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

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

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.

See also at cosheaf.

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

• 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

• nonabelian Poincaré duality

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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$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{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\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)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$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$\phantom{A}$

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)

• Owen Gwilliam, Kasia Rejzner, Comparing nets and factorization algebras of observables: the free scalar field, arxiv:1711.06674

and the beginnings of

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimension 2 and 4, talk at Geometric and Algebraic Structures in Mathematics, Stony Brook (2011) [arXiv:1111.4234]

• Kevin Costello, Claudia 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:

• 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

• Araminta Amabel, Notes on Factorization Algebras and TQFTs [arXiv:2307.01306]

  (with relation to functorial field theory)

Further review:

• Kevin Costello, Owen Gwilliam, Factorization algebra, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2310.06137]

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

• Gregory Ginot, Notes on factorization algebras, factorization homology and applications, arxiv/1307.5213

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

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

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

• Emily Cliff, Universal factorization spaces and algebras, arxiv/1608.08122

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:

• Donald Yau, Homotopical quantum field theory [arxiv/1802.08101]

• Marco Benini, Giorgio Musante, Alexander Schenkel, Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory [arXiv:2212.02546]

Conformal nets exhibit factorization algebras

• Andre Henriques, Conformal nets are factorization algebras, arXiv:1611.05529

Relation with vertex algebras

• Yusuke Nishinaka, An algebraic construction of functors between vertex algebras and Costello–Gwilliam factorization algebras, arXiv:2408.00412

We construct functors between the category of vertex algebras and that of Costello-Gwilliam factorization algebras on the complex plane ℂ, without analytic structures such as differentiable vector spaces, nuclear spaces, and bornological vector spaces. We prove that this pair of functors is an adjoint pair and that the functor from vertex algebras to factorization algebras is fully faithful. Also, we identify the class of factorization algebras that are categorically equivalent to vertex algebras. To illustrate, we check the compatibility with the commutative structures and the factorization algebras constructed as factorization envelopes, including the Kac-Moody factorization algebra, the quantum observables of the βγ system, and the Virasoro factorization algebra.