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Functorial quantum field theory, one of the two approaches of axiomatizing quantum field theory. The other is AQFT. FQFT formalizes the Schrödinger picture of quantum mechanics, while AQFT formalizes the Heisenberg picture.
Much work in quantum field theory is based on arguments using the path integral. While in the physics literature this is usually not a well defined object, it is generally assumed to satisfy a handful of properties, notably the sewing laws. These say, roughly, that the path integral over a domain $\Sigma$ which decomposes into subdomains $\Sigma_1$ and $\Sigma_2$ is the same as the path integral over $\Sigma_1$ composed with that over $\Sigma_2$.
Accordingly it is the S-matrix that is manifestly incarnated in the Atiyah-Segal picture of functorial QFT:
Here a quantum field theory is given by a functor
from a suitable category of cobordisms to a suitable category of vector spaces.
To a codimension-1 slice $M_{d-1}$ of space this assigns a vector space $Z(M_{d-1})$ – the (Hilbert) space of quantum states over $M_{d-1}$;
to a spacetime/worldvolume manifold $M$ with boundaries $\partial M$ one assigns the quantum propagator which is the linear map $Z(M) : Z(\partial_{in} M) \to Z(\partial_{out} M)$ that takes incoming states to outgoing states via propagation along the spacetime/worldvolume $M$. This $Z(M)$ is alternatively known as the the scattering amplitude or S-matrix for propagation from $\partial_{in}M$ to $\partial_{out}M$ along a process of shape $M$.
Now for genuine topological field theories all spaces of quantum states are finite dimensional and hence we can equivalently consider the dual vector space (using that finite dimensional vector spaces form a compact closed category). Doing so the propagator map
equivalently becomes a linear map of the form
Notice that such a linear map from the canonical 1-dimensional complex vector space $\mathbb{C}$ to some other vector space is equivalently just a choice of element in that vector space. It is in this sense that $Z(M)$ is equivalently a vector in $Z(\partial_{out}M) \otimes Z(\partial_{in}M)^\ast = Z(\partial M)$.
In this form in physics the propagator is usually called the correlator or n-point function .
Segal’s axioms for FQFT (CFT in his case) were originally explicitly about the propagators/S-matrices, while Atiyah formulated it in terms of the correlators this way. Both perspectives go over into each other under duality as above.
Notice that this kind of discussion is not restricted to topological field theory. For instance already plain quantum mechanics is usefully formulated this way, that’s the point of finite quantum mechanics in terms of dagger-compact categories.
It was in
that it was realized that
this means that this property can be taken as the defining property of the path integral, thereby circumventing the problem of constructing it as an actual integral;
this property can be conveniently axiomatized by saying that the path integral is a functor from a suitable category whose morphisms are cobordisms to a category of vector spaces.
(Strictly speaking, Atiyah’s original article mentions this functor slightly indirectly only.)
All this was originally formalized in the context of topological quantum field theory only. This is the easiest case that already exhibits all the functoriality that is implied by “FQFT” but by far not the only case (see below).
A pedagogical exposition of how the physicist’s way of thinking about the path integral leads to its definition as a functor is given in
A pedagogical exposition of the notion of quantum field theory as a functor on cobordisms is in
and a review of much of the existing material in the literature is in
The influential work by Moore and Segal on open-closed 2d TFTs is available as
The same topic is studied by Lazariou:
This mostly concentrates on topological quantum field theories, those where the path integral depends only on the diffeomorphism class of the domain it is evaluated on. This is the simplest and by far best understood case. But the idea of functorial FQFT is not restricted to this case.
This was realized in
There the notion of 2-dimensional conformal field theory is axiomatized as a functor on a category of 2-dimensional cobordisms with conformal structure.
(Apparently a similar definition has been given by Kontsevich, but never published.) The details of the category of conformal cobordisms can get a bit technical and slight variations of Segal’s original definition may be necessary. The work by Huang and Kong can be regarded as a further refinement and maybe completion of Segal’s program
Yi-Zhi Huang, Geometric interpretation of vertex operator algebras, Proc. Natl. Acad. Sci. USA, Vol 88. (1991) pp. 9964-9968
Liang Kong, Open-closed field algebras Commun. Math. Physics. 280, 207-261 (2008) (arXiv).
A very concrete construction of functorial CFTs (for the special case of rational CFTs) is provided by the FFRS-formalism.
But one notices that the formalization of quantum field theory as a functor on cobordisms encodes only a small aspect of the full sewing law imagined to be satisfied by the path integral: In a 1-category of $n$-dimensional cobordisms these are glued along $(n-1)$-dimensional boundaries. One could imagine more generally a formalization where a given cobordism is allowed to be chopped into arbitrary parts of arbitrary co-dimension such that the path integral can still consistently be evaluated on each of these parts.
This leads to the notion of extended quantum field theory, which is taken to be an $\infty$-functor on an infinity category of extended cobordisms. Early ideas about a formalization of this approach were given in
Making this precise involves giving a precise definition of an $\infty$-category of cobordisms. Several approaches exist, such as
or
There is a long-term project by Stephan Stolz and Peter Teichner which originally tried to refine Segal’s 1-functorial formulation of conformal field theory to a 2-functorial extended FQFT, as indicated in
More recently, Mike Hopkins and Jacob Lurie claimed (Hopkins-Lurie on Baez-Dolan) to have found a complete coherent formalization of topological extended FQFT in the context of (infinity,n)-categories using an (infinity,n)-category of cobordisms. This is described in
An explicit account of this for the 2-dimensional case is presented in
see also
In this context Dan Freed is picking up again his old work on higher algebraic structures in quantum field theory, as described in
where he argued that and how the path integral should assign $n$-categorical objects to domains of codimension $n$, and is re-expressing this in the $\infty$-functorial context. (Freed speaks of multi-tiered QFT instead of extended QFT.)
Freed’s ideas on how an extended or multi-tiered QFT arises from a path integral coming from a given background field were further formalized in the context of “finite” QFTs in
Simon Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids (arXiv)
Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory, PhD thesis, Sheffield (2008) (arXiv)
There are indications that a complete picture of this involves groupoidification
and, more generally geometric function theory:
a big advancement in the understanding of extended $\sigma$-model QFTs is the discussion in
which realizes $\sigma$-models by homming cobordism cospans into the total spaces (realized as infinity-stack) of background fields and regarding the resulting spans as pull-push operators on suitable geometric functions.
A similar approach to bring the old work by Dan Freed mentioned above in contact with the picture of extended functorial QFT and the Baez-Dolan-Lurie structure theorem is
See
As usual, the problem of constructing FQFT becomes much more tractable when linear approximations are applied. In homological FQFT and in TCFT the Hom-spaces of the cobordism category (the moduli spaces of cobordisms with given punctures/boundaries) are approximated by complexes of chains on them. This leads to formalization of $\infty$-functorial QFT in the context of dg-algebra.
duality between algebra and geometry in physics: