Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-11-30
Commun.Math.Phys. 176 (1996) 163-192
Physics
High Energy Physics
High Energy Physics - Theory
36 pp harvmac, "b" mode; references added
Scientific paper
10.1007/BF02099367
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 ${\bf Q}(J)$. 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 $5\not{|}d$) that the degree $d$ instanton numbers $n_d$ are divisible by $5^3$ -- a fact first conjectured by Clemens.
Lian Bong H.
Yau Shing-Tung
No associations
LandOfFree
Arithmetic Properties of Mirror Map and Quantum Coupling does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Arithmetic Properties of Mirror Map and Quantum Coupling, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Arithmetic Properties of Mirror Map and Quantum Coupling will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-309258