  We find a one-dimensional protected subsector of $\mathcal{N}=4$ matter
theories on a general class of three-dimensional manifolds. By means of
equivariant localization, we identify a dual quantum mechanics computing BPS
correlators of the original model in three dimensions. Specifically, applying
the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that
this localizes on a one-dimensional action with support on the fixed-point
submanifold of suitable isometries. We first show that our approach reproduces
previous results obtained on $S^3$. Then, we apply it to the novel case of $S^2
\times S^1$ and show that the theory localizes on two noninteracting quantum
mechanics with disjoint support. We prove that the BPS operators of such models
are naturally associated with a noncommutative star product, while their
correlation functions are essentially topological. Finally, we couple the
three-dimensional theory to general $\mathcal{N}=(2,2)$ surface defects and
extend the localization computation to capture the full partition function and
BPS correlators of the mixed-dimensional system.
