Multiloop World-Line Green Functions from String Theory

TL;DR

This paper demonstrates how multiloop bosonic Green functions in string theory simplify to world-line Green functions in the degenerate limit, establishing a precise correspondence through local coordinate choices and Schottky parametrization.

Contribution

It provides a detailed derivation of the connection between string theory Green functions and particle diagram Green functions, including explicit rules for the Schottky parametrization.

Findings

Multiloop string Green functions reduce to world-line Green functions in the degenerate limit.

A set of rules for identifying the relevant moduli space regions corresponding to particle diagrams.

Explicit construction of local coordinates on degenerate string worldsheets.

Abstract

We show how the multiloop bosonic Green function of closed string theory reduces to the world-line Green function as defined by Schmidt and Schubert in the limit where the string world-sheet degenerates into a $\Phi^3$ particle diagram. To obtain this correspondence we have to make an appropriate choice of the local coordinates defined on the degenerate string world sheet. We also present a set of simple rules that specify, in the explicit setting of the Schottky parametrization, which is the corner of moduli space corresponding to a given multiloop $\Phi^3$ diagram.