Multipoint correlation functions in Liouville field theory and minimal Liouville gravity

We study n+ 3-point correlation functions of exponential fields in Liouville field theory with n degenerate and 3 arbitrary fields. An analytical expression for these correlation functions is derived in terms of Coulomb integrals. The application of these results to the minimal Liouville gravity is considered. 1 Liouville field theory Last years since the seminal paper of A. Polyakov [1] Liouville field theory (LFT) attracts a lot of attention mainly due to it deep relation with the theory of bosonic strings in noncritical dimensions. LFT gives an example of non-rational CFT, which has different physical applications. Some results derived in LFT and minimal Liouville gravity can be compared with the results coming from the matrix models [2–8]. For the special set of fields (so called degenerate fields) the correlation functions in LFT are simply related to the correlation functions in the minimal models of CFT, which describe critical behavior of many interesting two-dimensional statistical systems. Liouville field theory is described by local Lagrangian density L = 1 4π (∂aφ) 2 + μe (1.1) with holomorphic stress-energy tensor T (z) = −(∂zφ) 2 +Q∂ zφ, which ensures local conformal invariance of the theory with central charge cL, parameterized in terms of coupling constant b as cL = 1 + 6Q , (1.2) where Q = b+ b. The scale parameter μ in Eq (1.1) is called the cosmological constant. 1 Basic objects in this theory are the exponential fields parameterized by a continuous parameter α Vα(z, z̄) = e , (1.3) which are the primary fields of the Virasoro algebra generated by stress-energy tensor T (z) with the conformal dimensions ∆L(α) = α(Q− α). Here and later z is a complex coordinate on a plane z = x1 + ix2. To simplify the notations we write the primary field defined by Eq (1.3) simply as Vα(z) and denote dz = dx1dx2. The important property of LFT is that the fields Vα and VQ−α have the same conformal dimension and really represent the same conformal field. It means, that they are related by a linear transformation Vα = R(α)VQ−α, (1.4) with function R(α) = (πμγ(b)) b2 γ(2bα− b) γ(2− 2α/b+ 1/b2) , which is known as the reflection amplitude. Here and later we use the notation γ(x) = Γ(x)/Γ(1− x). (1.5) In this paper we consider the correlation functions in LFT, which contains three arbitrary and n degenerate fields. Degenerate fields in LFT correspond to the primary fields Vα(z) with the value of parameter α α = − mb 2 − nb 2 m,n = 0, 1, 2, . . . . (1.6) To do all formulae of this paper more transparent we consider only the case n = 0. The correlation functions with n 6= 0 have more tedious form and we suppose to consider them in other publication. In is well known, that four-point correlation function with degenerate field V−mb/2(z) in LFT satisfy m + 1 order differential equation in each of the variable z and z̄ [9]. The explicit form for this function determines the integral representation for the solution to this differential equation. The correlation functions with three arbitrary fields and more than one degenerate fields satisfy already the system of differential equations in partial derivatives and the possibility to write the solution to this system in terms of finite-dimensional integrals is not evident. The solution to the conformal bootstrap problem for these correlation functions in terms of finite dimensional Coulomb integrals is the main result of this paper. In the case when more than three fields are non-degenerate the correlation function contains an infinite number of conformal blocks and there is no reason to expect, that it can be written as finite-dimensional integral. In this paper we express the multipoint correlation functions in terms of the integrals over the whole plane. In the region of convergency these integrals define completely correlation functions. Outside this region they should be understood in the sense of analytical continuation. This continuation can be performed for example by rewriting integrals over plane in terms of contour integrals, as it was described in [10, 11] (see also the appendix B). The correlation functions in LFT with three arbitrary and m degenerate fields have many different applications. In LFT (or in minimal models of CFT) perturbed by degenerate field they give the possibility to express the perturbative corrections to the three-point correlation functions (structure constants of operator product expansion) in terms of finite-dimensional integrals. Even in the minimal Equation (1.6) should be understood modulo transformation (1.4). 2 models of CFT [9], where for all non-zero correlation functions the screening condition is satisfied, our results give independent integral representation (which in many cases is simpler), where the number of integrations does not depend on three arbitrary fields. These multipoint correlation functions also appear and play an important role in studying of the conformal Toda field theory. We suppose to discuss the application of multipoint correlation functions in LFT to perturbed CFT and conformal Toda field theory in other publication. Here in section 2 we briefly consider the application of these functions to the minimal Liouville gravity. Multipoint correlation functions in LFT are rather complicated objects, however as it was noticed in [13], any multipoint correlation function 〈Vα1(z1) . . . Vαm(zm)〉 exhibits a pole in the variable α = m ∑ k=1 αk (1.7) if the screening condition is satisfied α = Q− nb (1.8) with a residue being expressed in terms of 2n-dimensional Coulomb integral. Namely, res α=Q−nb 〈Vα1(z1) . . . Vαm(zm)〉 = (−πμ) n ∏ i<j |zi − zj | −4αiαj ∫ n ∏