Current Algebra on the Torus

TL;DR

This paper derives one-loop correlation functions for affine Kac-Moody algebra currents on a torus, expressing them as elliptic functions and constructing invariant tensors recursively, with applications in string theory.

Contribution

It provides a general method to compute elliptic functions for affine algebra currents and explicitly constructs invariant tensors using Young tableaux, advancing current algebra theory.

Findings

Derived N-point one-loop correlation functions as elliptic functions

Constructed invariant tensor functions recursively using Young tableaux

Linked the lowest tensors to affine algebra character formulas

Abstract

We derive the N-point one-loop correlation functions for the currents of an arbitrary affine Kac-Moody algebra. The one-loop amplitudes, which are elliptic functions defined on the torus Riemann surface, are specified by group invariant tensors and certain constant tau-dependent functions. We compute the elliptic functions via a generating function, and explicitly construct the invariant tensor functions recursively in terms of Young tableaux. The lowest tensors are related to the character formula of the representation of the affine algebra. These general current algebra loop amplitudes provide a building block for open twistor string theory, among other applications.