Algebraic $K$-theory of toric hypersurfaces

Doran, Charles F.; Kerr, Matt

doi:10.4310/cntp.2011.v5.n2.a3

Cited by 29 publications

(60 citation statements)

References 62 publications

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“…The expressions there makes explicit all the mass parameters. One remarkable fact is that the computation can be done using the existing technology of mirror symmetry developed in other physical [22][23][24] or mathematics [25] contexts. This analysis extends naturally to the higher loop sunset integrals [55].…”

Section: Resultsmentioning

confidence: 99%

“…But it has the disadvantage of hiding all the physical parameters in the geometry of the elliptic curve. The expression using the trilogarithm has the advantage of making all the mass parameters explicit and generalising to all loop orders since the expansion of the higher-loop sunset graphs around p 2 = ∞ is expected to involve polylogarithms of order l at l-loop order [25,55].…”

Section: Resultsmentioning

confidence: 99%

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Feynman Integrals, Toric Geometry and Mirror Symmetry

Vanhove

2019

Texts &Amp; Monographs in Symbolic Computation

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This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals. We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series. We explain how this allows to determine the minimal differential operator acting on the Feynman integrals. We illustrate the method on sunset integrals in two dimensions at various loop orders. The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces. Therefore the sunset family is a natural home for mirror symmetry techniques. We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm. The equivalence between these two expressions is a consequence of 1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and 2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface. * IPHT-t18/096 arXiv:1807.11466v2 [hep-th]

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Feynman Integrals, Toric Geometry and Mirror Symmetry

Vanhove

2019

Texts &Amp; Monographs in Symbolic Computation

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“…The classical dilogarithm enters as a classical limit of the quantum dilogarithm involved in the kernel of the relevant operator. It would be very interesting to prove (3.41) by using the powerful techniques of [64].…”

Section: 'T Hooft Expansion Of the Fermionic Tracesmentioning

confidence: 99%

Operators and higher genus mirror curves

Codesido

Mariño

2017

J. High Energ. Phys.

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Abstract:We perform further tests of the correspondence between spectral theory and topological strings, focusing on mirror curves of genus greater than one with nontrivial mass parameters. In particular, we analyze the geometry relevant to the SU(3) relativistic Toda lattice, and the resolved C 3 /Z 6 orbifold. Furthermore, we give evidence that the correspondence holds for arbitrary values of the mass parameters, where the quantization problem leads to resonant states. We also explore the relation between this correspondence and cluster integrable systems.

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“…Dimensional reduction of the Hori-Vafa mirror. In this subsection, we describe the precise relations among the following 3-dimensional, 2-dimensional, and 1-dimensional integrals when q ∈ U ǫ : (3d) period integrals of the holomorphic 3-form Ω q over 3-cycles in the Hori-Vafa mirrorX q , (2d) integral of the holomorphic 2-form dX X ∧ dY Y on (C * ) 2 over relative 2-cycles of the pair ((C * ) 2 , C q ), and (1d) integrals of a Liouville form along 1-cycles in the mirror curve C q , The references of this subsection are [36] and [19]; see also [63].…”

Section: 4mentioning

confidence: 99%

“…Lemma 4.11. Suppose that u 1 , u 2 are real numbers such that such that w ′ i (τ, σ) is a nonzero real number for any flag (τ, σ) and for any i ∈ {1, 2, 3}, so that f = u 2 u 2 is generic and (36) and (52), for any flag (τ, σ), we may write…”

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confidence: 99%

On the remodeling conjecture for toric Calabi-Yau 3-orbifolds

Fang

Liu

Zong

2019

J. Amer. Math. Soc.

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The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti (BKMP) relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds.In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semi-projective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera. 4.5. The equivariant small quantum cohomology 34 4.6. Dimensional reduction of the equivariant Landau-Ginzburg model 35 4.7. Action by the stacky Picard group 39 5. Geometry of the Mirror Curve 40 5.1. Riemann surfaces 40 5.2. The Liouville form 41 5.3. Differentials of the first kind and the third kind 41 5.4. Toric degeneration 42 5.5. Degeneration of 1-forms 44 5.6. The action of the stacky group on on the central fiber 45 5.7. The Gauss-Manin connection and flat sections 46 5.8. Vanishing cycles and loops around punctures 47 5.9. B-model flat coordinates 48 5.10. Differentials of the second kind 51 6. B-model Topological Strings 51 6.1. Canonical basis in the B-model: θ 0 σ and [Vσ(τ )]. 51 6.2. Oscillating integrals and the B-model R-matrix 52 6.3. The Eynard-Orantin topological recursion and the B-model graph sum 55 6.4. B-model open potentials 57 6.5. B-model free energies 58 7. All Genus Mirror Symmetry 58 7.1. Identification of A-model and B-model R-matrices 58 7.2. Identification of graph sums 59 7.3. BKMP Remodeling Conjecture: the open string sector 60 7.4. BKMP Remodeling Conjecture: the free energies 61 References 65On the B-model side, by the result in [37], the Eynard-Orantin recursion is equivalent to a graph sum formula. So the B-model potentialF g,n can also be expressed as a graph sum:The all genus open-closed Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds. The Crepant Transformation Conjecture, proposed by Ruan [82,83] and later generalized by others in various situations, relates GW theories of K-equivalent smooth varieties, orbifolds, or Deligne-Mumford stacks. To establish this equivalence, one may need to do change of variables, analytic continuation, and symplectic transformation for the GW potential. In general, the higher genus Crepant Transformation Conjecture is difficult to formulate and prove. Coates-Iritani introduced the Fock sheaf formalism and proved all genus Crepant Transformation Conjecture for compact toric orbifolds [29]. The Remodeling Conjecture leads to simple formulation and proof of all genus Crepant Transformation Conjecture for semi-projective toric CY 3-orbifolds (which are always non-compact). The key point here is that our higher genus B-model, defined in terms of Eynard-Orantin invariants of the mirror curve, is global and analyti...

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Algebraic $K$-theory of toric hypersurfaces

Cited by 29 publications

References 62 publications

Feynman Integrals, Toric Geometry and Mirror Symmetry

Feynman Integrals, Toric Geometry and Mirror Symmetry

Operators and higher genus mirror curves

On the remodeling conjecture for toric Calabi-Yau 3-orbifolds

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