Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization

Abstract

We investigate the exact WKB method for the quantum Seiberg-Witten curve of 4d N = 2 pure SU (3) Yang-Mills, in the language of abelianization. The relevant differential equation is a third-order equation on CP 1 with two irregular singularities. Exact WKB analysis leads us to consider new Darboux coordinates on a moduli space of flat SL(3,C)-connections. In particular, in the weak coupling region we encounter coordinates of higher length-twist type generalizing Fenchel-Nielsen coordinates. The Darboux coordinates are conjectured to admit asymptotic expansions given by the formal quantum periods series; we perform numerical analysis supporting this conjecture.

WKB periods for higher order ODE and TBA equations

“…The exact WKB method for general higher order ODE has been developed in [55,56], where the Stokes phenomena of the WKB solutions are extensively studied. In [57][58][59][60], using the spectral network [61], the relation between the WKB periods and the TBA equations is examined for the Hitchin system. One can identify the cycle of the WKB period associated with the Y-function from the Stokes graph.…”

Quantum periods and TBA equations for $\mathcal{N}=2\ SU(2)\ N_f=2$ SQCD with flavor symmetry

“…The exact WKB analysis and the TBA equations for the quantum Seiberg-Witten curves are also studied in the context of the abelianization (e.g. [29,30]).…”

A Panorama Of Physical Mathematics c. 2022

“…The relation between Abelian and Nonabelian connections given by spectral networks has a qdeformed analog [281,282] which has been beautifully developed and extended in the recent works [283][284][285]. This subject should be further developed and should lead to useful insights about quantization of character varieties, connecting to the works [286][287][288] on quantum cluster varieties on the one hand, and the two [289,290] and four dimensional [291] approaches to quantization of these spaces, on the other.…”