Modular Constraints on Conformal Field Theories with Currents

TL;DR

This paper explores how modular invariance constrains the spectrum of 2D conformal field theories with currents, revealing bounds, kinks, and realizations of rational CFTs through semi-definite programming.

Contribution

It introduces new bounds on the twist gap in CFTs with currents, highlighting the impact of holomorphic currents and analyzing various W-algebras at small central charge.

Findings

Bounds on the twist gap depend on the presence of holomorphic currents.

Numerical boundaries correspond to known rational CFTs like WZW models and Monster CFT.

Bounds vary with different W-algebras at small central charge.

Abstract

We study constraints coming from the modular invariance of the partition function of two-dimensional conformal field theories. We constrain the spectrum of CFTs in the presence of holomorphic and anti-holomorphic currents using the semi-definite programming. In particular, we find the bounds on the twist gap for the non-current primaries depend dramatically on the presence of holomorphic currents, showing numerous kinks and peaks. Various rational CFTs are realized at the numerical boundary of the twist gap, saturating the upper limits on the degeneracies. Such theories include Wess-Zumino-Witten models for the Deligne's exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by $\mathcal{W}$-algebras of various type and observe that the bounds on the gap depend on the choice of $\mathcal{W}$-algebra in the small central charge region.