Mirror symmetry for concavex vector bundles on projective spaces

Elezi, Artur

doi:10.1155/s0161171203112136

Cited by 14 publications

(21 citation statements)

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“…Mariño and Vafa [27] have done the same for higher-genus, multiple-boundary invariants using Chern-Simons duals. In this note we will perform localization calculations to provide an explicit "A-model" verification of these predictions, and use an equivariant mirror theorem [8] to compute. The results we obtain match those authors' perfectly, including a dependence on an additional Z-valued parameter.…”

Section: The Physicsmentioning

confidence: 99%

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Open-string Gromov-Witten invariants: calculations and a mirror “theorem”

Graber¹,

Zaslow²

2002

Orbifolds in Mathematics and Physics

133

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We propose localization techniques for computing Gromov-Witten invariants of maps from Riemann surfaces with boundaries into a CalabiYau, with the boundaries mapped to a Lagrangian submanifold. The computations can be expressed in terms of Gromov-Witten invariants of one-pointed maps. In genus zero, an equivariant version of the mirror theorem allows us to write down a hypergeometric series, which together with a mirror map allows one to compute the invariants to all orders, similar to the closed string model or the physics approach via mirror symmetry. In the noncompact example where the Calabi-Yau is K P 2 , our results agree with physics predictions at genus zero obtained using mirror symmetry for open strings. At higher genera, our results satisfy strong integrality checks conjectured from physics.

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Section: The Physicsmentioning

confidence: 99%

“…(See [8] or [26] for this formula, but note that in their calculations, a different linearization on K P 2 is used.) In implementing this change of variables, one must not neglect that q = e t 1 should be replaced by e t 1 +I 1 = qe I 1 (q) on the left side.…”

Section: Equivariant Mirror Theoremmentioning

confidence: 99%

Open-string Gromov-Witten invariants: calculations and a mirror “theorem”

Graber¹,

Zaslow²

2002

Orbifolds in Mathematics and Physics

133

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“…It is also true that π * n+1 (E d ) = E d [6]. Suffices then to prove the theorem for 0-pointed stable maps.…”

Section: Virtual Class Of the Zero Locimentioning

confidence: 99%

Virtual class of zero loci and mirror theorems

Elezi

Luo²

2003

Advances in Theoretical and Mathematical Physics

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Let Y be the zero loci of a regular section of a convex vector bundle E over X. We provide a proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to Y . This in turn yields the expected relationship between Gromov-Witten theories of Y and X which together with Mirror Theorems allows for the calculation of enumerative invariants of Y inside of X. e-print archive: http://lanl.arXiv.org/abs/Math.AG/0307386

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“…If |a| ≤ n and |a|−ℓ − (a) ≤ n−2, (1.8) also holds with Z n;a (x, , Q) replaced by Z GW n;a (x, , Q); see [9, Theorem 9.1] for the ℓ − (a) = 0 case and [7,Theorem 5.1] for the ℓ − (a) ≥ 1 case. Thus, Corollary 1 is an immediate consequence of Theorem 1.…”

Section: Corollarymentioning

confidence: 99%

“…It takes the form Z GW n;a (x, , Q) = e −Jn;a(q)x/ Y n;a (x, , q) I n;a (q) , where Q = q · e Jn;a(q) , ( 1.11) for an explicit power series J n;a (q) ∈ q · Q[[q]]; see [9,Theorem 11.8] for the ℓ − (a) = 0 case and [7,Theorem 5.1] for the ℓ − (a) = 1 case. Along with (1.11), Theorem 1 immediately implies the ℓ − (a) ≤ 1 case of (1.9); the ℓ − (a) ≥ 2 case of (1.9), where J n;a (q) = 0, follows from Corollary 1.…”

Section: Corollarymentioning

confidence: 99%

Mirror symmetry for stable quotients invariants

Cooper¹,

Zinger²

2014

Michigan Math. J.

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The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular projective variety and carries a natural virtual class. We show that the analogue of Givental's J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the corresponding Gromov-Witten invariants (which arise from the moduli space of stable maps), but without the mirror transform (in the Calabi-Yau case). This implies that the stable quotients and Gromov-Witten twisted invariants agree if there is enough "positivity", but not in all cases. As a corollary of the proof, we show that certain twisted Hurwitz numbers arising in the stable quotients theory are also described by a fundamental object associated with this hypergeometric series. We thus completely answer some of the questions posed by Marian-Oprea-Pandharipande concerning their invariants. Our results suggest a deep connection between the stable quotients invariants of complete intersections and the geometry of the mirror families. As in Gromov-Witten theory, computing Givental's J-function (essentially a generating function for genus 0 invariants with 1 marked point) is key to computing stable quotients invariants of higher genus and with more marked points; we exploit this in forthcoming papers.

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Mirror symmetry for concavex vector bundles on projective spaces

Cited by 14 publications

References 20 publications

Open-string Gromov-Witten invariants: calculations and a mirror “theorem”

Open-string Gromov-Witten invariants: calculations and a mirror “theorem”

Virtual class of zero loci and mirror theorems

Mirror symmetry for stable quotients invariants

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