Abstract
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 Mathieu and Lam'e equations, respectively. This correspondence is usually expressed via an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative effects, the nature of which is very different in the electric, magnetic and dyonic regions. In the gauge theory dyonic region the spectral expansions are divergent, and indeed are not Borel-summable, so they are more properly described by resurgent trans-series in which perturbative and non-perturbative effects are deeply entwined. In the gauge theory electric region the spectral expansions are convergent, but nevertheless there are non-perturbative effects due to poles in the expansion coefficients, and which we associate with worldline instantons. This provides a concrete analog of a phenomenon found recently by Drukker, Marino and Putrov in the large N expansion of the ABJM matrix model, in which non-perturbative effects are related to complex space-time instantons. In this paper we study how these very different regimes arise from an exact WKB analysis, and join smoothly through the magnetic region. This approach also leads to a simple proof of a resurgence relation found recently by Dunne and Unsal, showing that for these spectral systems all non-perturbative effects are subtly encoded in perturbation theory, and identifies this with the Picard-Fuchs equation for the quantized elliptic curve.
Highlights
Parametrized by the scalar condensate u = Tr Φ2
This provides a concrete analog of a phenomenon found recently by Drukker, Marino and Putrov in the large N expansion of the ABJM matrix model, in which non-perturbative effects are related to complex space-time instantons
The dyonic region is characterized by divergent perturbative expansions described by resurgent transseries [11, 12, 46,47,48] that systematically unify perturbative and non-perturbative physics; the electric region has convergent perturbative expansions but there are nonperturbative effects associated with poles of the expansion coefficients. (This provides a concrete analog of a phenomenon found recently by Drukker, Marino and Putrov [49, 50] in the large N expansion of the ABJM matrix model, in which non-perturbative effects are related to complex space-time instantons, and which were subsequently related to poles in the ’t Hooft expansion coefficients [51, 52].) The magnetic region is a cross-over region in which non-perturbative effects are large, and the spectral bands and gaps are of equal width
Summary
We review relevant facts about the spectrum of the Mathieu equation [76, 78,79,80], translated into notation that makes explicit the relation to the Nekrasov partition function. This ambiguous imaginary non-perturbative term is cancelled by an identical term coming from an instanton gas analysis of the instanton/antiinstanton interaction [11, 12, 46, 47] This leading cancellation, at the two-instanton level, is just the tip of the iceberg: the cancellations between imaginary terms produced by lateral Borel summation of perturbation theory, and those coming from the multi-instanton sectors, occur at all orders, and these cancellations are encoded in relations between the coefficients of the resurgent trans-series expansion. This expression is obtained by substituting a Fourier mode ansatz for the Mathieu function, and equating coefficients [76,77,78] The second term on the right hand side is due to the perturbative part of the prepotential
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