Line Operators in 3d Holomorphic QFT: Meromorphic Tensor Categories and dg-Shifted Yangians

We study line operators and their OPE’s in perturbative 3d holomorphic-topological QFT’s, including holomorphic-topological twists (quarter-BPS sectors) of 3d $N=2$ theories. In particular, we develop the representation theory of the category $C$ of perturbative line operators and its chiral tensor product, by generalizing techniques introduced by Costello and collaborators. We argue that lines are equivalent to modules for an $A_{\mathrm{\infty }}$ algebra $A^{!}$ that’s Koszul-dual to bulk local operators. We further establish a non-renormalization theorem for the OPE’s of lines in a large class of theories (dubbed quasi-linear), allowing an exact resummation of quantum corrections. Based on physics arguments, we propose axioms for the full algebraic structure on $A^{!}$, calling it a “dg-shifted Yangian,” which controls the OPE of lines. A key part of the structure is a Maurer-Cartan element $r(z)\in A^{!}\otimes A^{!}(\text{\hspace{0.17em}⁣}(z^{-1})\text{\hspace{0.17em}⁣})$ that satisfies an $A_{\mathrm{\infty }}$ generalization of the Yang-Baxter equation. As examples, we consider 3d $N=2$ gauge theories with arbitrary Chern-Simons levels, linear matter, and superpotential, and explicitly compute 1) perturbative bulk local operators (as $A_{\mathrm{\infty }}$-chiral algebras); and 2) the Koszul-duals $A^{!}$ (proving they are dg-shifted Yangians).