# nLab factorization algebra

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Quantum field theory

functorial quantum field theory

# 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 $D \subset X$ 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

$V_{D_1} \otimes V_{D_2} \otimes \cdots \otimes V_{D_n} \to V_{D}$

such that composition of such operations is suitably respected.

In Euclidean (Wick rotated) field theory factorization algebras serve to axiomatize the operator product expansion.

## Definition

### Prefactorization algebra

###### Definition

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

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

• the hom-set $Fact_X(\{U_i\}_i, V)$ 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 $End(C)$ 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

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

These definitions appear in “Factorization algebras in quantum field theory” by Costello and Gwilliam.

### 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 $P I$ for the set of finite subsets $\alpha \subset I$ such that for $j,j' \in \alpha$ we have $U_j \cap U_{j'} = \emptyset$.

Given a prefactorization algebra $\mathcal{F}$ and $\alpha \in P I$ write

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

and for $\alpha_1, \cdots, \alpha_k \in P I$ write

$\mathcal{F}(\alpha_1, \cdots, \alpha_k) = \bigotimes_{(j_1, \cdots, j_k) \in \alpha_1 \times \cdots \times \alpha_k} \mathcal{F}(U_{j_1} \cap \cdots \cap U_{j_k}) \,.$

For each $1 \leq i \leq k$ there is a canonical morphism

$p_i : \mathcal{F}(\alpha_1,\cdots, \alpha_k) \to \mathcal{F}(\alpha_1, \cdots, \alpha_{i-1}, \alpha_{i+1}, \cdots, \alpha_k) \,.$
###### Definition

A prefactorization algebra $\mathcal{F} : Fact_X \to End(C)$ 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

$\bigoplus_{\alpha_1, \alpha_2 \in P I} \mathcal{F}(\alpha_1, \alpha_2) \stackrel{p_1 - p_2}{\to} \bigoplus_{\beta \in P I} \mathcal{F}(\beta) \to \mathcal{F}(U) \to 0$

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] $\mathcal{F} : Fact_X \to End(X)$ is a homotopy factorization algebra if for all factorizing covers $\{U_i \to U \subset X\}_{i \in I}$ the canonical morpshim

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

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

Factorization algebras have some similarity with

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

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

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

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

and the beginnings of

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4, arXiv:1111.4234
• K. Costello, C. Scheimbauer, Lectures on mathematical aspects of (twisted) supersymmetric gauge theories, pp. 57-88 in: Mathematical aspects of QFTs, D. Calaque, T. Strobl editors, Springer 2015

Lecture notes include

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

A comparison with FQFT for TFTs is presented in

• C. Scheimbauer, A factorization view on states/observables in topological field theories youtube 19 min, string-math 2017, Hamburg

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

A construction of chiral differential operators via quantization of $\beta\gamma$ system in BV formalism with an intermediate step using factorization algebras:

• Vassily Gorbounov, Owen Gwilliami, Brian Williams, Chiral differential operators via Batalin-Vilkovisky quantization, pdf
• Brian Williams, The Virasoro vertex algebra and factorization algebras on Riemann surfaces, Lett. Math. Phys. 107:12, 2189–2237 (2017) doi

A version of bosonic string theory related to factorization algebras is presented in

• Owen Gwilliam, Brian Williams, The holomorphic bosonic string?, arxiv/1711.05823

A higher-dimensional generalization of vertex algebras is suggested in the framework of factorization algebras in

We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an 'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.

Relation to homotopical algebraic quantum field theory: