Minimally Extended Current Algebras of Toroidal Conformal Field Theories

TL;DR

This paper introduces a universal method to decompose partition functions of rational toroidal conformal field theories into characters of minimal extensions of $$ current algebras, enhancing understanding of their algebraic structure.

Contribution

It provides an explicit, universal construction for decomposing partition functions into minimal current algebra characters for rational toroidal CFTs of any dimension.

Findings

Decomposition method applies to rational toroidal CFTs with arbitrary-dimensional target tori.

Explicit examples for 2D and 3D target space tori demonstrate the method.

Decompositions are not unique but universally applicable.

Abstract

It is well-known that families of two-dimensional toroidal conformal field theories possess a dense subset of rational toroidal conformal field theories, which makes such families an interesting testing ground about rationality of conformal field theories in families in general. Rational toroidal conformal field theories possess an extended chiral and anti-chiral algebra known as W-algebras. Their partition functions decompose into a finite sum of products of holomorphic and anti-holomorphic characters of these W-algebras. Instead of considering these characters, we decompose the partition functions into products of characters of minimal extensions of $\widehat{\mathfrak{u}}(1)$ current algebras, which already appear for rational conformal field theories with target space $S^1$. We present an explicit construction that determines such decompositions. While these decompositions are not unique, they are universal in the sense that any rational toroidal conformal field theory with a target space torus of arbitrary dimension admits such decompositions. We illustrate these decompositions with a few representative examples of rational toroidal conformal field theories with two- and three-dimensional target space tori.