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Integrability and cycles of deformed \(\mathcal{N} = 2\) gauge theory. (English) Zbl 1435.81129

Summary: To analyse pure \(\mathcal{N} = 2\) 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 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 one-cycle periods 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 periods \((Y(\theta, \pm u)\) 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.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81R12 Groups and algebras in quantum theory and relations with integrable systems
81R40 Symmetry breaking in quantum theory
81T60 Supersymmetric field theories in quantum mechanics
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