The Orbifold Topological Vertex

Bryan, Jenny; Cadman, Charles; Young, Ben

doi:10.48550/arxiv.1008.4205

Cited by 8 publications

(9 citation statements)

References 18 publications

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“…Its dual diagram is the toric diagram (or web diagram) of X, which is shown in blue. For a good pedagogical introduction to web diagrams and fan triangulations of toric Calabi-Yau orbifolds, see for instance Appendix B in[11].…”

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confidence: 99%

Mirror symmetry for orbifold Hurwitz numbers

Bouchard¹,

Serrano²,

Liu³

et al. 2014

J. Differential Geom.

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Abstract. We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the r-Lambert curve. We argue that the r-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of [C 3 /(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.

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confidence: 99%

Mirror symmetry for orbifold Hurwitz numbers

Bouchard¹,

Serrano²,

Liu³

et al. 2014

J. Differential Geom.

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“…Since the Schur functions are symmetric functions, one can do such regrouping and reordering of arguments freely. The partition functions of the orbifold models are sums of these Boltzmann weights over all partitions: geometry presented by Bryan et al [13].…”

Section: Orbifold Melting Crystal Models 21 Partition Functionsmentioning

confidence: 99%

“…The operator q W 0 /2 in (3.5) is an avatar of "framing factors" in the topological vertex construction [14]. In the orbifold case, those framing factors are replaced by "fractional" ones [13]. We shall use such a variant of (3.5) in the subsequent calculations.…”

Section: Deformations By External Potentialsmentioning

confidence: 99%

“…Such a melting crystal model can be found in the work of Bryan et al [13] on an orbifold version of the method of topological vertex [14]. They illustrated their method with two examples.…”

Section: Introductionmentioning

confidence: 99%

“…This affects the correspondence between the coupling constants t k , tk of the models and the time variables of the 2D Toda hierarchy as well. From a technical point of view, this characteristic is related to fractional framing factors in the orbifold topological vertex construction [13]. We shall encounter their avatars in our calculations.…”

Section: Introductionmentioning

confidence: 99%

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Orbifold melting crystal models and reductions of Toda hierarchy

Takasaki

2015

J. Phys. A: Math. Theor.

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Orbifold generalizations of the ordinary and modified melting crystal models are introduced. They are labelled by a pair a, b of positive integers, and geometrically related to Z a × Z b orbifolds of local CP 1 geometry of the O(0) ⊕ O(−2) and O(−1) ⊕ O(−1) types. The partition functions have a fermionic expression in terms of charged free fermions. With the aid of shift symmetries in a fermionic realization of the quantum torus algebra, one can convert these partition functions to tau functions of the 2D Toda hierarchy. The powers L a ,L −b of the associated Lax operators turn out to take a special factorized form that defines a reduction of the 2D Toda hierarchy. The reduced integrable hierarchy for the orbifold version of the ordinary melting crystal model is the bi-graded Toda hierarchy of bi-degree (a, b). That of the orbifold version of the modified melting crystal model is the rational reduction of bi-degree (a, b). This result seems to be in accord with recent work of Brini et al. on a mirror description of the genus-zero Gromov-Witten theory on a Z a × Z b orbifold of the resolved conifold.

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Torus Knots and the Topological Vertex

Jockers¹,

Klemm²,

Soroush³

2014

Lett Math Phys

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We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S 3 . The key role is played by the SL(2, Z) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Mariño. Comparing to the recent proposal by Aganagic and Vafa for knots on S 3 , we demonstrate that the disk amplitude of the A-brane associated to any knot is sufficient to reconstruct the entire Bmodel spectral curve. Finally, the construction of toric Lagrangian A-branes is generalized to other local toric Calabi-Yau geometries, which paves the road to study knots in other three-manifolds such as lens spaces.

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The Orbifold Topological Vertex

Cited by 8 publications

References 18 publications

Mirror symmetry for orbifold Hurwitz numbers

Mirror symmetry for orbifold Hurwitz numbers

Orbifold melting crystal models and reductions of Toda hierarchy

Torus Knots and the Topological Vertex

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