Abstract
Conformal blocks and their AGT relations to LMNS integrals and Nekrasov functions are best described by “conformal” (or Dotsenko–Fateev) matrix models, but in non-Gaussian Dijkgraaf–Vafa phases, where different eigenvalues are integrated along different contours. In such matrix models, the determinant representations and integrability are restored only after a peculiar Fourier transform in the numbers of integrations. From the point of view of conformal blocks, this is Fourier transform w.r.t. the intermediate dimensions, and this explains why such quantities are expressed through tau-functions in Miwa parametrization, with external dimensions playing the role of multiplicities. In particular, these determinant representations provide solutions to the Painlevé VI equation. We also explain how this pattern looks in the pure gauge limit, which is described by the Brezin–Gross–Witten matrix model.
Highlights
AGT relations [1], identifying the conformal blocks and the Nekrasov functions, possess different interpretations
We give a little more details about the interplay between the bilinear and Painleve equations in discussion of a simpler Painleve III example in sec.9 below, which is associated with the pure gauge limit (PGL) of conformal blocks
Since integrability behind the conformal blocks (15) gets explicit only after the Fourier transform in the internal α-parameter, one can expect the same to happen in the pure gauge limit
Summary
AGT relations [1], identifying the conformal blocks and the Nekrasov functions, possess different interpretations. Matrix models possess a lot of other nice properties, which can be transmitted to either the Nekrasov functions or to the conformal blocks, especially at c = 1 (i.e. β = 1) Among these features are the closely related integrability and determinant representations, see [9] for comprehensive reviews and references. As explained in detail in [5], by a suitable adjustment of Dotsenko-Fateev (DF) trick [2] (applying it to holomorphic quantities and making use of open rather than closed integration contours), conformal blocks can be converted into the matrix-model form.
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