Deformation Spaces

What is the Jacobian of a Riemann Surface with Boundary? (S. 52-53)

Thomas M. Fiore and Igor Kriz Abstract.

We de?ne the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of "open abelian varieties" which satis?es gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of "conformal ?eld theory" to be de?ned on this space. We further prove that chiral conformal ?eld theories corresponding to even lattices factor through this moduli space of open abelian varieties.

1. Introduction

The main purpose of the present note is to generalize the notion of the Jacobian of a Riemann surface to Riemann surfaces with real-analytically parametrized boundary (or, in other words, conformal ?eld theory worldsheets). The Jacobian of a closed surface is an abelian variety. What structure of "open abelian variety" captures the relevant data in the "Jacobian" of a CFT worldsheet? If we considered Riemann surfaces with punctures instead of parametrized boundary components, the right answer could be easily phrased in terms of mixed Hodge structures.

But in worldsheets, we see more structure, and some of it is in?nite-dimensional. For example, even to a disk with analytically parametrized boundary, one naturally assigns an in?nite-dimensional symplectic form and a restricted maximal isotropic space (cf. [7]). Any structure we propose should certainly include such data. Additionally, in worldsheets, boundary components can have inbound or outbound orientation, and an inbound and outbound boundary component can be glued to produce another worldsheet. So another test of having the right notion of "open abelian variety" is that it should enjoy a similar gluing structure.

We should point out that it is actually a remarkably strong requirement that a structure such as a (closed) abelian variety could somehow be "glued together" from "genus 0" data similar to the situation we described above for a disk. One quickly convinces oneself that naive approaches based on modelling somehow the 1-forms on a Riemann surface, together with mixed Hodge-type integral structure data, fail to produce the required gluing. In fact, in some sense, the desired structure must be "pure" rather than "mixed".

Note that there is no way of "gluing" a pure Hodge structure out of a mixed Hodge structure which does not already contain it: in the case of a closed Riemann surface with punctures, the mixed Hodge structure on its ?rst cohomogy contains the pure Hodge structure of the original closed surface, so no gluing is involved. Clearly, the situation is di?erent when we are gluing a non-zero genus surface from a genus 0 surface with parametrized boundary.

There is, however, a yet stronger test. When L is an even lattice (together with a Z/2-valued bilinear form b satisfying a suitable condition), one has a notion of conformal ?eld theory associated with L ([9, 4]). It could be argued that the de?nition only uses additive data, so the lattice conformal theories should "factor through open abelian varieties". In some sense, if one considers the conjectured space of open abelian varieties to be the "Jacobian" of the moduli space of worldsheets (with all its structure), then one could interpret this as a sort of "Abelian Langlands correspondence" for that space.