Leszek Hadasz

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

Classical geometry from the quantum Liouville theory

\mathcal{N} = 2

Modular bootstrap in Liouville field theory

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

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.

Braiding and fusion properties of the Neveu-Schwarz super-conformal blocks

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.

Classical Liouville action on the sphere with three hyperbolic singularities

A few years ago Zamolodchikov and Zamolodchikov proposed an expression for the four-point classical Liouville action in terms of the three-point actions and the classical conformal block [Nucl. Phys. B 477, 577 (1996)]. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on n-punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.

Braiding properties of the N = 1 super-conformal blocks (Ramond sector)

We construct, generalizing appropriately the method applied by J. Teschner in the case of the Virasoro conformal blocks, the braiding and fusion matrices of the Neveu-Schwarz super-conformal blocks. Their properties allow for an explicit verification of the bootstrap equation in the NS sector of the N = 1 supersymmetric Liouville field theory.