Ron Donagi (University of Pennsylvania)

Abstract: The moduli space of vector bundles on a variety $X$ which
admits an elliptic fibration $f: X \to B$ (with a section $\sigma: B \to
X$) can be described in terms of data on the base $B$ via the
Fourier-Mukai transform, or the spectral construction. This is an
"elliptic" version of the various integrable systems of meromorphic Higgs
fields on $B$: instead of Higgs fields with values in the canonical bundle
(or any other vector bundle) on $B$, one considers Higgs fields with
values in the family $f$ of elliptic curves.

The extension of this result to the case where $f: X \to B$ is a genus 1
fibration (having no section $\sigma: B \to X$) leads to some surprising
new features, including the appearance of gerbes, or non-commutative
structures, on $X$. In a sense, the non-commutativity of such a structure
is dual to the non-existence of a section. I will formulate a still partly
conjectural duality of derived categories on these gerbes, and will show
how it incorporates the Fourier-Mukai transform and other known results.

Mirror symmetry between Calabi-Yau threefolds is conjectured (by
Strominger, Yau and Zaslow) to be realized geometrically by a duality
quite similar to Fourier-Mukai, but involving special Lagrangian tori
instead of elliptic curves. The CY analogue of our results suggests a
modification and strengthening of the SYZ conjecture. In particular, there
should be a real integrable system on the stringy moduli space of
Calabi-Yaus (i.e. the moduli of the complex structure, Kahler structure,
and B-field) on which mirror symmetry acts.

This talk is based on joint work with Tony Pantev.