The Grassmannian VOA

TL;DR

This paper introduces a new family of vertex operator algebras based on Grassmannian cosets, providing detailed representation analysis, simple character formulas, and a gluing framework to construct complex algebras, including supersymmetric variants.

Contribution

It develops a comprehensive theory of Grassmannian VOAs, including character formulas, truncation conjectures, and a gluing method to build advanced algebraic structures, extending the understanding of coset VOAs.

Findings

Derived simple character formulas for generic parameters.

Proposed conjectural truncation curves for the algebra.

Demonstrated gluing technique to construct supersymmetric Grassmannian VOAs.

Abstract

We study the 3-parametric family of vertex operator algebras based on the unitary Grassmannian coset CFT $\mathfrak{u}(M+N)_k/(\mathfrak{u}(M)_k \times \mathfrak{u}(N)_k)$. This VOA serves as a basic building block for a large class of cosets and generalizes the $\mathcal{W}_\infty$ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the $\mathcal{N}=2$ supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specialization of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.