Split Heun functions via blown-up surface defects

Abstract

We study resonant solutions of the Heun equation and its confluent limits that arise in the Nekrasov-Shatashvili (NS) limit of four-dimensional $\mathcal{N}=2$ $\mathrm{SU}(2)$ gauge theories with fundamental hypermultiplets. At the resonant loci $2a/\hbar\in\mathbb{Z}$ in the Coulomb branch parameter $a$, the Floquet multipliers coalesce and the instanton expansions of the bulk and surface defect NS functions develop poles of increasing order. We derive blow-up equations involving exclusively NS functions and use them to resum these singular expansions. The resulting resummed bulk and surface defect NS functions reveal the analytic structure of the gauge-theoretic solutions near the resonant loci, including the branch structure of the accessory parameter and of the Floquet solutions that is obscured by the term-by-term instanton expansion. At resonance, the resummed accessory parameters and suitably normalized defect wavefunctions admit finite limits that describe periodic or antiperiodic solutions at the edges of spectral gaps and allow us to construct their logarithmic companions. We then identify distinct nested mass loci governing gap closure and semisimple resonant monodromy. On the larger locus the band-edge accessory parameters coalesce, while on the smaller locus two independent resonant (anti)periodic Floquet solutions survive. We develop the general resummation procedure for $N_f=(n_0,n_1)$ theories with $n_i\leq 2$ $(i=0,1)$, and demonstrate it explicitly for the $N_f=(1,1)$ theory.