Gauge theories can often be formulated in different but physically equivalent ways, a concept referred to as duality. Using a formalism based on graded geometry, we provide a unified treatment of all parent theories for different types of standard and exotic dualizations. Our approach is based on treating tensor fields as functions of a certain degree on graded supermanifolds equipped with a suitable number of odd coordinates. We present a universal two-parameter first order action for standard and exotic electric/magnetic dualizations and prove in full generality that it yields two dual second order theories with the desired field content and dynamics. Upon choice of parameters, the parent theory reproduces (i) the standard and exotic duals for p-forms and (ii) the standard and double duals for (p, 1) bipartite tensor fields, such as the linearized graviton and the Curtright field. Moreover, we discuss how deformations related to codimension-1 branes are included in the parent theory.

Motivated by the analogy between a weak field expansion of general relativity and Maxwell’s laws of electrodynamics, we explore physical consequences of a parity violating $$\theta $$ θ term in gravitoelectromagnetism. This is distinct from the common gravitational $$\theta $$ θ term formed as a square of the Riemann tensor. Instead it appears as a product of the gravitoelectric and gravitomagnetic fields in the Lagrangian, similar to the Maxwellian $$\theta $$ θ term. We show that this sector can arise from a quadratic torsion term in nonlinear gravity. In analogy to the physics of topological insulators, the torsion-induced $$\theta $$ θ parameter can lead to excess mass density at the interface of regions where $$\theta $$ θ varies and consequently it generates a correction to Newton’s law of gravity. We discuss also an analogue of the Witten effect for gravitational dyons.

We perform an in‐depth analysis of the transformation rules under duality for couplings of theories containing multiple scalars, p‐form gauge fields, linearized gravitons or (p, 1) mixed symmetry tensors. Following a similar reasoning to the derivation of the Buscher rules for string background fields under T‐duality, we show that the couplings for all classes of aforementioned multi‐field theories transform according to one of two sets of duality rules. These sets comprise the ordinary Buscher rules and their higher counterpart; this is a generic feature of multi‐field theories in spacetime dimensions where the field strength and its dual are of the same degree. Our analysis takes into account topological theta terms and generalized B‐fields, whose behavior under duality is carefully tracked. For a 1‐form or a graviton in 4D, this reduces to the inversion of the complexified coupling or generalized metric under electric/magnetic duality. Moreover, we write down an action for linearized gravity in the presence of θ‐term from which we obtain previously suggested on‐shell duality and double duality relations. This also provides an explanation for the origin of theta in the gravitational duality relations as a specific additional sector of the linearized gravity action.

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Exotic duality suggests a link between gauge theories for differential p-forms and tensor fields of mixed symmetry [D − 2, p] in D spacetime dimensions. On the other hand, standard Hodge duality relates p-form to (D − p − 2)-form gauge potentials by exchanging their field equations and Bianchi identities. Following the methodology and the recent proposal of Henneaux et al. that the double dual of the free graviton is algebraically related to the original graviton and does not provide a new, independent description of the gravitational field, we examine the status of exotic duality for p-forms. We find that the exotic dual is algebraically related to the standard dual of a differential form and therefore they provide equivalent descriptions as free fields. Introducing sources then leads to currents being proportional. This relation is extended in a straightforward way for higher exotic duals of the mixed symmetry type [