2010 Chern-Simons Master-Class Series

October 25, 2010

Professor Yan Soibelman, Kansas State University, Manhattan, Kansas, will talk about Motivic Donaldson-Thomas invariants and wall-crossing formulas on Monday through Friday of the week of November 1.

Lecturer:Professor Yan Soibelman (Kansas State University, Manhattan, Kansas)

Schedule: Monday, 1-3pm, 740 Evans; Tue, 12-2pm, 4020LC LeCont; Wednesday, 1-3pm, 740 Evans; Thursday, 12-2pm, 4020LC LeCont; Friday 1-3pm, 56 Hildebrand.

Title: Motivic Donaldson-Thomas invariants and wall-crossing formulas.

Donaldson-Thomas invariants of 3-dimensional Calabi-Yau manifolds are related to the (properly defined) count of various geometric objects like: special Lagrangian manifolds, semistable vector bundles, ideal sheaves, etc. It turns out that the unifying framework is the one of 3-dimensional Calabi-Yau categories endowed with Bridgeland stability condition. Then we count the ``number" of semistable objects with the fixed class in K-group.
Corresponding theory was developed in a series of our papers with Maxim Kontsevich. It gives also a mathematical approach to BPS invariants (both enumerative and refined) in gauge and string theory. Similarly to geometric story, our invariants change on the real codimension one ``walls" in the space of stability conditions. Categorically, the wall crossing formulas show how the ``motive of semistable objects" changes across the wall (this explains the term ``motivic").
There are two different approaches to the theory of motivic DT-invariants. I plan to discuss mostly a recent one based on moduli spaces of representations of quiver with potential. Applications include cluster transformations, complex integrable systems and new invariants of 3-dimensional manifolds.