Arithmetic Properties of Mirror Map and Quantum Coupling

Abstract

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 $\mathbf{Q}(J)$. 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 $modp$. Under the mirror hypothesis and an integrality assumption, we derive $modp$ congruences for the Fourier coefficients. For the quintics, we deduce (at least for $5\nmid d$) that the degree $d$ instanton numbers $n_d$ are divisible by $5^3$ -- a fact first conjectured by Clemens.

36 pp harvmac, "b" mode; references added

References in corpus (3)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces
• Topological Field Theory and Rational Curves

Cited by in corpus (61)

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• Rational Conformal Field Theories and Complex Multiplication
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• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
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• On the integrality of the Taylor coefficients of mirror maps
• Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau threefolds
• Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two Kähler Forms
• Exceptional String: Instanton Expansions and Seiberg-Witten Curve
• Critère pour l'intégralité des coefficients de Taylor des applications miroir
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• Topological strings and Wilson loops
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• Special Polynomial Rings, Quasi Modular Forms and Duality of Topological Strings
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• Differential Rings from Special Kähler Geometry
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• Period- and mirror-maps for the quartic K3
• Families of K3 surfaces over curves satisfying the equality of Arakelov-Yau's type and modularity
• Analytic Periods via Twisted Symmetric Squares
• The sixth Painleve transcendent and uniformization of algebraic curves
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• Closed Sub-Monodromy Problems, Local Mirror Symmetry and Branes on Orbifolds
• BCOV ring and holomorphic anomaly equation
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• Collapsing D-branes in one-parameter models and small/large radius duality
• Number theory casting a look at the mirror
• K3 mirror symmetry, Legendre family and Deligne's conjecture for Fermat quartic
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• The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
• The Master Equation for the Prepotential-Pub
• Prepotential, Mirror Map and F-Theory on K3
• Intégralité des coefficients de Taylor de racines d'applications miroir
• The Mori fan of the Dolgachev-Nikulin-Voisin family in genus $2$
• Prepotentials from Symmetric Products
• Non Perturbative Check of N=2, D=4 Heterotic/Type II Duality
• Genus Zero Modular Functions
• Mirror Symmetry and Projective Geometry of Fourier-Mukai Partners
• Harper operators, Fermi curves, and Picard-Fuchs equations
• Parabolic Higgs bundles, $t{t}^{\ast }$ connections and opers
• Fano-Mathieu correspondence
• Moduli Space of Quasi-Maps from P^{1} with Two Marked Points to P(1,1,1,3) and j-invariant
• Mirror Symmetry for Plane Cubics Revisited
• Apéry's irrationality proof, Beukers's modular forms and mirror symmetry
• Autoequivalences of Derived Category of A K3 Surface and Monodromy Transformations
• Calabi-Yau threefolds of type K (II): Mirror symmetry
• Mahler Measure of 3D Landau-Ginzburg Potentials
• Modularity of Feynman Integrals and Factorization of Appell F2 Systems
• Modular-type functions attached to Calabi-Yau varieties: integrality properties
• Algebraic structure of $t{t}^{\ast }$ equations for Calabi-Yau sigma models