Three-point function in the minimal Liouville gravity

Abstract

The problem of the structure constants of the operator product expansions in the minimal models of conformal field theory is revisited. We rederive these previously known constants and present them in the form particularly useful in the Liouville gravity applications. Analytic relation between our expression and the structure constant in Liouville field theory is discussed. Finally we present in general form the three-and two-point correlation numbers on the sphere in the minimal Liouville gravity. Liouville gravityBy gravity usually we imply a dynamic theory of the metric structure on certain manifold. In the two dimensional case the latter is supposed to be a two-dimensional surface Σ (compact or non-compact) of certain topology and equipped with a Riemann metric g ab (x). To avoid problems with moduli in this paper I always imply that Σ is a sphere. Also I consider the euclidean version of the gravity, i.e., g ab is always non-degenerate and has positive signature. In the path integral approach the problem is reduced to the evaluation of a functional integral over all Riemann metric D[g] modulo the diffeomorphism equivalent g ab (x). E.g., the gravitational partition function of a sphere is formally written as(1.1)1 On leave of absence from Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117259 Moscow, Russia.2 Laboratoire Associé au CNRS URA-768Here A eff [g] is supposed to be an effective action induced by some generally covariant "matter" field theory living on the surface. General covariance ensures that A eff [g] is invariant under diffeomorphisms.In general case of massive matter A eff [g] is a non-local and quite complicated functional of the metric. The problem (1.1) appears very complicated. There is a drastic simplification however, if all the matter inducing A eff [g] is "critical", i.e., described by a conformal field theory (CFT). In this case the form of A eff [g] is very universal and simple, being called the Liouville action. This fact was first discovered by A.Polyakov in 1981 [1] by direct computations with free fields. In general CFT this statement simply follows from the form of conformal anomaly. Moreover, conceptually this form of effective action can be taken (with few additional assumptions) as the very definition of CFT.Due to the diffeomorphism invariance of A eff [g] there is a gauge fixing problem in (1.1). One of the most convenient gauge choices is the conformal gauge where the coordinates on Σ are (locally) chosen in the way that (this is always possible in two dimensions) g ab (x) = e 2bφ(x) δ ab (1.2)The scale factor here is described by a quantum field φ(x) called the Liouville field (see below for the definition of the parameter b). One can also fix the gauge in a covariant way choosing an arbitrary metric g (0) ab (x) as the reference one and requiring g ab (x) = e 2bφ(x) g (0) ab (x) (1.3)In the latter approach the Liouville field φ is a usual scalar under coordinate transformations. Gauge (1.2) then implies a particular choice of coord...

A common limit of super Liouville theory and minimal models

“…The expression (A.16) coincides with the formula given in [43]. The normalised structure constants D are invariant under permutation of the variables d i .…”

“…Recently this limit was investigated in [42], and it was argued that only for p = 1 a physically sensible three-point function is obtained. The relation of the resulting theory to minimal models or generalised minimal models [43,44] is unclear until now. The continuation of Liouville theory to other, non-rational central charges c < 1 is more singular, and it is clear that the structure constants do not have a well-defined limit as functions of the field parameters, though it might be that one can find a sensible limit in terms of distributions.…”

N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theories on S2 × S2 and Liouville Gravity

“…In the following we will show that indeed three-point number and conformal blocks of this CFT arise as building blocks of the supersymmetric partition function of N = 2 U(2) gauge theory on S 2 × S 2 . Liouville Gravity (LG) [29,30] is a well-known two-dimensional theory of quantum gravity composed of three CFT sectors 1. Liouville theory sector, which has a central charge…”

“…The former have been computed in [29] (eq.7.9) for three generic dressed operators U α i and can be written in terms of γ function, see eq. (E.11), as…”

Four-dimensional N $$ \mathcal{N} $$ = 2 supersymmetric theory with boundary as a two-dimensional complex Toda theory

“…To this end, note that in [4], a supersymmetric theory was constructed on S 2 × S 2 , and its partition function was computed using the technique of localization. The authors observed that the building blocks of the partition function contained three-point functions and conformal blocks of Liouville gravity [5,6], and so they conjectured a novel AGT-like correspondence between the 4d supersymmetric theory on S 2 × S 2 and 2d Liouville gravity.…”