Modular Fibers And Illumination Problems

Seifert纤维化同调3-球的同调块的模变换

Modular transformations of homological blocks for Seifert fibered homology $3$-spheres

本文针对任意Seifert纤维化整同调3-球，给出了同调块的显式模变换公式，并基于此推导出Witten-Reshetikhin-Turaev不变量的渐近展开公式，为Andersen版本的Witten渐近猜想提供了新证明。

4-流形的纤维和公式、拓扑模形式与$6d\mathcal{N}=(1,0)$理论

Fiber sum formulas for 4-manifolds, topological modular forms and $6d\ \mathcal{N}=(1,0)$ theories

本研究借助六维方法下四维流形与拓扑模形式的关联，推导了与6d (1,0) SCFT紧化相关的无穷族光滑自旋四维流形的纤维和公式，发现自由理论也有非平凡公式且形式依赖理论与流形参数，为Stolz-Teichner猜想补充了证据。

Quantum Invariants, Modular Forms, and Lattice Points II

We study the SU(2) Witten--Reshetikhin--Turaev invariant for the Seifert fibered homology spheres with M-exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 1/2 and 3/2. By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large-N limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in N ->\infinity, is related to the number of integral l…

Moduli spaces of branched covers of Veech surfaces I: d-symmetric differentials

We give a description of asymptotic quadratic growth rates for geodesic segments on covers of Veech surfaces in terms of the modular fiber parameterizing coverings of a fixed Veech surface. To make the paper self contained we derive the necessary asymptotic formulas from the Gutkin-Judge formula. As an application of the method we define and analyze d-symmetric elliptic differentials and their modular fibers F^{sym}_d. For given genus g, g-symmetric elliptic differentials (with fixed base lattice) provide a 2-dimensional family of translation surfaces. We calculate several asymptotic constants…

Spaces of elliptic differentials

We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers F_d(1,1) are itself connected torus covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we calculate the Euler Characteristic and the parity of the spin structure of the quadratic differential (F_d(1,1)/(-id),q_d). We state and apply formulas for the asymptotic quadratic growth rates of various types of geodesic segments on a surface S "contained" in F_d(1,1). The quadratic grow…

Parallel connections and bundles of arrangements

Let \A be a complex hyperplane arrangement, and let $X$ be a modular element of arbitrary rank in the intersection lattice of \A. We show that projection along $X$ restricts to a fiber bundle projection of the complement of \A to the complement of the localization $\A_X$ of \A at $X$. The fiber is the decone of a realization of the complete principal truncation of the underlying matroid of \A along the flat corresponding to $X$. This result gives a topological realization of results of Stanley, Brylawsky, and Terao on modular factorization. We show that (generalized) parallel connection of mat…