#### Volume 25, issue 2 (2021)

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The theory of $N$–mixed-spin-$P$ fields

### Huai-Liang Chang, Shuai Guo, Jun Li and Wei-Ping Li

Geometry & Topology 25 (2021) 775–811
##### Abstract

This is the first part of our project toward proving the Bershadsky–Cecotti–Ooguri–Vafa Feynman graph sum formula of all genera Gromov–Witten invariants of quintic Calabi–Yau threefolds. We introduce the notion of $N$–mixed-spin-$\phantom{\rule{-0.17em}{0ex}}P$ fields, construct their moduli spaces, their virtual cycles and their virtual localization formulas, and obtain a vanishing result associated with irregular graphs.

##### Keywords
Gromov–Witten, mirror symmetry, high genus, mixed-spin-$P$ fields, cosection localization
##### Mathematical Subject Classification 2010
Primary: 14D23, 14J33, 14N35