Currently this entry consists mainly of notes taken live in a talk by John Francis at ESI Program on K-Theory and Quantum Fields (2012), without as yet, any double-checking or polishing. So handle with care for the moment.
symmetric monoidal (∞,1)-category of spectra
Factorization homology is a notion of homology theory for framed -dimensional manifolds with coefficients in En-algebras, due to (Francis). It is similar in spirit to factorization algebras, blob homology and topological chiral homology. In fact the definition of factorization homology turns out to be equivalent to that of topological chiral topology (Francis b).
Write for the category of manifolds with embeddings as morphisms. This is naturally a topological category, hence regard it as an (infinity,1)-category. Regard it furthermore as a symmetric monoidal (∞,1)-category with tensor product given by disjoint union.
Restriction along this inclusion gives an (∞,1)-functor
This turns out to be an equivalence of (∞,1)-categories. The inverse is defined to be factorization homology
This inverse sends an -disk algebra
to the functor that sends a manifold to the
This is equivalent to topological chiral homology, to be thought of as a topological version of chiral algebras. A version with values in homotopy types instead of chain complexes was given by Salvatore and Graeme Segal.
From a functor we get an extended TQFT with values in -linear -categories
From a -algebra we obtain the corresponding delooping which is a -linear (infinity,n)-category that is a fully dualizable object. The cobordism hypothesis identifies this with cobordism representations, and the claim is that this identification is compatible factorization homology.
A -algebra in is equivalently a differential graded algebra.
The value of the corresponding on the circle is the Hochschild homology of
Given a topological space we get a -algebra
Where is the n-fold loop space of .
Theorem (Salvatore and Lurie)
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
The definition appears in section 3 of
A detailed account is in
Application to higher Hochschild cohomology is discussed in