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 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 ⋄

Quantum geometry of resurgent perturbative/nonperturbative relations

“…These number theoretic functions are also associated with topological c = 3 Landau-Ginzburg models [104,105] and certain superconformal quantum field theories [106]. In addition, the special class of Chebyshev models also has an interesting geometric interpretation in terms of mirror maps in weighted projective spaces [108][109][110][111][112][113]. These correspond to the normal forms of functions with unimodular singularities [114].…”

“…Indeed, it is well known [107] that ratios of solutions to the hypergeometric equation describe conformal maps of spherical triangles. These form the simplest nontrivial example of a mirror map [108][109][110][111][112]:…”

“…The N f = 3 case corrsponds to the projective space P 3 (1, 1, 1, 1) discussed in section 5 of [109].…”

Calabi-Yau manifolds over finite fields, II

“…We proceed as before with a matrix that is now The null space of the matrix is two dimensional and we need to choose in this space a basis for the Mori cone. It is straightforward to do this in terms of combinatorial properties of the matrix however since this is not our main interest we can guess the answer by examining the large complex structure coordinates which for this case are λ = − 2φ (8ψ) 4 and µ = 1 (2φ) 2 .…”

“…The large complex structure limit is the limit ψ, ℑ(t) → ∞ and in this limit the exponential terms, which express the quantum corrections, tend to zero and y ttt (ψ) → 5, the classical value. On the other hand it has been shown [4] that 5 3 |n k k 3 for each k so that we can in fact write…”