The Grassmannian VOA

Abstract

We study the 3-parametric family of vertex operator algebras based on the unitary Grassmannian coset CFT u(M +N) k /(u(M) k ×u(N) k ). This VOA serves as a basic building block for a large class of cosets and generalizes the W ∞ 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 noncoset algebras. We demonstrate the power of this technology with some examples and show in particular that the 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 4parametric 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. 8 Four-parametric family of algebras 64 9 Discussion and Conclusions 70 A Character background 75 A.1 The cycle index and the Pólya enumeration theorem 75 A.2 Pólya enumeration theorem 75 A.3 The vacuum character of the Grassmannian 76 A.4 Alternative character counting 77 B OPE structure constants 80 B.1 Equations from OPE bootstrap 80 B.2 Universal structure constants 82

On Bethe equations of 2d conformal field theory

“…A parallel construction in the Yangian language was studied in [74][75][76][77][78]. But there are other important vertex algebras which seem to require more complicated basic elements [45,79]. One such generalization is to consider the Grassmannian vertex operator algebra as a more general building block.…”

“…Just as W 1+∞ is a two-parametric family of algebras, the unitary Grassmannian coset algebras depend on three complex parameters. But there are indications from the representation theory [79] that these algebras are part of fourparametric family of algebras. Performing a OPE bootstrap for Grassmannian analogously to [19,20] is much more difficult due to significantly larger number of local fields, so it would be good to find other approaches to study these algebras.…”