Tensor Networks as Geometry

The multiscale entanglement renormalization ansatz (MERA) is a tensor network that can efficiently approximate ground states of critical spin chains --that is, lattice versions of 1+1 CFTs. Its network structure extends in an additional dimension corresponding to renormalization group scale. Accordingly, MERA has has been proposed to be a discrete realization of the AdS/CFT correspondence. While a first proposal speculated that MERA = discrete hyperbolic plane (time slice of AdS3), a second proposal conjectured that MERA = discrete 1+1 de Sitter. In this talk I will attach a geometry to MERA from the perspective of a CFT path integral. Surprisingly, the corresponding metric does not have euclidean nor lorentzian signature, but is instead degenerate. I will also describe how MERA can be modified to represent either the hyperbolic plane or 1+1 de Sitter.