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We apply exact WKB methods to the study of the partition function of pure N = 2 $$ \mathcal{N}=2 $$ ϵ i -deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in ϵ 2 /ϵ 1 (i.e. at large central charge) and expansion in ϵ 1 . We find corrections of the form ~ exp − S W periods ϵ 1 $$ \left[-\frac{\mathrm{SW}\;\mathrm{periods}}{\upepsilon_1}\right] $$ to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of ϵ 1 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

Pure N = 2 $$ \mathcal{N}=2 $$ super Yang-Mills and exact WKB