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Dominic Joyce

2010-09-06
11:10:00 - 12:10:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Calabi-Yau manifolds are compact Kahler manifolds with trivial canonical bundle. They have a rich geometric structure. Calabi-Yau 3-folds are of interest in String Theory and are the subject of "Mirror Symmetry" -- a family of conjectures relating the complex geometry of one Calabi-Yau 3-fold X with the symplectic geometry of a different Calabi-Yau 3-fold X*.
The "Donaldson-Thomas invariants" of a Calabi-Yau 3-fold X, introduced by Richard Thomas in 1998, are integers which "count" Gieseker (semi)stable coherent sheaves on X in a fixed topological class (Chern character). Coherent sheaves are algebro-geometric objects which generalize holomorphic vector bundles. Donaldson-Thomas invariants have the nice property that they are unchanged under deformations of the complex structure of X. They have attracted attention recently through the "MNOP Conjecture", which relates Donaldson-Thomas invariants counting rank 1 sheaves (ideal sheaves) to Gromov-Witten invariants counting curves on X.
Thomas' original definition worked only for topological classes in which there are no strictly semistable sheaves. Also, the dependence of the invariants on the stability condition (polarization / Kahler class) was not understood. We will describe a new generalization of Donaldson-Thomas invariants, with the following properties:
* they take values in the rationals;
* they are defined for all topological classes, and are equal to Donaldson-Thomas invariants when these are defined;
* they are unchanged under deformations of the complex structure of X; and
* they transform according to a known wall-crossing formula under change of stability condition.
This is related to the 2008 paper by Kontsevich and Soibelman. Joint work with Yinan Song.