# Localization theory for W-algebras

### APA

Dhillon, G. (2018). Localization theory for W-algebras. Perimeter Institute. https://pirsa.org/18090037

### MLA

Dhillon, Gurbir. Localization theory for W-algebras. Perimeter Institute, Sep. 10, 2018, https://pirsa.org/18090037

### BibTex

@misc{ pirsa_PIRSA:18090037, doi = {10.48660/18090037}, url = {https://pirsa.org/18090037}, author = {Dhillon, Gurbir}, keywords = {Mathematical physics}, language = {en}, title = {Localization theory for W-algebras}, publisher = {Perimeter Institute}, year = {2018}, month = {sep}, note = {PIRSA:18090037 see, \url{https://pirsa.org}} }

**Collection**

**Subject**

The localization theorem, which has played a central role in representation theory since its discovery in the 1980s, identifies a regular block of Category O for a semisimple Lie algebra with certain D-modules on its flag variety. In this talk we will explain work in progress which produces a similar picture for the Virasoro algebra and more generally for affine W-algebras. Some new purely algebraic input is (i) a version of the Feigin-Fuchs duality between Verma modules for Vir at central charges c and 26 - c, which applies to all smooth representations and other affine W-algebras, and (ii) a linkage principle for representations in category O of a W-algebra. As geometric input, we will explain how to (i) adapt the Beilinson-Drinfeld construction of vertex algebras via factorization spaces to also produce representations and in particular (ii) develop a factorizable version of affine Borel--Weil--Bott.