The recursive relation for the 1-point conformal block on a torus is derived and used to prove the identities between conformal blocks recently conjectured by Poghossian in [1]. As an illustration of the efficiency of the recurrence method the modular invariance of the 1-point Liouville correlation function is numerically analyzed.
Four-point super-conformal blocks for the N = 1 Neveu-Schwarz algebra are defined in terms of power series of the even super-projective invariant. Coefficients of these expansions are represented both as sums over poles in the "intermediate" conformal weight and as sums over poles in the central charge of the algebra. The residua of these poles are calculated in both cases. Closed recurrence relations for the block coefficients are derived.
Zamolodchikov's recursion relations are used to analyze the existence and approximations to the classical conformal block in the case of four parabolic weights. Strong numerical evidence is found that the saddle point momenta arising in the classical limit of the DOZZ quantum Liouville theory are simply related to the geodesic length functions of the hyperbolic geometry on the 4-punctured Riemann sphere. Such relation provides new powerful methods for both numerical and analytical calculations of these functions. The consistency conditions for the factorization of the 4-point classical Liouville action in different channels are numerically verified. The factorization yields efficient numerical methods to calculate the 4-point classical action and, by the Polyakov conjecture, the accessory parameters of the Fuchsian uniformization of the 4-punctured sphere.
Using recursive relations satisfied by Nekrasov partition functions and by irregular conformal blocks we prove the AGT correspondence in the case of N = 2 superconformal SU(2) quiver gauge theories with N f = 0, 1, 2 antifundamental hypermultiplets.
Whittaker modules for two families of Whittaker pairs related to the subalgebras of the Virasoro algebra generated by L r , . . . , L 2r and L 1 , L n are analyzed. The structure theorems for the corresponding universal Whittaker modules are proved and some of their consequences are derived. All the Gaiotto [e-print arXiv:0908.0307] and the Bonelli-Maruyoshi- Tanzini [J. High Energy Phys. 1202, 031 (2012)] states in an arbitrary Virasoro algebra Verma module are explicitly constructed. C 2012 American Institute of Physics. [http://dx.
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