Topological Correlators and Surface Defects from Equivariant Cohomology

Abstract

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.

arXiv comment40 pages, 3 figures; v2: minor revision, published version

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• More on holographic correlators: Twisted and dimensionally reduced structures
• One-Dimensional Sectors From the Squashed Three-Sphere
• The topological line of ABJ(M) theory
• Twisted holography of defect fusions
• Comments on chiral algebras and $\Omega$-deformations
• Constant maps in equivariant topological strings and geometric modeling of fluxes
• Intersecting Surface defects and 3d Superconformal indices
• Algebras, traces, and boundary correlators in $\mathcal{N}=4$ SYM
• ${\text{AdS}}_4$ Holography and the Hilbert Scheme
• Schur correlation functions from q-deformed Yang-Mills
• Generating functions for Higgs/Coulomb branch operators from 1d-3d cohomological equivalence