Crepant resolutions and open strings

Brini, Andrea; Cavalieri, Renzo; Ross, Dustin

doi:10.1515/crelle-2017-0011

Cited by 23 publications

(48 citation statements)

References 72 publications

(155 reference statements)

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“…This is a K 0 T (pt)-lattice in the space of flat sections for the equivariant quantum connection on X which is isomorphic to the integral equivariant K-group K 0 T (X): it generalizes the integral structure for non-equivariant quantum cohomology constructed by Iritani [50] and Katzarkov-Kontsevich-Pantev [55]. Similar structures have been studied by Okounkov-Pandharipande [66] in the case where X is a Hilbert scheme of points in C 2 , and by Brini-Cavalieri-Ross [17] in the case where X is a 3-dimensional toric Calabi-Yau stack. We define the integral structure in §3.1.…”

Section: Equivariant Gamma-integral Structurementioning

confidence: 54%

The Crepant Transformation Conjecture for toric complete intersections

Coates

Iritani

Jiang

2018

Advances in Mathematics

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Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y , with a Fourier-Mukai transformation between the K-groups of X and Y , via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y : they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne-Mumford stacks and toric complete intersections, and the Mellin-Barnes method for analytic continuation of hypergeometric functions.

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Section: Equivariant Gamma-integral Structurementioning

confidence: 54%

The Crepant Transformation Conjecture for toric complete intersections

Coates

Iritani

Jiang

2018

Advances in Mathematics

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“…by degeneration techniques [BG08], one should probably work harder to see the Toda spectral setup and the topological recursion emerge. Perhaps the best route to follow here will hinge on deriving the S-and R-calibrations of the quantum cohomology of Y Γ emerge from the steepest descent asymptotics of the Toda spectral data, as in [BCR13], and then retrieve the topological recursion from Givental's R-action on the associated cohomological field theory [DBOSS14]. This would lead to a proof of the remodeling conjecture beyond the toric case.…”

Section: Implications For Gw Theorymentioning

confidence: 99%

Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

Borot

Brini

2018

Advances in Theoretical and Mathematical Physics

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We consider the Gopakumar-Ooguri-Vafa correspondence, relating U(N ) Chern-Simons theory at large N to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients S Γ = Γ\S 3 of the three-sphere by the free action of a finite isometry group. Guided by string theory dualities, we propose a large N dual description in terms of both A-and B-twisted topological strings on (in general non-toric) local Calabi-Yau threefolds. The target space of the B-model theory is obtained from the spectral curve of Toda-type integrable systems constructed on the double Bruhat cells of the simply-laced group identified by the ADE label of Γ. Its mirror Amodel theory is realized as the local Gromov-Witten theory of suitable ALE fibrations on P 1 , generalizing the results known for lens spaces. We propose an explicit construction of the family of target manifolds relevant for the correspondence, which we verify through a large N analysis of the matrix model that expresses the contribution of the trivial flat connection to the Chern-Simons partition function. Mathematically, our results put forward an identification between the 1/N expansion of the sl N +1 LMO invariant of S Γ and a suitably restricted Gromov-Witten/Donaldson-Thomas partition function on the A-model dual Calabi-Yau. This 1/N expansion, as well as that of suitable generating series of perturbative quantum invariants of fiber knots in S Γ , is computed by the Eynard-Orantin topological recursion.

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“…Another line of developement consists of extending the remodeling proposal of [19] to the non-toric setting at hand for type D and E. A first step here would be to fully spell out the computation of the disk functions as in [21,22] for the case at hand, and then derive the topological recursion from the analysis of the descendent theory. A promising route would be to derive the J-and R-calibrations for the quantum cohomology of X Γ from the steepest descent analysis of oscillating integrals of the Toda differential, as in [23], and then retrieve the topological recursion from Givental's R-action on the associated cohomological field theory [36,44]. This would lead to a proof of the remodeling conjecture on an important class of examples, beyond the toric case.…”

Section: Discussionmentioning

confidence: 99%

On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds

Brini¹

2018

Proceedings of Symposia in Pure Mathematics

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Gopakumar, Ooguri and Vafa famously proposed the existence of a correspondence between a topological gauge theory on one hand -U(N ) Chern-Simons theory on the three-sphere -and a topological string theory on the other -the topological A-model on the resolved conifold. On the physics side, this duality provides a concrete instance of the large N gauge/string correspondence where exact computations can be performed in detail; mathematically, it puts forward a triangle of striking relations between quantum invariants (Reshetikhin-Turaev-Witten) of knots and 3-manifolds, curve-counting invariants (Gromov-Witten/Donaldson-Thomas) of local Calabi-Yau 3-folds, and the Eynard-Orantin recursion for a specific class of spectral curves. I here survey recent results on the most general frame of validity of this correspondence and discuss some of its implications. 1 arXiv:1711.07826v1 [hep-th] 20 Nov 2017asymptotic expansion of the Reshetikhin-Turaev invariant associated to the quantum group U q (sl N ) at large N and fixed q N equates the formal Gromov-Witten potential of X in the genus expansion. This has a B-model counterpart due to recent developments in higher genus toric mirror symmetry a [19,42], where the same genus expansion can be phrased in terms of the Eynard-Orantin topological recursion [41] on the Hori-Iqbal-Vafa mirror curve of X [58,59].The GOV correspondence has had a profound impact for both communities involved. In Gromov-Witten / Donaldson-Thomas theory, it has laid the foundations of the use of large N dualities to solve the topological string on toric backgrounds [4,6,77] and to obtain all-genus results that went well beyond the existing localisation computations at the time, as well as some striking results for the intersection theory on moduli spaces of curves [75] and, via the relation of Chern-Simons theory to random matrices, an embryo of the remodeling proposal [19,71,73]. In the other direction, the integral structure of BPS invariants leads to non-trivial constraints for the structure of quantum knot invariants [66,68,74].Since the original correspondence of [47,81] was confined to the case where the gauge group is the unitary group U(N ), the base manifold is S 3 , and the knot is the trivial knot, a natural question to ask is whether a similar connection could be generalised to other classical gauge groups [17,18,89], knots other than the unknot [24,65,66] (see also [3] for a significant generalisation, in a rather different setting), as well as categorified/refined invariants of various types [7,51]. A further natural direction would be to seek the broadest generalisation of the correspondence in its original form beyond the basic case of the three-sphere; this would require to provide a description of the string dual of Chern-Simons both in terms of Gromov-Witten theory and of the Eynard-Orantin theory on a specific spectral curve setup. This program was initiated in [5] (see also [25,52,53]) for the case of lens spaces, and what is presumably its widest frame of validity has been recently ...

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Crepant resolutions and open strings

Cited by 23 publications

References 72 publications

The Crepant Transformation Conjecture for toric complete intersections

The Crepant Transformation Conjecture for toric complete intersections

Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds

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