Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

Young, Benjamin; Bryan, Jenny

doi:10.1215/00127094-2010-009

Cited by 63 publications

(104 citation statements)

References 30 publications

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“…Let us mention that, in the literature, Γ L± (x) is often denoted Γ ± (x), see e.g. [Oko01], while Γ R± (x) is sometimes denoted Γ ± (x) [You10,BCC14]. Observe that we have…”

Section: 1mentioning

confidence: 99%

Dimers on rail yard graphs

Boutillier¹,

Bouttier²,

Chapuy³

et al. 2017

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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Abstract. We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson-fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions.Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick's theorem.Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimer model. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.

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“…Let us mention that, in the literature, Γ L± (x) is often denoted Γ ± (x), see e.g. [Oko01], while Γ R± (x) is sometimes denoted Γ ± (x) [You10,BCC14]. Observe that we have…”

Section: 1mentioning

confidence: 99%

Dimers on rail yard graphs

Boutillier¹,

Bouttier²,

Chapuy³

et al. 2017

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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“…Thus, we will use this case to test our formula (3.4). For C 3 /Z 2 × Z 2 , the counting in the non-commutative DT chamber has been done in [29]. Homology 2-cycles of the generalized conifold correspond to the simple roots α 1 , .…”

Section: Toric Cy Without Compact 4-cyclesmentioning

confidence: 99%

Wall Crossing and M-Theory

Aganagic¹,

Ooguri²,

Vafa³

et al. 2011

Publ. Res. Inst. Math. Sci.

123

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We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a CalabiYau 3-fold X. When X has no compact 4-cycles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in M-theory compactified down to 5 dimensions by a Calabi-Yau 3-fold X. The generating function of the D-brane bound states is expressed as a reduction of the square of the topological string partition function, in all chambers of the Kähler moduli space.2010 Mathematics Subject Classification: 14N35. Keywords: string theory, bound state, Fock space, Calabi-Yau manifold. §1. IntroductionThe topological string theory gives solutions to a variety of counting problems in string theory and M-theory. From the worldsheet perspective, the A-model topological string partition function Z top generates the Gromov-Witten invariants, which count holomorphic curves in a Calabi-Yau (CY) 3-fold X. On the other hand, from the target space perspective, Z top computes the Gopakumar-Vafa (GV) invariants, which count BPS states of spinning black holes in 5 dimensions constructed from M2 branes in M-theory on X [9]. Moreover, the absolute-valueThis is a contribution to the special issue "The golden jubilee of algebraic analysis".Communicated by M. Kashiwara.

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“…• J. Brian and Young [2010] generalized the Szendrői-Young formula for X L ,0 and X ‫)ޚ2/ޚ(‬ 2 . The method is different from the one used in [Young 2009]: they use vertex operator method.…”

Section: Introductionmentioning

confidence: 99%

Refined open noncommutative Donaldson–Thomas invariants for small crepant resolutions

Nagao¹

2011

Pacific J. Math.

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We study analogs of noncommutative Donaldson-Thomas invariants corresponding to the refined topological vertex for small crepant resolutions of toric Calabi-Yau 3-folds. We give three definitions of the invariants which are equivalent to each others and provide "wall-crossing" formulas for the invariants. In particular, we get normalized generating functions which are unchanged under wall-crossing.

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Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

Cited by 63 publications

References 30 publications

Dimers on rail yard graphs

Dimers on rail yard graphs

Wall Crossing and M-Theory

Refined open noncommutative Donaldson–Thomas invariants for small crepant resolutions

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