[Paper Review] Curve counting on elliptic Calabi-Yau threefolds via derived categories

Georg Oberdieck, Junliang Shen|arXiv (Cornell University)|Aug 25, 2016

Advanced Algebra and Geometry40 references16 citations

TL;DR

This paper proves that the generating series of Pandharipande–Thomas invariants for an elliptic Calabi–Yau threefold with a reduced class in the base satisfies the elliptic transformation law of Jacobi forms, establishing a deep modular structure. Using a derived auto-equivalence and wall-crossing techniques, the authors express PT invariants in terms of Gromov–Witten invariants and universal Jacobi forms, proving the series is a quasi-Jacobi form of weight -10 for K3×E and providing strong evidence for the Igusa cusp form conjecture.

ABSTRACT

We prove the elliptic transformation law of Jacobi forms for the generating series of Pandharipande--Thomas invariants of an elliptic Calabi--Yau 3-fold over a reduced class in the base. This proves part of a conjecture by Huang, Katz, and Klemm. For the proof we construct an involution of the derived category and use wall-crossing methods. We express the generating series of PT invariants in terms of low genus Gromov--Witten invariants and universal Jacobi forms. As applications we prove new formulas and recover several known formulas for the PT invariants of $\mathrm{K3} imes E$, abelian 3-folds, and the STU-model. We prove that the generating series of curve counting invariants for $\mathrm{K3} imes E$ with respect to a primitive class on the $\mathrm{K3}$ is a quasi-Jacobi form of weight -10. This provides strong evidence for the Igusa cusp form conjecture.

Motivation & Objective

• To understand how derived auto-equivalences constrain curve counting invariants on Calabi–Yau threefolds.
• To verify the Huang–Katz–Klemm conjecture on the modular properties of generating series of Pandharipande–Thomas invariants.
• To establish a connection between derived categories and Jacobi forms in the context of curve counting.
• To provide new computational tools for PT invariants using wall-crossing and Gromov–Witten theory.
• To offer strong evidence for the Igusa cusp form conjecture via the case of K3×E.

Proposed method

• Construct an involution on the derived category of coherent sheaves on the elliptic Calabi–Yau threefold.
• Use wall-crossing techniques to relate Pandharipande–Thomas invariants to Gromov–Witten invariants.
• Express the generating series of PT invariants as a ratio of modular forms and Jacobi forms.
• Apply the theory of Jacobi forms to derive the elliptic transformation law for the generating series.
• Use degeneration arguments and Gromov–Witten theory to compute invariants in abelian threefolds and K3×E.
• Leverage the GW/PT correspondence and known formulas for universal Jacobi forms to derive explicit expressions.

Experimental results

Research questions

• RQ1 How do derived auto-equivalences affect curve counting invariants on Calabi–Yau threefolds?
• RQ2 Does the generating series of Pandharipande–Thomas invariants for an elliptic Calabi–Yau threefold exhibit modular properties as predicted by Huang, Katz, and Klemm?
• RQ3 Is the generating series of PT invariants for K3×E with a primitive class on K3 a quasi-Jacobi form of weight -10?
• RQ4 Can wall-crossing and derived categories be used to compute PT invariants in terms of Gromov–Witten invariants?
• RQ5 Do the results support the Igusa cusp form conjecture?

Key findings

• The generating series of PT invariants for a reduced class H in the base of an elliptic Calabi–Yau threefold satisfies the elliptic transformation law of Jacobi forms of index h−1, where h is the arithmetic genus of H.
• The series Z_H(q,t) = PT_H(q,t)/PT_0(q,t) is a quasi-Jacobi form of weight -10 for K3×E with a primitive class on K3, providing strong evidence for the Igusa cusp form conjecture.
• For K3×E with a primitive class H_1 of genus 1, the generating series is PT_H_1(q,t) = 24 · Δ(t)⁻¹ · ℘(q,t), where Δ(t) is the modular discriminant and ℘(q,t) is a Jacobi form.
• For abelian threefolds A×E with a class H of genus 2, the generating series is PT^A×E_2(q,t) = φ_{-2,1}(q,t), a standard Jacobi form of weight -2 and index 1.
• For genus 3 classes on A×E, assuming the GW/PT correspondence, the series is PT^A×E_3(q,t) = 12·℘(q,t)·φ_{-2,1}(q,t)² − ϑ_{D4}(t)·φ_{-2,1}(q,t)², where ϑ_{D4}(t) is a theta function.
• The Euler characteristic version of the generating series on A×E for genus 3 is not modular, indicating that modularity does not extend to the naive Euler characteristic.