On a Conjecture of Givental

TL;DR

This paper investigates Givental's conjecture relating higher genus Gromov-Witten invariants to genus-0 data, focusing on the projective line case, and provides preliminary checks supporting the conjecture.

Contribution

It offers initial evidence and analysis for Givental's conjecture specifically in the context of the projective line, a case with well-understood invariants.

Findings

Supported Givental's conjecture through simple checks

Analyzed higher genus invariants in the projective line case

Identified puzzles and findings related to the conjecture

Abstract

These brief notes record our puzzles and findings surrounding Givental's recent conjecture which expresses higher genus Gromov-Witten invariants in terms of the genus-0 data. We limit our considerations to the case of a projective line, whose Gromov-Witten invariants are well-known and easy to compute. We make some simple checks supporting his conjecture.