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The spectral problem of the ABJ Fermi gas

Johan Kallen

Abstract

The partition function on the three-sphere of ABJ theory can be rewritten into a partition function of a non-interacting Fermi gas, with an accompanying one-particle Hamiltonian. We study the spectral problem defined by this Hamiltonian. We determine the exact WKB quantization condition, which involves quantities from refined topological string theory, and test it successfully against numerical calculations of the spectrum.

The spectral problem of the ABJ Fermi gas

Abstract

The partition function on the three-sphere of ABJ theory can be rewritten into a partition function of a non-interacting Fermi gas, with an accompanying one-particle Hamiltonian. We study the spectral problem defined by this Hamiltonian. We determine the exact WKB quantization condition, which involves quantities from refined topological string theory, and test it successfully against numerical calculations of the spectrum.

Paper Structure

This paper contains 14 sections, 140 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The ABJ matrix model and the spectral problem
  3. Review of the solution of the spectral problem for $M=0$
  4. The quantum volume for $M>0$
  5. Seiberg like duality
  6. Testing the WKB quantization condition
  7. The partition function of ABJ
  8. Borel resummation of $\mathcal{R}_\ell(k,M)$
  9. Comparing coefficients for integer values of $k$
  10. Summary and outlook
  11. Polynomial part of the quantum volume
  12. $M$ even
  13. $M$ odd
  14. Mellin transform