Rational conformal field theory with matrix level and strings on a torus

TL;DR

This paper explores the connection between matrix-level affine algebras and rational conformal field theories describing strings on complex-multiplication tori, revealing their dense presence in the moduli space of string theories.

Contribution

It establishes a link between the algebra $U_{2,K}$ and rational CFTs on complex-multiplied tori, highlighting their equivalence in characters and partition functions.

Findings

Rational CFTs on complex-multiplied tori have identical characters and partition functions to $U_{m,K}$ algebras.

Rational theories are dense in the moduli space of strings on $T^m$.

The connection may facilitate new insights in string theory and conformal field theory.

Abstract

Study of the matrix-level affine algebra $U_{m,K}$ is motivated by conformal field theory and the fractional quantum Hall effect. Gannon completed the classification of $U_{m,K}$ modular-invariant partition functions. Here we connect the algebra $U_{2,K}$ to strings on 2-tori describable by rational conformal field theories. As Gukov and Vafa proved, rationality selects the complex-multiplication tori. We point out that the rational conformal field theories describing strings on complex-multiplication tori have characters and partition functions identical to those of the matrix-level algebra $U_{m,K}$. This connection makes obvious that the rational theories are dense in the moduli space of strings on $T^m$, and may prove useful in other ways.