  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
$\Sigma^{(d)}\times \mathbb{R}$, where $\Sigma^{(d)}$ is a $d$-dimensional
compact space and $\mathbb{R}$ is time. Compactness of $\Sigma^{(d)}$ 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
$tt^*$ equations), and when it is not. We also mention a relation to the
boundary anomalies and boundary states on the Euclidean $\Sigma^{(d)} \times
\mathbb{R}_{\geq 0}$. We then work out the examples of a free 3D Dirac fermion
and a 3D $\mathcal{N}=2$ chiral multiplet. Finally, we consider 3D theories on
$\mathbb{T}^2\times \mathbb{R}$, where the space $\mathbb{T}^2$ 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.
