ScalingStacks

We expect that our constructions (in particular Theorem 1.3.1) will be part of a $4$-dimensional TQFT.

This article is the first step towards a higher representation-theoretic reconstruction of Heegaard Floer theory, which would be a fulfilment of the program of Crane and Frenkel for ${\mathfrak{gl}}(1|1)$. This article focuses on dimensions $1$ and $2$, where homotopic phenomena can be avoided.

It is natural to ask if our constructions extend to dimension $0$. Work in progress of the first author and Reeshad Arian [ArMa] shows this is possible at the decategorified level.

We are pursuing two directions for the extension to dimension $3$.

In a work in preparation, the first author extends the tensor product functoriality to the $A_{\infty}$-setting. This is a key step to construct morphisms of $2$-representations from Heegaard Floer diagrams by cutting them into pieces.

Work in preparation of the second author [Rou4] provides a construction of invariants of links in $S^{3}$. Appropriate $t$-structures are used to handle the homotopic phenomena.

Note that Ellis, Pevtkova and Vértesi in [ElPeVe] construct homological invariants of tangles in the setting of bordered Heegaard Floer theory in which ${\mathfrak{gl}}(1|1)$-categorifications appear. Note also that [DouLiMa] considered $3$-manifolds with codimension-$2$ corners in the setting of [DouMa].

We expect that our algebraic constructions will provide a blueprint for the construction of higher categorical structures and $3$ and $4$-dimensional invariants associated to (ordinary) simple Lie algebras.