In this thesis, we relate geometric field theories with classical, homotopical invariants of algebraic objects. We begin by defining an abstract setting in which to model the local observables of field theories depending on a Riemannian structure. We then introduce a family of examples whose input is implicit in quantum mechanical systems. Following standard higher categorical procedures, we produce an extension of these local constructions to general Riemannian manifolds.
Using abstract homotopy theory, factorization algebras, and factorization homology, we relate the observables of a field theory on a circle (of fixed size) to Hochschild homology with coefficients in a module. The action maps of this module depend explicitly on the geometry on which the field theory habitates. We end with a general discussion of how these techniques may be used in more general contexts.