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Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds

Arkadij Bojko

Abstract

Gross-Joyce-Tanaka arXiv:2005.05637 proposed a wall-crossing conjecture for Calabi-Yau fourfolds. Assuming it, we prove the conjecture of Cao-Kool arXiv:1712.07347 for 0-dimensional sheaf-counting invariants on projective Calabi-Yau 4-folds. From it, we extract the full topological information contained in the virtual fundamental classes of Hilbert schemes of points which turns out to be equivalent to the data of all descendent integrals. As a consequence, we can express many generating series of invariants in terms of explicit universal power series. i) On $\mathbb{C}^4$, Nekrasov proposed invariants with a conjectured closed form arXiv:1712.08128. We show that an analog of his formula holds for compact Calabi-Yau 4-folds satisfying the wall-crossing conjecture. ii) We notice a relationship to corresponding generating series for Quot schemes on elliptic surfaces which are also governed by a wall-crossing formula. This leads to a Segre-Verlinde correspondence for Calabi-Yau fourfolds.

Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds

Abstract

Gross-Joyce-Tanaka arXiv:2005.05637 proposed a wall-crossing conjecture for Calabi-Yau fourfolds. Assuming it, we prove the conjecture of Cao-Kool arXiv:1712.07347 for 0-dimensional sheaf-counting invariants on projective Calabi-Yau 4-folds. From it, we extract the full topological information contained in the virtual fundamental classes of Hilbert schemes of points which turns out to be equivalent to the data of all descendent integrals. As a consequence, we can express many generating series of invariants in terms of explicit universal power series. i) On , Nekrasov proposed invariants with a conjectured closed form arXiv:1712.08128. We show that an analog of his formula holds for compact Calabi-Yau 4-folds satisfying the wall-crossing conjecture. ii) We notice a relationship to corresponding generating series for Quot schemes on elliptic surfaces which are also governed by a wall-crossing formula. This leads to a Segre-Verlinde correspondence for Calabi-Yau fourfolds.

Paper Structure

This paper contains 29 sections, 40 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. Wall-crossing conjecture for Calabi--Yau 4-folds
  3. Tautological insertions
  4. Segre series of all ranks
  5. Nekrasov genus
  6. Segre--Verlinde correspondence
  7. 4D-2D correspondence
  8. Vertex algebras in algebraic geometry and topology
  9. Vertex algebras
  10. Axioms of vertex algebras on homology of stacks
  11. Point-canonical orientations
  12. Vertex algebras over pairs
  13. Wall-crossing conjecture for Calabi--Yau fourfolds
  14. Topological construction of Joyce's vertex algebras
  15. Insertions
  16. ...and 14 more sections

Key Result

Theorem \oldthetheorem

Let $\alpha,\alpha_1,\ldots,\alpha_M \in G^0(X)$, $a=\textnormal{rk}(\alpha), a_i=\textnormal{rk}(\alpha_i)$, then assuming Conjecture conjecture WC we have for point-canonical orientations. Here $z$ is the unique solution to Moreover, in the same setting we have the explicit expressions

Theorems & Definitions (107)

  • Conjecture \oldthetheorem: Cao--Kool CK1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem: Joyce Joycehall
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Cao--Gross--Joyce CGJ
  • Definition \oldthetheorem
  • ...and 97 more