Mirror symmetry, variations of Hodge structure, and K-theory
by Charles DORAN (University of Washington)
Abstract:
A cohomological ("D-brane charge") perspective on Kontsevich's homological
mirror symmetry conjecture results in a predicted equivalence between the
middle dimensional integral cohomology and even topological K-theory of a
mirror pair of Calabi-Yau threefolds. Remarkably enough, the equivalence
apparently extends far beyond just an isomorphism of Z-modules to an
identification of natural skew-forms on both sides and even a dictionary
between integral monodromy in the variation of Hodge structures of one
Calabi-Yau manifold and special automorphisms of K-theory of the other.
The speaker will provide a gentle introduction to the subject, starting with
some basic algebraic geometry (pencils of hypersurfaces in projective
space), simple differential equations (generalized hypergeometric ODEs),
an introduction to Hodge theory and toric geometry, and culminating in
(the statement of) joint results with John Morgan establishing these
predictions for a large class of Calabi-Yau threefolds with b_3 = 4.