Quantum groups and quantum cohomology
By: Davesh Maulik & Andrei Okounkov
Contributors: Okounkov, Andrei, 1969- author
Series: Astérisque -- 408; Astérisque -- 408.
Publisher: Société Mathématique de France
Published: 2019
Place of publication: Paris
Language: eng
Number of pages: 209
ISBN-13: 9782856299005
Copyright date: 2019
Subjects: Quantum groups; Cohomology operations; 31.23 rings, algebras
Notes: "In this paper, the authors study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, they construct a Hopf algebra YQ, the Yangian of Q, acting on the cohomology of these varieties, and show several results about their basic structure theory. The authors 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 YQ; equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of YQ. A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection. In the second part, the authors give an extended example of the general theory for moduli spaces of sheaves on C2, 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. The authors show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, the authors 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."- publisher

Includes bibliographical references (pages 201-209).

Text in English with an abstract also in French.
Table of contents: title: General theory; title: Nakajima varieties; title: Stable envelopes; title: Properties of R-matrices; title: Yangians; title: Further properties of the Yangians; title: Quantum multiplication; title: Shift operators; title: Minuscule shifts and R-matrices; title: Quantum multiplication by divisors; title: Cotangent bundles of Grassmannians; title: Instanton moduli; title: Classical r-matrix and gld(1); title: Free bosons; title: The full R-matrix; title: Quantum multiplication for M(r, n); title: Gamma functions; title: Core Yangian modulo h; title: The Yangian of gld(1); title: Yangian and vertex algebras.