Cohomological Hall algebras, vertex algebras and instantons

Abstract

We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner ${\mathcal{W}}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.

arXiv comment72 pages, 4 figures, v2: Added some clarifications and updated references 73 pages, 4 figures, v3: Corrected typos and clarified some minor points

References in corpus (63)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Brane Dimers and Quiver Gauge Theories
• Gauge Fields and D-branes
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• Dimer models and toric diagrams
• Triality in Minimal Model Holography
• Quantum Groups and Quantum Cohomology
• On the algebras of BPS states
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters
• Quivers, Tilings, Branes and Rhombi
• Irregular singularities in Liouville theory
• Non-commutative Donaldson-Thomas theory and the conifold
• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of ${\mathbb{A}}^2$
• Quantum Algebraic Approach to Refined Topological Vertex
• W-symmetry, topological vertex and affine Yangian
• Crystal Melting and Toric Calabi-Yau Manifolds
• BPS/CFT correspondence II: Instantons at crossroads, Moduli and Compactness Theorem
• Lectures on Nakajima's Quiver Varieties
• BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters
• Quiver W-algebras
• BPS/CFT correspondence IV: sigma models and defects in gauge theory
• Webs of W-algebras
• Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I
• The cohomological Hall algebra of a preprojective algebra
• Plane partitions with a "pit": generating functions and representation theory
• Notes on Ding-Iohara algebra and AGT conjecture
• Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Spiked Instantons from Intersecting D-branes
• Higher Spins and Yangian Symmetries
• Exploring ${\mathcal{W}}_{\infty }$ in the quadratic basis
• Yangian realisations from finite W algebras
• The BRST formulation of G/H WZNW models
• Holography and Koszul duality: the example of the $M2$ brane
• On W algebras commuting with a set of screenings
• Irreducible Generic Gelfand-Tsetlin Modules of $\mathfrak{gl}(n)$
• Quiver elliptic W-algebras
• Twin-plane-partitions and $\mathcal{N}=2$ affine Yangian
• On two cohomological Hall algebras
• Shifted Yangians and finite W-algebras
• Classification of irreducible modules of W_3 algebra with c = -2
• BPS/CFT correspondence V: BPZ and KZ equations from qq-characters
• Branes, Black Holes and Topological Strings on Toric Calabi-Yau Manifolds
• On cohomological Hall algebras of quivers : Yangians
• Branching rules for quantum toroidal gl(n)
• W-superalgebras as truncation of super-Yangians
• W-Algebras of Negative Rank
• Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras
• On shifted super Yangians and a class of finite W-superalgebras
• Finite W-superalgebras and truncated super Yangians
• Exponential system-size dependence of the lifetime of transient spiral chaos in excitable and oscillatory media
• Plane Partition Realization of (Web of) W-algebra Minimal Models
• Crystals and intersecting branes
• AGT relations for sheaves on surfaces
• Nekrasov's Partition Function and Refined Donaldson-Thomas Theory: the Rank One Case
• The Sebastiani-Thom Isomorphism in the Derived Category
• VOA[M4]
• Lectures on perverse sheaves on instanton moduli spaces
• Relative Donaldson-Thomas theory for Calabi-Yau 4-folds
• Irregular Singularities in the H3+ WZW Model
• Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for $2$-dimensional Calabi-Yau categories (with an appendix by Ben Davison)

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• Quiver Yangian from Crystal Melting
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Trialities of $\mathcal{W}$-algebras
• Shifted Quiver Yangians and Representations from BPS Crystals
• Duality of subregular W-algebras and principal W-superalgebras
• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• A slow review of the AGT correspondence
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Instanton R-matrix and W-symmetry
• On Extensions of $\widehat{\mathfrak{gl}(m|n)}$ Kac-Moody algebras and Calabi-Yau Singularities
• Gauge/Bethe correspondence from quiver BPS algebras
• A Note on Quiver Quantum Toroidal Algebra
• Higgsed network calculus
• Superspin chains and supersymmetric gauge theories
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Mixed network calculus
• BPS Hall Algebra of Scattering Hall States
• Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
• Quiver Yangians and Crystal Melting: A Concise Summary
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• The local motivic DT/PT correspondence
• On Quiver W-algebras and Defects from Gauge Origami
• Gauge origami and quiver W-algebras
• Crystal Melting, BPS Quivers and Plethystics
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• BPS States Meet Generalized Cohomology
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• More on Affine Dynkin Quiver Yangians
• Charging solid partitions
• Wall-Crossing Effects on Quiver BPS Algebras
• Algorithms for representations of quiver Yangian algebras
• Higher rank FZZ-dualities
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• Towards the theory of Yangians
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• Supersymmetric polynomials and algebro-combinatorial duality
• Cohomological Hall algebras and vertex algebras
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• The Grassmannian VOA
• Orbifolds of Gaiotto-Rapčák $Y$-algebras
• Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and W-algebras
• DT invariants from vertex algebras
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations
• Weyl Mutations in Quiver Yangians
• Feigin-Semikhatov conjecture and related topics
• Quiver matrix model of ADHM type and BPS state counting in diverse dimensions
• Reduced quiver quantum toroidal algebras
• Vertex Algebras and Commutative Algebras