Quantum Groups and Quantum Cohomology

Abstract

In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q, the Yangian of Q, acting on the cohomology of these varieties, and show several results about their basic structure theory. We prove a formula for quantum multiplication by divisors in terms of this Yangian action. The quantum connection can be identified with the trigonometric Casimir connection for Y_Q; equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of Y_Q. A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection. In the second part, we give an extended example of the general theory for moduli spaces of sheaves on C^2, framed at infinity. Here, the Yangian action is analyzed explicitly in terms of a free field realization; the corresponding R-matrix is closely related to the reflection operator in Liouville field theory. We show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, we obtain an action of the W-algebra W(gl(r)) on the equivariant cohomology of rank $r$ moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

277 pages, to appear in Asterisque

Cited by in corpus (107)

• The affine Yangian of ${\mathfrak{gl}}_1$ revisited
• On Three Dimensional Quiver Gauge Theories and Integrability
• Supersymmetric gauge theory and the Yangian
• Baxter's Relations and Spectra of Quantum Integrable Models
• Ding-Iohara-Miki symmetry of network matrix models
• Cohomological Hall algebras, vertex algebras and instantons
• Coproduct for the Yangian of an affine Kac-Moody algebra
• Webs of W-algebras
• Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)
• The Crepant Transformation Conjecture for Toric Complete Intersections
• Quantum Integrability and Generalised Quantum Schubert Calculus
• Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety
• Etingof conjecture for quantized quiver varieties
• Combinatorial constructions of derived equivalences
• Baxter Q-operator from quantum K-theory
• On quantum separation of variables
• Anomaly in RTT relation for DIM algebra and network matrix models
• Motivic Chern classes and K-theoretic stable envelopes
• M-theory in the Omega-background and 5-dimensional non-commutative gauge theory
• Holomorphic blocks and the 5d AGT correspondence
• Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
• A slow review of the AGT correspondence
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems
• Triality in Little String Theories
• Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras
• Instanton R-matrix and W-symmetry
• Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology
• 3d Mirror Symmetry and Elliptic Stable Envelopes
• On Extensions of $\widehat{\mathfrak{gl}(m|n)}$ Kac-Moody algebras and Calabi-Yau Singularities
• Instanton moduli spaces and $\mathcal{W}$-algebras
• Quivers with loops and perverse sheaves
• A Note on Quiver Quantum Toroidal Algebra
• Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory
• On AGT-W Conjecture and q-Deformed W-Algebra
• The Stringy Instanton Partition Function
• Chern-Schwartz-MacPherson classes of degeneracy loci
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• On the number of points of nilpotent quiver varieties over finite fields
• Compactification on the Ω-background and the AGT correspondence
• On categories O for quantized symplectic resolutions
• A brief review of the 2d/4d correspondences
• BPS/CFT correspondence V: BPZ and KZ equations from qq-characters
• Superspin chains and supersymmetric gauge theories
• qKZ/tRS Duality via Quantum K-Theoretic Counts
• A note on the algebraic engineering of 4D $\mathcal{N}=2$ super Yang-Mills theories
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• On cohomological Hall algebras of quivers : Yangians
• Vertex Algebras at the Corner
• Homomorphisms between different quantum toroidal and affine Yangian algebras
• Yangian associated with 2D $\mathcal{N}=1$ SCFT
• $q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation
• Rectangular superpolynomials for the figure-eight knot
• More lectures on Hilbert schemes of points on surfaces
• 3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties
• Etingof conjecture for quantized quiver varieties II: affine quivers
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Finite ${\epsilon }_2$-corrections to the $\mathcal{N}=2$ SYM prepotential
• Polynomials associated with fixed points on the instanton moduli space
• On the K-theory stable bases of the Springer resolution
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• Recursive Method for Nekrasov partition function for classical Lie groups
• Elliptic quantum groups and their finite-dimensional representations
• Quiver varieties and tensor products, II
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Toroidal q-Opers
• Equivariant quantum cohomology and Yang-Baxter algebras
• 3d N=2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians
• Bethe subalgebras in Yangians and the wonderful compactification
• Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)
• Moduli spaces of framed symplectic and orthogonal bundles on P2 and the K-theoretic Nekrasov partition functions
• AGT relation in the light asymptotic limit
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• A pedagogical introduction to quantum integrability, with a view towards theoretical high-energy physics
• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• Donaldson-Thomas theory of $[{\mathbb{C}}^2/{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$
• Introduction to quiver varieties -- for ring and representation theoriests
• Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves
• On even spin $W_{\infty }$
• $\widehat{\mathfrak{sl}}(n{)}_N$ WZW conformal blocks from $SU(N)$ instanton partition functions on ${\mathbb{C}}^2/{\mathbb{Z}}_n$
• Yang-Baxter algebras as convolution algebras: The Grassmannian case
• Quantum crystals, Kagome lattice and plane partitions fermion-boson duality
• The meromorphic R-matrix of the Yangian
• Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• The Zoo of Opers and Dualities
• Quantum $K$-theory of toric varieties, level structures, and 3d mirror symmetry
• Quiver varieties and symmetric pairs
• Type $\hat{\mathrm{C}}$ Brauer loop schemes and loop model with boundaries
• Frobenius manifolds and quantum groups
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Grassmann-Grassmann conormal varieties, integrability, and plane partitions
• Elliptic Stable Envelopes and Finite-dimensional Representations of Elliptic Quantum Group
• Loop Models and $K$-Theory
• Hypertoric geometry and Gromov-Witten theory
• Quivers with loops and Lagrangian subvarieties
• Symmetric deformed 2D/3D Hurwitz-Kontsevich model and affine Yangian of gl(1)
• Stable bases of the Springer resolution and representation theory
• A projection formula for the ind-Grassmannian
• Shuffle products for elliptic stable envelopes of Nakajima varieties
• Vanishing cycles on Poisson varieties
• The Crepant Transformation Conjecture implies the Monodromy Conjecture
• Bethe/Gauge correspodence: a short review on an aspect of the integrability nature of supersymmetric gauge theories
• An update of quantum cohomology of homogeneous varieties
• Singular Vector of Ding-Iohara-Miki Algebra and Hall-Littlewood Limit of 5D AGT Conjecture
• Convolution Products and R-Matrices for KLR Algebras of Type $B$
• Virtual Coulomb branch and vertex functions