The Grassmannian VOA

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.

ancillary Mathematica notebook for character calculations included

References in corpus (69)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• The D1/D5 System And Singular CFT
• Infinite Chiral Symmetry in Four Dimensions
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• The Topological Vertex
• An AdS_3 Dual for Minimal Model CFTs
• The Worldsheet Dual of the Symmetric Product CFT
• Triality in Minimal Model Holography
• Deriving the ${\text{AdS}}_3/{\text{CFT}}_2$ Correspondence
• Higher Spins & Strings
• Tinkertoys for Gaiotto Duality
• On the Landau-Ginzburg Description of $N=2$ Minimal Models
• Vertex operator algebras, Higgs branches, and modular differential equations
• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• Para-Liouville/Toda central charges from M5-branes
• Large N=4 Holography
• W-symmetry, topological vertex and affine Yangian
• A holographic dual for string theory on ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathrm{S}}^3\times {\mathrm{S}}^1$
• Ding-Iohara-Miki symmetry of network matrix models
• Explicit examples of DIM constraints for network matrix models
• Counting Tensor Model Observables and Branched Covers of the 2-Sphere
• BPS spectrum on AdS${}_3\times$S${}^3\times$S${}^3\times$S${}^1$
• Instanton moduli spaces and bases in coset conformal field theory
• ABJ Triality: from Higher Spin Fields to Strings
• String Theory as a Higher Spin Theory
• Chiral Algebras for Trinion Theories
• The Search for a Holographic Dual to AdS(3)xS(3)xS(3)xS(1)
• Cohomological Hall algebras, vertex algebras and instantons
• Webs of W-algebras
• (p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters
• Extended higher spin holography and Grassmannian models
• Strings on ${\text{AdS}}_3\times {\text{S}}^3\times {\text{S}}^3\times {\text{S}}^1$
• Exploring ${\mathcal{W}}_{\infty }$ in the quadratic basis
• Even spin minimal model holography
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Ramond Sector Characters and N=2 Landau-Ginzburg Models
• Anomaly in RTT relation for DIM algebra and network matrix models
• On spectrum of ILW hierarchy in conformal field theory
• Parafermionic Liouville field theory and instantons on ALE spaces
• $\mathcal{W}$-algebra Modules, Free Fields, and Gukov-Witten Defects
• Instanton R-matrix and W-symmetry
• W-algebras with set of primary fields of dimensions (3, 4, 5) and (3,4,5,6)
• Explicit generators in rectangular affine W-algebras of type A
• On the coset duals of extended higher spin theories
• On Extensions of $\widehat{\mathfrak{gl}(m|n)}$ Kac-Moody algebras and Calabi-Yau Singularities
• The supersymmetric affine Yangian
• Casimir eigenvalues for universal Lie algebra
• $\mathcal{N}=2$ Minimal Models: A Holographic Needle in a Symmetric Orbifold Haystack
• Twin-plane-partitions and $\mathcal{N}=2$ affine Yangian
• Rectangular W-algebras, extended higher spin gravity and dual coset CFTs
• Affine sl(2|1) and D(2|1;alpha) as Vertex Operator Extensions of Dual Affine sl(2) Algebras
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• Higher spin algebras and large $\mathcal{N}=4$ holography
• Rectangular W-(super)algebras and their representations
• On spectrum of ILW hierarchy in conformal field theory II: coset CFT's
• A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories
• The MacMahon R-matrix
• Superconformal Coset Equivalence from Level-Rank Duality
• Maulik-Okounkov's R-matrix from Ding-Iohara-Miki algebra
• Yangian associated with 2D $\mathcal{N}=1$ SCFT
• Gluing two affine Yangians of ${\mathfrak{gl}}_1$
• Gluing affine Yangians with bi-fundamentals
• Grassmannian Topological Kazama-Suzuki Models and Cohomology
• $\widehat{\mathfrak{sl}}(n{)}_N$ WZW conformal blocks from $SU(N)$ instanton partition functions on ${\mathbb{C}}^2/{\mathbb{Z}}_n$
• On even spin $W_{\infty }$
• Symmetry Algebras of Stringy Cosets
• $n$-th parafermion ${\mathcal{W}}_N$ characters from $U(N)$ instanton counting on ${\mathbb{C}}^2/{\mathbb{Z}}_n$

Cited by in corpus (1)

• Adding Complex Fermions to the Grassmannian-like Coset Model