Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems

摘要: The Nekrasov-Shatashvili limit for the low-energy behavior of $$ \mathcal{N}=2 $$ and \mathcal{N}={2}^{*} supersymmetric SU(2) gauge theories is encoded in spectrum Mathieu Lame equations, respectively. This correspondence usually expressed via an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative effects, nature which very different electric, magnetic dyonic regions. In theory region spectral expansions are divergent, indeed not Borel-summable, so they more properly described by resurgent trans-series perturbative effects deeply entwined. electric convergent, nevertheless there due to poles expansion coefficients, we associate with worldline instantons. provides a concrete analog phenomenon found recently Drukker, Marino Putrov large N ABJM matrix model, related complex space-time paper study how these regimes arise from exact WKB analysis, join smoothly through region. approach also leads simple proof resurgence relation Dunne Unsal, showing that systems all subtly perturbation theory, identifies Picard-Fuchs equation quantized elliptic curve.