Zbigniew Jaskólski

Recursive representation of the torus 1-point conformal block

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.

Proving the AGT relation for N f = 0, 1, 2 antifundamentals

Using recursive relations satisfied by Nekrasov partition functions and by irregular conformal blocks we prove the AGT correspondence in the case of

Whittaker pairs for the Virasoro algebra and the Gaiotto - BMT states

\mathcal{N} = 2

Whittaker pairs for the Virasoro algebra and the Gaiotto-Bonelli-Maruyoshi-Tanzini states

superconformal SU(2) quiver gauge theories with Nf = 0, 1, 2 antifundamental hypermultiplets.

Classical geometry from the quantum Liouville theory

A flow in the formulation and proof of Lemma 2.7 of E. Felinska, Z. Jaskolski, and M. M. Kosztolowicz, J. Math. Phys. 53, 033504 (2012) is fixed in Sec. I of this Erratum. This has no consequences for the rest of the paper. An essential error was made in Theorems 3.5, 3.6, and Corollary 3.7 of Sec. III of E. Felinska, Z. Jaskolski, and M. M. Kosztolowicz, J. Math. Phys. 53, 033504 (2012). As it was pointed out by V. Mazorchuk and K. Zhao [“Simple Virasoro modules which are locally finite over positive part,” e-print arXiv:1205.5937v2 [math.RT]] they are valid only in the very limited case of n = 3. In order to introduce necessary correction we have rewritten a part of Sec. III after Theorem 3.3. This is presented in Sec. II of this Erratum.In this paper we analyze Whittaker modules for two families of Wittaker pairs related to the subalgebras of the Virasoro algebra generated by Lr, . . . , L2r and L1, Ln. The structure theorems for the corresponding universal Whittaker modules are proved and some of their consequences are derived. All the Gaiotto [17] and the Bonelli-Maruyoshi-Tanzini [34] states in an arbitrary Virasoro algebra Verma module are explicitly constructed. e-mail: letjenen@gmail.com e-mail: jask@ift.uni.wroc.pl e-mail: mikosz@ift.uni.wroc.pl

Modular bootstrap in Liouville field theory

Whittaker modules for two families of Whittaker pairs related to the subalgebras of the Virasoro algebra generated by Lr, …, L2r and L1, Ln 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)10.1007/JHEP02(2012)031] states in an arbitrary Virasoro algebra Verma module are explicitly constructed.

Elliptic recurrence representation of the N = 1 Neveu-Schwarz blocks

Abstract Zamolodchikovs 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.

Elliptic recurrence representation of the N = 1 superconformal blocks in the Ramond sector

Abstract The modular matrix for the generic 1-point conformal blocks on the torus is expressed in terms of the fusion matrix for the 4-point blocks on the sphere. The modular invariance of the toric 1-point functions in the Liouville field theory with DOZZ structure constants is proved.

Liouville theory and uniformization of four-punctured sphere

Abstract We apply a suitably generalized method of Al. Zamolodchikov to derive an elliptic recurrence representation of the Neveu–Schwarz superconformal blocks.

The integration ofG-invariant functions and the geometry of the Faddeev-Popov procedure

The structure of the 4-point N = 1 super-conformal blocks in the Ramond sector is analyzed. The elliptic recursion relations for these blocks are derived.