  Dualities play a central role in both quantum field theories and condensed
matter systems. Recently, a web of dualities has been discovered in 2+1
dimensions. Here, we propose in particular a generalization of the Son's
fermion-fermion duality to 3+1 dimensions. We show that the action of charged
Dirac fermions coupled to an external electromagnetic field is dual to an
action of neutral fermions minimally coupled to an emergent vector gauge field.
This dual action contains also a further tensor (Kalb-Ramond) gauge field
coupled to the emergent and electromagnetic vector potentials. We firstly
demonstrate the duality in the massive case. We then show the duality in the
case of massless fermions starting from a lattice model and employing the
slave-rotor approach already used in the 2+1-dimensional duality [Burkov, Phys.
Rev. B 99, 035124 (2019)]. We finally apply this result to 3D Dirac semimetals
in the low-energy regime. Besides the implications in topological phases of
matter, our results shed light on the possible existence of a novel web of
dualities in 3+1-dimensional (non-supersymmetric) quantum field theories.
