Quiver Varieties and Hilbert Schemes

Abstract

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the Γ-equivariant Hilbert scheme X Γ[n] and the Hilbert scheme X[n] Γ (where X = C 2 , Γ ⊂ SL(C 2 ) is a finite subgroup, and X Γ is a minimal resolution of X/Γ) are quiver varieties for the affine Dynkin graph, corresponding to Γ via the McKay correspondence, the same dimension vectors, but different parameters ζ (for earlier results in this direction see [4,12,13]). In particular, it follows that the varieties X Γ[n] and X [n] Γ are diffeomorphic. Computing their cohomology (in the case Γ = Z/dZ) via the fixed points of (C * × C * )-action we deduce the following combinatorial identity: the number U CY (n, d) of uniformly coloured in d colours Young diagrams consisting of nd boxes coincides with the number CY (n, d) of collections of d Young diagrams with the total number of boxes equal to n. I was partially supported by RFFI grants 99-01-01144 and 99-01-01204.

Moduli spaces of framed sheaves and quiver varieties

“…So, they are quiver varieties, although not in Nakajima's sense (except for n = 2), but in a more general one (see [21] and Section 3). When n = 2, Tot(O P 1 (−2)) is the ALE space A 1 , and indeed our description coincides with that one obtained by Kuznetsov in [34].…”

Hilbert schemes of points of OP1(−n) as quiver varieties

“…Comparison with Nakajima quiver varieties. We focus now on the case n = 2; in particular, as a consequence of Theorem 4.5, we recover a result for ALE spaces due to Kuznetsov [22]. We recall that, according to Nakajima [26], any quiver Q with vertex set I is associated with a quiver variety…”

N=2 gauge theories, instanton moduli spaces and geometric representation theory

“…In the rank one case N = 1, this morphism is an isomorphism: In this instance M u,n, w (X k ) ∼ = Hilb n (X k ) is the Hilbert scheme of n points on X k for all u and w [26], which is a Nakajima quiver variety by [69].…”