Se p 20 06 Braid cobordisms , triangulated categories , and flag varieties Mikhail Khovanov and Richard

2 3 M ay 2 00 7 Braid cobordisms , triangulated categories , and flag varieties Mikhail Khovanov and Richard

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

0 Mirror symmetry and actions of braid groups on derived categories

After outlining the conjectural relationship between the conjec-tural mirror symmetry programmes of Kontsevich and Strominger-Yau-Zaslow, I will describe some natural consequences of this which are proved from scratch in joint work with Mikhail Khovanov and Paul Seidel. Namely, actions of braid groups are found on derived categories of coherent sheaves, dual to Seidel's braid group of symplecti...

Braiding via Geometric Lie Algebra Actions

We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov-Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s...

Rouquier Complexes are Functorial over Braid Cobordisms

Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.

An invariant of tangle cobordisms via subquotients of arc rings