Classical irregular block, N=2 pure gauge theory and Mathieu equation

TL;DR

This paper explores the deep connections between classical irregular conformal blocks, N=2 pure gauge theories, and the Mathieu equation, providing new insights and explicit formulas within the AGT and Bethe/gauge frameworks.

Contribution

It identifies the classical irregular block as a missing element in the triple correspondence and derives explicit relations to gauge theory superpotentials and Mathieu eigenvalues.

Findings

Classical irregular block can be recovered from classical blocks on torus and sphere.

Exact correspondence between irregular block and gauge theory superpotential established.

New formulas relate Mathieu eigenvalues to Bethe-like equations.

Abstract

Combining the semiclassical/Nekrasov-Shatashvili limit of the AGT conjecture and the Bethe/gauge correspondence results in a triple correspondence which identifies classical conformal blocks with twisted superpotentials and then with Yang-Yang functions. In this paper the triple correspondence is studied in the simplest, yet not completely understood case of pure SU(2) super-Yang-Mills gauge theory. A missing element of that correspondence is identified with the classical irregular block. Explicit tests provide a convincing evidence that such a function exists. In particular, it has been shown that the classical irregular block can be recovered from classical blocks on the torus and sphere in suitably defined decoupling limits of classical external conformal weights. These limits are "classical analogues" of known decoupling limits for corresponding quantum blocks. An exact correspondence between the classical irregular block and the SU(2) gauge theory twisted superpotential has been obtained as a result of another consistency check. The latter determines the spectrum of the 2-particle periodic Toda (sin-Gordon) Hamiltonian in accord with the Bethe/gauge correspondence. An analogue of this statement is found entirely within 2d CFT. Namely, considering the classical limit of the null vector decoupling equation for the degenerate irregular block a celebrated Mathieu's equation is obtained with an eigenvalue determined by the classical irregular block. As it has been checked this result reproduces a well known weak coupling expansion of Mathieu's eigenvalue. Finally, yet another new formulae for Mathieu's eigenvalue relating the latter to a solution of certain Bethe-like equation are found.