Four Point Functions in the SL(2, R) WZW Model

We consider winding conserving four point functions in the SL(2, R) WZW model for states in arbitrary spectral flow sectors. We compute the leading order contribution to the expansion of the amplitudes in powers of the cross ratio of the four points on the worldsheet, both in the m− and x−basis, with at least one state in the spectral flow image of the highest weight discrete representation. We also perform certain consistency check on the winding conserving three point functions. [email protected] [email protected] The SL(2, R) WZW model describes strings moving in AdS3 and has important applications to gravity and black hole physics in two and three dimensions. It is also an interesting subject by itself, as a step beyond the well known rational conformal field theories. Actually, unlike string propagation in compact target spaces, where the spectrum is in general discrete and the model can then be studied using algebraic methods, the analysis of the worldsheet theory in the non-compact AdS3 background requires the use of more intricate analytic techniques. String theory on AdS3 contains different sectors characterized by an integer number w, the spectral flow parameter or winding number [1]. The short string sectors correspond to maps from the worldsheet to a compact region in AdS3 and the states in this sector belong to discrete representations of SL(2, R) with spin j ∈ R and unitarity bound 1 2 < j < k−1 2 . Other sectors contain long strings at infinity, near the boundary of spacetime, described by continuous representations of SL(2, R) with spin j = 1 2 + is, s ∈ R. Several correlation functions have been computed in various sectors [2, 3]. In particular, four point functions of w = 0 states were computed in [2] analytically continuing previous results in the SL(2, C)/SU(2) coset model [4] which corresponds to the Euclidean H3 background. In this letter we consider four point functions of states in arbitrary w sectors, a crucial ingredient to determine the consistency of the theory through factorization. Different basis have been used in the literature to compute correlation functions in this theory. Vertex operators and expectation values for the spectral flow representations were constructed in [1, 2] in the m−basis, where the generators (J 0 , J̄ 3 0 ) of the global SL(2, C) isometry are diagonalized. The m−basis has the advantage that all values of w can be treated simultaneously. In particular, all winding conserving N−point functions have the same coefficient in this basis, for a given N , and they differ only in the worldsheet coordinate dependence [2, 5] which reflects the change in the conformal weight of the states ∆(j) → ∆(j) = ∆(j)− wm− k 4 w , (1) where ∆(j) = − j(j−1) k−2 is the dimension of the unflowed operators. Alternatively, the x−basis refers to the SL(2, R) isospin parameter which can be interpreted as the coordinate of the boundary in the context of the AdS/CFT correspondence. The operators Φj(x, x̄) in the x−basis and Φj;m,m̄(z, z̄) in the m−basis are related by the following transformation Φj;m,m̄(z, z̄) = ∫ dx |x|2 xx̄Φj(x, x̄; z, z̄) . (2) Finally, the μ−basis was found convenient to relate correlation functions in Liouville and SL(2, C)/SU(2) models [5, 6]. In this letter we extend the results for the four point function of unflowed states in SL(2, R) given in [2] to the case of winding conserving four point functions for states in arbitrary spectral flow sectors. This is accomplished by transforming the 1 x−basis expression found in [4] to the m−basis in order to exploit the fact that the coefficient of all winding conserving correlators is the same (for a given number of external states). In order to simplify the calculations we consider first the four point function in which one of the original unflowed operators is a highest weight and then analyze the more general case in which it is replaced by a global SL(2, R) descendant. Finally we transform the result back to the x−basis. Actually the explicit expression computed in [4] and further analyzed in [2] corresponds to the leading order in the expansion of the four point function in powers of the cross ratio of the worldsheet coordinates. It was pointed out in [4] that higher orders in the expansion may be determined iteratively once the lowest order is fixed as the initial condition. We will further discuss this topic for four point functions involving spectral flowed states. Let us start recalling the result for the four point function of unflowed states originally computed for the SL(2, C)/SU(2) model in [4] and later analytically continued to SL(2, R) in [2], namely 〈Φj1(x1, z1)Φj2(x2, z2)Φj3(x3, z3)Φj4(x4, z4)〉 = ∫ dj C(j1, j2, j) B(j) C (j, j3, j4)F(z, x) F̄(z̄, x̄) × |x43| |x42| |x41| |x31| 2(j4−j1−j2−j3) × |z43| |z42| |z41| |z31| 2(∆4−∆1−∆2−∆3) , (3) where the integral is over j = 1 2 + is with s a positive real number. Here B and C are the coefficients corresponding to the two and three point functions of unflowed operators respectively (see [2] for the explicit expression in our conventions), and F is a function of the cross ratios z = z21z43 z31z42 , x = x21x43 x31x42 , with a similar expression for the antiholomorphic part. The function F is obtained by requiring (3) to be a solution of the Knizhnik-Zamolodchikov (KZ) equation [7, 8]. Expanding F in powers of z as follows F(z, x) = zj12 x12 ∞