Surgery and Matter for 3d Theories

Abstract: We try to give a geometric construction for 3d $\mathcal{N}=2$ gauge theories using three-manifolds and Dehn surgeries. We follow the story that wrapping M5-branes on plumbing three-manifolds leads to 3d theories with mixed Chern-Simons levels. This construction can be decorated by adding non-compact Lagrangian submanifolds in the cotangent bundles of three-manifolds. The M5-branes wrapping on these submanifolds through M-theory/IIB string duality lead to flavor D5-branes that engineer chiral multiplets. In this note, we only consider unknotted matter circles, which are intersections between these non-compact M5-branes and plumbing manifolds. Then various dualities of 3d theories can be interpreted as Kirby moves and equivalent surgeries. We also find the dictionary between geometric structures of three-manifolds and physical aspects of gauge theories.