The Fukaya category of X physically arises by topologically twisting an N=2 SCFT with target X and considering a category whose objects are the D branes of the theory (or so I am told). Unlike the B-twisted theory (which gives rise to DbCoh), the A-twisted theory (the Fukaya category) is usually not local in the target in a manifestly useable way. However, a conjecture of Kontsevich (proven in large generality by Ganatra-Pardon-Shende) identifies a class of targets X for which a version of the Fukaya category does satisfy a useful local-to-global principle: The category can be computed as global sections of a cosheaf on a skeleton of X. In this talk I'll propose that there is a universal stack of local Liouville structures giving rise to such a (co)sheaf on any skeleton, and compute it in the real-2-dimensional case. The resulting stack is a stack of broken paracycles, and one can completely classify its sheaves as precisely semicyclic objects; moreover, the symplectic invariance of the wrapped category recovers the 2-Segal condition of Dyckerhoff-Kapranov.

Another outcome is that one can model the K-theory of categories through the Floer theory of Lagrangian cobordisms---for example, if one knew everything about Lagrangian cobordisms in Euclidean space, one could compute the K-groups of the integers. (But it's not clear that Lagrangian cobordisms and their Floer theory is any easier than K-theory!)