Math Department at MIT | Contacts: Pavel Etingof, Victor Kac

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Spring 2011 | Fridays 3:00 - 5:00pm at 2-136

This talk is about the 2-dimensional case. A rudimentary nonabelian multiplicative surface integration was known since the 1920's (work of Schlesinger). I will present a much more sophisticated construction. My main result is a 3-dimensional nonabelian Stokes Theorem. This result is completely new; only a special case of it was predicted (without proof) in papers in mathematical physics.

The talk is fairly elementary, requiring only some knowledge of Lie groups and their Lie algebras. And there are many color pictures!

My motivation for this work has to do with a problem in twisted deformation quantization. I will say a few words about this at the end of the talk.

For full details see the lecture notes or the preprint on arXiv.

It can be proved that a uniqueness of DG enhancements exists for a large class of triangulated categories. This class includes all derived categories of quasi-coherent sheaves, bounded derived categories of coherent sheaves and category of perfect complexes on quasi-projective schemes, as well as on a noncommutative varieties.

This result shows that triangulated categories which have a geometric nature largely distinguished among all of triangulated categories; for which this property does not hold in general.

One consequence of these results is a theorem asserting that an existence of a fully faithful functor between such categories implies an existence of a fully faithful functor between them that has integral form, i.e. that is represented by an object on the product.

Moreover, for projective varieties there is a strong uniqueness for DG enhancements, which implies that any fully faithful functor from the category of perfect complexes on the projective variety to another category of this type has integral type, i.e. it is represented by an object on a product.

These results have also application to deformation theory of objects in derived categories and to homological mirror symmetry.

This is a joint paper with Valera Lunts.

An algebraic approach to Boundary Conformal Field Theory has then been developed (K.-H. Rehren and R.L.). It turns out that Boundary Conformal Field Theory on the half-plane is described a (non necessarily local) conformal net on the real line.

In a recent work, starting with a local conformal net A on the real line, one considers a unitary semigroup E(A) associated with A. Each element V of this semigroup gives a new Boundary Quantum Field Theory on the half-plane (E. Witten and R.L.). The computation of second quantization elements of E(A) is obtained by an analog of the Beurling-Lax theorem theorem convening shift invariant subspaces of H^{\infty} of the disk. The corresponding subgroup of E(A) is isomorphic to the semigroup of symmetric, holomorphic inner functions on the disk.

A very recent description of Boundary QFT on the interior of the Lorentz hyperboloid is explained (K.-H. Rehren and R.L.). A different, but surprisingly isomorphic, semigroup plays a role here. The natural states are thermal states at Hawking temperature rather than ground states.

In the second talk I will explain an application of our results to the classical problem of computing Goldie ranks for primitive ideals. The study of Goldie ranks was initiated by Joseph some 30 years ago.

All necessary information about W-algebras and Goldie ranks will be explained. Warning: speaker's opinion on what is necessary may be different from yours.