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Z-extremization and F-theorem in Chern-Simons matter theories

Antonio Amariti, Massimo Siani

Abstract

The three dimensional exact R symmetry of N=2 SCFTs extremizes the partition function localized on a three sphere. Here we verify this statement at weak coupling. We give a detailed analysis for two classes of models. The first one is an SU(N)_k gauge theory at large k with both fundamental and adjoint matter fields, while the second is a flavored version of the ABJ theory, where the CS levels are large but they do not necessarily sum up to zero. We study in both cases superpotential deformations and compute the R charges at different fixed points. When these fixed points are connected by an RG flow we explicitly verify that the free energy decreases at the endpoints of the flow between the fixed points, corroborating the conjecture of an F-theorem in three dimensions.

Z-extremization and F-theorem in Chern-Simons matter theories

Abstract

The three dimensional exact R symmetry of N=2 SCFTs extremizes the partition function localized on a three sphere. Here we verify this statement at weak coupling. We give a detailed analysis for two classes of models. The first one is an SU(N)_k gauge theory at large k with both fundamental and adjoint matter fields, while the second is a flavored version of the ABJ theory, where the CS levels are large but they do not necessarily sum up to zero. We study in both cases superpotential deformations and compute the R charges at different fixed points. When these fixed points are connected by an RG flow we explicitly verify that the free energy decreases at the endpoints of the flow between the fixed points, corroborating the conjecture of an F-theorem in three dimensions.

Paper Structure

This paper contains 15 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. $\mathcal{Z}$ extremization and $F$ theorem
  3. $SU(N)_k$ YM CS with fundamental and adjoint fields
  4. R charges from two loop in field theory
  5. R charges from the partition function
  6. RG flow and $F$-theorem
  7. Higgsing flow and the $F$-theorem
  8. Superpotential Flow
  9. Flavored ABJ
  10. Two loop computation
  11. $\mathcal{Z}$ extremization
  12. $F$-theorem
  13. Conclusions and discussion
  14. Useful formulas and conventions
  15. Integrals and Series for the computation of the partition function

Figures (3)

  • Figure 1: One--loop diagrams for gauge propagators.
  • Figure 2: Two--loop divergent diagrams contributing to the matter propagators.
  • Figure 3: Quiver of the ABJ flavored model