Year: 2019
Type: preprint
Abstract: We study the Schwarzian derivative of the Seiberg--Witten period ratio in the rigid limit of the two-parameter mirror family associated with a degree-$12$ Calabi--Yau hypersurface in $\mathbb{P}^4_{1,1,2,2,6}$. The leading tangential Yukawa coefficient is proportional to the holomorphic cubic tensor of rigid special Kähler geometry. Using the mixed normal components and a prescribed constant-coefficient connection rule, we reconstruct the Schwarzian quadratic differential, whose coefficient in the effective coupling $τ$ is $π^2θ_2(τ/2)^8/8$. With the weak-coupling frame, dynamical scale, and perturbative normalization fixed, we prove that the holomorphic germ of this coefficient determines the prepotential up to an additive constant within a convergent local class, and vanishes identically if and only if all instanton corrections vanish. We then use the genus-zero mirror correspondence in the chosen chamber to relate the first two Schwarzian Fourier coefficients to compact Gromov--Witten invariants. The Picard--Fuchs system and full Yukawa tensor give the generating functions at base degrees one and two and their asymptotics as the fiber degree tends to infinity. Locally uniform period estimates fix the rigid-limit normalization and justify extraction of the gauge-instanton coefficients. This yields an explicit asymptotic relation between the compact invariants and the normalized second Schwarzian coefficient. The extension to higher base degree remains conditional.
Source: arXiv (Cornell University)
Author: Yang, Wenzhe
Institution: Xingtai University
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2994594802