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F-Theorem without Supersymmetry

Igor R. Klebanov, Silviu S. Pufu, Benjamin R. Safdi

TL;DR

The paper provides non-supersymmetric tests of the F-theorem in three dimensions by analyzing perturbative RG flows of slightly relevant operators, large-N double-trace deformations, and free or CS-type theories on S^3. It derives a perturbative relation between the sphere free energy F and the beta function, demonstrates monotonic decrease of the rescaled quantity $\tilde F$ along RG flows, and extends the analysis to various concrete examples including O(N) vector models and fermionic double-trace deformations. Explicit calculations for free conformal fields and CS theories yield quantitative checks consistent with F-theorem expectations, while the discussion of a coupling-space metric hints at a broader, gradient-flow structure. The work thus strengthens evidence for the F-theorem beyond SUSY contexts and suggests a framework for its potential nonperturbative proof in odd dimensions, with analogous behavior in $(-1)^{(d-1)/2}\log|Z|$ across $S^d$.

Abstract

The conjectured F-theorem for three-dimensional field theories states that the finite part of the free energy on S^3 decreases along RG trajectories and is stationary at the fixed points. In previous work various successful tests of this proposal were carried out for theories with {\cal N}=2 supersymmetry. In this paper we perform more general tests that do not rely on supersymmetry. We study perturbatively the RG flows produced by weakly relevant operators and show that the free energy decreases monotonically. We also consider large N field theories perturbed by relevant double trace operators, free massive field theories, and some Chern-Simons gauge theories. In all cases the free energy in the IR is smaller than in the UV, consistent with the F-theorem. We discuss other odd-dimensional Euclidean theories on S^d and provide evidence that (-1)^{(d-1)/2} \log |Z| decreases along RG flow; in the particular case d=1 this is the well-known g-theorem.

F-Theorem without Supersymmetry

TL;DR

The paper provides non-supersymmetric tests of the F-theorem in three dimensions by analyzing perturbative RG flows of slightly relevant operators, large-N double-trace deformations, and free or CS-type theories on S^3. It derives a perturbative relation between the sphere free energy F and the beta function, demonstrates monotonic decrease of the rescaled quantity along RG flows, and extends the analysis to various concrete examples including O(N) vector models and fermionic double-trace deformations. Explicit calculations for free conformal fields and CS theories yield quantitative checks consistent with F-theorem expectations, while the discussion of a coupling-space metric hints at a broader, gradient-flow structure. The work thus strengthens evidence for the F-theorem beyond SUSY contexts and suggests a framework for its potential nonperturbative proof in odd dimensions, with analogous behavior in across .

Abstract

The conjectured F-theorem for three-dimensional field theories states that the finite part of the free energy on S^3 decreases along RG trajectories and is stationary at the fixed points. In previous work various successful tests of this proposal were carried out for theories with {\cal N}=2 supersymmetry. In this paper we perform more general tests that do not rely on supersymmetry. We study perturbatively the RG flows produced by weakly relevant operators and show that the free energy decreases monotonically. We also consider large N field theories perturbed by relevant double trace operators, free massive field theories, and some Chern-Simons gauge theories. In all cases the free energy in the IR is smaller than in the UV, consistent with the F-theorem. We discuss other odd-dimensional Euclidean theories on S^d and provide evidence that (-1)^{(d-1)/2} \log |Z| decreases along RG flow; in the particular case d=1 this is the well-known g-theorem.

Paper Structure

This paper contains 16 sections, 102 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Perturbed Conformal Field Theory
  3. Beta function and the running coupling
  4. Free energy on $S^d$
  5. Towards a more general proof of the F-theorem
  6. $F$-coefficients for free conformal fields
  7. Free conformal scalar field
  8. Free massless fermion field
  9. Chern-Simons Theory
  10. Double trace deformations
  11. Bosonic double trace deformation
  12. RG flows in $O(N)$ vector models
  13. Fermionic double trace deformation
  14. Comments on Massive Free Fields
  15. Free massive scalar field
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The change in free energy $\delta F_\Delta =F_\text{IR}- F_\text{UV}$ when the UV theory is perturbed by a relevant double trace operator $O^2$, where $O$ has dimension $\Delta$.
  • Figure 2: The change in free energy $\delta F_\Delta = F_\text{IR}- F_\text{UV}$ when the UV theory is perturbed by a relevant double trace operator $\bar{\chi} \chi$, where $\chi$ is a fermionic operator of dimension $\Delta$.