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The gravity dual of supersymmetric gauge theories on a two-parameter deformed three-sphere

Dario Martelli, Achilleas Passias

Abstract

We present rigid supersymmetric backgrounds for three-dimensional N=2 supersymmetric gauge theories, comprising a two-parameter U(1)xU(1)-invariant deformed three-sphere, and their gravity duals. These are described by supersymmetric solutions of four-dimensional N=2 gauged supergravity with a self-dual metric on the ball and different instantons for the graviphoton field. We find two types of solutions, distinguished by their holographic free energies. In one type the holographic free energy is constant, whereas in another type it depends in a simple way on the parameters and is generically complex. This leads to a conjecture for the localized partition function of a class of N=2 supersymmetric gauge theories on these backgrounds.

The gravity dual of supersymmetric gauge theories on a two-parameter deformed three-sphere

Abstract

We present rigid supersymmetric backgrounds for three-dimensional N=2 supersymmetric gauge theories, comprising a two-parameter U(1)xU(1)-invariant deformed three-sphere, and their gravity duals. These are described by supersymmetric solutions of four-dimensional N=2 gauged supergravity with a self-dual metric on the ball and different instantons for the graviphoton field. We find two types of solutions, distinguished by their holographic free energies. In one type the holographic free energy is constant, whereas in another type it depends in a simple way on the parameters and is generically complex. This leads to a conjecture for the localized partition function of a class of N=2 supersymmetric gauge theories on these backgrounds.

Paper Structure

This paper contains 16 sections, 104 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction and summary
  2. Supergravity solutions
  3. Regularity on the boundary
  4. Regularity in the bulk
  5. Regularity of the gauge field
  6. Supersymmetry
  7. Parameter space
  8. Holographic free energy
  9. Discussion
  10. Integrability conditions
  11. Integrability condition of the bulk Killing spinor equation
  12. Integrability condition of the boundary Killing spinor equation
  13. More on the Killing spinors
  14. $\chi$, $\chi^c$ and $\tilde{\chi}$
  15. Coordinate $\theta$ and the $a=0$ case
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Solutions exist for all parameters inside the wedge. Solutions for parameters below the parabola are always real. Solutions for parameters above the parabola are complex in the Type II cases. The black dot represents Euclidean AdS$_4$, with round three-sphere boundary.