Softcover ISBN:  9782856299005 
Product Code:  AST/408 
List Price:  $67.00 
AMS Member Price:  $53.60 

Book DetailsAstérisqueVolume: 408; 2019; 212 ppMSC: Primary 17; 14;
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 Walgebra \(\mathcal{W}(\mathfrak{gl}(r))\) on the equivariant cohomology of rank \(r\) moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.ReadershipGraduate students and research mathematicians.

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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 Walgebra \(\mathcal{W}(\mathfrak{gl}(r))\) on the equivariant cohomology of rank \(r\) moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.
Graduate students and research mathematicians.