Nakajima Varieties Research Articles

Framed cohomological Hall algebras and cohomological stable envelopes

There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver Q to the Yangian Y^{Q}_textrm{MO} by Maulik–Okounkov, whose construction is based on the notion of stable envelopes of Nakajima varieties. In this article, we introduce the cohomological Hall algebra of the moduli stack of framed representations of a quiver Q (framed CoHA), and we show that the equivariant cohomology of the disjoint union of the Nakajima varieties {mathcal {M}}_Q(text {v},text {w}) for all dimension vectors text {v} and framing vectors text {w} has a canonical structure of subalgebra of the framed CoHA. Restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map. As a corollary, we deduce an explicit inductive formula to compute stable envelopes in terms of tautological classes.

Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A

In type A we find equivalences of geometries arising in three settings: Nakajima's (“framed”) quiver varieties, conjugacy classes of matrices and loop Grassmannians. These are all given by explicit formulas. In particular, we embedd the framed quiver varieties into Beilinson-Drinfeld Grassmannians. This provides a compactification of Nakajima varieties and a decomposition of affine Grassmannians into Nakajima varieties. As an application we provide a geometric version of symmetric and skew (GL(m),GL(n)) dualities.

Quantum difference equation for Nakajima varieties

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in $${{\,\mathrm{Pic}\,}}(X)\otimes {\mathbb {C}}$$ . We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.

Elliptic stable envelopes

We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of Maulik and Okounkov [Astérisque 408 (2019), ix+209]. We apply them to the computation of the monodromy of q q -difference equations arising in the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik–Zamolodchikov equations.

Quiver varieties and symmetric pairs

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the typeAAcase, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft–Procesi row/column removal reductions.

Quasimap Counts and Bethe Eigenfunctions

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.

Quivers with loops and generalized crystals

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

Quasimap Counts and Bethe Eigenfunctions

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.

Nakajima Varieties and Repetitive Algebras

We realize certain graded Nakajima varieties of finite Dynkin type as orbit closures of repetitive algebras of Dynkin quivers. As an application, we show that the perverse sheaves introduced by Nakajima to describe irreducible characters of quantum loop algebras are isomorphic to the intersection cohomology sheaves of these orbit closures.

A geometric realization of spin representations and Young diagrams from quiver varieties

Applying the techniques of an earlier paper with Frenkel, we develop a geometric realization of spin representations and Clifford algebras. In doing so, we give an explicit parametrization of the irreducible components of Nakajima varieties of type D in terms of Young diagrams. We explicitly compute the geometric action of the Lie algebra and are able to extend the geometric action to the entire Clifford algebra used in the classical construction of the spin representations.

Homological realization of Nakajima varieties and Weyl group actions

We give a realization of Nakajima varieties and the action of the Weyl group on them using certain canonical structures of homological algebras and their natural generalization, which we develop in this paper. We consider in detail the case of an affine quiver, where we present a simple homological characterization of Nakajima varieties and its relation to moduli of sheaves on the projective plane.