Remarks on Berry connection in QFT, anomalies, and applications

Abstract

Berry connection has been recently generalized to higher-dimensional QFT, where it can be thought of as a topological term in the effective action for background couplings. Via the inflow, this term corresponds to the boundary anomaly in the space of couplings, another notion recently introduced in the literature. In this note we address the question of whether the old-fashioned Berry connection (for time-dependent couplings) still makes sense in a QFT on \Sigma^{(d)}× \mathbb{R}Σ(d)×ℝ, where \Sigma^{(d)}Σ(d) is a dd-dimensional compact space and \mathbb{R}ℝ is time. Compactness of \Sigma^{(d)}Σ(d) relieves us of the IR divergences, so we only have to address the UV issues. We describe a number of cases when the Berry connection is well defined (which includes the tt^*tt* equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean \Sigma^{(d)} × \mathbb{R}_{≥ 0}Σ(d)×ℝ≥0. We then work out the examples of a free 3D Dirac fermion and a 3D \mathcal{N}=2𝒩=2 chiral multiplet. Finally, we consider 3D theories on \mathbb{T}^2× \mathbb{R}𝕋2×ℝ, where the space \mathbb{T}^2𝕋2 is a two-torus, and apply our machinery to clarify some aspects of the relation between 3D SUSY vacua and elliptic cohomology. We also comment on the generalization to higher genus.

A Note on Real Line Bundles with Connection and Real Smooth Deligne Cohomology

“…In the future we hope that this will help to provide a classification of Real vector bundles together with a Real connection on manifolds with an involution, similar to the classification of bundles with a connection on smooth or complex manifolds via Deligne cohomology as in [13]. An important example of a Real connection is provided by the Berry connection which may be viewed as a link between quantum mechanics and topology as formulated in [3] and [23] (see also [6]). The Grassmann-Berry connection on the Bloch bundle has also been studied in [10,Section II D].…”