nLab
factorization algebra

Contents

Idea

A factorization algebra is an algebra over an operad where the operad in question is like the little disk operad, but with each disk embedded into a given manifold X.

This way a factorization algebra is an assignment of a chain complex V D to each ball DX embedded in X, and for each collection of non-intersecting embedded balls D 1,,D nDX sitting inside a bigger embedded ball D in X a morphism

V D 1V D 2V D nV DV_{D_1} \otimes V_{D_2} \otimes \cdots \otimes V_{D_n} \to V_{D}

such that composition of such operations is suitably respected.

Definition

Prefactorization algebra

Definition

For X a topological space write Fact X be the colored operad in Set whose

This specifies composition uniquely.

Definition

For (C,) a symmetric monoidal abelian category let End(C) be its endomorphism operad. A prefactorization algebra on X with values in C is an algebra over an operad over Fact X in C, hence a morphism of operads

:Fact XEnd(C).\mathcal{F} : Fact_X \to End(C) \,.

These definitions appear (here).

Factorization algebras

Definition

For X a topological space and UX an open subset, a open cover {U iU} iI is called a factorizing cover if for every finite set of points {x 1,,x k}U there is a finite subset {U i j} jJI of pairwise disjoint open subsets such that each point is contained in their union.

Remark

Every Hausdorff space admits a factorizing cover.

Notation

For a factorizing cover {U iU} iI write PI for the set of finite subsets αI such that for j,jα we have U jU j=.

Given a prefactorization algebra and αPI write

(α):= jαF(U j)\mathcal{F}(\alpha) := \otimes_{j \in \alpha} F(U_j)

and for α 1,,α kPI write

(α 1,,α k)= (j 1,,j k)α 1××α k(U j 1U j k).\mathcal{F}(\alpha_1, \cdots, \alpha_k) = \bigotimes_{(j_1, \cdots, j_k) \in \alpha_1 \times \cdots \times \alpha_k} \mathcal{F}(U_{j_1} \cap \cdots \cap U_{j_k}) \,.

For each 1ik there is a canonical morphism

p i:(α 1,,α k)(α 1,,α i1,α i+1,,α k).p_i : \mathcal{F}(\alpha_1,\cdots, \alpha_k) \to \mathcal{F}(\alpha_1, \cdots, \alpha_{i-1}, \alpha_{i+1}, \cdots, \alpha_k) \,.
Definition

A prefactorization algebra :Fact XEnd(C) is called a factorization algebra if for every open subset UX and every factorizing cover {U iU} iI the sequence

α 1,α 2PI(α 1,α 2)p 1p 2 βPI(β)(U)0\bigoplus_{\alpha_1, \alpha_2 \in P I} \mathcal{F}(\alpha_1, \alpha_2) \stackrel{p_1 - p_2}{\to} \bigoplus_{\beta \in P I} \mathcal{F}(\beta) \to \mathcal{F}(U) \to 0

is an exact sequence.

These definitions appear here.

Homotopy factorization algebras

Let now (C,) specifically be a category of chain complexes.

Definition

A prefactorization algebra :Fact XEnd(X) is a homotopy factorization algebra if for all factorizing covers {U iUX} iI the canonical morpshim

k0 α 1,,α kPI(α 1,,α k)[k1](U)\bigoplus_{k \geq 0} \bigoplus_{\alpha_1, \cdots, \alpha_k \in P I} \mathcal{F}(\alpha_1, \cdots, \alpha_k)[k-1] \to \mathcal{F}(U)

is a quasi-isomorphism, where the differential on the left is defined by (…).

Remark

This is the analogue of a descent condition for simplicial presheaves.

These definitions appear here.

Factorization algebras have some similarity with

References

This may be regarded as a slight variation on the concept chiral algebra originally introduced by Beilinson and Drinfeld.

A definition appears in section 4.1 Topological Chiral Homology of

There it is demonstrated how factorization algebras can be used to construct extended FQFTs.

Concrete constructions of formal algebras for familiar quantum field theories are described in

This can also be found mentioned in the talk notes of the Northwestern TFT Conference 2009, see in particular

More is at

There seems to be a close relation between the description of quantum field theory by factorization algebras and the proposal presented in

The relation of locally constant factorization algebras to higher order Hochschild homology is in

  • Gregory Ginot, Thomas Tradler, Mahmoud Zeinalian, Derived higher Hochschild homology, topological chiral homology and factorization algebras, arxiv/1011.6483

An (infinity,1)-category theoretic treatment of higher factorization algebras is in