Combinatorial 2d topological conformal field theory from a local cyclic $A_\infty$ algebra

I will explain a construction of a combinatorial 2d TCFT, assigning partition functions to triangulated cobordisms (as chain maps between spaces of states), in such a way that a Pachner flip induces a $Q$-exact change. More generally, the partition function becomes a nonhomogeneous closed cochain on the “flip complex.” One has a combinatorial counterpart of the BV operator $G_{0,-}$ arising from evaluating the theory on a special 1-cycle on the flip complex of the cylinder. The local input for the model is a cyclic $A_\infty$ algebra, with the operation $m_3$ playing the role of the BRST-primitive $G$ of the stress-energy tensor $T=Q(G)$.
I will also describe a way to incorporate invariance-up-to-homotopy with respect to the second 2d Pachner move (stellar subdivision/aggregation). This version of the model is based on secondary polytopes of Gelfand-Kapranov-Zelevinsky and uses a certain enhancement (by extra homotopies) version of a cyclic $A_\infty$ algebra as input
The talk is based on a joint work with Andrey Losev and Justin Beck, https://arxiv.org/pdf/2402.04468.pdf.