Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Groups and Quantum Cohomology

Davesh Maulik, Andrei Okounkov

TL;DR

This work establishes a structural bridge between the equivariant quantum cohomology of Nakajima quiver varieties and Yangian symmetries encoded through geometric R-matrices. By constructing the Yangian $\mathsf{Y}_Q$ and identifying Baxter subalgebras with the operators of quantum multiplication by tautological divisors, the authors show that the quantum connection coincides with the trigonometric Casimir connection, while divisor operators correspond to elements in Baxter subalgebras. The theory is concretely realized for the instanton moduli $\mathcal{M}(r,n)$, where a free-field realization yields a Liouville-reflection-type $R$-matrix and a W-algebra action $\mathcal{W}(\mathfrak{gl}(r))$ on the cohomology, with implications for AGT-type conjectures. They also develop a robust framework of stable envelopes and $R$-matrices that underpins the link between geometry and representation theory, and outline powerful extensions to K-theory, DT theory, and higher-rank phenomena. Overall, the paper provides a unifying perspective that connects quantum cohomology, integrable systems, and geometric representation theory for Nakajima varieties, with broad consequences in mathematical physics and beyond.

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.

Quantum Groups and Quantum Cohomology

TL;DR

This work establishes a structural bridge between the equivariant quantum cohomology of Nakajima quiver varieties and Yangian symmetries encoded through geometric R-matrices. By constructing the Yangian and identifying Baxter subalgebras with the operators of quantum multiplication by tautological divisors, the authors show that the quantum connection coincides with the trigonometric Casimir connection, while divisor operators correspond to elements in Baxter subalgebras. The theory is concretely realized for the instanton moduli , where a free-field realization yields a Liouville-reflection-type -matrix and a W-algebra action on the cohomology, with implications for AGT-type conjectures. They also develop a robust framework of stable envelopes and -matrices that underpins the link between geometry and representation theory, and outline powerful extensions to K-theory, DT theory, and higher-rank phenomena. Overall, the paper provides a unifying perspective that connects quantum cohomology, integrable systems, and geometric representation theory for Nakajima varieties, with broad consequences in mathematical physics and beyond.

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 moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

Paper Structure

This paper contains 183 sections, 139 theorems, 1198 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fundamental structures and conjectures
  3. Baxter subalgebras and quantum multiplication
  4. Quantum multiplication by divisors
  5. Shift operators and qKZ
  6. Yangian of $\widehat{\mathfrak{gl}(1)}$ and instanton moduli
  7. Further directions
  8. K-theory
  9. Monodromy of QDE and categorification
  10. Higher rank Donaldson-Thomas theory
  11. Hilbert Schemes of points of general surfaces
  12. K-theoretic DT theory
  13. Update
  14. Acknowledgments
  15. General Theory
  16. ...and 168 more sections

Key Result

Theorem 1.3.1

The operators of cohomological degree $2$ in the Baxter subalgebra are the operators of modified quantum multiplication by elements of $H^2_\mathsf{G}(\mathcal{M})_\textup{taut}$

Figures (1)

  • Figure 1: From the YB equation and $\left[g\otimes g, R\right]=0$ we deduce that $R_{F_2,F_1}$ conjugates $g_{F_2} R_{F_2,W} g_{F_1} R_{F_1,W}$ to the product in the opposite order. Taking the trace over $F_2 \otimes F_1$ gives \ref{['e_Baxter']}.

Theorems & Definitions (254)

  • Conjecture 1
  • Theorem 1.3.1
  • Theorem 1.3.2
  • Theorem 1.3.3
  • Theorem 1.4.1
  • Definition 2.1.1
  • Proposition 2.2.1
  • Proposition 2.2.2
  • proof
  • Lemma 2.2.3
  • ...and 244 more