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

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 ($Y(θ,\pm u)$ or their square roots, $Q(θ,\pm u)$). 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.

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