Talbot 2008: Affine Lie algebras and chiral structures

Mentored by Dennis Gaitsgory.

March 30 to April 5, 2008.
Plymouth, Massachusetts.


The notes are still in very rough form, but are available upon request from Sheel Ganatra. (E-mail ganatra AT math DOT mit). There are, however, notes from the chiral algebras seminar led by Mike Hopkins, which also has a nice list of references.

Talk Schedule

Monday Tuesday Wednesday Thursday Friday


Affine grassmannian and factorization


Reps of quantum groups = FS

Constructing the functor

Intro to quantum groups

Factorization algebras and $E_2$ algebras

Twistings and twisted D-modules

More affine grassmannian

Proof of equivalence

Rep theory for quantum groups

$E_2$ modules and factorization modules

Chiral categories

discussion session


Drinfeld doubles

Koszul duality, $E_2$ algebras, and Drinfeld doubles

FS category

The twisted Whittaker category

discussion session

-- Monday --

1. Introduction/Overview. [Dennis Gaitsgory]

2. Overview of quantum groups. [Ian]
Definition of quantum universal enveloping algebras $U_q(g)$; the structure of its category of representations -- braided and ribbon tensor structure and R-matrix; the big and small quantum group, at and away from roots of unity.

3. Representation theory for quantum groups. [Travis]

4. Drinfeld doubles. [Nick]
Show the equivalence $DD(U_q(n_+)) = \operatorname{Rep} U_q(g)$ away from roots of unity, and fully faithful embedding at roots of unity.

-- Tuesday --

5. Affine grassmannian and factorization structures. [Owen]
Definition of the affine grassmannian. How it gives a factorization space.

6. Factorization algebras and E2 algebras. [John or Jacob]

7. E2 modules and factorization modules. [John or Jacob]

8. Koszul duality, E2 algebras and Drinfeld doubles. [Jacob]

-- Wednesday --

9. D-modules. [Zhiwei]

10. Twistings and twisted D-modules. [Reimundo]

11. Chiral categories. [Jacob or John]

12. Factorizable sheaves. [Carl]

-- Thursday --

13. Why FS is the same as chiral modules for the partial Koszul dual of Uq(n+). [Dennis Gaitsgory]

14. More affine grassmannian. [Vivek]

15. Discussion.

16. The twisted Whittaker category. [Xinwen]
Definition of the twisted Whittaker category $\mathrm{Whit}^c$ of the affine Grassmannian.

-- Friday --

17. Construction of the functor $\mathbf{Whit}^c$ → FS. [Scott]
What it says.

18. Proof that the functor is an equivalence. [Richard]

19. TBD.