Localization theory for W-algebras


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


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


          @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}}

Gurbir Dhillon Stanford University


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.