Direct Proof of Mirror Theorem of Projective Hypersurfaces up to degree 3 Rational Curves

TL;DR

This paper provides a direct proof of the mirror theorem for projective hypersurfaces up to degree 3, using Gromov-Witten invariants and residue integrals, with potential for generalization to higher degrees.

Contribution

It introduces a direct derivation method for mirror transformations of projective hypersurfaces up to degree 3, simplifying previous approaches and enabling generalization.

Findings

Derived mirror transformation formulas up to degree 3

Compared localization formula with residue integral representation

Method can be extended to higher degrees with manageable complexity

Abstract

In this paper, we directly derive generalized mirror transformation of projective hypersurfaces up to degree 3 genus 0 Gromov-Witten invariants by comparing Kontsevich localization formula with residue integral representation of the virtual structure constants. We can easily generalize our method for rational curves of arbitrary degree except for combinatorial complexities.