Witten's Vertex Made Simple

TL;DR

This paper simplifies the diagonalization of Witten's vertex matrices, enabling explicit eigenvalue calculations for various scale dimensions and providing insights into their behavior, including for fractional values.

Contribution

It introduces a straightforward method using SL(2,R) techniques and Watson-Sommerfeld transformation to diagonalize Witten's vertex matrices for all scale dimensions.

Findings

Eigenvalues of Neumann matrices are explicitly calculated for all s.

Eigenfunctions for s=1 include a p term, with x replaced by midpoint position.

Method applies to matter and ghost sectors, including fractional s.

Abstract

The infinite matrices in Witten's vertex are easy to diagonalize. It just requires some SL(2,R) lore plus a Watson-Sommerfeld transformation. We calculate the eigenvalues of all Neumann matrices for all scale dimensions s, both for matter and ghosts, including fractional s which we use to regulate the difficult s=0 limit. We find that s=1 eigenfunctions just acquire a p term, and x gets replaced by the midpoint position.