) by Ej~=J~, E~=Ji~i1, E~=J~, E~=Jj, H1=J~, H0=~k—J~.

(4.5)

The q-deformed quantum algebra has in place of (4.3) the relations [Hi, EJ±] ±A1JEJ~ =

2

[Er,

El]

—

q~Il~/2

511 qH~/

(adE±)A)Ej± 0, =

i

*1,

(4.6)

where the ranges of the subscripts i, j are as above for the non-affine and affine cases, and the q-deformed ad operation is described in ref. [5]. In particular for SUq(2) the last relation in (4.6) is [27] Ej3E~ (q + 1 —

+

q’)(E~2E~Ej

—

E~EOEI2) —E~E~3 =

0

(4.7)

with similar relations having the indices 0 and 1 interchanged, and also with the superscript replaced with +. A representation of an affine algebra has an integer level k ~ 0. For the case of SUq(2) the representation is specified in addition by a highest weight I, where for integrable representations 0 s~j~ k/2, j an integer or half integer. The states in the representation are generated by the lowering operators E~,Ej, subject to the Serre relation (4.7). The formal module thus generated has “null vectors” which are not actually states in the representation. These null vectors are generated by the following two basic ones —



(4.8)

It may be checked that (4.8) give zero-norm states for the q-deformed algebra as well, for the inner product in which the generators satisfy their natural hermiticity relations. *

We have chosen to write the state as a ket rather than a bra vector to obtain agreement with our convention that the state at the origin is a ket.

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4.3. REALISATION OF QUANTUM AFFINE SYMMETRY We now give the rules governing the occurrences of contours of e’~4 and e~4 in the blocks of the perturbed integrable RCFT. Rule (1). In a perturbed RCFT block the charge Jeaclosed (Z) lies in the range 0 ~Je~0ut~ ~ k/2, for I z I not equalling I z I for any singular point Z 1 in the block. Proof Let (Z,} give the locations of external vertex operators, and (L~} the4 locations of operators the —IT curvature points. be the locations of the eta± and e~”~4 respectively that Let are (yk), to be{9,} integrated along contours. We have to consider the integral

fill d2yk 111 d29

4(Yk, Yk)

4(Z

1(

1~IChia±

1,

ill e~+

2~)

]] e_Ia+4(9 1,

vi)>, (4.9)

where the integrals over Yk’ y~are in the metric having curvature 2ii- at the z~,—IT at the locations v1 and being flat elsewhere. One may first perform the integrals over the Yk, getting a RCFT correlation function. Recall from subsect. 4.1 that the screening charge e~~’does not “notice” the curvatures as it is an integral of a one-form over a one cycle. Thus at this stage of the integration we have from subsect. 3.2 that 0 ~1eaclosed
We would like to interpret the result (1) as follows. A perturbed theory I to a charge sector I’. Because the charges of e”~~’and e_1(1~ balance each other, an infinite number of “screened vertex operator” links a charge sector

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451

screened vertex operators connect any two given charge sectors, though only a finite number occur at any fixed order of perturbation theory. (The order of perturbation equals the number of e1dl~~ contours.) Thus we obtain graded fusion rules, which give the number of ways a charge sector I maps to a sector j’ by an operator of charge I”. With e~a~contours being allowed, one might think that a charge sector j could be mapped to an arbitrarily high charge sector at a suitably high order of perturbation theory. The result (1) says that such is not the case; blocks mapping outside the charge range 0 s~j< k/2 vanish or arise with zero coefficient in the solution of (4.9) by contours. In other words, no “new” intermediate fields are produced in the perturbed theory blocks. Rule (2). The screening charges E~ fdz e~4(z),

E~

fdZ e_Ia+4(Z)

(4.10),(4.11)

satisfy the Serre relation (4.7). Proof The proof follows an analogous argument in ref. [5] for the case of the conformal WZW theories. Each monomial in (4.7) corresponds to a set of four nested contours, with the innermost contour corresponding to the leftmost operator. We convert such a nested set to an “ordered” set, where the arguments still run along a circle of fixed radius, but with a particular choice of ordering. Phases are encountered when connecting one ordering to another. As an illustration, the term with the argument of e”~4 less than all three arguments of the e1t~*~1 receives from the four monomials in (4.7) contributions equalling (1 + 2q + 2q2 + q3) times 1, q~, q2, q3, respectively. These contributions cancel given the weighting coefficients in (4.7). Similarily, the coefficients of the other terms vanish, and result follows. Rule (3). Let the charge enclosed at radius I z I be j, and the curvature enclosed Denote such a state arising at radius I z I by
Kjl(E~)2~~=0, KjI(E~)~2~~’=0,

(4.12),(4.13)

where k=p’—2, a~=2p/p’. Proof Eq. (4.12) is just a restatement of (i) in subsect. 3.2; it says that the BRST charge annihilates a state formed by screened vertex operators. (The screening charge fdZ e~~4’(z)does not notice the curvature, as it is the integral of a one-form.) To motivate (4.13) note that if (II were generated by a single vertex operator ehJdI+l~(Z 2IT at Z 0) with curvature 0 then we have a cancellation of contours due to phase factors. Following a calculation analogous to (3.10), but with

452

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/ Kac—Moody symmetry

e_1a~contours instead of eIa~contours, and noting the phases given by (4.2), we get

I

C

UdYjKV_joa±(0)U Va

(Yj)V_jia

(Z

1)...)

=

,=l

fl (1— q230+J+l)(

~‘

q”)

j=l

x fZl

fl Va

fldyiKVja(O)

y~<...
(Yj)

Vja

(Z1)...).

(4.14)

+ vanishes, establishing rule (4) for the For r1 k 2] + 1 the factor 1 special case where K] I is produced by a single vertex operator. To establish (3) in the general case where Ki I is produced by more than one (screened) vertex operator we proceed as follows. We introduce the notation 2J0+T

=

— q

—

El

...

E~V_j•a

(Z)

ffldZ~

e’t2’~(Z~)Vj-a(Z)

(4.15)

where Sk equals 0 or 1, and the factor (2sk 1) in the exponential on the r.h.s. gives e~4(Z) for Ej and e’~4(z) for E~.The contour integrals over the 4 run in circles of constant radius from Z to Z, and are nested in the order I 4 I < I 42 I for k 1
~

(4.16)

Next, we observe 2~~1flEskV_j~,a+(Zm)...) KjI~ =const.xKjI~2~’~flE$kV_ja±(Zm)...),(4.17) where

I z’ I

=

zI

f III rn-I

jPfl(Z~) =

dz;Va(Z;)

(4.18)

with the contours analogous to those in (3.11). Holding fixed the arguments of the integrals in Q in (4.16), we find that the charge enclosed at radius I z~I —E is k + 1 —j, thus exceeding k/2. Thus even before integration over these arguments the block would vanish by (iii) in subsect. 3.2, provided that the vertex operators at

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453

I Z I > I Zm I give a RCFT state or, more generally, give an integral over the operators in an RCFT state. We now show that such is locations of some e1l~±4 the case. Notice that Eg~flEs~V_ja~(Zm) =fdZ’V_a+(Z’)IJESkV_ja+(Zm),

EjE~JIIEskV_j,*a*(Zm)

=

const.X +

(4.19)

fdZ’Via±(Z’) JIJESkV...j,,,a+(Zm)

const’.XE~E~UF2sk~”_j,,,a+(Zrn),

(4.20)

k

E~E~E~ lllEskV_jn,a+(Zrn)

=

const. x fdZ’V~a*(Z’) JIJESkV_jma+(Zm) +

const’. X E~EjEj

I~~l ESkV_ja( Zm)

+const”.XE~E~Ej UESkV...ja(Zm).

(4.21)

The operator ~ has spin I 1, SO it can carry zero, one or two screenings only; (4.19), (4.20) and (4.21) correspond respectively to using these three possiblities for re-expressing contours. Further, the product of operators E~E~EjE~ can be reordered using eq. (4.7), whereupon eqs. (4.19)—(4.21) can be applied again, iteratively. In this manner we can reduce the contours in the perturbed theory block to those for conformal theory blocks, with integrals over the locations of some e”÷~.But as mentioned above, such conformal theory blocks vanish identically by (iii) of subsect. 3.2, and we obtain our desired result. El =

Rule (4). Denote by (0 I the state created by a point carrying curvature and no charge. Then we have the null relations (0 I _~.(E~) 0, =

(DI ...,,.(Efl2

=

0.

—

IT

(4.22),(4.23)

Thus (0 I corresponds to a level-i representation of the affine algebra generated by Ej, E~. The proof follows from the phase considerations in subsect. 4.1 and arguments similar to the ones in the proof of rule (3) above. The identification of level is done by noting that the sum of the heights of the two basic null vectors in affine SU(2) equals the level plus 2. Rule (5). The homogeneous grading on the affine algebra representation generated by (4.10) and (4.11) is the grading by order of perturbation theory.

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This is obvious, since the grade is changed only by E~=Ji~, which is the perturbing operator, and not by E~ =

Let us summarise what we have found. In a conformal theory the quantum algebra structure manifests itself in the fusion of blocks [6] or alternatively in the placings of contours in a Feigin—Fuchs construction of the blocks [3]. We find that the latter description gets extended to correspond to an affine algebra for the perturbed theory where the contours arise from both the screening charge and the perturbation. For example, a primary field ~ corresponds to a highest weight j (m 1)/2 of the quantum affine algebra, and the set of contours that may encircle it in a non-vanishing block generate an integrable highest-weight representation with this highest weight. It is interesting to recall here that just as the conformal algebra is the commutant (in the bosonic Hilbert space) of the screening charge, the integrals of the deformed theory are the commutants of the pair of charges (4.10) and (4.11) [28,29]. —

=

4.4.

ANALYSIS OF RESULTS; FUSION, BRAIDING

We have examined the perturbation series generated by the deformation A413 in the c < 1 rational conformal theories. The theory was examined in a metric that was flat almost everywhere, with curvature only at the locations of external operator insertions and at a finite number of additional points. We restricted ourselves to the topology of a punctured sphere; we note however in sect. 5 some conjectures for higher-genus surfaces. 4’and It maye~1+/1 be checked that ultraviolet diverinsertions) produce only a gences (from the coincidence of e”~’ renormalisation of the identity. Infrared divergences occur along the semi-infinite cylinders and are absorbed into the definition of the operators placed at their ends. Using the quantum affine structure organising the holomorphic blocks of the perturbation series, one may write down the general term of this series in terms of the Clebsch—Gordan coefficients of the quantum affine algebra and the inner product on its integrable representation spaces. We defer examining this structure, and the fusion and braiding relations of the blocks, to a later paper. We comment briefly, however, on the significance of our curvature considerations for these operations. The presence of the curvature 2IT at the insertion of the external vertex operator is essential in obtaining the null relations (4.12) and (4.13). If we put zero curvature at these insertions, (4.12) is unaltered, while we find the change k k + 2 =p’ in (4.13). This changed value of the level does not correctly give the “primary” state content of the theory, which has in a natural way only fields with spins j 0, 1/2,.. k/2. As mentioned in subsect. 4.1, we interpret the correct curvature choice as giving a metric in which the “equal time” surfaces are of —~

=

.,

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455

constant length, so that the “screening charges” act on time-independent Hilbert spaces. The requirement of such a curvature has been noted before in the context of string field theory vertices, and in topological field theory where it generates contact interactions [30]. Having obtained the basic “primary fields” in the perturbed theory as highestweight states of integrable representations of a level-k quantum algebra, we face an apparent paradox when considering “fusions” of such operators. In the conformal theory fusions correspond to tensoring quantum group representations; in the pertubed theory fusions would correspond to tensoring quantum affine representations. But the tensoring of two level-k representations gives a level-2k representation, so that our set of “primary” operators appears to be not closed under fusions. To see that such is not actually the case consider more carefully the geometry of fusions on the two-dimensional surface. Each operator insertion carries a curvature of 2ir, so “fusing” two such operators generates a curvature 4IT. To obtain again an “operator insertion through a cylinder” we must also fuse in two of the points carrying no charge but curvature —IT. (Two such points corresponded to each operator insertion; now that there will be one insertion less, two of them must be swallowed up.) But by rule (4) of subsect. 4.3, each of these curvature points corresponds to a representation of level 1. Thus we are really fusing two representations of level k and two of level 1, all together, getting a level 2k + 2 k + (k + 2) k +p’. But from eqs. (4.6) and (4.5) we see that for a quantum algebra with q e2’~°” the level is defined only mod p’! Thus fusion of two level-k primaries yields another level-k primary, in the fashion — IT

=

=

=

x Vk x V1 x V’

—‘

Vk,

(4.24)

where V’ is the module for a level-I representation. One may alternatively rewrite this as (VkXVIXVI)x(YXVIXV1)_(VkxVtxVt),

(4.25)

which corresponds to fusion of two level k + 2 0 representations yielding another such representation. We recall that level-0 representations also arise in the quantum affine R-matrices considered by Jimbo [31]in constructing solutions of the Yang—Baxter equation. In our problem it appears likely that the representations (V”~x V1 x V’) (describing operator insertions together with their negative curvature points) would satisfy braiding relations given by the quantum affine R-matrix [211. (Level 0 is important to have this matrix reduce to that for the conformal case when restricted to the conformal blocks.) We find it satisfying that one has this dual picture: a level-k algebra giving integrable representations corresponding to the “primary” fields, and a level k + 2 =p’ 0 algebra permitting closure under braiding and fusion. It would be =

p’

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Kac — Moody symmetry

interesting to know the braiding relations between the Vk spaces alone, and between the Vk and V’ spaces. From an algebraic viewpoint it is more elegant to consider insertion of fields at points carrying curvature which is an arbitrary integer multiple of IT. Extending the calculations and discussion following (4.2) to this case we find that the level of the representation is —(curvature/IT) mod p’. Thus the two parameters I and k characterising a representation of SLJq(2) are connected to the charge and the curvature at the insertion respectively. If we “fuse” such insertion points (without this time including any other curvature points in the definition of fusion) then the curvature adds giving that the levels add mod p’, while the charge varies over a finite range due to the possibility of insertion of screening charges, as in our preceding derivations. The “decoupling rules” for field in the operator product expansion for this general case will be presented elsewhere. One might be concerned that the dimensions of primary fields used in computing the phase correspond to the stress tensor of the action (1.3), and yet obtained a structure symmetrical in the two generators (4.10) and (4.11). To show that this symmetry is really natural to the problem, we indicate how a more symmetrical action without boundary charge would give the same results. The situation is best illustrated by an example. Consider the two-point function of two 41,2 fields on the cylinder, placed at w1, w2. The holomorphic block for this two-point function is 2~I*4(wi) e’~2’~4(w f dw (et~/ 2) e1a*~(w)).We compute the correlation function in this expression in two ways, corresponding respectively to using the actions with and without the boundary charge: (i) Compute the three-point function of the three-field exponentials on the plane and transfer to the cylinder using 2a+4(w (e_I/ =

2a*4(w 1) e_1~’

(Z(w~))

-

4(w))cy1jflder

2)e~’~

(Z(w2))

.

Z(W)

x(e_1/2a±4(Z(wl)) eI~’2a*4(z(w 2)) x e”~÷’~(Z(w)))piaae.

(4.26) 21*+4 with the action (1.3);

Here h12 3/8a~—1/2 the dimension of e”~1’is 1.is the dimension of the e_1/ (ii) Put a charge e’~2~”~ at on the cylinder and a compensating charge e~”2~ ±a_)~ at Compute as before the correlator on the plane (with the extra charges inserted this time) but transfer to the cylinder assuming a dimension a~/8for e~~’2~~b and a~/2for el~~+l~. These dimensions for the exponentials correspond to the action (1.2). One checks that the same block is obtained in both ways. The action without boundary charge is more satisfying conceptually because of the symmetrical =

—~

~.

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Kac — Moody symmetry

457

treatment of the two exponentials, while the action (1.3) (giving (1.4) for the perturbed theory) is more useful for computations as one can then carry over the usual language of conformal theory in many proofs. But in either case for the purposes of the perturbed theory the change of language only corresponds to a change in the charge placed at the end of the semi-infinite cylinder for producing a given primary field insertion.

5. Discussion We have shown that the perturbed integrable theory arising from a c < 1 RCFT is characterised by a quantum affine algebra in the same way that the conformal theory was characterised by a quantum algebra. It appears natural to expect that a complete theory of such integrable systems can be constructed to parallel the approach to RCFTs. We restricted our attention to charges of type ja + in the theory, ignoring the a_ charges. Considering both types of charges together one finds two different quantum affine algebras, characterised by q e’~ and ~ e”~, respectively. This is all still in the holomorphic sector, and there is a corresponding pair of algebras in the antiholomorphic sector. One might fear that perturbation theory with both e~~’and e”-4’ charges would lead to an infinity of terms in the unperturbed RCFT itself, as the relation p ‘a + + Pa 0 permits charge balance to be obtained in a given correlator in an infinite number of ways. But in fact all terms save those from the lowest nonvanishing order of perturbation theory give zero in a tree-level correlator. This is because an extra set of {p, p’) screening charges generate a BRST charge at one or more points in each emerging block, causing the block to vanish. Higher orders of perturbation theory do appear to be relevant for correlators at higher genus, where the extra screening charges put BRST operators on the handles and thereby characterise the fields flowing through those handles. The author does not, however, have a complete understanding of such a picture on arbitrary-genus surfaces. One should be able to write the general term of the perturbation series for any correlation function using the Clebsch—Gordan coefficients and inner-product structure on quantum group representations; this study will be taken up elsewhere. It is possible that the infinitely many blocks used in this description arise from expanding some non-linear differential equation about a point giving the conformal theory It is interesting to note that the null vector (4.12) has a conformal theory origin, where it says that the primary K] I is BRST closed. But the other null vector (4.13) has an order related to the number of primaries accessible from (I I by fusion with =

=

— =

~.

*

This possibility arose in a discussion with T. Eguchi.

S.D. Mathur

458

/

Kac — Moody symmetry

the perturbation. This suggests the existence of a larger BRST-type complex (cf. the complex for the SU(N) conformal theories constructed in ref. [5])where these two null vectors play a symmetrical role. Quantum symmetry and quantum affine symmetry have been discussed before in the context of asymptotic states arising in the (truncated) sine-Gordon theories [14,15,32]. In these approaches there do not appear independent “left” and “right” symmetries. We on the other hand are considering “off-shell” Green functions for the perturbed theory, which manifest a larger symmetry structure. Thus it seems that the extra affine symmetries we observe relate to the detailed structure of the solitons, which is obscured in treating the solitons as asymptotic particle-like states. The ultraviolet limit (A 0) of our theory obviously yields the RCFT; it would be interesting to see how the “infrared” limit gives the reduced symmetry found in the asymptotic states. It would also be interesting to know the relation of our algebras to the Yanginans observed in massive current algebras [33]. We have discussed only the negative root generators of the quantum affine algebra, as these are enough to obtain a highest-weight representation starting with a highest-weight state. But it should also be possible to obtain the positive root generators as “contour removing operators”, as discussed in ref. [341for the case of conformal theories. This would enable a check of the value k =p’ 2 of the level suggested by the representation spaces. It should also be possible to extend our results to all W-algebra minimal models arising from simply laced groups. In our approach, there appears to be the following relation between perturbed c < 1 RCFTs and the sine-Gordon theories. In the more general context of Toda theories based on a group G it was found [35] that the simple roots of G could be decomposed in various specific ways into those for a subgroup H (giving a conformal theory) and those giving an integrable perturbation of this conformal theory From our present point of view, the conformal theory roots have a grade 0 while the perturbation roots have a grade 1 in the affine algebra. Thus the two-exponentials in (1.4) would correspond to Ji~i1,J11 for sine-Gordon, but to —,

—

~.

—

J1~,J11 for the perturbed RCFT. Given the canonical hermiticity structure on the algebra, this suggests different null vector sets and different left-right combinations of blocks in considering the two types of theories. It appears plausible that the blocks emerging in the perturbed integrable RCFT would help to generalise the mathematical structures and results obtained from the study of RCFTs. One might obtain new structures in knot theory and associated subjects, by studying the braiding of blocks on arbitrary-genus surfaces. It is important to note here that the block structure of the perturbed theory is graded by the order of the perturbation, so that a finite set of blocks can be *

Unitarity gives the constraint that G/H be a symmetric space.

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459

filtered out for study instead of dealing with the entire infinite representation at once. It would be interesting to find integrable lattice models based on the infinite-dimensional representations that we have found here. The sine-Gordon theory is a scaling limit of the eight-vertex model, with the mass parameter (which is related to the perturbation strength A in our problem) giving the direction of deformation from the six-vertex model to the eight-vertex model. Thus the structures we find might be connected to some aspect of the eight-vertex model, though the connection is not at the R-matrix level One may also be able to obtain further understanding of the structure of integrable representations of quantum affine algebras by studying the associated physical theory. ~.

I would like to thank D. Olive and J.-L. Gervais for patiently explaining many of their results to me. I am grateful to T. Eguchi, P. Fendley, P. Feng, M. Goulian, M. Grisaru, J. McCarthy, H. Riggs, M. Rinaldi, H. Saleur, A. Sen, N.P. Warner and B. Zwiebach for many enlightening discussions. I am grateful to ITP, Santa Barbara, for a visit where some of this work was done. This work is supported by DOE grant DE-FGO2-88ER-25065.

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I am debted to H. Saleur and P. Fendley for discussions on this point.

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Kac — Moody symmetry

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