Quantising Chiral Bosons On Riemann Surfaces

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.

💡 Research Summary

The paper develops a consistent quantum theory of chiral bosons on arbitrary two‑dimensional manifolds by extending Sen’s action to a bi‑metric formulation. In Sen’s original construction a physical chiral scalar (A) couples to the spacetime metric (g) while a second “shadow’’ scalar (C) couples to a flat metric (\eta). The shadow field decouples from the physical sector, but the presence of a second metric allows a geometric definition of the tensor (M) that mixes the two metrics.

Hull and Lambert generalise this to allow the shadow scalar to couple to an arbitrary second metric (\bar g). The resulting action contains a single non‑trivial component (M_{-}^{-}) of the tensor (M), which can be written explicitly in terms of the zweibein parametrisations of the two metrics. Choosing
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