Conservation of energy in lattice field theories

A basis for numerical analysis is discretization, that is, approximation of continuum objects by finite ones. Discretizations exhibiting exact (not just approximate) conservation laws have proved to be most successful for computational purposes.
Usually conservation laws are obtained from symmetries using the Noether theorem. In particular, conservation of energy is obtained from translational symmetry, which is necessarily broken during discretization. There was a folklore belief that no conserved discrete energy-momentum tensor exists; e.g., in 2016, D. Chelkak, A. Glazman, and S. Smirnov introduced a “halfway” conserved tensor.
But we construct an exactly conserved discrete energy-momentum tensor, approximating the continuum one at least for free fields. The construction has a topological nature and involves a certain 'projection' of a cochain cross-product. The topological meaning of the projection is not clear so far.
Most part of the talk is elementary. No knowledge of physics is required.