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Grassmann tensor network renormalization and fermionic topological quantum field theory: a new route towards quantum gravity

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Recently, the development of tensor network renormalization approach has provided us a powerful tool to construct new classes of topological quantum field theories(TQFTs) in discrete space-time. For example, the Turaev Viro’s states sum constructions are fixed point tensor networks representing a special class of 2+1D TQFTs. Interestingly, the Grassmann variable generalization of tensor network renormalization approach leads to new classes of TQFTs for interacting fermion systems, namely, the fermionic TQFTs. In this talk, I will start with a fermionic topological nonlinear sigma model and discuss its corresponding new mathematics - group super-cohomology theory. Then I will explain the fermionic generalization of Dijkgraaf -Witten gauge theory by using group supercohomology theory. Finally I will show examples beyond fermionic Dijkgraaf -Witten gauge theory and discuss possible new routes towards quantum gravity.