DLCQ strings and branched covers of torii

TL;DR

This paper reviews how discrete light-cone quantization of strings constrains the Riemann surfaces in the path integral to branched covers of a torus, with implications for matrix string theory and explicit genus 1 checks.

Contribution

It establishes a connection between DLCQ string path integrals and branched covers of tori, providing explicit genus 1 verification and exploring links to matrix string theory.

Findings

Path integral of thermal string free energy constrained to branched covers of a torus.

Explicit genus 1 check supporting the theoretical constraint.

Potential relation between branched covers and matrix string model limits.

Abstract

In this lecture I will review some results about the discrete light-cone quantization (DLCQ) of strings and some connections of the results with matrix string theory. I will review arguments which show that, in the path integral representation of the thermal free energy of a string, the compactifications which are necessary to obtain discrete light-cone quantization constrains the integral over all Riemann surfaces of a given genus to the set of those Riemann surfaces which are branched covers of a particular torus. I then review an explicit check of this result at genus 1. I discuss the intriguing suggestion that these branched covers of a torus are related to those which are found in a certain limit of the matrix string model