**RESEARCH IN PAIRS**

**Affine Slodowy Variety and Beilinson-Bernstein Theorem Localization for Affine W-Algebras**

October 12 - 23, 2015

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 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. |