A survey on mixed spin P-fields

TL;DR

This survey reviews the theory of mixed spin P-fields, a framework connecting Gromov-Witten and Landau-Ginzburg theories for quintic Calabi-Yau threefolds, highlighting key ideas, properties, and computational techniques.

Contribution

It provides a comprehensive overview of MSP fields, including their moduli, properties, and applications to computing invariants, advancing understanding of phase transitions in string theory.

Findings

Polynomial relations among GW and FJRW invariants derived from torus actions

Effective algorithms for computing low-genus GW invariants

Properties like cosection localisation and properness of the moduli space

Abstract

This is a survey on the mixed spin P-fields (MSP fields for short) theory which provides a platform to understand the phase transition between Gromov-Witten theory of quintic CY 3-folds and Landau-Ginzburg theory of the corresponding quintic polynomials. It discusses key ideas that lead to the definition of MSP fields and how moduli of stable maps to the quintic and that of 5-spin curves appear in the moduli of MSP fields. It also explains some properties of the moduli of MSP fields such as the cosection localisation, the properness of the degeneracy locus, and a torus action on the moduli.. Some vanishings arising from the torus action provide polynomial relations among GW-invarants and FJRW-invaraints which give an effective algorithm for the computation of those invariants. Some examples of computations of genus 1 low degree of GW invariants are provided.