Alina Marian

The level-rank duality for non-abelian theta functions

We prove that the strange duality conjecture of Beauville–Donagi–Tu holds for all curves. We establish first a more extended level-rank duality, interesting in its own right, from which the standard level-rank duality follows by restriction.

Generic strange duality for

Strange duality is shown to hold over generic K3 surfaces in a large number of cases. The isomorphism for elliptic K3 surfaces is established first via Fourier-Mukai techniques. Applications to Brill-Noether theory for sheaves on K3s are also obtained. The appendix written by Kota Yoshioka discusses the behavior of the moduli spaces under change of polarization, as needed in the argument.

K3

In the prequel to this paper, two versions of Le Potiers strange duality conjecture for sheaves over abelian surfaces were studied. A third version is considered here. In the current setup, the isomorphism involves moduli spaces of sheaves with fixed determinant and fixed determinant of the Fourier-Mukai transform on one side, and moduli spaces where both determinants vary, on the other side. We first establish the isomorphism in rank one using the representation theory of Heisenberg groups. For product abelian surfaces, the isomorphism is then shown to hold for sheaves with fiber degree 1 via Fourier-Mukai techniques. By degeneration to product geometries, the duality is obtained generically for a large number of numerical types. Finally, it is shown in great generality that the Verlinde sheaves encoding the variation of the spaces of theta functions are locally free over moduli.

surfaces

We give a brief exposition of the 2d TQFT that captures the structure of the GL Verlinde numbers, following Witten.