Quantum Groups and Quantum Cohomology
By Davesh Maulik, A. Okounkov. 2012
511 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/946bcd69f9cc961d876cf123c1f3e5e1861a2805
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Citations in the corpus
- Kirwan surjectivity for quiver varieties (2016, result): “r, the expectation expressed in Section 2.2.2 of [AO]|that their map (9) is an embedding near the origin of E T|follows. We note the applicability of the above results in other, similar contexts (cf. [MO]). Analogues of Corollary 1.5 can also be proven for K-theory and elliptic cohomology equivarian”
- Coproduct for Yangians of affine Kac–Moody algebras (2017, methodology): “Then the coproduct in [MO12] and the non-standard coproduct are related by the stable envelope, which is a ‘half’ of the universal R-matrix, and is a key object introduced in [MO12].”
- The affine Yangian of gl1 revisited (2014, unknown): “gian of gl1 has recently appeared simultaneously in the work of Maulik-Okounkov [MO] and Schiffmann-Vasserot [SV2] in connection with the AldayGaiotto-Tachikawa conjecture. While the presentation from [MO] is purely geometric, the algebraic presentation in [SV2] is quite involved. In this article, we”
- BAXTER'S RELATIONS AND SPECTRA OF QUANTUM INTEGRABLE MODELS (2013, background): “y be realized in equivariant K-theory of quiver varieties [N1], and similarly, finite-dimensional representations of the Yangian are realized in equivariant cohomology of these varieties [V] (see also [MO]). In the Yangian case, the Baxter polynomial is expected to be equal to the operator of (quantu”