Blowing-up the edge: connection formulae and stability chart of the Lam\'e equation

TL;DR

This paper develops a novel approach to analyze the stability of spectral problems related to the Lamé equation using connections with supersymmetric gauge theories and conformal field theory, revealing the structure of the spectrum's edges.

Contribution

It introduces a systematic method for analyzing semi-classical Virasoro blocks and resumming Nekrasov partition functions to characterize spectral stability charts of quantum integrable systems.

Findings

Branch cuts in partition functions mark spectrum edges.

Spectrum analysis of Lamé and related equations.

Agreement with isomonodromic deformation results.

Abstract

We study periodic spectral problems through their connection with supersymmetric gauge theories and two-dimensional conformal field theory. To characterize the associated stability chart, we develop a novel and systematic approach for analyzing semi-classical Virasoro blocks near their poles. Via the AGT correspondence, these blocks correspond to SU(2) Nekrasov partition functions in the Nekrasov-Shatashvili limit, which we propose to resum using an appropriate limit of blow-up equations. We show that the analytic structure of the resulting resummed partition functions features branch cuts located precisely at the edges between bands and gaps in the spectrum of the associated quantum integrable system with periodic potential. We examine the Nekrasov partition functions of $\mathcal{N}=2$ SQCD with $N_f \le 4$ flavors and of the $\mathcal{N}=2^*$ theory, which are related to the Heun equation, its confluent forms, and the Lam\'e equation. In the latter case, we analyze the spectrum in detail and solve the associated connection problem. Finally, we compare our results with those obtained via isomonodromic deformation techniques and the computation of orbifold surface defect partition functions in the $\mathcal{N}=2^*$ gauge theory, finding perfect agreement.