Quantum groups and quantum cohomology

by Maulik, Davesh

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Title Statements

Full title :Quantum groups and quantum cohomology / Davesh Maulik & Andrei Okounkov.

Edition Statements

Publication Date:©2019

Publication Statements

Publisher(s):Société Mathématique de France
Place of Publication:Paris
Language:eng
Publication Date:©2019

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

The Physical Object

Description:ix, 209 pages : : illustrations ; : 24 cm

Edition Identifiers

ISBN:9782856299005, 2856299008
oclc:on1102680467
LC Classification:QC20.7.G76
Dewey Decimal:530.1522

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