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

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.

47 pages, 3 figures

References in corpus (6)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Entanglement Entropy at Large Central Charge
• Asymptotically free N=2 theories and irregular conformal blocks
• Deforming SW curve
• Riemann-Hilbert treatment of Liouville theory on the torus
• Compactification in deconstructed gauge theory with topologically non-trivial link fields

Cited by in corpus (4)

• Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach