Relative quantum cohomology under birational transformations

TL;DR

This paper investigates how relative quantum cohomology changes under birational transformations, establishing direct identifications and analytic continuations of relative $I$-functions for toric complete intersections, with connections to extremal transitions and FJRW theory.

Contribution

It provides new results on the behavior of relative quantum cohomology under birational transformations, including explicit identifications and analytic continuation of $I$-functions for specific toric cases.

Findings

Relative $I$-functions can be directly identified for certain toric complete intersections with normal crossings divisors.

Relative $I$-functions are related by analytic continuation for smooth divisors.

Connections established between relative quantum cohomology, extremal transitions, and FJRW theory.

Abstract

We study how relative quantum cohomology, defined by Tseng--You and Fan--Wu--You, varies under birational transformations. For toric complete intersections with simple normal crossings divisors that contain the loci of indeterminacy, we prove that their respective relative $I$-functions can be directly identified. For toric complete intersections with smooth divisors, we prove that their respective relative $I$-functions are related by analytic continuation. We also study some connections with extremal transitions and FJRW theory.