Classical limit of irregular blocks and Mathieu functions

TL;DR

This paper establishes a detailed connection between the Mathieu equation and two-dimensional conformal field theory (2d CFT), enabling the use of 2d CFT techniques to analyze Mathieu operator spectra, especially in the classical limit.

Contribution

It provides a full correspondence between the Mathieu equation and 2d CFT irregular blocks, including eigenvalues and eigenfunctions, and discusses potential extensions to other coupling regimes.

Findings

Mathieu eigenvalues match classical irregular block expansions

Eigenfunctions derived from irregular blocks reproduce Mathieu exponents

Sketch of proof for asymptotic behavior hypotheses of irregular blocks

Abstract

The Nekrasov-Shatashvili limit of the N=2 SU(2) pure gauge (Omega-deformed) super Yang-Mills theory encodes the information about the spectrum of the Mathieu operator. On the other hand, the Mathieu equation emerges entirely within the frame of two-dimensional conformal field theory (2d CFT) as the classical limit of the null vector decoupling equation for some degenerate irregular block. Therefore, it seems to be possible to investigate the spectrum of the Mathieu operator employing the techniques of 2d CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its realization within 2d CFT has to be established. In our previous paper [1], we have found that the expression of the Mathieu eigenvalue given in terms of the classical irregular block exactly coincides with the well known weak coupling expansion of this eigenvalue in the case in which the auxiliary parameter is the noninteger Floquet exponent. In the present work we verify that the formula for the corresponding eigenfunction obtained from the irregular block reproduces the so-called Mathieu exponent from which the noninteger order elliptic cosine and sine functions may be constructed. The derivation of the Mathieu equation within the formalism of 2d CFT is based on conjectures concerning the asymptotic behaviour of irregular blocks in the classical limit. A proof of these hypotheses is sketched. Finally, we speculate on how it could be possible to use the methods of 2d CFT in order to get from the irregular block the eigenvalues of the Mathieu operator in other regions of the coupling constant.