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Counting surfaces on Calabi-Yau 4-folds II: $\mathrm{DT}$-$\mathrm{PT}_0$ correspondence

Younghan Bae, Martijn Kool, Hyeonjun Park

Abstract

This is the second part in a series of papers on counting surfaces on Calabi-Yau 4-folds. In this paper, we introduce $K$-theoretic $\mathrm{DT}, \mathrm{PT}_0, \mathrm{PT}_1$ invariants and conjecture a $\mathrm{DT}$-$\mathrm{PT}_0$ correspondence. For certain tautological insertions, we derive Lefschetz principles in both the compact and toric case allowing reductions to 3-dimensional $\mathrm{DT}, \mathrm{PT}$ invariants. We also develop a topological vertex and conjecture a $\mathrm{DT}$-$\mathrm{PT}_0$ vertex correspondence. These methods enable us to verify our conjectures in several examples.

Counting surfaces on Calabi-Yau 4-folds II: $\mathrm{DT}$-$\mathrm{PT}_0$ correspondence

Abstract

This is the second part in a series of papers on counting surfaces on Calabi-Yau 4-folds. In this paper, we introduce -theoretic invariants and conjecture a - correspondence. For certain tautological insertions, we derive Lefschetz principles in both the compact and toric case allowing reductions to 3-dimensional invariants. We also develop a topological vertex and conjecture a - vertex correspondence. These methods enable us to verify our conjectures in several examples.
Paper Structure (33 sections, 51 theorems, 355 equations, 10 figures)

This paper contains 33 sections, 51 theorems, 355 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. PT0 pairs on Calabi-Yau 4-folds
  4. Vertex formalism
  5. Virtual Lefschetz principle
  6. Relations to physics
  7. Open questions
  8. Notation and conventions
  9. Preliminaries
  10. Basic properties of stable pairs
  11. Moduli spaces and virtual structures
  12. Equivariant case
  13. Virtual localization
  14. Invariants and correspondences
  15. Tautological invariants
  16. ...and 18 more sections

Key Result

Theorem 1.4

Let $(F,s)$ be a $(\mathbb C^*)^4$-fixed $\mathrm{PT}_0$ pair on $\mathbb C^4$ with scheme theoretic support $Z$. Let $Z_1 \subset \mathbb C^4$ be the pure part of $Z$.

Figures (10)

  • Figure 1: Adding boxes to $S_1 \cup S_2$.
  • Figure 2: The solid partition of Example \ref{['ex:solidpart']}.
  • Figure 3: The solid partition of Example \ref{['ex:solidpart2']}.
  • Figure 4: The scheme $W$ of Example \ref{['ex:PT0=PT1ex1']}.
  • Figure 5: The solid partition and scheme $W$ of Example \ref{['ex:PT0=PT1ex2']}.
  • ...and 5 more figures

Theorems & Definitions (147)

  • Definition 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Example 1.9
  • Proposition 1.10
  • Example 1.11
  • ...and 137 more