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Generalized Bethe expansions of superconformal indices

Alejandro Cabo-Bizet, Wei Li

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.

Generalized Bethe expansions of superconformal indices

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 SYM. For or , 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.

Paper Structure

This paper contains 40 sections, 270 equations, 3 tables.

Table of Contents

  1. Introduction and summary
  2. Summary of main results
  3. Generalized Bethe expansions
  4. Contribution of continuous solutions
  5. A solution-generating method
  6. Application on SYM
  7. Bethe expansions
  8. Quasi-periodicities of superconformal indices
  9. Bethe expansion and Bethe solutions, for special Bethe contour
  10. Bethe contour
  11. Bethe expansion of indices
  12. Isolated and continuous Bethe solutions
  13. General Bethe contours
  14. Ambiguity in the choice of Bethe integrand
  15. Contribution from continuous solutions
  16. ...and 25 more sections

Theorems & Definitions (8)

  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • Definition 4.5
  • Definition 4.6
  • Definition 4.7
  • Definition 4.8