Gauge Theories Labelled by Three-Manifolds

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional ${mathcal{N} = 2}$ gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that ${S^{3}_{b}}$ partition functions of two mirror 3d ${mathcal{N} = 2}$ gauge theories are equal. Three-dimensional ${mathcal{N} = 2}$ field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional ${mathcal{N} = 2}$ SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.