The Disk Partition Function in String Theory

Understand

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.