A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

Abstract

Let $G$ be a finite subgroup of $\mathrm{SU}(4)$ whose elements have age not larger than one. In the first part of this paper, we define $K$-theoretic stable pair invariants on the crepant resolution of the affine quotient ${\mathbb{C}}^4/G$, and conjecture closed formulae for their generating series, expressed in terms of the root system of $G$. In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case and conjecture closed formulae for the quotient stacks $[{\mathbb{C}}^4/{\mathbb{Z}}_r]$, $[{\mathbb{C}}^4/{\mathbb{Z}}_2\times {\mathbb{Z}}_2]$. Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.

arXiv comment41 pages. Published version

References in corpus (34)

• The intrinsic normal cone
• Curve counting via stable pairs in the derived category
• The orbifold Chow ring of toric Deligne-Mumford stacks
• Enumerative geometry of Calabi-Yau 4-folds
• Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds
• A Monograph on the Classification of the Discrete Subgroups of SU(4)
• Lectures on K-theoretic computations in enumerative geometry
• Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Riemann-Roch theorems and elliptic genus for virtually smooth Schemes
• Magnificent Four with Colors
• Zero-dimensional Donaldson-Thomas invariants of Calabi-Yau 4-folds
• $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• Counting sheaves on Calabi-Yau 4-folds, I
• Orientability for gauge theories on Calabi-Yau manifolds
• Higher rank K-theoretic Donaldson-Thomas theory of points
• Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
• Curve counting via stable objects in derived categories of Calabi-Yau 4-folds
• The Quantum McKay Correspondence for polyhedral singularities
• Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via descendent insertions
• Stable pair invariants of local Calabi-Yau 4-folds
• ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds
• Tautological stable pair invariants of Calabi-Yau 4-folds
• Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds
• Counting perverse coherent systems on Calabi-Yau 4-folds
• Curve counting theories via stable objects II: DT/ncDT flop formula
• Quasimaps and stable pairs
• Double Quiver Gauge Theory and BPS/CFT Correspondence
• Donaldson-Thomas theory of $[{\mathbb{C}}^2/{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$
• Noncommutative Instantons in Diverse Dimensions
• On the M2-Brane Index on Noncommutative Crepant Resolutions
• Enumerating coloured partitions in 2 and 3 dimensions

Cited by in corpus (1)

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