The crepant resolution conjecture

Bryan, Jenny; Graver, Tom

doi:10.1090/pspum/080.1/2483931

Cited by 115 publications

(228 citation statements)

References 32 publications

(43 reference statements)

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“…Following the notation of [31] and [34], the Fock space description of the equivariant cohomology of the Hilbert scheme of points of C 2 is given in terms of creation-annihilation operators α k , k ∈ Z obeying the Heisenberg algebra…”

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confidence: 99%

The stringy instanton partition function

Bonelli

Sciarappa

Tanzini

et al. 2014

J. High Energ. Phys.

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Abstract:We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A 1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C 2 /Z 2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P 1 × C 2 .

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confidence: 99%

The stringy instanton partition function

Bonelli

Sciarappa

Tanzini

et al. 2014

J. High Energ. Phys.

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“…In this article, we compare the equivariant orbifold Gromov-Witten theory of the symmetric products of A r with the equivariant Gromov-Witten theory of the crepant resolutions in the spirit of Bryan and Graber's Crepant Resolution Conjecture [4].…”

Section: Resultsmentioning

confidence: 99%

“…OESym n .A r / 2 Q.t 1 ; t 2 /OEu; s 1 ; : : : ; s r ; which encode 3-point extended Gromov-Witten invariants of OESym n .A r / (see (2)(3)(4)(5)(6)). …”

Section: Resultsmentioning

confidence: 99%

“…In fact, Theorems 0.2 and 0.3, in conjunction with the results of Maulik and Oblomkov [13; 14; 15], establish the following equivalences for divisor operators. Before the study of the Gromov-Witten theory of OESym n .A r /, the case of the affine plane C 2 was the only known example for the above tetrahedron to hold for all operators; see Bryan and Graber [4], Bryan and Pandharipande [5] and Okounkov and Pandharipande [17; 18]. If the generation conjecture (Conjecture 5.1) of Maulik and Oblomkov is assumed, these four theories will be equivalent in our case of A r as well.…”

Section: Tetrahedron Of Equivalencesmentioning

confidence: 99%

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Orbifold Gromov–Witten theory of the symmetric product of 𝒜_r

Cheong

Gholampour

2012

Geom. Topol.

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“…Since birational Calabi-Yau 3-folds or orbifolds should be derived equivalent (cf. [Bri02], [BKR01], [Kaw05]), Question 1.1 (i) for GW theory is related to the analytic continuation problem of quantum cohomologies discussed in [Rua83], [BG09], [CIT09]. Also we expect that Question 1.1 (ii) is related to the modularity problem of partition functions of GW invariants, as the action of autoequivalences on the derived category should correspond to the monodromy action under the mirror symmetry.…”

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confidence: 99%

Stable pair invariants on Calabi–Yau threefolds containing ℙ2

Toda

2016

Geom. Topol.

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Abstract. We relate Pandharipande-Thomas stable pair invariants on Calabi-Yau 3-folds containing the projective plane with those on the derived equivalent orbifolds via wall-crossing method. The difference is described by generalized Donaldson-Thomas invariants counting semistable sheaves on the local projective plane, whose generating series form theta type series for indefinite lattices. Our result also derives non-trivial constraints among stable pair invariants on such Calabi-Yau 3-folds caused by Seidel-Thomas twist.1. Introduction 1.1. Motivation. It is an important subject to count algebraic curves on Calabi-Yau 3-folds, or more generally on CY3 orbifolds On the other hand, the derived category of coherent sheaves D b Coh(X) on a Calabi-Yau 3-fold X is also an important mathematical subject, due to its role in Kontsevich's Homological mirror symmetry conjecture [Kon95]. It was suggested by Pandharipande-Thomas [PT09] that the derived category also plays a crucial role in curve counting, as their stable pair invariants count two term complexeswhere F is a pure one dimensional sheaf and s is surjective in dimension one. In this paper, we concern how symmetries in the derived categories affect stable pair invariants. More precisely, we are interested in the following questions:Question 1.1.(i) How stable pair invariants on two Calabi-Yau 3-folds or orbifolds are related, if they have equivalent derived categories ?1 In this paper, an orbifold means a smooth Deligne-Mumford stack.

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The crepant resolution conjecture

Cited by 115 publications

References 32 publications

The stringy instanton partition function

The stringy instanton partition function

Orbifold Gromov–Witten theory of the symmetric product of 𝒜_r

Stable pair invariants on Calabi–Yau threefolds containing ℙ2

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The crepant resolution conjecture

Cited by 115 publications

References 32 publications

The stringy instanton partition function

The stringy instanton partition function

Orbifold Gromov–Witten theory of the symmetric product of 𝒜r

Stable pair invariants on Calabi–Yau threefolds containing ℙ2

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Product

Resources

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Orbifold Gromov–Witten theory of the symmetric product of 𝒜_r