Accessory parameters for Liouville theory on the torus

Abstract

We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.

The Polyakov relation for the sphere and higher genus surfaces

“…The normalization of the action S we use in this paper is the one adopted in [5]; it is related to the one used in [17,30,31] which we call S T by S T = 2πS and to the one used in [2,3] and in [14,15] which we call S CM S by S CM S = 4πS.…”

“…For the torus with one source a much stronger result was proven in [17], i.e. that the acces-sory parameter is a real-analytic function of the coupling and of the modulus everywhere except for a zero measure set.…”

Virasoro vacuum block at next-to-leading order in the heavy-light limit

“…of (B.3) is due to a normalization factor z 2h 1 that we must take into account when matching the monodromy computation with our definition of the Virasoro block. 6 Rigorous results that are somewhat related to our problem are presented in [37][38][39] with applications in [40].…”

Holographic duals of large-c torus conformal blocks

“…In the sphere CFT case, it turns out that momenta are holographically related to the accessory parameters of the monodromy method and this observation is instrumental in proving the holographic duality between classical blocks and geodesic lengths in the npoint case [6,10,38]. In this respect, the monodromy method on the torus [39][40][41][42] which is essentially based on Virasoro symmetry provides an interesting possibility to go beyond the sl(2) Casimir equation analysis of [19].…”