Are all identically conserved geometric tensors metric variations of actions? A status report

Abstract

Noether's theorem, that local gauge variations of gauge invariant actions are identically conserved (more tautologically, that gauge variations of gauge invariants vanish) was established a century ago. Its converse, in the geometric context: are all identically conserved local symmetric tensors variations of some coordinate invariant action? remains unsolved to this day. We survey its present state and discuss some of our concrete attempts at a solution, including a significant improvement. For notational simplicity, details are primarily given in D=2, but we discuss generic D as well.