Moduli spaces of flat tori and elliptic hypergeometric functions

Abstract

In the genus one case, we make explicit some constructions of Veech [76] on flat surfaces and generalize some geometric results of Thurston [71] about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne and Mostow [11], which concern Appell-Lauricella hypergeometric functions. 2 S. GHAZOUANI AND L. PIRIOTable of Contents 1. Introduction 2 2. Notations and preliminary material 23 3. Twisted (co-)homology and integrals of hypergeometric type 26 4. An explicit expression for Veech's map and some consequences 50 5. Flat tori with two conical points 92 6. Miscellanea 105 Appendix A : 1-dimensional complex hyperbolic conifolds 128 Appendix B : the Gauß-Manin connection associated to Veech's map 135 References 152

Advanced tools for basis decompositions of genus-one string integrals

“…These connections suggest that twisted de Rham theory could provide a unified and robust framework to comprehend genus-one string integrals, their reductions, monodromy relations, and open-closed string relations [1-4, 27, 76-91]. Existing mathematical frameworks that leverage twisted cohomology setups have already demonstrated that meromorphic Kronecker-Eisenstein series form a basis under certain conditions, and a generalization of this could substantiate the conjectures presented herein [37,46,47,[92][93][94][95].…”