Opers and TBA

TL;DR

This work analyzes the conformal limit of the TBA equations governing the hyperkähler moduli spaces of 4d ${\cal N}=2$ theories on a circle, showing that the limit yields a complex Lagrangian submanifold ${\cal L}_{\epsilon}$ whose image in the conformal limit ${\cal M}_{\epsilon}$ is the oper manifold for class ${\cal S}$ theories. In the ${\cal S}[A_1]$ case, ${\cal L}_{\epsilon}$ corresponds to SL(2) opers realized as Schrödinger operators with rational potentials, with X_{\gamma}$ identified as Fock coordinates or Wronskians on a WKB triangulation; this is tested via several local models (harmonic oscillator, intricate local model, three-punctured sphere, cubic, and pure SU(2)). The paper also builds a Hitchin-moduli-space perspective, showing that the conformal-limit equations provide flat sections of the Hitchin connection and that the oper condition arises from a line subbundle stabilized by the Higgs field, reinforced by an explicit Airy example. Finally, an inverse-scattering-like parameterization is proposed to reconstruct periods from prescribed cross-ratios, enabling direct matching between scattering data and conformal-limit coordinates. This framework links conformal blocks, opers, and BPS spectra through a unified TBA/Hitchin formalism with potential extensions to quantized blocks.

Abstract

In this note we study the "conformal limit" of the TBA equations which describe the geometry of the moduli space of four-dimensional N=2 gauge theories compactified on a circle. We argue that the resulting conformal TBA equations describe a generalization of the oper submanifold in the space of complex flat connections on a Riemann surface. In particular, the conformal TBA equations for theories in the A1 class produce solutions of the Schrödinger equation with a rational potential.