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Tensor network description of 3D Quantum Gravity and Diffeomorphism Symmetry

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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.