Handle Operators in String Theory

TL;DR

This paper introduces handle operators to incorporate topological features of Riemann surfaces into string amplitudes, providing an exact, gauge-invariant formalism applicable to arbitrary worldsheet topologies.

Contribution

The authors explicitly construct handle operators for string amplitudes, ensuring gauge invariance and reparametrisation invariance across all topologies, with detailed consistency checks.

Findings

All loop amplitudes are gauge-invariant.

One- and two-point sphere amplitudes are computed explicitly.

Reproduction of the four-point sphere amplitude from glued three-point amplitudes.

Abstract

We derive how to incorporate topological features of Riemann surfaces in string amplitudes by insertions of bi-local operators called handle operators. The resulting formalism is exact and globally well-defined in moduli space. After a detailed and pedagogical discussion of Riemann surfaces, complex structure deformations, global vs local aspects, boundary terms, an explicit choice of gluing-compatible and global (modulo U(1)) coordinates (termed `holomorphic normal coordinates'), finite changes in normal ordering, and factorisation of the path integral measure, we construct these handle operators explicitly. Adopting an offshell local coherent vertex operator basis for the latter, and gauge fixing invariance under Weyl transformations using holomorphic normal coordinates (developed by Polchinski), is particularly efficient. All loop amplitudes are gauge-invariant (BRST-exact terms decouple up to boundary terms in moduli space), and reparametrisation invariance is manifest, for arbitrary worldsheet curvature and topology (subject to the Euler number constraint). We provide a number of complementary viewpoints and consistency checks (including one-loop modular invariance, we compute all one- and two-point sphere amplitudes, glue two three-point sphere amplitudes to reproduce the exact four-point sphere amplitude, etc.).