Aspects of Berry phase in QFT

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 $ \mathcal{N}=\left(2,2\right) $ and 4d $ \mathcal{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.