Topological Correlators and Surface Defects from Equivariant Cohomology

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.