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factorization algebra

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Idea
Related concepts
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Idea

A factorization algebra is like an algebra over an operad where the operad in question is like the E-k operad, but with each disk embedded into a given manifold $X$ .

This way a factorization algebra is an assignment of a object (in the standard case a vector space) $V_{D}$ to each ball $D \subset X$ embedded in $X$ , and for each collection of non-intersecting embedded balls $D_{1}, \dots, D_{n} \subset D \subset X$ sitting inside a bigger embedded ball $D$ in $X$ a morphism

V_{D_{1}} \otimes V_{D_{2}} \otimes \dots \otimes V_{D_{n}} \to V_{D}

such that for every collection of further nested balls inside balls, all the different ways to bracket

V_{D_{11}} \otimes V_{D_{12}} \otimes \dots \otimes V_{D_{1 n}} \otimes V_{D_{21}} \otimes V_{D_{22}} \otimes \dots \to V_{D} .

This is pretty much like saying that the factorization algebra is an extended FQFT of genus 0 cobordisms that are embedded into $X$ .

Indeed, such factorization algebras serve to describe quantum field theories on $X$ , pretty much in a way that generalizes the familiar way that 2-dimensional conformal field theory is given by vertex operator algebras. See also the comments on the references below.

Related concepts

Factorization algebras have some similarity with

blob homology
local nets in AQFT.

References

This may be regarded as a slight variation on the concept chiral algebra originally introduced by Beilinson and Drinfeld.

A definition appears in section 4.1 Topological Chiral Homology of

Jacob Lurie, On the Classification of Topological Field Theories

There it is demonstrated how factorization algebras can be used to construct extended FQFTs.

Concrete constructions of formal algebras for familiar quantum field theories are described in

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

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)

Revised on April 6, 2010 13:23:18 by Urs Schreiber (131.211.235.55)

nLab factorization algebra

Contents

Idea

Related concepts

References

nLab
factorization algebra