Quantum cohomology of [ℂN/μr]

Bayer, Arend; Cadman, Charles

doi:10.1112/s0010437x10004793

Cited by 16 publications

(19 citation statements)

References 25 publications

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“…However, we believe that our techniques are applicable more generally, allowing one to extract invariants in the vicinity of other special points in the moduli space. For instance, we expect that the expansion about an orbifold point in the quantum Kähler moduli space computes orbifold Gromov-Witten invariants [85][86][87], see Appendix A for the quintic partition function expanded around its Landau-Ginzburg orbifold point.…”

Section: Discussionmentioning

confidence: 99%

Two-Sphere Partition Functions and Gromov–Witten Invariants

Jockers¹,

Kumar

Lapan

et al. 2014

Commun. Math. Phys.

130

315

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Many N = (2, 2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N = (2, 2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -recently computed via localization by Benini et al. and Doroud et al. -yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in α ′ . We compute these quantities for the quintic and for Rødland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P 7 , recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.1 The work of Böhm [24, 25] provides a promising proposal, but we have been unable to implement it well enough to produce a mirror for this example.2 Local special Kähler manifolds are also often called projective special Kähler manifolds and are distinct from special Kähler manifolds -see, e.g., [32].3 This geometric structure gives rise to a Hodge filtration F 3 ⊂ F 2 ⊂ F 1 ⊂ F 0 of weight 3, with F 3 ≃ L, F 2 ≃ V, F 1 ≃ L ⊥ , and F 0 ≃ V C , where L ⊥ is the subspace of V C perpendicular to L with respect to the symplectic pairing ·, · -see, e.g., [33,34].The last expression involves the periods of Ω,B J Ω(ξ) , I, J = 0, . . . , h 2,1 , (2.6) with respect to a canonical symplectic basis (A I , B J ) of H 3 (Y ξ , Z) satisfying A I , B J = δ I J , A I , A J = B I , B J = 0 . (2.7) 2.3 The quantum Kähler moduli space M KählerThe main player of this note is the quantum-corrected Kähler moduli space M Kähler of a Calabi-Yau threefold Y , which is defined as the corresponding space of chiral-antichiral and antichiralchiral moduli of the underlying SCFT. It is also a local special Kähler manifold, parametrizing 15 We thank C. Vafa for pointing this out to us. 16 Note that in spacetime, some of these fields become quaternionic and are subject to further gs corrections.

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Section: Discussionmentioning

confidence: 99%

Two-Sphere Partition Functions and Gromov–Witten Invariants

Jockers¹,

Kumar

Lapan

et al. 2014

Commun. Math. Phys.

130

315

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“…In [1] this was used to calculate generating functions of orbifold GromovWitten invariants in the case of the C 3 /Z 3 orbifold, which corresponds to a phase in the moduli space of local P 2 , its crepant resolution. The predictions obtained in this way have been later verified mathematically in orbifold Gromov-Witten theory [11,17,20,25], and other examples have been recently calculated [18,26].…”

Section: Introductionmentioning

confidence: 89%

Topological Open Strings on Orbifolds

Bouchard

Klemm

Mariño

et al. 2010

Commun. Math. Phys.

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We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C 3 /Z 3 orbifold point of local P 2 , where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold GromovWitten invariants of C 3 /Z 3 , or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.

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“…Orbifold Gromov-Witten theory, constructed in symplectic category by Chen-Ruan [18] and in algebraic category by Abramovich, Graber and Vistoli [3], [2], has been an area of active research in recent years. Calculations of orbifold Gromov-Witten invariants in examples present numerous new challenges, see [21], [19], [40], and [12] for examples.…”

Section: Introductionmentioning

confidence: 99%

Gromov-Witten theory of root Gerbes I: Structure of genus 0 moduli spaces

Andreini¹,

Jiang²,

Tseng³

2015

J. Differential Geom.

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Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r > 1 one defines the stack r L/X of r-th roots of L. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus 0 twisted stable maps to r L/X. Our main results are explicit constructions of moduli stacks of genus 0 twisted stable maps to r L/X starting from moduli stack of genus 0 stable maps to X. As a consequence, we prove an exact formula expressing genus 0 Gromov-Witten invariants of r L/X in terms of those of X.Date: November 1, 2018. 1 2 ELENA ANDREINI, YUNFENG JIANG, AND HSIAN-HUA TSENG References 36The Gromov-Witten theoretic version of the decomposition conjecture, which can be formulated for arbitrary G-gerbes more general than G-banded gerbes, states that Gromov-Witten theory of the G-gerbe is equivalent to certain twist of the Gromov-Witten theory of someétale cover of the base. A detailed discussion of the conjecture in full generality can be found in [38]. For G-banded gerbes this conjecture states that the Gromov-Witten theory of a G-banded gerbe over X is equivalent to (certain twists of) the Gromov-Witten theory of the disjoint union of |Conj(G)| copies 1 of X after a change of variables. Computations of Gromov-Witten invariants ofétale gerbes is thus intimated connected to the decomposition conjecture.The simplest examples of G-gerbes are trivial gerbes. The trivial G-gerbe over a Deligne-Mumford stack X is the product X × BG. In [7] the computation of Gromov-Witten invariants of X × BG is handled as a special case of a general product formula for orbifold Gromov-Witten invariants of product Deligne-Mumford stacks X × Y. As a consequence the decomposition conjecture is proven for trivial G-gerbes.An interesting class of non-trivial gerbes is provided by root gerbes associated to line bundles. This is the first of two papers in which we study Gromov-Witten theory of root gerbes of line bundles over smooth projective varieties, with the decomposition conjecture in mind. The present paper is devoted to study the genus 0 Gromov-Witten theory of root gerbes.Let X be a smooth complex projective variety and L → X a line bundle. Given an integer r > 0, let G := r L/X → X 1 Here Conj(G) is the set of conjugacy classes of G.

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Quantum cohomology of [ℂ^N/μ_r]

Cited by 16 publications

References 25 publications

Two-Sphere Partition Functions and Gromov–Witten Invariants

Two-Sphere Partition Functions and Gromov–Witten Invariants

Topological Open Strings on Orbifolds

Gromov-Witten theory of root Gerbes I: Structure of genus 0 moduli spaces

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