Curve counting and DT/PT correspondence for Calabi-Yau 4-folds

Abstract

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold $X$. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of $X$ defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of $X$ using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of $X$. After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds using a localization formula. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions.

arXiv comment28 pages. Published version

References in corpus (17)

• The Topological Vertex
• The intrinsic normal cone
• Curve counting via stable pairs in the derived category
• Enumerative geometry of Calabi-Yau 4-folds
• The 3-fold vertex via stable pairs
• Reduced classes and curve counting on surfaces I: theory
• Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Magnificent Four with Colors
• Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
• Donaldson-Thomas theory for Calabi-Yau 4-folds
• Orientability for gauge theories on Calabi-Yau manifolds
• Curve counting via stable objects in derived categories of Calabi-Yau 4-folds
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds II: Fano 3-folds
• Counting conics on sextic 4-folds

Cited by in corpus (23)

• $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds
• Counting sheaves on Calabi-Yau 4-folds, I
• Higher rank K-theoretic Donaldson-Thomas theory of points
• Curve counting via stable objects in derived categories of Calabi-Yau 4-folds
• Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via descendent insertions
• Stable pair invariants of local Calabi-Yau 4-folds
• Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds
• Tautological stable pair invariants of Calabi-Yau 4-folds
• Counting perverse coherent systems on Calabi-Yau 4-folds
• The Origin of Calabi-Yau Crystals in BPS States Counting
• Noncommutative Instantons in Diverse Dimensions
• Charging solid partitions
• Enumerative invariants and wall-crossing formulae in abelian categories
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• Cosection Localization and Vanishing for Virtual Fundamental Classes of D-Manifolds
• A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
• Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V_5 and V_22
• Equivariant Segre and Verlinde invariants for Quot schemes