Characters of tangent spaces at torus fixed points and 3 d -mirror symmetry
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Characters of tangent spaces at torus fixed points and 3d-mirror symmetry Andrey Smirnov1 · Hunter Dinkins1 Received: 17 August 2019 / Revised: 10 January 2020 / Accepted: 4 May 2020 © Springer Nature B.V. 2020
Abstract Let X be a Nakajima quiver variety and X its 3d-mirror. We consider the action of the Picard torus K = Pic(X ) ⊗ C× on X . Assuming that (X )K is finite, we propose a procedure for obtaining the K-character of the tangent spaces at the fixed points in terms of certain enumerative invariants of X known as vertex functions. Keywords 3d mirror symmetry · K -theory of quiver varieties · Vertex functions Mathematics Subject Classification 14N35
1 Introduction In this paper, we study symplectic varieties which appear as Higgs and Coulomb branches of certain three-dimensional gauge theories with N = 4 supersymmetry. These theories were considered, for example, in [4,8,11,12]. We assume that the physical theories under consideration are of quiver type, in which case the Higgs branches are known as Nakajima quiver varieties, see [10,17] and Section 2 in [15] for an introduction.1 In this subsection we recall the most basic facts about these varieties important in the constructions below. The Nakajima varieties are smooth quasi-projective symplectic varieties equipped with a natural action of an algebraic torus T. The torus T acts on a Nakajima variety X by scaling the symplectic form ω ∈ H 2 (X , C). We denote by ∈ char(T) the character of the one-dimensional T-module Cω. We denote by A = ker() ⊂ T the subtorus
B
Hunter Dinkins [email protected] Andrey Smirnov [email protected]
1
University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
1 We believe that our main conjecture holds in full generality. We restrict the exposition to the quiver
varieties for the sake of simplicity of the exposition.
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preserving the symplectic form and by C× := T/A the corresponding one-dimensional factor. The Nakajima quiver varieties are examples of symplectic resolutions, and thus their cohomology are even [13]. The Nakajima varieties are defined as quotients by a group G=
G L(vi )
i∈I
where I denotes the (finite) set of vertices of the corresponding quiver. This means that X is naturally equipped with a set of rank vi tautological vector bundles Vi . alg
top
Theorem 1.1 ([16]) If X is a Nakajima variety then K T (X ) = K T (X ) is generated by Schur functors of tautological bundles Vi , i ∈ I . We will use K T (X ) to denote the T-equivariant K -theory ring of X . Due to the last theorem, we do not distinguish between the algebraic and the topological versions. This theorem implies that: K T (X ) = Z[xi,±1j , a± , ±1 ]/R where xi, j , i ∈ I , j = 1, . . . , vi denote the Grothendieck roots of the tautological bundles, a, stand for the equivariant parameters of T and R denotes a certain ideal. If X T is finite, then the ideal R can be described as the ideal of Laurent polynomials whose restrictions (11) to all fixed points in X T vanish. As
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