A mathematical definition of 3d indices

          @misc{ pirsa_PIRSA:17020021,
            doi = {10.48660/17020021},
            url = {https://pirsa.org/17020021},
            author = {Dimofte, Tudor},
            keywords = {Mathematical physics},
            language = {en},
            title = {A mathematical definition of 3d indices},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {feb},
            note = {PIRSA:17020021 see, \url{https://pirsa.org}}
          }
          

Abstract

3d field theories with N=2 supersymmetry play a special role in the evolving web of connections between geometry and physics originating in the 6d (2,0) theory. Specifically, these 3d theories are associated to 3-manifolds M, and their vacuum structure captures the geometry of local systems on M. (Sometimes M arises as a cobordism between two surfaces C, C', in which case the 3d theories encode some functorial relation between the geometry of Hitchin systems on C and C'.) I would like to explain some of the mathematics of 3d N=2 theories. In particular, I would like to explain how Hilbert spaces in these theories arise as Dolbeault cohomology of certain moduli spaces of bundles. One application is a homological interpretation of the "pentagon relation" relating flips of triangulation on a surface.