$\mathcal{F}$-harmonic 3-forms and the Type IIA flow on 6-dimensional symplectic Lie algebras

TL;DR

The paper addresses the problem of characterizing $\mathcal{F}$-harmonic 3-forms and the long-time behavior of the Type IIA flow on 6-dimensional symplectic manifolds by restricting to left-invariant data on symplectic Lie groups. It derives explicit algebraic and ODE reductions using Lie algebra and symplectic cohomology frameworks, enabling concrete analysis via finite-dimensional systems. Through two explicit examples (nilpotent and solvable), it computes cohomology bases, identifies $\mathcal{F}$-harmonic loci, and analyzes the flow dynamics: the nilpotent case exhibits behavior connected to Lagrangian fibrations and harmonic structures, while the solvable case generically develops finite-time singularities with harmonic limits after rescaling. Overall, the work demonstrates how the Type IIA flow encodes geometric structures in non-Kähler settings and highlights distinct dynamical regimes across Lie algebra types.

Abstract

In this paper, we investigate the problem of $\mathcal{F}$-harmonic forms and the long-time behavior of the Type IIA flow on certain nilpotent and solvable symplectic Lie algebras (and corresponding nilmanifolds and solvmanifolds). In particular, we demonstrate that the Type IIA flow in these cases are very useful to detect desired geometric structures including Lagrangian torus fibrations and harmonic almost complex structures.

Theorems & Definitions (18)

• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Remark 3.5
• Proposition 4.1
• Proposition 4.2
• Definition 4.3
• Remark 4.4