Combinatorial 2d higher topological quantum field theory from a local cyclic \(A_\infty \) algebra

We construct combinatorial analogs of 2d higher topological quantum field theories. We consider triangulations as vertices of a certain CW complex \(\Xi \). In the “flip theory,” cells of \(\Xi _\textrm{flip}\) correspond to polygonal decompositions obtained by erasing the edges in a triangulation. These theories assign to a cobordism \(\Sigma \) a cochain Z on \(\Xi _\textrm{flip}\) constructed as a contraction of structure tensors of a cyclic \(A_\infty \) algebra V assigned to polygons. The cyclic \(A_\infty \) equations imply the closedness equation \((\delta +Q)Z=0\). In this context, we define combinatorial BV operators and give examples with coefficients in \(\mathbb {Z}_2\). In the “secondary polytope theory,” \(\Xi _\textrm{sp}\) is the secondary polytope (due to Gelfand–Kapranov–Zelevinsky) and the cyclic \(A_\infty \) algebra has to be replaced by an appropriate refinement that we call an \(\widehat{A}_\infty \) algebra. We conjecture the existence of a good Pachner CW complex \(\Xi \) for any cobordism, whose local combinatorics is described by secondary polytopes and the homotopy type is that of Zwiebach’s moduli space of complex structures. Depending on this conjecture, one has an “ideal model” of combinatorial 2d HTQFT determined by a local \(\widehat{A}_\infty \) algebra.