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Generalized Homological Mirror Symmetry and Applications

Objective

Mirror symmetry arose originally in physics, as a duality between $N = 2$ superconformal field theories. Witten formulated a more mathematically accessible version, in terms of topological field theories. Both conformal and topological field theories can be defined axiomatically, but more interestingly, there are several geometric ways of constructing them. A priori, the mirror correspondence is not unique, and it does not necessarily remain within a single class of geometric models. The classical case relates $\sigma$-models, but in a more modern formulation, one has mirror dualities between different Landau-Ginzburg models, as well as between such models and $\sigma$-models; orbifolds should also be included in this. The simplest example would be the function $W: \C \rightarrow \C$, $W(x) = x^{n+1}$, which is self-mirror (up to dividing by the $\bZ/n+1$ symmetry group, in an orbifold sense). While the mathematics of the $\sigma$-model mirror correspondence is familiar by now, generalizations to Landau-Ginzburg theories are only beginning to be understood. Today it is clear that Homologcal Mirror Symmetry (HMS) as a categorical correspondence works and it is time for developing direct geometric applications to classical problems - rationality of algebraic varieties and Hodge conjecture. This the main goal of the proposal. But in order to attack the above problems we need to generalize HMS and explore its connection to new developments in modern Hodge theory. In order to carry the above program we plan to further already working team Vienna, Paris, Moscow, MIT.

Field of science

  • /natural sciences/mathematics

Call for proposal

ERC-2008-AdG
See other projects for this call

Funding Scheme

ERC-AG - ERC Advanced Grant

Host institution

UNIVERSITAT WIEN
Address
Universitatsring 1
1010 Wien
Austria
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 060 800
Principal investigator
Ludmil Katzarkov (Prof.)
Administrative Contact
Helmut Schaschl (Dr.)

Beneficiaries (1)

UNIVERSITAT WIEN
Austria
EU contribution
€ 1 060 800
Address
Universitatsring 1
1010 Wien
Activity type
Higher or Secondary Education Establishments
Principal investigator
Ludmil Katzarkov (Prof.)
Administrative Contact
Helmut Schaschl (Dr.)