On the decategorification of some higher actions in Heegaard Floer homology

TL;DR

This paper decategorifies higher actions in bordered Heegaard Floer homology, linking them to actions on exterior powers of surface homology and providing an open-closed TQFT interpretation with a gluing formula.

Contribution

It introduces a decategorification of higher actions in bordered Heegaard Floer strands algebras and connects them to surface homology actions within an open-closed TQFT framework.

Findings

Decategorification aligns with actions on exterior powers of surface homology.

Provides a new interpretation of these actions in open-closed TQFT.

Establishes a gluing formula for the decategorified actions.

Abstract

We decategorify the higher actions on bordered Heegaard Floer strands algebras from recent work of Rouquier and the author and identify the decategorifications with certain actions on exterior powers of homology groups of surfaces. We also suggest an interpretation for these actions in the language of open-closed TQFT, and we prove a corresponding gluing formula.