The Disk Partition Function in String Theory

TL;DR

This paper addresses the challenge of computing the disk partition function in open string theory by resolving gauge volume issues, leading to a straightforward method for calculating D-brane tensions.

Contribution

It introduces a novel approach to handle the residual gauge group volume in the disk partition function calculation, simplifying the derivation of D-brane tensions.

Findings

Demonstrates that PSL(2,R) behaves like a finite negative volume group in the path integral.

Provides a simple prescription for the disk partition function calculation.

Re-derives D-brane tensions using the new method.

Abstract

We investigate the disk partition function for the open string. This is a subtle problem because of the presence of a residual gauge group $\mathrm{PSL}(2,\mathbb{R})$ on the worldsheet even after fixing the conformal gauge. It naively has infinite volume and leads to a vanishing answer. We use different methods that all demonstrate that $\mathrm{PSL}(2,\mathbb{R})$ effectively behaves like a group with finite negative volume in the path integral, which leads to a simple prescription for the computation of the disk partition function. We apply our findings to give a simple rederivation of the D-brane tensions.