Quantising Chiral Bosons On Riemann Surfaces

TL;DR

This paper develops a method to quantize chiral bosons on arbitrary Riemann surfaces, computes their partition functions, and applies the results to heterotic string theory ensuring modular invariance and anomaly freedom.

Contribution

It generalizes Sen's action to arbitrary metrics on Riemann surfaces and derives the partition function for chiral scalars in this setting, including on higher genus surfaces.

Findings

Partition function expressed as products of holomorphic functions for rational tori.

Single holomorphic product for chiral bosons on even self-dual lattices.

Ensures modular invariance and anomaly cancellation in heterotic string world-sheet theory.

Abstract

Sen's action in two dimensions governs a chiral boson coupled to a two-dimensional metric together with a second chiral boson that couples to a flat two-dimensional metric. This second scalar decouples from the physical degrees of freedom. The generalisation of this action to one in which the second chiral scalar couples to an arbitrary second metric is used to formulate the theory on an arbitrary two-dimensional manifold. We use this action with both metrics Riemannian (or complex) to formulate the path integral on any Riemann surface. We calculate the partition function in this way and check the result with that calculated using canonical quantisation, and then extend this to multiple chiral bosons. The partition function for chiral scalars taking values on a rational torus is a sum of terms, each of which is the product of two holomorphic functions, one a function of the modulus of the first metric and the other a function of the modulus of the second metric. In particular, for the case of chiral bosons moving on a torus defined by an even self-dual lattice, the partition function is a single product of two such holomorphic functions, not a sum of such terms. This is applied to the heterotic string to give a world-sheet action whose quantisation is modular invariant and free from anomalies. We discuss modular invariance for the moduli of both metrics and the extension to higher genus Riemann surfaces.