  When continuous parameters in a QFT are varied adiabatically, quantum states
typically undergo mixing---a phenomenon characterized by the Berry phase. We
initiate a systematic analysis of the Berry phase in QFT using standard quantum
mechanics methods. We show that a non-trivial Berry phase appears in many
familiar QFTs. We study a variety of examples including free electromagnetism
with a theta angle, and certain supersymmetric QFTs in two and four spacetime
dimensions. We also argue that a large class of QFTs with rich Berry properties
is provided by CFTs with non-trivial conformal manifolds. Using the
operator-state correspondence we demonstrate in this case that the Berry
connection is equivalent to the connection on the conformal manifold derived
previously in conformal perturbation theory. In the special case of chiral
primary states in 2d N=(2,2) and 4d N=2 SCFTs the Berry phase is governed by
the tt* equations. We present a technically useful rederivation of these
equations using quantum mechanics methods.
