Quantum Groups and Quantum Cohomology (1211.1287v3)

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.

• The paper presents a novel Yangian construction via a geometric R-matrix framework that bridges quantum groups with the quantum cohomology of Nakajima quiver varieties.
• It provides an explicit formula for quantum multiplication by divisors, advancing representation theory and the algebraic structure of quiver varieties.
• The study demonstrates the action of W-algebras on rank r moduli spaces, hinting at universal structures in integrable systems and algebraic geometry.

Quantum Groups and Quantum Cohomology

The paper "Quantum Groups and Quantum Cohomology," authored by Davesh Maulik and Andrei Okounkov, advances the mathematical understanding of the interplay between quantum groups and the equivariant cohomology of Nakajima quiver varieties. This deep, methodical exploration introduces and rigorously analyzes a Hopf algebra termed the Yangian, denoted as $Y_Q$, associated with a quiver $Q$. This paper bridges the domains of algebraic geometry, representation theory, and mathematical physics, and it expands upon foundational work in the field through a geometric $R$-matrix approach.

Overview of Main Concepts

The central thrust of the paper is the development of a coherent structural theory for the quantum cohomology of Nakajima quiver varieties, marking significant progress toward understanding these varieties' quantum deformation. Nakajima quiver varieties, parameterized by complex numbers $\theta$ and $\zeta$, embody rich geometric structures and symmetries represented algebraically through quivers—graphs denoting relations between vector spaces.

By adopting a geometric $R$-matrix framework, the authors construct the Yangian $Y_Q$, which acts on the cohomology of these varieties. A noteworthy result is the demonstrated alignment of quantum multiplication with elements of certain Baxter subalgebras of $Y_Q$. The geometric shift operators, crucial for this construction, relate closely to the quantum Knizhnik-Zamolodchikov (qKZ) connection, further illustrating the deep integrative nature of the quantum framework introduced.

Technical Contributions

1. Yangian Construction: The Yangian $Y_Q$ is formed through a geometric construction involving the concept of equivariant symplectic resolutions and uses the $R$-matrix to provide an equivariant lift of the quantum cohomology structure. This framework enables the formulation of a quantum connection identified with the trigonometric Casimir connection.
2. Quantum Multiplication: An explicit formula is given for quantum multiplication by divisors on quiver varieties within this Yangian framework. This result highlights significant advancements in representation theory, providing concrete algebraic structures and operations on the underlying varieties.
3. Case Studies in Moduli Spaces: The paper explores explicit examples, including the moduli spaces of sheaves on $\mathbb{C}^2$, framed at infinity. It utilizes free field realizations to analyze the Yangian actions, providing connections to physical theories like Liouville field theory.
4. W-algebra Action: A valuable corollary of this construction is the action of the W-algebra $W(\mathfrak{gl}(r))$ on the equivariant cohomology of rank $r$ moduli spaces, addressing conjectures in topological field theory.

Implications and Future Work

The paper's results have wide-ranging implications for both algebraic geometry and mathematical physics. Practically, they pave the way for more efficient computation of quantum cohomologies in increasingly general setups. Theoretically, the work hints at deep universal structures shared across symplectic varieties, integrable systems, and quantum groups.

Further exploration will likely focus on extending these results to a broader class of varieties and examining dualities between geometric settings and physical models of quantum integrable systems. Of particular interest is the conjectured relationship between quantum multiplication operators and Baxter subalgebras, proposing fascinating algebraic geometry-rich exploration paths.

Overall, this work advances the foundational understanding of quantum cohomology in conjunction with quantum groups, laying groundwork for a richer narrative to emerge across the fields of geometry and physics. Future research speculations point to extending the framework laid down in this paper to understand more complex algebraic and topological structures in higher-dimensional varieties.