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Local B-model and Mixed Hodge Structure

Hold Date
2009-11-02 15:30〜2009-11-02 17:00
Ito Campus, Faculty of Mathematics building, seminar room 2
Object person
Yukiko KONISHI(Kyoto Univ.)

Local mirror symmetry is a variant of mirror symmetry derived from mirror symmetry of  toric Calabi--Yau hypersurfaces.
Its statement is as follows. Take a 2-dimensional reflexive polyhedron (e.g the convex hull of (1,0),(0,1),(-1,-1) ).
On one side, one can associate to this a toric surface whose fan is generated by integral points of the polyhedron  (e.g. P^2),
and its local Gromov--Witten invariants (local A-model).
On other side , one can associate an affine hypersurface C in 2-dimensional algebraic torus T^2 whose defining equation  is the sum of Laurent monomials corresponding to integral points of the  given polyhedron, and  the relative cohomology group H^2(T^2, C)  (local B-model).
The both of them are closely related to a system of differential equations associated to the polyhedron called the A-hypergeometric system due to Gel'fand, Kapranov, Zelevinsky.
As to the local B-model,the (V)MHS of  H^2(T^2,C)  has been studied by  Batyrev and Stienstra.
In the joint work with Satoshi Minabe (arXiv:0907.4108),
we defined, using their results,  an analogue of the Yukawa coupling whose direct definition was not known so far.
In this talk, I explain these mixed Hodge theoretic aspects of the local B-model.