Handle operators in string theory

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 physical 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 oneand two-point sphere amplitudes, glue two three-point sphere amplitudes to reproduce the exact four-point sphere amplitude, etc.). (c) 2020 Published by Elsevier B.V.

Automated summary

In this work, the authors discuss how to incorporate topological features of Riemann surfaces in string amplitudes by inserting handle operators, and explicitly construct these operators. The use of an offshell local coherent vertex operator basis and gauge fixing invariance under Weyl transformations using holomorphic normal coordinates are particularly efficient for ensuring gauge invariance and reparametrization invariance. Various complementary viewpoints and consistency checks are provided, including one-loop modular invariance and calculations of one- and two-point sphere amplitudes for validation.