Localization in quiver moduli spaces

Weist, Thorsten

doi:10.1090/s1088-4165-2013-00436-3

Cited by 41 publications

(80 citation statements)

References 28 publications

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“…This classification of the fixed points is already known in the mathematical literature [4]. These fixed points are depicted in the diagram:…”

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confidence: 85%

“…Therefore, a lot of interesting formulae, which compute the number of the BPS bound states or topological invariants, have been derived. In mathematics, topological properties of the quiver moduli spaces are investigated in [3,4] for example, and they come to fruition of the so-called "wall crossing formula" by Joyce and Song [5], and Kontsevich and Soibelman [6]. The wall crossing formula is physically interpreted and rederived in the Coulomb branch of the quiver quantum mechanics [7,8].…”

Section: Jhep11(2014)123mentioning

confidence: 99%

“…This is a consequence of the orthogonality between the Higgs branch (Z = 0 and q = 0) and the Coulomb branch (Z = 0 and q = 0) from the F-term condition. 4 The Wilson loop for the non-Abelian part is obtained by a trace over a representation R. In the diagonal gauge, it is written in terms of a symmetric polynomial of u 2,i , that is, a Schur polynomial associated with the representation (Young diagram) R. If we denote the polynomial by s R (u 2,1 , u 2,1 , . . .…”

Section: Jhep11(2014)123mentioning

confidence: 99%

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Exact results in quiver quantum mechanics and BPS bound state counting

Ohta

Sasai

2014

J. High Energ. Phys.

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We exactly evaluate the partition function (index) of N = 4 supersymmetric quiver quantum mechanics in the Higgs phase by using the localization techniques. We show that the path integral is localized at the fixed points, which are obtained by solving the BRST equations, and D-term and F-term conditions. We turn on background gauge fields of R-symmetries for the chiral multiplets corresponding to the arrows between quiver nodes, but the partition function does not depend on these R-charges. We give explicit examples of the quiver theory including a non-coprime dimension vector. The partition functions completely agree with the mathematical formulae of the Poincaré polynomials (χ y -genus) and the wall crossing for the quiver moduli spaces. We also discuss exact computation of the expectation values of supersymmetric (Q-closed) Wilson loops in the quiver theory.

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“…This classification of the fixed points is already known in the mathematical literature [4]. These fixed points are depicted in the diagram:…”

mentioning

confidence: 85%

Section: Jhep11(2014)123mentioning

confidence: 99%

Section: Jhep11(2014)123mentioning

confidence: 99%

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Exact results in quiver quantum mechanics and BPS bound state counting

Ohta

Sasai

2014

J. High Energ. Phys.

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“…to eachQ (k) := Q(k−1) withQ (1) :=Q. Now it is straightforward to check that there exist natural surjective morphisms f k : Q →Q (k) which become injective on finite subquivers if k ≫ 0, see also [18,Section 3.4]. Since the support of X is finite as a representation ofQ, we can find k ≥ 0 such that the full subquiver with vertices supp(X) ⊆Q (k+1) 0 is a tree.…”

Section: Proofmentioning

confidence: 99%

Cell decompositions for rank two quiver Grassmannians

Rupel

Weist

2019

Math. Z.

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We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians. Quiver Covering TheoryWe refer to [12] for an introduction to covering theory. Let Q be an acyclic quiver with vertices Q 0 and arrows Q 1 which we denote by α : s(α) → t(α). A C-representation X of Q consists of a collection of C-vector spaces X i for i ∈ Q 0 and a collection of C-linear maps X α : X s(α) → X t(α) for α ∈ Q 1 . Given

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“…can be lifted to ψ : T → G α [19]. We split ψ to components ψ i : T → GL(M i ), i ∈ Q 0 and decompose every M i with respect to the character group Λ of T…”

Section: Motivic Vanishing Cyclementioning

confidence: 99%

On the motivic Donaldson–Thomas invariants of quivers with potentials

Mozgovoy

2013

Mathematical Research Letters

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Abstract. We study motivic Donaldson-Thomas invariants for a class of quivers with potentials using the strategy of Behrend et al. [1]. This class includes quivers with potentials arising from consistent brane tilings and quivers with zero potential. Our construction is an alternative to the constructions of Kontsevich and Soibelman [8,9]. We construct an integration map from the equivariant Hall algebra to the quantum torus and show that our motivic Donaldson-Thomas invariants are images of the natural elements in the equivariant Hall algebra. We show that the inegration map is an algebra homomorphism and use this fact to prove the Harder-Narasimhan relation for the motivic Donaldson-Thomas invariants.

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Localization in quiver moduli spaces

Cited by 41 publications

References 28 publications

Exact results in quiver quantum mechanics and BPS bound state counting

Exact results in quiver quantum mechanics and BPS bound state counting

Cell decompositions for rank two quiver Grassmannians

On the motivic Donaldson–Thomas invariants of quivers with potentials

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