Summing Over Inequivalent Maps in the String Theory Interpretation of Two Dimensional QCD

TL;DR

This paper investigates the string-theoretic interpretation of 2D QCD on a torus, demonstrating that the free energy can be expressed as a sum over inequivalent branched surface maps, with evidence supporting the validity of this approach for various coverings and configurations.

Contribution

It provides a rigorous analysis of the sum over inequivalent maps in 2D QCD, including proofs for prime coverings and conjectures for general cases, advancing the understanding of string descriptions of gauge theories.

Findings

The free energy corresponds to a sum over branched surface maps.

The sum is consistent for all smooth maps with prime coverings.

Leading terms align with contributions from maps with branch points.

Abstract

Following some recent work by Gross, we consider the partition function for QCD on a two dimensional torus and study its stringiness. We present strong evidence that the free energy corresponds to a sum over branched surfaces with small handles mapped into the target space. The sum is modded out by all diffeomorphisms on the world-sheet. This leaves a sum over disconnected classes of maps. We prove that the free energy gives a consistent result for all smooth maps of the torus into the torus which cover the target space $p$ times, where $p$ is prime, and conjecture that this is true for all coverings. Each class can also contain integrations over the positions of branch points and small handles which act as ``moduli'' on the surface. We show that the free energy is consistent for any number of handles and that the first few leading terms are consistent with contributions from maps with branch points.