An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

Iritani, Hiroshi

doi:10.1016/j.aim.2009.05.016

Cited by 218 publications

(375 citation statements)

References 57 publications

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“…Indeed, they are abelian and therefore far simpler. Their relevance seem to be new, although they are very closely related to the Γ-factors which appear in quantum cohomology [34,45].…”

Section: The Construction Of the Inverse Functormentioning

confidence: 99%

Yangians, quantum loop algebras, and abelian difference equations

Gautam

Laredo

2015

J. Amer. Math. Soc.

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Abstract. Let g be a complex, semisimple Lie algebra, and Y (g) and Uq(Lg) the Yangian and quantum loop algebra of g. Assuming that is not a rational number and that q = e πi , we construct an equivalence between the finite-dimensional representations of Uq(Lg) and an explicit subcategory of those of Y (g) defined by choosing a branch of the logarithm. This equivalence is governed by the monodromy of the abelian, additive difference equations defined by the commuting fields of Y (g). Our results are compatible with q-characters, and apply more generally to a symmetrisable Kac-Moody algebra g, in particular to affine Yangians and quantum toroïdal algebras. In this generality, they yield an equivalence between the representations of Y (g) and Uq(Lg) whose restriction to g and Uqg respectively are integrable and in category O.

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“…Indeed, they are abelian and therefore far simpler. Their relevance seem to be new, although they are very closely related to the Γ-factors which appear in quantum cohomology [34,45].…”

Section: The Construction Of the Inverse Functormentioning

confidence: 99%

Yangians, quantum loop algebras, and abelian difference equations

Gautam

Laredo

2015

J. Amer. Math. Soc.

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“…This appearance ofΓ c class has been foretold from various explicit computations via mirror symmetry [10][11][12][13].…”

Section: Jhep02(2014)103mentioning

confidence: 99%

“…As discussed above, we impose the projection conditions 11) in order to introduce a minimal coupling of a charged chiral multiplet. It implies that…”

Section: Parity On the Mirrormentioning

confidence: 99%

Exact partition functions on $ \mathbb{R}{{\mathbb{P}}^2} $ and orientifolds

Kim

Lee

2014

J. High Energ. Phys.

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We consider gauged linear sigma models (GLSM) on RP 2 , obtained from a parity projection of S 2 . The theories admit squashing deformation, much like GLSM on S 2 , which allows us to interpret the partition function as the overlap amplitude between the vacuum state and crosscap states. From these, we extract the central charge of Orientifold planes, and observe that the Gamma class makes a prominent appearance as in the recent D-brane counterpart. We also repeat the computation for the mirror Landau-Ginzburg theory, which naturally brings out the θ-dependence as a relative sign between two holonomy sectors on RP 2 . We also show how our results are consistent with known topological properties of D-brane and Orientifold plane world-volumes, and discuss what part of the wrapped D-brane/Orientifold central charge should be attributed to the quantum volumes.

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“…This symplectic transformation encodes all information on the relationships between the genus zero GromovWitten theories of X and Z. We refer to Iritani [13] for details and references on this still-evolving conjecture. At a lower level, the conjecture implies the cohomological crepant resolution conjecture; that is, the quantum corrected cohomology ring of Z (deformed by Gromov-Witten invariants computed on curves contracted by ρ) is isomorphic to the orbifold (Chen-Ruan) cohomology ring of X , after evaluation of the quantum parameters to roots of the unity.…”

Section: The Cohomological Crepant Resolution Conjecturementioning

confidence: 99%

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

Boissière¹,

Mann²,

Perroni³

2011

Nagoya Math. J.

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Abstract. We prove a formula computing the Gromov-Witten invariants of genus zero with three marked points of the resolution of the transversal A3-singularity of the weighted projective space P(1, 3, 4, 4) using the theory of deformations of surfaces with An-singularities. We use this result to check Ruan's conjecture for the stack P(1, 3, 4, 4). §1. IntroductionThe main results of this paper concern the weighted projective space P(1, 3, 4, 4). The singular locus of its coarse moduli space |P(1, 3, 4, 4)| is the disjoint union of an isolated singularity of type (1/3)(1, 1, 1) (we use Reid's notation in [17]) and a transversal A 3 -singularity. Using toric methods, we construct a crepant resolution Z of |P(1, 3, 4, 4)|. In Theorem 3.3.1, we determine a formula for certain Gromov-Witten invariants of Z over the A 3 -singularity using the theory of deformations of surfaces with rational double points and the deformation invariance property of Gromov-Witten invariants. We then apply this result in Theorem 5.2.1 to construct a ring isomorphism-predicted in [18] by Ruan's cohomological crepant resolution conjecture -between the quantum corrected cohomology ring of Z and the Chen-Ruan orbifold cohomology of P(1, 3, 4, 4), after evaluating the quantum parameters related to the transversal A 3 -singularity to a fourth root of the unity and putting the last parameter to zero. This last evaluation is quite surprising (in [2], we show that this parameter can be evaluated to 1). To confirm this property, we show in Proposition 5.3.1 that, for all weighted projective spaces P(1, . . . , 1, n) with n weights equal to 1 which have only

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An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

Cited by 218 publications

References 57 publications

Yangians, quantum loop algebras, and abelian difference equations

Yangians, quantum loop algebras, and abelian difference equations

Exact partition functions on $ \mathbb{R}{{\mathbb{P}}^2} $ and orientifolds

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

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