  We study the role of "$\theta$ terms" in the action for three-dimensional
$U(1)$ symmetric tensor gauge theories, describing quantum phases of matter
hosting gapless higher-spin gauge modes and gapped subdimensional particle
excitations, such as fractons. In Maxwell theory, the $\theta$ term is a total
derivative which has no effect on the gapless photon, but has two important,
closely related consequences: attaching electric charge to magnetic monopoles
(the Witten effect) and leading to a Chern-Simons theory on the boundary. We
will find that a similar story holds in the higher-spin $U(1)$ gauge theories.
These theories admit generalized $\theta$ terms which have no effect on the
gapless gauge mode, but which bind together electric and magnetic charges (both
of which are generally subdimensional) in specific combinations, in a
higher-spin manifestation of the Witten effect. We derive the corresponding
Witten quantization condition. We find that, as in Maxwell theory, imposing
time-reversal invariance restricts $\theta$ to certain discrete values. We also
find that these new $\theta$ terms imply a non-trivial boundary structure. The
boundaries host fracton excitations coupled to a tensor $U(1)$ gauge field with
a Chern-Simons-like action, in both chiral and non-chiral varieties. These
boundary theories open a door to the study of $U(1)$ fracton phases described
by tensor Chern-Simons theories, not only on boundaries of three-dimensional
systems, but also in strictly two spatial dimensions. We explicitly work
through three examples of bulk and boundary theories, the principles of which
can be readily extended to arbitrary higher-spin theories.
