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A degeneration formula of Donaldson-Thomas theory on Calabi-Yau 4-folds

Yalong Cao, Gufang Zhao, Zijun Zhou

Abstract

We prove a degeneration formula for Donaldson-Thomas theory on Calabi-Yau 4-folds, and apply it to compute zero dimensional invariants on $\mathbb{C}^4$ and on any local curve.

A degeneration formula of Donaldson-Thomas theory on Calabi-Yau 4-folds

Abstract

We prove a degeneration formula for Donaldson-Thomas theory on Calabi-Yau 4-folds, and apply it to compute zero dimensional invariants on and on any local curve.
Paper Structure (40 sections, 56 theorems, 460 equations)

This paper contains 40 sections, 56 theorems, 460 equations.

Table of Contents

  1. Introduction
  2. Background/Motivation
  3. Expanded degenerations and family invariants
  4. Expanded pairs and relative invariants
  5. Degeneration formulae
  6. Applications
  7. Related works and some future direction
  8. Recollection of expanded pairs and degenerations
  9. Expanded pairs
  10. Expanded degenerations
  11. Topological data
  12. Relative Hilbert stacks
  13. Shifted symplectic structures
  14. Shifted symplectic structure on $\textbf{Hilb}(X/{\mathbb A}^1)/\mathcal{C}$
  15. Lagrangian structure on $(\textbf{Hilb} (Y, D)\to \textbf{Hilb} ({\mathcal{D}}))/{\mathcal{A}}$
  16. ...and 25 more sections

Key Result

Theorem 1.1

$\mathop{\mathrm{Hilb}}\nolimits^{P_t }(X/{\mathbb A}^1)$ has a derived enhancement $\textbf{Hilb}^{P_t }(X/{\mathbb A}^1)$ with a $(-2)$-shifted symplectic structure over ${\mathcal{C}}^{P_t}$. Moreover its classical truncation is orientable in the sense of Definition ori on even cy.

Theorems & Definitions (150)

  • Theorem 1.1: Theorem \ref{['thm on sft symp str']}, Theorem \ref{['ori of cy4 family']}
  • Theorem 1.2: Family invariants, Proposition \ref{['iso cond 1']}, Theorem \ref{['thm pb on family']}
  • Theorem 1.3: Theorem \ref{['thm on lag']}, Theorem \ref{['relative ori of cy4 family']}
  • Theorem 1.5: Relative invariants, Proposition \ref{['iso cond 2']}, Theorem \ref{['thm on relative invs']}
  • Remark 1.6
  • Theorem 1.7: Degeneration formula, Theorem \ref{['thm on dege formula']}
  • Theorem 1.8: Theorem \ref{['thm on ck conj']}
  • Theorem 1.9: Theorem \ref{['thm on local curve']}
  • Definition 2.1
  • Proposition 2.2
  • ...and 140 more