Bialgebras, and Lie monoid actions in Morse and Floer theory, I

TL;DR

This work develops a forest-based analytic framework to study Lie monoid actions on Morse and Floer theories by introducing moduli spaces of biforests, namely the forest biassociahedra $K^{\underline{k}}_{\underline{l}}$ and forest bimultiplihedra $J^{\underline{k}}_{\underline{l}}$. It defines an $f$-bialgebra structure on Morse and Floer data, organized by multi-indices that index ascending (product) and descending (coproduct) forests, and uses fibered and intersection constructions to derive coherent, partially compactified moduli spaces whose boundary behavior yields algebraic relations and morphisms. The paper situates these ideas within the broader context of $A_\infty$-bialgebras, Fukaya categories, and Wehrheim–Woodward-type field theories, and outlines conjectures linking the Morse and Floer chain-level structures to group actions and functors between (decorated) categories. By providing explicit geometric realizations and a robust combinatorial backbone, the work offers a path toward a higher-categorical treatment of symmetry actions in Morse/Floer theory and their manifestations in wrapped and cornered Floer theories.

Abstract

We introduce a new family of oriented manifolds with boundaries called the forest biassociahedra and forest bimultiplihedra, generalizing the standard biassociahedra. They are defined as moduli spaces of ascending-descending biforests and are expected to act as parameter spaces for operations defined on Morse and Floer chains in the context of compact Lie group actions. We study the structure of their boundary, and derive some algebraic notions of ``$f$-bialgebras'', as well as related notions of bimodules, morphisms and categories. This allows us to state some conjectures describing compact Lie group actions on Morse and Floer chains, and on Fukaya categories.

Figures (4)

• Figure 1: The moduli space $K^2_2$. It consists in two intervals, parametrized by the height between vertices. These two intervals overlap where these heights are zero.
• Figure 2: The moduli space $K^3_2$. Each of the six chambers is parametrized by two heights. When a height goes to zero, one crosses a wall (interior dashed lines) and moves to an adjascent chamber. When a height goes to infinity, the graph breaks: one reaches an exterior boundary of the hexagon.
• Figure 3: Graphs as intersections of trees. Observe that the green graph corresponds to the middle left chamber of $K^3_2$.
• Figure 4: A fibered product boundary face of $K^4_3$.

Theorems & Definitions (8)

• Conjecture 1: Lie monoid actions in Morse theory
• Remark 1.1
• Conjecture 2: Lie monoid actions in Floer theory
• Remark 1.2
• Remark 1.3
• Conjecture 3: Hamiltonian actions in Floer theory
• Remark 1.4
• Definition 2.1