Integrability, susy $SU(2)$ matter gauge theories and black holes

TL;DR

This work extends the ODE/IM correspondence to 4D $\mathcal{N}=2$ $SU(2)$ gauge theories with $N_f=1,2$ fundamental matter by linking quantum Seiberg–Witten periods to integrable-model data via $Q$, $T$, and $Y$ functions and their TBAs. It establishes gauge–integrability identifications at both leading and exact levels, proving that $a$ and $a_D$ encode, respectively, Floquet and QNM information through precise $T$ and $Y$ relations, and that TBAs reproduce the NS-instanton expansions. The paper then applies these results to black-hole perturbations, showing quasinormal modes are governed by Baxter $Q$-function zeros and by $A_D$-period quantization, with gravity duals corresponding to generalized extremal charged black holes. Altogether, the authors provide a unified, non-perturbative framework connecting 4D $\mathcal{N}=2$ gauge theories, 2D integrable models, and BH physics, enabling exact calculations of spectra and non-perturbative observables. The approach offers new insights into wall-crossing, $\mathbb{Z}_2/\mathbb{Z}_3$ symmetries in the gauge sector, and potential extensions to broader BH backgrounds and quiver theories.

Abstract

We show that previous correspondence between some (integrable) statistical field theory quantities and periods of $SU(2)$ $\mathcal{N}=2$ deformed gauge theory still holds if we add $N_f=1,2$ flavours of matter. Moreover, the correspondence entails a new non-perturbative solution to the theory. Eventually, we use this solution to give exact results on quasinormal modes of black branes and holes.

Figures (12)

• Figure 2.1: Plot of the solution $\varepsilon(\theta)$ of the integrability TBAs \ref{['int-TBA-full-1']}, \ref{['int-TBA-full-2']}, (the colored continuous curves) vs the Riccati numeric solution as in \ref{['lnQPi']}, \ref{['Q2++']} (the black dots).
• Figure 2.2: Plot of $\ln Q(\theta)$, using the TBA solution as in \ref{['TBAQ1']}, \ref{['TBAQ2']} (the colored continuous curves) vs the Riccati numeric solution as in \ref{['lnQPi']}, \ref{['Q2++']} (the black dots).
• Figure 3.1: Comparison of the solution of the gauge TBA \ref{['ga-TBA']}, \ref{['ga-TBA2']} (the colored continuous curves) vs. Riccati ODE numeric integration (the black dots), for $N_f=1$ and $N_f=2$ on the left and right respectively.
• Figure 3.2: A strip of the $y$ complex plane, where in blue we show the contour of integration of SW differential for the $SU(2)$$N_f=1$ theory, and in red its branch cuts.
• Figure 3.3: A strip of the $y$ complex plane, where in blue we show the contour of integration of SW differential for the $SU(2)$$N_f=2$ theory, and in red its branch cuts.