The holographic F theorem

TL;DR

This paper investigates the holographic realization of the F theorem in three-dimensional quantum field theories, revealing conditions under which the F quantity increases along RG flows, thus challenging the strongest form of the theorem.

Contribution

It constructs holographic RG flows for deformations of 3D CFTs and analyzes the behavior of the F quantity, showing violations of the strongest F theorem in certain cases.

Findings

F decreases from UV to IR fixed points in some flows.

F can increase along RG flows for operators with dimension between 3/2 and 5/2.

The strongest version of the F theorem is generally violated.

Abstract

The F theorem states that, for a unitary three dimensional quantum field theory, the F quantity defined in terms of the partition function on a three sphere is positive, stationary at fixed point and decreases monotonically along a renormalization group flow. We construct holographic renormalization group flows corresponding to relevant deformations of three-dimensional conformal field theories on spheres, working to quadratic order in the source. For these renormalization group flows, the F quantity at the IR fixed point is always less than F at the UV fixed point, but F increases along the RG flow for deformations by operators of dimension $3/2 < \Delta < 5/2$. Therefore the strongest version of the F theorem is in general violated.