The String Partition Function for QCD on the Torus

TL;DR

The paper investigates a string-theoretic formulation of pure glue QCD$_2$ on a torus, showing that a strong/weak gauge coupling duality yields modular covariance in the partition function. It develops a generating framework for the genus-$g$ free energies $F_g^+$, computing exact expressions up to $g=8$ in terms of modular forms, and reveals a surprisingly tame, non-divergent behavior in the small-area limit. The results highlight a modular structure reminiscent of topological strings, with $F_g^+$ of weight $6g-6$ after holomorphic completion, yet no simple holomorphic anomaly equation governs the full theory. These findings advance the program of deriving a four-dimensional QCD string by exploiting two-dimensional solvable models and motivate further study of the worldsheet dynamics and higher-genus extensions.

Abstract

We study the free energy of the pure glue QCD string with a torus target space and the gauge groups $SU(N)$ and (chiral) $U(N)$. It is highly constrained by a strong/weak gauge coupling duality which results in modular covariance. The string free energy is computed exactly in terms of modular forms for worldsheet genera 1 - 8. It has a surprisingly mild singularity in the weak gauge coupling/small area limit.