Relative mirror symmetry, theta functions and the Gamma conjecture

TL;DR

This paper advances the understanding of relative mirror symmetry for Fano varieties with snc anticanonical divisors, proving a new mirror theorem, studying theta functions, and confirming a version of the Gamma conjecture.

Contribution

It introduces a relative mirror theorem for snc pairs without nef assumptions, develops theta functions for these pairs, and applies these tools to prove a Gamma conjecture variant.

Findings

Proved a relative mirror theorem for snc pairs.

Developed theta functions associated with the pair $(X,D)$.

Established a version of the Gamma conjecture using these new tools.

Abstract

Let $X$ be a Fano variety, and $D\subset X$ be an snc anticanonical divisor. We study relative mirror symmetry for the log Calabi--Yau pair $(X,D)$. (1) We prove a relative mirror theorem for snc pairs without assuming the divisors are nef. (2) We study theta functions associated with the pair $(X,D)$. (3) We introduce functions on the mirror that are obtained from the higher-degree part of the big relative quantum cohomology. As an application, we use these new ingredients in relative mirror symmetry to prove a version of the mirror symmetric Gamma conjecture for $X$ for $\mathcal O_X$ and $\mathcal O_{\operatorname{pt}}$ in this setting, where the Landau--Ginzburg potential is defined as a sum of theta functions.