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Remarks on A2 Toda Field Theory 1 Introduction

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Remarks on A2 Toda Field Theory 1 Introduction View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Remarks on A2 Toda Field Theory S. A. Apikyan1 Department of Theoretical Physics Yerevan State University Al. Manoogian 1, Yerevan 375049, Armenia C. Efthimiou2 Department of Physics University of Central Florida Orlando, FL 32816, USA Abstract We study the Toda field theory with finite Lie algebras using an extension of the Goulian-Li technique. In this way, we show that, af- ter integrating over the zero mode in the correlation functions of the exponential fields, the resulting correlation function resembles that of a free theory. Furthermore, it is shown that for some ratios of the charges of the exponential fields the four-point correlation functions which contain a degenerate field satisfy the Riemann ordinary differ- ential equation. Using this fact and the crossing symmetry, we derive a set of functional equations for the structure constants of the A2 Toda field theory. 1 Introduction The Toda field theory (TFT) provides an extremely useful description of a large class of two-dimensional integrable quantum field theories. For this reason these models have attracted a considerable interest in recent years and many outstanding results in various directions have been established. TFTs are divided in three broad categories: finite Toda theories (FTFTs) for which the underlying Kac-Moody algebra [1, 2] is a finite Lie algebra, affine Toda theories (ATFTs) for which the underlying Kac-Moody algebra [email protected] [email protected] 1 is an affine algebra and indefinite Toda theories (ITFTs) for which the under- lying Kac-Moody algebra is an indefinite Kac-Moody algebra. The classes of FTFTs and ATFTs are well-studied and known to be integrable. In addition, the FTFTs enjoy conformal invariance. A review of the most interesting de- velopments in ATFTs is presented in Ref. [3] where there is also a list of references to the original papers. The class of ITFTs is the least studied as there are still many open questions regarding the indefinite Kac-Moody algebras. A special class of the ITFTs, namely the hyperbolic Toda Theo- ries (HTFTs), for which the underlying Kac-Moody algebra is a hyperbolic Kac-Moody algebra were studied in Ref. [4] and it was shown that they are conformal but not integrable. However, despite all progress in TFTs, there still remain many unresolved questions and problems. For example, one may ask what the structure con- stants of the conformally invariant TFTs are. In this paper, we address this question. We focus on FTFs and, in particular, on the A2 FTT. In Sec. 2 the A2 FTFT is introduced, some notations are fixed, and then we continue to show how the correlation function of exponential fields in the FTFT reduces to correlation functions of a free field theory with conformal W -symmetry [5, 6, 7, 8]. In Sec. 3 we prove that, for some special cases of the exponential fields, the four-point correlation functions which contain a “degenerate” primary field satisfy the Riemann ordinary differential equa- tion. Then, in Sec. 4 the conformal bootstrap technique is applied to derive a set of functional equations for the structure constants of the A2 FTFT. 2 A2 Finite Toda Field Theory We consider the finite conformal Toda field theory associated with the simply- laced Lie algebra A2 described by the action 2 2 1 2 be ' R S = d x (∂') + µ e i· + Q ' . (1) 8π 4π · " i=1 # Z X In the above equation, ei i =1, 2 are the simple roots of Lie algebra A2. These define the fundamental weights wi of the Lie algebra by the equation e w = δ . i · j ij 2 The background charge Q is proportional to the Weyl vector ρ: 2 Q =(b +1/b) ρ , ρ = wi . i=1 X The local conformal invariance of the FTFT with central charge c =2+12Q2 is ensured by the existence of the holomorphic and antiholomorphic energy- momentum tensors T (z)= 1 (∂')2 + Q ∂2' , − 2 · T (¯z)= 1 (∂¯')2 + Q ∂¯2' . − 2 · It is well-known that the FTFTs possess, besides the standard confor- mal symmetry, an additional W -symmetry. In particular, the A2 FTFT we are studying in the present paper contains the additional holomorphic and antiholomorphic currents W (z), W (¯z) with spin 3, which generate the W3 algebra. The vertex operators 2a '(x) Va(x)=e · are spinless primary fields of the W -algebra. Let Ln, Wn be the Fourier modes of the holomorphic fields T (z), W (z). Then L0Va =∆(a) Va ,W0Va = w(a) Va , LnVa =0,WnVa =0,n>0 , where the conformal dimension ∆(a)isgivenby ∆(a)=2a (Q a) . · − The correlation function of N vertex operators is formally defined by the functional integral N 2a '(x ) S['] G (x ,...,x )= ' e i· i e− . (2) a1;:::;an 1 n D i=1 Y 3 We introduce the following orthogonal decomposition of the field ': '(x)='0 + '˜ (x) , where '0 is the zero mode and '˜ denotes the part of the field that is orthog- onal to the zero mode: d2x '˜(x)=0. Z Now, the integration of the functional integral (2) over the zero mode '0 can be done in a similar fashion to the Liouville case [9] to find µ s1+s2 1 G (x ,...,x )= Γ( s )Γ( s ) a1;:::;an 1 n 8π b2 det e − 1 − 2 | | N s1 s2 2a '~ (x ) 2 be1 '~ 2 be2 '~ S0['~] '˜ e i i d xe · d xe · e− , (3) × D i=1 Z Z Y where S0 is the action of the free field theory, 1 R S = d2x (∂'˜)2 + Q '˜ , 0 8π 4π · Z and 1 s =(b det e )− [ Qe + k e k e ] , 1 ij − 22 1 22 − 2 21 1 s =(b det e )− [ Qe + k e k e ] , 2 ij − 12 2 11 − 1 12 N k =2 ai , Q =(Q, 0) . i=1 X Assuming that s1 and s2 are both positive integers, then the remaining functional integral in expression (3) can be reduced to the correlation function of the W3 minimal model [7, 8]. Unfortunately, the situation is much more complicated, i.e., in general, s1 and s2 are not positive integers. However, the solution of the problem is hidden in the previous observation: supposing that we know the exact expressions of the structure constants for the W3 minimal model, then we can recover the expressions for the structure constants of the A2 FTFT by analytic continuation (similarly to the Liouville case) [10, 11]. 4 3 Four-Point Correlation Functions Now, let’s concentrate on the following 4-point correlation function: V (z)V (z )V (z )V (z ) = G (z, z ,z ,z ) , (4) h a+ a1 1 a2 2 a3 3 i a+a1a2a3 1 2 3 where the special vertex operator 2a+ ' V (z)=e · , a = b, b/√3 a+ + − satisfies the null vector equation 2 [∆+(5∆+ +1)W 2 12w+L 1 +6w+(∆+ +1)L 2]Va+ =0. (5) − − − − Taking into account the last equation and the explicit representation of the current W in terms of the field ∂' (see Ref. [8]), we find that the selected 4-point correlation function satisfies the differential equation ∂2 (∆ +1) V (z)V (z )V (z )V (z ) + ∂z2 h a+ a1 1 a2 2 a3 3 i 3 ∆ + δ 1 ∂ 2 i i + V (z)V (z )V (z )V (z ) − (z z )2 z z ∂z h a+ a1 1 a2 2 a3 3 i i=1 i i i X h − − i 3 A +4 i V ∂ϕ V z z h a+ ··· 1 ai ···i i=1 i X − 3 B +4 i V ∂ϕ V =0, (6) z z h a+ ··· 2 ai ···i i=1 i X − where δ = 2√2i[2a (a2 a2 )+2a (a2 a2 ) i − +2 i2 − i1 i2 +1 − +2 + a a (4a Q) a a (4a Q)] , +2 i1 +1 − − +1 i2 i1 − Ai =2√2i(a+2ai1 + a+1ai2) , B =2√2i(a a a a ) . i +1 i1 − +2 i2 Moreover, for the special ratios a a a 2 i2 = +2 1+ +2 (7) ai1 −a+1 ± s a+1 5 of the charges ai, equation (6) can be further reduced to the equation ∂2 (∆ +1) V (z)V (z )V (z )V (z ) + ∂z2 h a+ a1 1 a2 2 a3 3 i 3 ∆ + δ 1+A ∂ 2 i i + V (z)V (z )V (z )V (z ) =0, (8) − (z z )2 (z z ) ∂z h a+ a1 1 a2 2 a3 3 i i=1 i i i X h − − i where A = 2√2i a2 + a2 . ± +1 +2 It is well-known that in the case ofq the four-point functions, the partial differential equation (8), using the projective Ward identities [12], can be reduced to the Riemann ordinary differential equation 1 d2 3 1+A d ∆ + δ (∆ +1) + i i 2 + dz2 z z dz − (z z )2 i=1 i i X h − − i 3 ∆ +∆ +(1+A) + ij V (z)V (z )V (z )V (z ) =0, (9) (z z )(z z )h a+ a1 1 a2 2 a3 3 i i<j i j X − − where ∆ =∆ +∆ ∆ ,(k = i, j), (i, j, k =1, 2, 3). ij i j − k 6 4 Functional Equations for Structure Con- stants Now any four-point function can be explicitly decomposed in terms of the three-point function G (z, z¯)= V (z )V (z )V (z )V (z ) a1a2a3a4 h a1 1 a2 2 a3 3 a4 4 i 2 a1a2 = C(a1, a2, Q a)C(a, a3, a4) Fa (z, z¯) .
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