Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

Abstract

We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on R 4 and with two-dimensional conformal field theory. We construct a stacky compactification of the minimal resolution X k of the A k−1 toric singularity C 2 /Z k , which is a projective toric orbifold X k such that X k \ X k is a Z k -gerbe. We construct moduli spaces of torsion free sheaves on X k which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasi-projective variety, compute its dimension, and classify its fixed points under the natural induced toric action. We use this construction to compute the partition functions and correlators of chiral BPS operators for N = 2 quiver gauge theories on X k with nontrivial holonomies at infinity. The partition functions are computed with and without couplings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas, which relate them to the corresponding Nekrasov partition functions on the affine toric subsets of X k . We compare our new partition functions with previous computations, explore their connections to the representation theory of affine Lie algebras, and find new constraints on fractional instanton charges in the coupling to fundamental matter. We show that the partition functions in the low energy limit are characterised by the Seiberg-Witten curves, and in some cases also by suitable blowup equations involving Riemann theta-functions on the Seiberg-Witten curve with characteristics related to the nontrivial holonomies.

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

“…; here and in the following we shall usually implicitly assume that k is odd for simplicity (see [26] for the general case), although many of our conclusions hold more generally. Note that for k = 2, the divisor [ℓ ∞ /Z 2 ] ∼ = P 1 × BZ 2 corresponding to the vector b ′ 3 is a trivial Z 2 -gerbe (the quotient stack of P 1 by the trivial Z 2 -action), where BZ 2 is the quotient groupoid Spec(C)/Z 2 .…”

“…A natural class of complex surfaces X on which these considerations may be extended consists of orbifolds of C 2 and their resolutions. We begin by describing ALE spaces of type A k−1 regarded as toric varieties, following [51,30,26]; in this paper all cones are understood to be strictly convex rational polyhedral cones in a real vector space. For any non-negative integer i, define the lattice vector in L by…”

“…As it is difficult (for us) to work directly over the Nakajima quiver varieties, one needs to independently develop a means for studying gauge theories with instanton moduli associated to the chamber C ∞ . Such a theory was developed by [26] in the framework of framed sheaves on a suitable orbifold compactification X k = X k ∪ D ∞ , which is a smooth projective toric orbifold. The compactification divisor D ∞ is a Z k -gerbe over a football which as a toric Deligne-Mumford stack has a presentation as a global quotient stack…”

“…, w j−1 ), and it contains the moduli space of U (N ) instantons on X k of first Chern class i u i c 1 (R i ) and holonomy at infinity ρ = j w j ρ j [42]. One furthermore has [26,42] Theorem 2.13 There is a birational morphism…”

Blowup relations on $\mathbb{C}^2/\mathbb{Z}_2$ from Nakajima-Yoshioka blowup relations

“…Appropriate relations from the gauge theory side of AGT possibly could be obtained from the blowup relations on C 2 /Z 2 (possibly modified by 5th compact dimension). Namely, in [BMT11] (see also [BPSS13]) 4d blowup formula was proved…”