# Quantum Groups and Quantum Cohomology

Share this page
*Davesh Maulik; Andrei Okounkov*

A publication of the Société Mathématique de France

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 \(\mathbf{Y}_Q\), 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 \(\mathbf {Y}_Q\);
equivalently, the divisor operators correspond to certain elements of
Baxter subalgebras of \(\mathbf{Y}_Q\). A key role is played
by geometric shift operators which can be identified with the quantum
KZ difference connection.

In the second part, the authors give an extended example of the
general theory for moduli spaces of sheaves on
\(\mathbb{C}^2\), 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. 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
\(\mathcal{W}(\mathfrak{gl}(r))\) on the equivariant cohomology
of rank \(r\) moduli spaces, which implies certain conjectures
of Alday, Gaiotto, and Tachikawa.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians.