Generalized Bethe expansions of superconformal indices

TL;DR

This paper demonstrates the existence of infinitely many Bethe expansions for 4D superconformal theories and introduces a systematic analytic method to find all solutions, revealing intricate cancellations in specific gauge theories.

Contribution

It provides a new systematic analytic approach to obtain all Bethe solutions for superconformal indices, including continuous solutions, and explores their contributions in specific gauge theories.

Findings

Infinite Bethe expansions exist for superconformal theories.

Continuous solutions contribute via isolated points on their manifolds.

Cancellation occurs between tachyon contributions from different solutions.

Abstract

We show the existence of infinitely many Bethe expansions for general four dimensional superconformal theories. We then propose an analytic method to systematically obtain all the Bethe solutions, both isolated and continuous, for general superconformal theories. In particular, we show that the contribution from the continuous manifold of solutions to the index comes from isolated points on this manifold. We check our proposals on $\mathcal{N}=4$ SYM. For $U(3)$ or $SU(3)$, which are the first examples with continuous solutions, we demonstrate the non-trivial cancellation between the tachyon contributions from the previously known isolated solutions and those from the continuous solutions.