Abstract
We apply exact WKB methods to the study of the partition function of pure $$ \mathcal{N}=2 $$ ϵi-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in ϵ2/ϵ1 (i.e. at large central charge) and expansion in ϵ1. We find corrections of the form ~ exp $$ \left[-\frac{\mathrm{SW}\;\mathrm{periods}}{\upepsilon_1}\right] $$ to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of ϵ1 when passing from the Nekrasov form of the partition function to the topological string theory inspired form. We conjecture that such corrections should be computable from a worldsheet perspective on the partition function. Our results follow upon the determination of the Stokes graphs associated to the Mathieu equation with complex parameters and the application of exact WKB techniques to compute the Mathieu characteristic exponent.
Highlights
Non-perturbative completions of string perturbation theory have been proposed in various backgrounds
Topological string theory is a promising framework within which to tackle this problem, as the nature of the perturbation theory is the same as that of the full string theory — giving rise to a generically non-convergent genus expansion explicitly linked to computations on underlying Riemann surfaces
For a large class of backgrounds, the topological string theory partition function Ztop is fully computable in various series expansions
Summary
Non-perturbative completions of string perturbation theory have been proposed in various backgrounds. The conformal field theory technique at the heart of the analysis in this paper relies on null vector decoupling It permits the computation of conformal blocks, mapped to Zinst under the 2d/4d correspondence, via solution of a differential equation. We will perform an exact WKB analysis of this equation, determining the Stokes regions for various complex values of the parameters, and incorporating Stokes phenomena in the computation of the characteristic exponent of its solutions This procedure introduces corrections of the order exp[− 1 ] in the relation between the characteristic exponent, linked to the vacuum expectation value a of the scalar field in gauge theory, and a certain complex parameter u of the equation, which is related to the gauge theory partition function via a Matone-style relation.
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