F-theorem without supersymmetry

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 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.arXiv:1105.4598v4 [hep-th]

$ \mathcal{Z} $ extremization in chiral-like Chern-Simons theories

“…In this case the free energy is a function of the R charge, and the properties of the N = 2 theories require the knowledge of the exact R charge current appearing in the superconformal algebra. Surprisingly, as first observed in [5], the exact R charges have the special property of extremizing the free energy F. 1 In every known example this extremization is a maximization, and identifying the free energy as a good candidate for counting the number of degrees of freedom led to conjecture a three-dimensional c-theorem, known as F-theorem [7][8][9]. In [8,10], the F-maximization was found to be valid along the whole RG flows of a large class of weakly coupled theories, even without supersymmetry [8], corroborating the validity of an F-theorem.…”

“…Surprisingly, as first observed in [5], the exact R charges have the special property of extremizing the free energy F. 1 In every known example this extremization is a maximization, and identifying the free energy as a good candidate for counting the number of degrees of freedom led to conjecture a three-dimensional c-theorem, known as F-theorem [7][8][9]. In [8,10], the F-maximization was found to be valid along the whole RG flows of a large class of weakly coupled theories, even without supersymmetry [8], corroborating the validity of an F-theorem. Other properties like the relation with the supergravity computations and the volumes of the AdS dual geometry [7,[11][12][13] and the relation between the Lagrange multipliers enforcing the marginality of the superpotential terms and the coupling constants along the RG flows [10] were then observed.…”

Holographic tunneling wave function

“…Along the AdS branch of the saddle points the asymptotic expansion of the scalar field is then φ =αe −Hy +βe −2Hy , Before doing so, however, we first consider a mass deformation of the simpler, minimal O(N ) vector model. The partition function of this is related to that of the critical model by a double trace RG flow [5,29]. In the minimal model a mass deformation is sourced bỹ β multiplying an operator of conformal dimension one.…”

“…First we consider the action of the minimal O(N ) vector model on a three sphere of radius r and a finite constant mass deformation µ 2 , given by [29] …”

Fake gaps in AdS3/CFT2

“…12 The components for h ww and hww of the metric are independent of δφ and can be solved easily provided a warping factor e 2A(y) is specified. Using (B.14), the scalar equation (B.15) reduces to…”