Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology

Abstract

We find a counterpart of the classical fact that the regular representation R ( G ) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G 0 ⊂ G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra g of G, and has the structure of a g ⊕ g -module. The standard regular representation R ( G ^ ) of the affine group G ^ has two commuting actions of the Lie algebra g ^ with total central charge 0, and carries the structure of a conformal field theory. The modified versions R ′ ( G ^ ) and R ′ ( G ^ 0 ) , originating from the loop version of the Bruhat decomposition, have two commuting g ^ -actions with central charges shifted by the dual Coxeter number, and acquire vertex operator algebra structures derived from their Fock space realizations, given explicitly for the case G = SL ( 2 , C ) . The quantum Drinfeld–Sokolov reduction transforms the representations of the affine Lie algebras into their W -algebra counterparts, and can be used to produce analogues of the modified regular representations. When g = sl ( 2 , C ) the corresponding W -algebra is the Virasoro algebra. We describe the Virasoro analogues of the modified regular representations, which are vertex operator algebras with the total central charge equal to 26. The special values of the total central charges in the affine and Virasoro cases lead to the non-trivial semi-infinite cohomology with coefficients in the modified regular representations. The inherited vertex algebra structure on this cohomology degenerates into a supercommutative associative superalgebra. We describe these superalgebras in the case when the central charge is generic, and identify the 0th cohomology with the Grothendieck ring of finite-dimensional G-modules. We conjecture that for the integral values of the central charge the 0th semi-infinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.