Quantising Chiral Bosons On Riemann Surfaces

TL;DR

We study a bi‑metric generalisation of Sen's chiral boson action in two dimensions, allowing the shadow sector to couple to an arbitrary second metric and formulating the theory on general Riemann surfaces. Using a combination of path integration with complex metrics and canonical quantisation, we compute the partition function and show that on a torus with a rational Narain lattice it factorises into holomorphic pieces in the two independent moduli $\tau$ and $\rho$, while in general it mixes these moduli. The results yield modular invariant constructions and illuminate anomaly cancellation in heterotic‑string–like settings by coupling to shadow sectors; complete factorisation occurs at even self‑dual lattices, enabling modular invariance under separate torus moduli. The framework naturally extends to higher genus surfaces and provides a route to anomaly‑free, modular invariant worldsheet theories involving chiral bosons and shadow sectors.

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.