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In this paper, we derive the explicit formula for higher cup products onhypercubic lattices, based on the recently developed geometrical interpretationin the simplicial case. We illustrate how this formalism can elucidate latticeconstructions on hypercubic lattices for various models and deriving them fromspacetime actions. In particular, we demonstrate explicitly that the (3+1)D SPT$S = \frac{1}{2} \int w_2^2 + w_1^4$ (where $w_1$ and $w_2$ are the first andsecond Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang modelconstructed on the cubic lattice by Burnell-Chen-Fidkowski-Vishwanath. Otherexamples include the double-semion model, and also the `fermionic' toric codein arbitrary dimensions on hypercubic lattices. In addition, we extend previousconstructions of exact boson-fermion dualities and the Gu-Wen GrassmannIntegral to arbitrary dimensions. Another result which may be of independentinterest is a derivation of a cochain-level action for the generalizeddouble-semion model, reproducing a recently derived action on the cohomologylevel.

Higher cup products on hypercubic lattices: application to lattice models of topological phases