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

TL;DR

This paper extends Feigin-Frenkel duality at the critical level to complex rank, connecting affine vertex algebras and classical W-algebras through interpolation and Poisson structures.

Contribution

It introduces a novel interpolation of Feigin-Frenkel duality to complex rank, unifying constructions of centers and W-algebras in Deligne categories.

Findings

Interpolated Feigin-Frenkel duality at complex rank.

Constructed classical W-algebras for complex Lie algebras.

Recovered classical duality at positive integer ranks.

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.