This paper builds a clean finite-dimensional model for the global “zero-mode” part of a class of gauge theories and shows that the resulting partition function does not depend on a natural choice that one must make to compute it. Concretely, the author constructs a Batalin–Vilkovisky (BV) structure on the space of zero modes of a quadratic, elliptic higher abelian gauge theory on a closed oriented Riemannian manifold, and proves that the associated finite-dimensional BV integral is independent of the choice of maximal admissible Lagrangian under mild hypotheses.

To describe fields globally, the paper uses Cheeger–Simons differential cohomology. This language keeps track not only of the usual differential-form fields seen in perturbation theory, but also of integral and torsion pieces that encode topological data. The BV complex that organizes fields and their symmetries is built from a mapping cone construction induced by the equations of motion. The hypercohomology of that complex is the space of zero modes. The author packages the resulting algebraic and geometric data into what he calls an “abelian BV group”: a graded abelian group with a shifted symplectic pairing, compatible inner products, and a quadratic weight.

Given such an abelian BV group one must pick a Lagrangian subgroup to perform a finite-dimensional BV integral. The paper gives a precise definition of admissible and maximal admissible Lagrangians and then defines a zero-mode partition function as a ratio of finite-dimensional integrals over even and odd parts. The main theorem says that when the abelian BV group satisfies extra regularity conditions (called “nice”: a reduced unimodular lattice condition and positivity of the induced quadratic form) this zero-mode partition function does not depend on which maximal admissible Lagrangian was chosen.

The result yields an explicit factorization of the zero-mode contribution into three pieces: a graded volume of the connected zero modes, a finite torsion factor coming from discrete topology, and a theta-like sum over degree-zero topological sectors. This finite-dimensional contribution is the global part of the full field-theory partition function and is meant to be taken separately from the familiar analytic determinants and analytic torsion that come from nonzero modes. The paper works out the construction and formulas in examples including p-form Maxwell theory, abelian Chern–Simons theory, and Maxwell–Chern–Simons theory. For p-form Maxwell theory the author notes the formula is similar in form to an earlier result by Moore and Saxena, but differs in the torsion factor.