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Irregular vanishing on $\mathbb{P}^2 \times \mathbb{P}^2$

Huai-Liang Chang, Sanghyeon Lee, Jun Li

TL;DR

This work develops Mixed-Spin-P (MSP) field theory for the Calabi–Yau hypersurface $X_{3,3} \subset \mathbb{P}^2 \times \mathbb{P}^2$, establishing a $\mathbb{C}^*$-equivariant framework with a cosection-localized obstruction theory to study fixed loci under torus actions. It introduces the associated localization graphs and analyzes MSP fields over subcurves, providing a detailed decomposition of the torus-fixed moduli via flat graphs and the notions of regular versus irregular graphs. The main technical result is a vanishing theorem: the virtual contributions from irregular graphs (excluding pure loops) vanish under localization, enabling reduction to fixed loci with no $0$-$\infty$ edges and simplifying BCOV-type polynomiality analyses; this is extended to N-MSP fields as well. An appendix develops the hybrid Landau–Ginzburg theory at infinity, linking the local hybrid theory to global MSP computations and FJRW-type invariants. These results advance computational approaches to Gromov–Witten and FJRW-type invariants for CY3s realized as complete intersections in products of projective spaces, with potential applications to BCOV Feynman rules and polynomiality properties.

Abstract

In this paper, we describe Mixed-Spin-P(MSP) fields for a smooth CY 3-fold $X_{3,3} \subset \mathbb{P}^2 \times \mathbb{P}^2$. Then we describe $\mathbb{C}^* -$fixed loci of the moduli space of these MSP fields. We prove that any virtual localization term coming from the fixed locus corresponding to an irregular graph does not contribute to the invariant if the graph is not a pure loop, and also prove this vanishing property for the moduli space of N-MSP fields.

Irregular vanishing on $\mathbb{P}^2 \times \mathbb{P}^2$

TL;DR

This work develops Mixed-Spin-P (MSP) field theory for the Calabi–Yau hypersurface , establishing a -equivariant framework with a cosection-localized obstruction theory to study fixed loci under torus actions. It introduces the associated localization graphs and analyzes MSP fields over subcurves, providing a detailed decomposition of the torus-fixed moduli via flat graphs and the notions of regular versus irregular graphs. The main technical result is a vanishing theorem: the virtual contributions from irregular graphs (excluding pure loops) vanish under localization, enabling reduction to fixed loci with no - edges and simplifying BCOV-type polynomiality analyses; this is extended to N-MSP fields as well. An appendix develops the hybrid Landau–Ginzburg theory at infinity, linking the local hybrid theory to global MSP computations and FJRW-type invariants. These results advance computational approaches to Gromov–Witten and FJRW-type invariants for CY3s realized as complete intersections in products of projective spaces, with potential applications to BCOV Feynman rules and polynomiality properties.

Abstract

In this paper, we describe Mixed-Spin-P(MSP) fields for a smooth CY 3-fold . Then we describe fixed loci of the moduli space of these MSP fields. We prove that any virtual localization term coming from the fixed locus corresponding to an irregular graph does not contribute to the invariant if the graph is not a pure loop, and also prove this vanishing property for the moduli space of N-MSP fields.

Paper Structure

This paper contains 19 sections, 10 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Description of MSP fields
  3. The Landau-Ginzburg model
  4. Moduli space of Mixed-Spin-P fields
  5. $\mathbb{C}^*$--action on MSP fields, equivariant perfect obstruction theory, and $\mathbb{C}^*$--fixed loci
  6. Decomposition of the domain curve
  7. Localization graph corresponding to a $\mathbb{C}^*$--fixed MSP field
  8. Linearization of $\mathbb{C}^*$--action on $\mathbb{P}^1$ to a line bundle on $\mathbb{P}^1$
  9. Linearization of $\mathbb{C}^*$--action on $\mathbb{P}(1,3)$ to a line bundle on $\mathbb{P}(1,3)$
  10. Description of MSP fields over subcurves
  11. Flat graphs and decomposition of the fixed loci
  12. Regular and irregular graphs
  13. Proof for irregular vanishing
  14. Vanishing for the special case
  15. Decoupling and trimming of edges $E_{01}$ and $E_{0 \infty}$
  16. ...and 4 more sections

Key Result

Lemma 2.3

Let $w_i$ be the weight of the $\mathbb{C}^*$--representation on ${\mathcal{O}}(d)|_{p_i}$ for $i=0,1$. Then we have

Theorems & Definitions (25)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Proposition 2.9
  • ...and 15 more