Representation Homology, Lie Algebra Cohomology and Derived Harish-Chandra Homomorphism

We define and study a class of N = 2 vertex operator algebras W G labelled by complex reflection groups. They are extensions of the N = 2 super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group G. If G is a Coxeter group, the N = 2 super Virasoro algebra enhances to the (small) N = 4 superconformal algebra. With the exception of G = Z 2 , which corresponds to just the N = 4 algebra, these are non-deformable VOAs that exist only for a specific negative value of the central charge. We describe a free-field realization of W G in terms of rank(G) βγbc ghost systems, generalizing a construction of Adamovic for the N = 4 algebra at c = −9. If G is a Weyl group, W G is believed to coincide with the N = 4 VOA that arises from the four-dimensional super Yang-Mills theory whose gauge algebra has Weyl group G. More generally, if G is a crystallographic complex reflection group, W G is conjecturally associated to an N = 3 4d superconformal field theory. The free-field realization allows to determine the elusive "R-filtration" of W G , and thus to recover the full Macdonald index of the parent 4d theory.

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“…(x + x −1 ) τ − τ 2 χ g Adj (u) . (6.46) 34 In fact this is the main conjecture in [50]. 35 The super-determinant is defined in the standard way.…”

Section: The Hall-littlewood Limitmentioning

confidence: 92%