nLab
factorization algebra

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 $D\subset X$ embedded in $X$, and for each collection of non-intersecting embedded balls $D_1,\cdots ,D_n\subset D\subset X$ sitting inside a bigger embedded ball $D$ in $X$ a morphism

$$V_{D_1}\otimes V_{D_2}\otimes \cdots \otimes V_{D_n}\to V_D$$V_{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 ${\mathrm{Fact}}_X$ be the colored operad in Set whose

• objects are the connected open subsets of $X$;

• the hom-set ${\mathrm{Fact}}_X(\{U_i{\}}_i,V)$ is the singleton precisely if the $U_i$ are all in $V$ and are pairwise disjoint and is the empty set otherwise.

This specifies composition uniquely.

Definition

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

$$ℱ:{\mathrm{Fact}}_X\to \mathrm{End}(C).$$\mathcal{F} : Fact_X \to End(C) \,.

These definitions appear (here).

Factorization algebras

Definition

For $X$ a topological space and $U\subset X$ an open subset, a open cover $\{U_i\hookrightarrow U{\}}_{i\in I}$ is called a factorizing cover if for every finite set of points $\{x_1,\cdots ,x_k\}\subset U$ there is a finite subset $\{U_{i_j}{\}}_{j\in J\subset I}$ 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 $\{U_i\to U{\}}_{i\in I}$ write $PI$ for the set of finite subsets $\alpha \subset I$ such that for $j,j\prime \in \alpha$ we have $U_j\cap U_{j\prime }=\varnothing$.

Given a prefactorization algebra $ℱ$ and $\alpha \in PI$ write

$$ℱ(\alpha ):={\otimes }_{j\in \alpha }F(U_j)$$\mathcal{F}(\alpha) := \otimes_{j \in \alpha} F(U_j)

and for ${\alpha }_1,\cdots ,{\alpha }_k\in PI$ write

$$ℱ({\alpha }_1,\cdots ,{\alpha }_k)={⨂}_{(j_1,\cdots ,j_k)\in {\alpha }_1\times \cdots \times {\alpha }_k}ℱ(U_{j_1}\cap \cdots \cap U_{j_k}).$$\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 $1\le i\le k$ there is a canonical morphism

$$p_i:ℱ({\alpha }_1,\cdots ,{\alpha }_k)\to ℱ({\alpha }_1,\cdots ,{\alpha }_{i-1},{\alpha }_{i+1},\cdots ,{\alpha }_k).$$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 $ℱ:{\mathrm{Fact}}_X\to \mathrm{End}(C)$ is called a factorization algebra if for every open subset $U\subset X$ and every factorizing cover $\{U_i\to U{\}}_{i\in I}$ the sequence

$$\underset{{\alpha }_1,{\alpha }_2\in PI}{⨁}ℱ({\alpha }_1,{\alpha }_2)\stackrel{p_1-p_2}{\to }\underset{\beta \in PI}{⨁}ℱ(\beta )\to ℱ(U)\to 0$$\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 $(C,\otimes )$ specifically be a category of chain complexes.

Definition

A prefactorization algebra $ℱ:{\mathrm{Fact}}_X\to \mathrm{End}(X)$ is a homotopy factorization algebra if for all factorizing covers $\{U_i\to U\subset X{\}}_{i\in I}$ the canonical morpshim

$$\underset{k\ge 0}{⨁}\underset{{\alpha }_1,\cdots ,{\alpha }_k\in PI}{⨁}ℱ({\alpha }_1,\cdots ,{\alpha }_k)[k-1]\to ℱ(U)$$\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.

Examples

• beta-gamma system

Related concepts

Factorization algebras have some similarity with

• Hochschild homology;

• blob homology, factorization homology, topological chiral homology;

• local nets in AQFT.

• chiral algebra, vertex operator algebra

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
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

References

The notion of factorization algebra may be regarded as a slight variation on the concept chiral algebra originally introduced in

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras.

A definition formulated genuinely in Higher Algebra appears in section 4.1 Topological Chiral Homology of

• Jacob Lurie, On the Classification of Topological Field Theories

This discusses how locally constant factorization algebras obtained from En-algebras induce extended FQFTs.

A fairly comprehensive account of factorization algebras as a formalization of perturbative quantum field theory (see at factorization algebra of observables) is in

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory

• Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)

and the beginning of

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4 (arXiv:1111.4234)

Lecture notes include

• Kevin Costello (with Owen Gwilliam), Factorization algebras in perturbative quantum field theory in Strings, Field, Topology, Oberwolfach report No 28, 2009 (pdf)

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

  • notes by Christoph Wockel, Talk by Kevin Costello

  • notes by Evan Jenkins on the same talk: Factorization algebras in perturbative quantum gravity

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

• Stefan Hollands, arXiv:0802.2198

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

• Dennis Gaitsgory, John Francis Chiral Koszul duality (arXiv:1103.5803)