A commutant realization of W^(2)_n at critical level

TL;DR

This paper constructs a free field realization of a superalgebra at critical level, identifies its commutant as a bosonic algebra generated by n+1 fields, and relates it to W-algebras and invariant differential operators.

Contribution

It provides a new realization of W^{(2)}_n at critical level via commutants in free field systems, extending known cases and conjecturing a general pattern.

Findings

The commutant is purely bosonic and generated by n+1 fields.

The Zhu algebra of the commutant matches invariant differential operators.

For n ≤ 4, the commutant is isomorphic to W^{(2)}_n at critical level.

Abstract

For n\geq 2, there is a free field realization of the affine vertex superalgebra A associated to psl(n|n) at critical level inside the bc\beta\gamma system W of rank n^2. We show that the commutant C=Com(A,W) is purely bosonic and is freely generated by n+1 fields. We identify the Zhu algebra of C with the ring of invariant differential operators on the space of n\times n matrices under SL_n \times SL_n, and we classify the irreducible, admissible C-modules with finite dimensional graded pieces. For n\leq 4, C is isomorphic to the W_n^{(2)}-algebra at critical level, and we conjecture that this holds for all n.