Factorization homology and HKR Theorems for $E_n$ algebras

I will discuss work in progress with John Francis and
David Gepner that applies techniques from factorization homology
to prove analogs of the Hochschild-Kostant-Rosenberg theorem.
More explicitly, for a commutative algebra in characteristic
zero, we will express its $E_n$ Hochschild (co)homology in terms of
its commutative (co)tangent complex. Similar results are due to
Calaque-Willwacher and Toën, but our approach avoids formality.
I hope to emphasize connections with topological field theory.