nLab factorization algebra of observables

Contents

Context

Higher algebra

Quantum field theory

Idea
Examples
Related concepts
References

Idea

Given a local action functional on fields which are sections of a suitable field bundle, perturbation theory and renormalization allow to assign to open subsets of spacetime/worldvolume the corresponding quantum BV-complexes/Beilinson-Drinfeld algebras. This assignment is canonically equipped with the structure of a factorization algebra, the factorization algebra of observables of the theory.

This formalization of algebraic quantum field theory is similar to, but a bit different from, the notion of local net of observables.

Examples

Over a Riemann surface: vertex operator algebra (Gwilliam, section 6.3, section 6.5)
(…)

Related concepts

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 $\phantom{A}$

chiral algebra, topological chiral homology

References

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 dimension 2 and 4, talk at Geometric and Algebraic Structures in Mathematics, Stony Brook (2011) [arXiv:1111.4234]

Lecture notes include

Kevin Costello (with Owen Gwilliam), Factorizaton 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

Last revised on December 14, 2023 at 18:11:11. See the history of this page for a list of all contributions to it.

nLab factorization algebra of observables

Context

Higher algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Quantum field theory

Contents

Contents

Idea

Examples

Related concepts

References