Handle operators in string theory

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