Motivic invariants of quivers via dimensional reduction

Morrison, Andrew

doi:10.1007/s00029-011-0081-z

Cited by 13 publications

(29 citation statements)

References 17 publications

(28 reference statements)

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“…Using these formulas, absence of exotics for framed and unframed invariants reduces to absence of exotics for the framed asymptotic ones. The latter will then be proven shortly using the results of [116]. Therefore, in short, rationality of both framed and unframed invariants is established, granting the motivic wallcrossing formula of [106] for SU (N ) quivers.…”

Section: 5mentioning

confidence: 78%

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Geometric engineering of (framed) BPS states

Chuang

Diaconescu

Manschot

et al. 2014

Advances in Theoretical and Mathematical Physics

100

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BPS quivers for N = 2 SU (N ) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N , relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with "exotic" SU (2) R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determines the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations. arXiv:1301.3065v2 [hep-th] 15 Apr 2013 2 W.-Y. CHUANG, D.-E. DIACONESCU, J. MANSCHOT, G. W. MOORE, Y. SOIBELMAN 6.1. A mutation of the SU (3) quiver 72 6.2. A deceptive chamber 73 7. Line defects and framed BPS states 75 7.1. Geometric construction of magnetic line defects 75 7.2. Framed stability conditions 78 7.3. Framed BPS states, Donaldson-Thomas invariants, and wallcrossing 79 7.4. A recursion formula for unframed BPS states 83 7.5. Absence of exotics I 86 8. BPS states and cohomological Hall algebras 90 8.1. Cohomological Hall algebras 90 8.2. Framing and SL(2, C) spin action 92 8.3. Absence of exotics II 94 Appendix A. Exceptional collections and quivers for X N 95 Appendix B. Motives for pedestrians 101 Appendix C. Kronecker modules 103 C.1. Harder-Narasimhan filtrations 104 C.2. Application to representations of the SU (3) quiver 106 Appendix D. Background material on extensions 107 Appendix E. Classifications of fixed points 110 ReferencesThis section contains a detailed construction of a discrete family X N , N ≥ 2 of toric Calabi-Yau threefolds employed in geometric engineering [5,96,101,99,98]

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Section: 5mentioning

confidence: 78%

“…Motivic NCDT invariants can also be explicitly computed in certain examples using different techniques [130,121,117,118,116]. A computation based on [116] is outlined for SU (N ) quivers in Section 7.5.…”

Section: 4mentioning

confidence: 99%

Geometric engineering of (framed) BPS states

Chuang

Diaconescu

Manschot

et al. 2014

Advances in Theoretical and Mathematical Physics

100

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“…The cyclic quiver. The proof of Theorem 3.6 proceeds analogously to the conifold case, using dimensional reduction and appropriate splittings, reducing the calculation to the case q = 1 already done in [9,24]. Once again, we refer for the details to [10].…”

Section: 4mentioning

confidence: 99%

“…The results we are reporting on here are rather limited. In particular, we do not introduce any new techniques for computing DT invariants in this paper; rather we use the dimensional reduction technique already used in [6], and systematized in [24], to see what we can learn about the structure of the invariants and their deformation properties. Thus, the cases in which we have been able to compute the generating series for the motivic DT invariants are the cases in which the potential is linear with respect to one of the generators of the algebra.…”

Section: Introductionmentioning

confidence: 99%

“…We explore only the simplest deformations of potentials for some well-studied quivers of geometric origin: the three-loop quiver underlying the Hilbert scheme of points on threefolds; the conifold quiver; and, generalizing the first, the cyclic quiver with loops. In the case of undeformed potentials, corresponding to commutative Calabi-Yau threefolds, the motivic invariants in these examples were computed in [6], [25] and [9,24], respectively. We study some simple perturbations of these potentials, corresponding to certain deformation quantizations of the commutative threefolds, and compute or conjecture the corresponding motivic DT invariants.…”

Section: Introductionmentioning

confidence: 99%

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Motivic Donaldson–Thomas invariants of some quantized threefolds

Cazzaniga¹,

Morrison²,

Pym³

et al. 2017

J. Noncommut. Geom.

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This paper is motivated by the question of how motivic Donaldson-Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi-Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal An-singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, as plethystic exponentials of simple rational functions. While our results are limited by the standard dimensional reduction techniques that we employ, they nevertheless allow us to conjecture formulae for more interesting cases, such as the elliptic Sklyanin algebras.

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Instanton Counting and Wall-Crossing for Orbifold Quivers

Cirafici¹,

Sinkovics

Szabo

2012

Ann. Henri Poincaré

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Abstract. Noncommutative Donaldson-Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative N = 2 gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated to the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson-Thomas invariants in the Coulomb branch in terms of gauge theory variables and orbifold data. We introduce the notion of virtual instanton quiver associated with the natural symplectic charge lattice which governs the quantum wall-crossing behaviour of BPS states in this context. The McKay correspondence naturally connects our formalism with other approaches to wall-crossing based on quantum monodromy operators and cluster algebras.

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Motivic invariants of quivers via dimensional reduction

Cited by 13 publications

References 17 publications

Geometric engineering of (framed) BPS states

Geometric engineering of (framed) BPS states

Motivic Donaldson–Thomas invariants of some quantized threefolds

Instanton Counting and Wall-Crossing for Orbifold Quivers

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