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The Nekrasov-Shatashvili limit for the low-energy behavior of N=2 and N=2*supersymmetric SU(2) gauge theories is encoded in the spectrum of the Mathieuand Lam'e equations, respectively. This correspondence is usually expressed viaan all-orders Bohr-Sommerfeld relation, but this neglects non-perturbativeeffects, the nature of which is very different in the electric, magnetic anddyonic regions. In the gauge theory dyonic region the spectral expansions aredivergent, and indeed are not Borel-summable, so they are more properlydescribed by resurgent trans-series in which perturbative and non-perturbativeeffects are deeply entwined. In the gauge theory electric region the spectralexpansions are convergent, but nevertheless there are non-perturbative effectsdue to poles in the expansion coefficients, and which we associate withworldline instantons. This provides a concrete analog of a phenomenon foundrecently by Drukker, Marino and Putrov in the large N expansion of the ABJMmatrix model, in which non-perturbative effects are related to complexspace-time instantons. In this paper we study how these very different regimesarise from an exact WKB analysis, and join smoothly through the magneticregion. This approach also leads to a simple proof of a resurgence relationfound recently by Dunne and Unsal, showing that for these spectral systems allnon-perturbative effects are subtly encoded in perturbation theory, andidentifies this with the Picard-Fuchs equation for the quantized ellipticcurve.

Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems