I will discuss local operators in 3d N=2 theories with a holomorphic-topological twist, and compatible boundary conditions. In the bulk, local operators form a shifted-Poisson vertex algebra (in cohomology); while on a boundary condition one finds vertex-algebra modules for the bulk algebra. Being more careful, one would expect to find $L_\infty/A_\infty$ analogues of vertex algebras in the bulk/boundary. The bulk and boundary algebras “categorify” 3d N=2 indices and half-indices (respectively), both widely used in physics. They are also of great relevance to the 3d-3d correspondence. For a wide class of 3d N=2 Lagrangian gauge theories and boundary conditions, I will explain how to identify bulk and boundary algebras algorithmically, and give examples of how they match across dualities.

(Based on work with K. Costello and D. Gaiotto.)