Gauge theories labelled by three-manifolds
This paper describes a conjectural relation between three dimensional manifolds with a certain type of boundary data and three dimensional field theories. To a surface, one may associate the moduli of flat \(\text{SL}_2\mathbb{C}\) connections modulo gauge transformation. This space has a natural symplectic structure. The boundary data associated to a \(3\)-manifold is exactly a polarization of the moduli space of the boundary. This may be understood via an ideal triangulation of the \(3\)-manifold with boundary coming from unpaired faces of the tetrahedra, and from the links of the ideal tetrahedra. The theory is then specified by describing the edge parameters, and associated polarizations for one tetrahedron together with the gluing laws. It is not surprising that \(\text{SL}_2\mathbb{Z}\) acts on the space of polarizations. An analogous action on the space of Lagrangian densities leads to the set of field theories associated to \(3\)-manifolds. For example the \(T\) matrix replaces a Lagrangian \(\mathcal{L}\) with \(\mathcal{L}+\frac{1}{4\pi}A\wedge dA\). A specific field theory is associated to a single tetrahedron, and then a gluing procedure is given to describe new theories by gluing existing theories.
scientific article; zbMATH DE number 6249788
scientific article