Affine Slodowy Variety and Beilinson-Bernstein Theorem Localization for Affine W-Algebras
October 12 – 23, 2015

The affine W -algebras are certain vertex  algebras associated  with a pair (g, e), with g a finite-dimensional reductive Lie algebra and e a nilpotent element of g, which can be regarded as affinizations of finite W -algebras.  They can be also considered as generalizations of affine Kac-Moody algebras and the Visasoro algebra.  The study  of affine W -algebras began with the work of Zamolodchikov.  Mathematically, affine W -algebras are defined by the method of quantized  Drinfeld-Sokolov reduction  that  was discovered by Feigin  and  Frenkel.   The general definition of affine W -algebras were given by Kac, Roan and Wakimoto in 2003. The most recent developments  in representation theory  of affine W -algebras were done by Kac and Wakimoto  and the first author of this program.

In this research program,  we are interested in an affine analogue to the Slodowy variety, that  we call the  affine Slodowy variety.  We hope that  the affine Slodowy variety  can be used to obtain  an analogue  to the  Beilinson-Bernstein  localization  Theorem  for affine W – algebras,  and  thus  gives a geometrical  realization  of representations of affine W -algebras. The motivation < span style=""> comes from different  generalizations  of the  localization theorem:  for finite W -algebras (cf. works of Dodd-Kremnizer),  for affine Lie algebras (cf. e.g. works of Frenkel- Gaitsgory),  and for affine W -algebras at critical level (cf. works of the first author  together Kuwabara  and Malikov).

We also feel that  the study  of the affine Slodowy variety,  particularly its geometrical properties,  can be of independent interest.

Our project is joint with Toshiro Kuwabara.


Tomoyuki Arakawa (Kyoto University)
Anne Moreau (Université de Poitiers)