Localization and gluing of orbifold amplitudes: The Gromov-Witten orbifold vertex

Ross, Dustin

doi:10.1090/s0002-9947-2013-05835-7

Cited by 24 publications

(50 citation statements)

References 21 publications

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“…In [21,22] Lagrangian Floer theory is employed to study the case when the boundary condition is a fiber of the moment map. In the toric context, a mathematical approach [13,33,54,68] to construct operatively a virtual counting theory of open maps is via the use of localization [3,4,8,41,59], quantum knot invariants [47,62], and ordinary Gromov-Witten and DonaldsonThomas theory via "gluing along the boundary" [2,60,63]. Since Ruan's influential conjecture [69], an intensely studied problem in Gromov-Witten theory has been to determine the relation between GW invariants of target spaces related by a crepant birational transformation (CRC).…”

Section: 2mentioning

confidence: 99%

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-CortiIritani-Tseng and Ruan, we furthermore propose 1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and 2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold An-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov-Witten theory of this family of targets.

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Section: 2mentioning

confidence: 99%

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In [16], the local GW partition function at each torus fixed point was computed explicitly in terms of three-partition cyclic Hodge integrals on the moduli stack of stable maps into the classifying stacks BZ n , where n is a positive integer. These local contributions are indexed by triples of conjugacy classes (μ 1 , μ 2 , μ 3 ) in the generalized symmetric groups Z n S |μ i | .…”

Section: Theorem 11 (Theorem 21) In the Toric Setting Conjecture mentioning

confidence: 99%

“…We denote the disconnected vertex by [16]. For our current purposes, it is more convenient to work with a slight modification.…”

Section: Remark 23mentioning

confidence: 99%

Cyclic Hodge integrals and loop Schur functions

Ross

Zong

2015

Advances in Mathematics

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We conjecture an evaluation of three-partition cyclic Hodge integrals in terms of loop Schur functions. Our formula implies the orbifold Gromov-Witten/Donaldson-Thomas correspondence for toric Calabi-Yau threefolds with transverse A n singularities. We prove the formula in the case where one of the partitions is empty, and thus establish the orbifold Gromov-Witten/Donaldson-Thomas correspondence for local toric surfaces with transverse A n singularities.

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“…spaces which are locally modeled by finite quotients of C 3 [1,2,4]. Moreover, the topological vertex algorithm has been generalized to three dimensional toric orbifolds in both GW theory [14] and DT theory [2]. In GW theory, the orbifold vertex is a generating function of abelian Hodge integrals, whereas in the DT case it is a generating function of colored plane partitions.…”

Section: Context and Motivationmentioning

confidence: 99%