Tensor network description of 3D Quantum Gravity and Diffeomorphism Symmetry

          @misc{ pirsa_PIRSA:20110036,
            doi = {10.48660/20110036},
            url = {https://pirsa.org/20110036},
            author = {Dittrich, Bianca},
            keywords = {Quantum Fields and Strings},
            language = {en},
            title = {Tensor network description  of 3D Quantum Gravity and Diffeomorphism Symmetry},
            publisher = {Perimeter Institute},
            year = {2020},
            month = {nov},
            note = {PIRSA:20110036 see, \url{https://pirsa.org}}
          }
          

Abstract

In contrast to the 4D case, there are well understood theories of quantum gravity for the 3D case. Indeed, 3D general relativity constitutes a topological field theory (of BF or equivalently Chern-Simons type) and can be quantized as such. The resulting quantum theory of gravity offers many interesting lessons for the 4D case. In this talk I will discuss the quantum theory which results from quantizing 3D gravity as a topological field theory. This will also allow a derivation of a holographic boundary theory, together with a geometric interpretation of the boundary observables. The resulting structures can be interpreted in terms of tensor networks, which provide states of the boundary theory. I will explain how a choice of network structure and bond dimensions constitutes a complete gauge fixing of the diffeomorphism symmetry in the gravitational bulk system. The theory provides a consistent set of rules for changing the gauge fixing and with it the tensor network structure. This provides an example of how diffeomorphism symmetry can be realized in a tensor network based framework. I will close with some remarks on the 4D case and the challenges we face there.