Quantum Finite Element (QFE) seeks to generalize lattice field theory to any smooth Euclidean Riemann manifold by adapting the Finite Elements Method (FEM) in the classical limit and the geometrical characterization of Regge Calculus, supplemented by counter terms required to cancel the local scheme due to ultraviolet divergences at simplicial cut-off. High precision numerical test are presented for the 2D phi 4th theory at the Wilson-Fisher fixed point compared with the exact solution of the Ising CFT on the Riemann two sphere (S2). Progress toward further tests of 3D phi 4th theory on S3 and on the R x S2 boundary of AdS is discussed as well as the challenges required for including Fermionic and non-Abelian gauge fields.

Theoretical Developments