$K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds

Abstract

Recently, Nekrasov discovered a new "genus" for Hilbert schemes of points on ${\mathbb{C}}^4$. We conjecture a DT/PT correspondence for Nekrasov genera for toric Calabi-Yau 4-folds. We verify our conjecture in several cases using a vertex formalism. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi-Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the $K$-theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov-Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of $K$-theoretic stable pair invariants of the local resolved conifold. Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov-Okounkov-Osinenko.

47 pages, published version

References in corpus (15)

• The Refined Topological Vertex
• Magnificent Four with Colors
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• Equivariant K-theory and refined Vafa-Witten invariants
• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• Counting sheaves on Calabi-Yau 4-folds, I
• 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
• Stable pair invariants of local Calabi-Yau 4-folds
• Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via descendent insertions
• 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
• K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
• Refinements of Kool-Thomas Invariants via Equivariant $K$-theoretic invariants

Cited by in corpus (29)

• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• Counting sheaves on Calabi-Yau 4-folds, I
• Curve counting via stable objects in derived categories of 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
• Gauge origami and quiver W-algebras
• Double Quiver Gauge Theory and BPS/CFT Correspondence
• Tetrahedron instantons in Donaldson-Thomas theory
• BPS States Meet Generalized Cohomology
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• The homology of moduli stacks of complexes
• Enumerative invariants and wall-crossing formulae in abelian categories
• Double nested Hilbert schemes and the local stable pairs theory of curves
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Universal structures in $\mathbb{C}$-linear enumerative invariant theories
• Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
• Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• The geometry of double nested Hilbert schemes of points on curves
• A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds
• Cosection Localization and Vanishing for Virtual Fundamental Classes of D-Manifolds
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• An Overview of Crystals and Double Quiver Yangians
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
• Equivariant Segre and Verlinde invariants for Quot schemes
• Wall-crossing for punctual Quot-schemes