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On the ‘Dotsenko–Fateev’ representation of the toric conformal blocks

We demonstrate that the recent ansatz of Maruyoshi and Yagi (2010 arXiv: 1009.5553), inspired by the original remark of R Dijkgraaf and C Vafa, reproduces the toric conformal blocks in the same sense as the spherical blocks given by the integral representation of Mironov et al (2010 J. High Energy Phys. JHEP02(2010)030 (arXiv: 0911.5721)) and Mironov et al (2010 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 β-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.