Interpolating Feigin-Frenkel Duality at the Critical Level to Matrices of Complex Size

Abstract

In this paper, we extend Feigin-Frenkel duality at the critical level to complex rank by identifying two seemingly unrelated constructions in complex rank. On the affine side, we interpolate Molev's construction of higher Segal-Sugawara vectors and thereby describe the centers of universal affine vertex algebras at the critical level in Deligne's interpolating categories. On the $\mathcal{W}$-side, we construct the classical $\mathcal{W}$-algebras associated with Feigin's Lie algebras of complex rank ${\mathfrak{gl}}_{\lambda }$ and ${\mathfrak{po}}_{\lambda }$ as Poisson vertex algebras, realizing their Drinfeld-Sokolov reduction via an interpolated Adler-Gelfand-Dickey bracket. Upon specialization to positive integer rank in types A, B, and C, this recovers the usual Feigin-Frenkel duality at the critical level. As applications, we obtain a uniform construction of several families of higher Segal-Sugawara vectors for Lie superalgebras and recover a complex-rank analogue of the universal Bethe algebra.

Substantially revised and expanded version, with new applications to Lie superalgebras and interpolated Bethe algebras, a new appendix, and significant technical and expository revisions throughout. 74 Pages