A topological quantum field theory for $\mathrm{Spin}(7)$-instantons

We construct a topological quantum field theory based on the moduli space of $\mathrm{Spin}(7)$-instantons on 8-dimensional manifolds. Using the Mathai-Quillen formalism, we derive the action of the theory in purely geometric terms, which coincides with prior results in the literature. We then reformulate the theory within the AKSZ formalism, obtaining a Batalin-Vilkovisky action that, after gauge fixing, matches our Mathai-Quillen construction while making the BRST symmetry explicit and providing a natural framework for classical observables. We also show that the Batalin-Vilkovisky action can be elegantly recast as a Chern-Simons type theory.