Determinants for the Lightcone Worldsheet

TL;DR

This paper reviews methods for calculating determinants of Laplacians on complex two-dimensional regions, focusing on the effects of corners and boundary conditions, with applications to string scattering amplitudes involving D-branes.

Contribution

It generalizes existing techniques to include corners joining different boundary conditions and applies these to compute string scattering amplitudes with D-branes.

Findings

Corners affect the Laplacian determinants significantly.

Generalization to mixed boundary conditions is achieved.

Applications to D-brane scattering amplitudes are demonstrated.

Abstract

The evaluation of the determinant of the Laplacian defined on two dimensional regions of various shapes is an essential ingredient in calculating the scattering amplitudes of strings. In lightcone parameterization the regions are rectangular in shape with several slits of different length and location cut parallel to the $\tau$ axis of the rectangle. This paper offers a compendium of applications of the methods of Kac and McKean and Singer to the calculation of such worldsheet determinants. Particular attention is paid to the effect of corners on the determinants. The effect of corners joining edges with like boundary conditions is implicit in Kac's results. We discuss the generalization to a corner joining a Dirichlet edge to a Neumann edge, and apply it to a scattering amplitude involving D-branes.