Spin in $p$-adic AdS/CFT

We study the holographic dual of the simplest notion of spin in a $p$-adic field theory, namely Green's functions which involve non-trivial sign characters over the $p$-adic numbers. In order to recover these sign characters from bulk constructions, we find that we must introduce a non-dynamical $U(1)$ gauge field on the line graph of the Bruhat-Tits tree. Wilson lines of this gauge field on suitable paths yield the desired sign characters. We show explicitly how to start with complex scalars or fermions in the bulk, coupled to the $U(1)$ gauge field, and compute the holographic two-point functions of their dual operators on the boundary.