Integrability and cycles of deformed ${\cal N}=2$ gauge theory

TL;DR

This paper explores the integrability structure of deformed ${ m N}=2$ $SU(2)$ gauge theory using ODE/IM correspondence, revealing connections to Liouville CFT and deriving quantum analogs of classical symmetries.

Contribution

It establishes a novel link between deformed Seiberg-Witten cycles, Baxter functions, and Liouville CFT at the self-dual point, extending integrability methods to gauge theories with matter and higher rank.

Findings

Identifies the broken $bZ_2$ R-symmetry as a manifestation of discrete symmetry in the ODE/IM framework.

Connects deformed SW cycles to Baxter's $T$ and $Q$ functions of Liouville CFT.

Derives a TBA for the cycles and presents asymptotic expansion techniques.

Abstract

To analyse pure ${\cal N}=2$ $SU(2)$ gauge theory in the Nekrasov-Shatashvili (NS) limit (or deformed Seiberg-Witten (SW)), we use the Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence, and in particular its (broken) discrete symmetry in its extended version with {\it two} singular irregular points. Actually, this symmetry appears to be 'manifestation' of the spontaneously broken $\mathbb{Z}_2$ R-symmetry of the original gauge problem and the two deformed SW cycles are simply connected to the Baxter's $T$ and $Q$ functions, respectively, of the Liouville conformal field theory at the self-dual point. The liaison is realised via a second order differential operator which is essentially the 'quantum' version of the square of the SW differential. Moreover, the constraints imposed by the broken $\mathbb{Z}_2$ R-symmetry acting on the moduli space (Bilal-Ferrari equations) seem to have their quantum counterpart in the $TQ$ and the $T$ periodicity relations, and integrability yields also a useful Thermodynamic Bethe Ansatz (TBA) for the cycles ($Y(\theta,\pm u)$ or their square roots, $Q(\theta,\pm u)$). A latere, two efficient asymptotic expansion techniques are presented. Clearly, the whole construction is extendable to gauge theories with matter and/or higher rank groups.