Representation theory of $W$-algebras, II

Abstract

We study the representation theory of the W -algebra W k (ḡ) associated with a simple Lie algebra ḡ at level k. We show that the "−" reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k ∈ C. Moreover, we show that the character of each irreducible highest weight representation of W k (ḡ) is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra g of ḡ. As a consequence we complete (for the "−" reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of W -algebras.

The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras

“…An alternative construction of W -algebras is also possible using a quantum Hamiltonian reduction based on the works of Feigin and Frenkel [13], Kac et al [25], Kac and Wakimoto [24] and De Sole and Kac [36]. D'Andrea et al [36] and Arakawa [3] have shown that the two definitions of finite W -algebras are equivalent.…”

“…, p n ) of the diagram's row lengths, where p i is the number of bricks in the i-th row of the pyramid, so that 1 p 1 · · · p n . The figure illustrates the pyramid with columns (1, 3, 4, 2, 1) and rows (1,2,3,5):…”

Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks

“…These vertex algebras are our main computational tools. 4 In this paper we do not employ the full toolbox available to us in topologically twisted 4d gauge theory. In particular, this toolbox includes topological 3d interfaces (domain walls) and topological 2d surface defects.…”

“…Once we achieve that, the general machine of 4d gauge theory dualities will then allow us to construct specific collections of qGL dual twisted D-modules with matching properties 4 The resulting structure is a higher analogue of the structure which arises naturally in the study of 3d…”

W-algebras, higher rank false theta functions, and quantum dimensions

“…where g is the corresponding simple Lie algebra, W c p (g) is the affine W -algebra associated to g [7,15], and W 0 ( p) Q and W ( p) Q are certain vertex algebras defined below. In the special case of g = s 2 , we recover a more familiar embedding of vertex algebras studied for examples in [2,10,11]:…”