The theory of N-Mixed-Spin-P fields

TL;DR

This paper introduces N-Mixed-Spin-P fields, constructs their moduli spaces and virtual cycles, and establishes localization formulas and vanishing results, advancing the understanding of Gromov-Witten invariants for quintic Calabi-Yau threefolds.

Contribution

It defines N-Mixed-Spin-P fields and develops foundational tools like moduli spaces and localization formulas, crucial for proving BCOV's Feynman graph sum formula.

Findings

Constructed moduli spaces for N-Mixed-Spin-P fields

Derived virtual localization formulas for these moduli spaces

Proved vanishing results for irregular graphs

Abstract

This is the first part of the project toward proving the BCOV's Feymann graph sum formula of all genera Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, we introduce the notion of N-Mixed-Spin-P fields, construct their moduli spaces, their virtual cycles, their virtual localization formulas, and a vanishing result associated with irregular graphs.