$F$-extremization determines certain large-$N$ CFTs

TL;DR

This paper demonstrates that the conformal data of large-N melonic CFTs can be determined by extremizing the universal part of the sphere free energy, extending F-maximization to non-supersymmetric theories in continuous dimensions.

Contribution

It introduces a universal extremization principle for melonic CFTs using the sphere free energy, generalizing known maximization procedures to non-supersymmetric, continuous-dimensional theories.

Findings

Conformal data of melonic CFTs are fixed by extremizing .

The extremization extends F and a-maximization to non-supersymmetric theories.

Classifies melonic CFTs as mean field theories extremizing  with IR marginality.

Abstract

We show that the conformal data of a range of large-$N$ CFTs, the melonic CFTs, are specified by constrained extremization of the universal part of the sphere free energy $F=-\log Z_{S^d}$, called $\tilde{F}$. This family includes the generalized SYK models, the vector models (O$(N)$, Gross-Neveu, etc.), and the tensor field theories. The known $F$ and $a$-maximization procedures in SCFTs are therefore extended to these non-supersymmetric CFTs in continuous $d$. We establish our result using the two-particle irreducible (2PI) effective action, and, equivalently, by Feynman diagram resummation. $\tilde{F}$ interpolates in continuous dimension between the known $C$-functions, so we interpret this result as an extremization of the number of IR degrees of freedom, in the spirit of the generalized $c,F,a$-theorems. The outcome is a complete classification of the melonic CFTs: they are the conformal mean field theories which extremize the universal part of the sphere free energy, subject to an IR marginality condition on the interaction Lagrangian.