  When a quantum field theory is trivially gapped, its infrared fixed point is
an invertible field theory. The partition function of the invertible field
theory records the response to various background fields in the long-distance
limit. The set of background fields can include spacetime-dependent coupling
constants, in which case we call the corresponding invertible theory a
parameterized invertible field theory. We study such parameterized invertible
field theories arising from free Dirac fermions with spacetime-dependent mass
parameters using the Atiyah-Patodi-Singer index theorem for superconnections.
In particular, the response to an infinitesimal modulation of the mass is
encoded into a higher analog of the Berry curvature, for which we provide a
general formula. When the Berry curvature vanishes, the invertible theory can
still be nontrivial if there is a remaining torsional Berry phase, for which we
list some computable examples.
