Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

Hosono, Shinobu; Klemm, Albrecht; Theisen, Stefan; Yau, Shing–Tung

doi:10.1016/0550-3213(94)00440-p

Cited by 302 publications

(567 citation statements)

References 67 publications

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“…) It is known that the so-called mirror map and the quantum coupling have many interesting number theoretic properties based on numerical experiments -as previously observed by many [1] [3][4] [5]. For example the Fourier coefficients of the mirror map appears to be integral in all known cases.…”

Section: Introductionmentioning

confidence: 80%

“…Note that since N e has no prime divisor other than 2, 3, there is no other prime p dividing all n d , by Claim 1. The orbifold group is of type (10,5,2). In this case, we have N e = −2 6 .5 3 and K cl = 1.…”

Section: Examplesmentioning

confidence: 99%

“…The Picard-Fuchs equations for elliptic curves in a weighted projective space (For the general form of the Picard-Fuchs equation see [4] [5].) is of this form where z is a suitable coordinate for the complex structure moduli space of the curves.…”

Section: N=2 Schwarzian Equationmentioning

confidence: 99%

“…Let X be the Fermat quintics in P 4 . The orbifold group here is an abelian group of type (5,5,5)[12] [1]. In this case, we have N e = −5 3 .…”

Section: Examplesmentioning

confidence: 99%

“…First we discuss, along the lines of [4] [5], the construction of deformation coordinates based on which the mirror map is defined. Then We review the polynomial differential equations for the mirror maps, studied in [8] [9].…”

Section: Introductionmentioning

confidence: 99%

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Arithmetic properties of mirror map and quantum coupling

Lian

Yau²

1996

Commun.Math. Phys.

109

170

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We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of the J-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function field Q(J). This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced mod p. Under the mirror hypothesis and an integrality assumption, we derive mod p congruences for the Fourier coefficients.For the quintics, we deduce (at least for 5 |d) that the degree d instanton numbers n d are divisible by 5 3 -a fact first conjectured by Clemens.Revised 12/94 ⋄

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Section: Introductionmentioning

confidence: 80%

Section: Examplesmentioning

confidence: 99%

Section: N=2 Schwarzian Equationmentioning

confidence: 99%

“…Let X be the Fermat quintics in P 4 . The orbifold group here is an abelian group of type (5,5,5)[12] [1]. In this case, we have N e = −5 3 .…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Arithmetic properties of mirror map and quantum coupling

Lian

Yau²

1996

Commun.Math. Phys.

109

170

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Topological string theory and integrable structures

Klemm¹

2005

Fortschritte der Physik

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Black hole attractors and the entropy function in four‐ and five‐dimensional N = 2 supergravity

Perz

2007

Fortschritte der Physik

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In this overview of selected aspects of the black hole attractor mechanism, after introducing the necessary foundations, we examine the relationship between two ways to describe the attractor phenomenon in four‐dimensional N = 2 supergravity: the entropy function and the black hole potential. We also exemplify their practical application to finding solutions to the attractor equations for a conifold prepotential. Next we describe an extension of the original definition of the entropy function to a class of rotating black holes in five‐dimensional N = 2 supergravity based on cubic polynomials, exploiting a connection between four‐ and five‐dimensional black holes. This link allows further the derivation of five‐dimensional first‐order differential flow equations governing the profile of the fields from infinity to the event horizon and construction of non‐supersymmetric interpolating solutions in four dimensions by dimensional reduction. Finally, since four‐dimensional extremal black holes in N = 2 supergravity can be viewed as certain two‐dimensional string compactifications with fluxes, we discuss implications of the conifold example in the context of the entropic principle, which postulates as a probability measure on the space of these string compactifications the exponentiated entropy of the corresponding black holes.

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Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

Cited by 302 publications

References 67 publications

Arithmetic properties of mirror map and quantum coupling

Arithmetic properties of mirror map and quantum coupling

Topological string theory and integrable structures

Black hole attractors and the entropy function in four‐ and five‐dimensional N = 2 supergravity

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