Quantum Lefschetz theorem revisited

TL;DR

This paper revisits the quantum Lefschetz theorem for smooth Deligne-Mumford stacks, introducing admissible series and hypergeometric modifications to confirm predictions about genus zero quantum cohomology beyond convex cases.

Contribution

It introduces the concept of admissible series and extended variables, broadening the applicability of the quantum Lefschetz theorem beyond convexity assumptions.

Findings

Hypergeometric modifications lie on the Lagrangian cone of the complete intersection.

Confirms predictions about quantum Lefschetz theorem beyond convexity.

Introduces a new framework using extended variables for hypergeometric series.

Abstract

Let $X$ be any smooth Deligne-Mumford stack with projective coarse moduli, and $Y$ be a smooth complete intersection in $X$ associated with a direct sum of semi-positive line bundles. We will introduce a useful and broad class known as admissible series for discussing quantum Lefschetz theorem. For any admissible series on the Givental's Lagrangian cone of $X$, we will show that a hypergeometric modification of the series lies on the Lagrangian cone of $Y$. This confirms a prediction from Coates-Corti-Iritani-Tseng about the genus zero quantum Lefschetz theorem beyond convexity. In our quantum Lefschetz theorem, we use extended variables to formulate the hypergeometric modification, which may be of self-independent interest.