## Abstract

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.

## Details

**Talk Number**PIRSA:18090037

**Speaker Profile**Gurbir Dhillon

**Collection**Mathematical Physics

- Mathematical physics

**Scientific Area**

- Scientific Series

**Talk Type**