Intrinsic sign problems in topological quantum field theories

The sign problem is a widespread numerical hurdle preventing us from simulating the equilibrium behaviour of many interesting models, most notably the Hubbard model. Research aimed at solving the sign problem, via various clever manipulations, has been thriving for a long time with various recent exciting results. The complementary question, of whether some phases of matter forbid the existence of any sign-free microscopic model, has received attention only recently. In this talk, I'll review recent progress and discuss a novel, and quite general, criterion we obtained for when a topological quantum field theory (TQFT) has no sign-free local microscopic model. Most but not all TQFTs test positive under this criterion. Relations to sign problems in frustrated magnets will also be discussed.