On "Dotsenko-Fateev" representation of the toric conformal blocks

TL;DR

This paper shows that a recent ansatz accurately reproduces toric conformal blocks using beta-ensembles, extending previous integral representations from spherical to toric cases and providing evidence at finite N.

Contribution

It demonstrates that the Dotsenko-Fateev type integral representation applies to toric conformal blocks at finite N, not just in the large-N limit, and discusses potential generalizations.

Findings

The ansatz reproduces toric conformal blocks at finite N.

Explicit checks performed at the first two levels for 1-point functions.

Discussion of extensions to higher genus surfaces.

Abstract

We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal blocks in the same sense that the spherical blocks are given by the integral representation of arXiv:1001.0563 with a peculiar choice of open integration contours for screening insertions. In other words, we provide some evidence that the toric conformal blocks are reproduced by appropriate beta-ensembles not only in the large-N limit, but also at finite N. The check is explicitly performed at the first two levels for the 1-point toric functions. Generalizations to higher genera are briefly discussed.