Integrability and cycles of deformed N = 2 gauge theory

摘要

To analyse pureN=2SU(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 withtwosingular irregular points. Actually, this symmetry appears to be ‘manifestation’ of the spontaneously brokenZ2R-symmetry of the original gauge problem and the two deformed SW one-cycle periods are simply connected to the Baxter'sTandQfunctions, 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 brokenZ2R-symmetry acting on the moduli space (Bilal-Ferrari equations) seem to have their quantum counterpart in theTQand theTperiodicity relations, and integrability yields also a useful Thermodynamic Bethe Ansatz (TBA) for the periods (Y(θ,±u)or their square roots,Q(θ,±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.