Topological deformations of local gauge theories: a presymplectic BV-AKSZ approach
The same local degrees of freedom can be described by systems with different gauge structure and field content. Standard examples are the duality between the scalar field and the 2-form gauge field in four dimensions, and unimodular gravity, where the cosmological constant becomes a global degree of freedom. In a precise sense, the spaces of gauge-inequivalent formal solutions of such systems differ by a finite-dimensional sector. Moreover, because their gauge sectors are different, their consistent interactions and their sensitivity to the global structure of spacetime may differ substantially. We show that such systems are naturally related by what we call topological deformations: gauging a symmetry of generic degree couples the system to a BF-type sector without introducing new local degrees of freedom. The construction is carried out systematically within the presymplectic BV-AKSZ approach, where it amounts to a modification of the underlying presymplectic gauge PDE. Gauging degree-0 symmetries identifies solutions related by global symmetries, which reproduces the scalar--2-form and principal chiral model--Freedman--Townsend dualities. Gauging degree-1 symmetries, i.e. deformations, gives rise to new global degrees of freedom, such as the cosmological constant in unimodular gravity.