  It is customary to couple a quantum system to external classical fields. One
application is to couple the global symmetries of the system (including the
Poincar\'{e} symmetry) to background gauge fields (and a metric for the
Poincar\'{e} symmetry). Failure of gauge invariance of the partition function
under gauge transformations of these fields reflects 't Hooft anomalies. It is
also common to view the ordinary (scalar) coupling constants as background
fields, i.e. to study the theory when they are spacetime dependent. We will
show that the notion of 't Hooft anomalies can be extended naturally to include
these scalar background fields. Just as ordinary 't Hooft anomalies allow us to
deduce dynamical consequences about the phases of the theory and its defects,
the same is true for these generalized 't Hooft anomalies. Specifically, since
the coupling constants vary, we can learn that certain phase transitions must
be present. We will demonstrate these anomalies and their applications in
simple pedagogical examples in one dimension (quantum mechanics) and in some
two, three, and four-dimensional quantum field theories. An anomaly is an
example of an invertible field theory, which can be described as an object in
(generalized) differential cohomology. We give an introduction to this
perspective. Also, we use Quillen's superconnections to derive the anomaly for
a free spinor field with variable mass. In a companion paper we will study
four-dimensional gauge theories showing how our view unifies and extends many
recently obtained results.
