nLab
factorization algebra

Contents

Context

Higher algebra

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 XX.

This way a factorization algebra is an assignment of a chain complex V DV_D to each ball DXD \subset X embedded in XX, and for each collection of non-intersecting embedded balls D 1,,D nDXD_1 , \cdots, D_n \subset D \subset X sitting inside a bigger embedded ball DD in XX a morphism

V D 1V D 2V D nV 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.

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

Definition

Prefactorization algebra

Definition

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

  • objects are the connected open subsets of XX;

  • the hom-set Fact X({U i} i,V)Fact_X(\{U_i\}_i, V) is the singleton precisely if the U iU_i are all in VV and are pairwise disjoint and is the empty set otherwise.

This specifies composition uniquely.

Definition

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

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

These definitions appear here.

Factorization algebras

Definition

For XX a topological space and UXU \subset X an open subset, a open cover {U iU} iI\{U_i \hookrightarrow U\}_{i \in I} is called a factorizing cover if for every finite set of points {x 1,,x k}U\{x_1, \cdots, x_k\} \subset U there is a finite subset {U i j} jJI\{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 iU} iI\{U_i \to U\}_{i \in I} write PIP I for the set of finite subsets αI\alpha \subset I such that for j,jαj,j' \in \alpha we have U jU j=U_j \cap U_{j'} = \emptyset.

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

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

and for α 1,,α kPI\alpha_1, \cdots, \alpha_k \in P I 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 1ik1 \leq i \leq k 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)\mathcal{F} : Fact_X \to End(C) is called a factorization algebra if for every open subset UXU \subset X and every factorizing cover {U iU} iI\{U_i \to U\}_{i \in I} 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 herelass=‘newWikiWord’>Factorization%20algebra?</span>).

See also at cosheaf.

Homotopy factorization algebras

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

Definition

A [prefactorization algebra] :Fact XEnd(X)\mathcal{F} : Fact_X \to End(X) is a homotopy factorization algebra if for all factorizing covers {U iUX} iI\{U_i \to U \subset X\}_{i \in I} 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 herelass=‘newWikiWord’>Factorization%20algebra?</span>).

Examples

Factorization algebras have some similarity with

Isbell duality between algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}geometryA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\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}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\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 }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}superalgebraA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)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}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A}superalgebra
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}algebraA\phantom{A}A\phantom{A}geometryA\phantom{A}
A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}AQFTA\phantom{A}A\phantom{A}FQFTA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}

References

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 βγ\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.

Homotopy prefactorization algebras play also a major role in

Last revised on October 6, 2018 at 02:29:37. See the history of this page for a list of all contributions to it.