# Which QFTs were rigorously constructed?

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Which QFTs have mathematically rigorous constructions a la AQFT? I understand there are many such constructions in 2D, in particular 2D CFT has been extensively studied mathematically. But even in 2D there are many theories without known constructions e.g. non-linear sigma models in most curved target spaces. In higher dimensions the list of non-free examples is much shorter.

I'm looking for a complete list of QFTs constructed to-date with reference to each construction. Also, a good up-to-date review article of the entire subject would be nice.

EDIT: This question concerns QFTs in Minkowski (or at least Euclidean) spacetime, not spacetimes with curvature and/or non-trivial topology.

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retagged Mar 24, 2014
G. Scharf thinks he constructed QED rigorously in his "Finite Quantum Electrodynamics" (a realistic QFT) ;-)

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@Vladimir: Scharf provides a perturbative construction only, using the method of Epstein and Glaser. I think the question pertains to rigorous nonpertubative constructions.

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@Squark: what do you mean by "a la AQFT"? Do you only want a list of theories constructed using the methods of Algebraic QFT? or do you want the list of all theories constructed by whichever method yet satisfy the Wightman axioms of Axiomatic QFT?

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@Abdelmalek: I mean theories constructed by whichever methods. Anything that can be reasonably claimed to be a rigorous construction of a QFT. I think the Wightman axioms are probably too restrictive, but Haag-Kastler should probably apply to all reasonable examples

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Gents, so far all answers concern the 2D case only. I suppose there are _some_ interacting examples in 3D at least, no?

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@Squark: I don't think Wightman axioms are too restrictive. Haag-Kastler is just easier to work with. Under suitable technical assumptions one can go from one to the other and back.

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For CFT there are many examples. I will give some examples of local conformal nets on the circle (or real line). The Ising model Pieter mentions is the Virasoro net with $c=1/2$. The Virasoro net can be constructed for the discrete $c<1$ and $c>1$. See eg.

• Kawahigashi Y. Longo R. (2004) "Classification of local conformal nets. Case c<1" Ann. of Math. 160, p493-522

They furthermore classify all local conformal nets with central charge $c<1$. Positive energy representations of loop groups give conformal nets.

The conformal nets associated to lattices and its orbifolds are constructed in

• Dong & Xu. Conformal nets associated with lattices and their orbifolds. Advances in Mathematics (2006) Volume: 206, Issue: 1, Pages: 279-306

and in the same issue Kawahigashi and Longo have constructed the "moonshine" net.

• Kawahigashi & Longo. Local conformal nets arising from framed vertex operator algebras. Adv. Math. 206 (2006), 729-751.

For massive models in 2D Lechner constructed the factorizing S-matrix models in which are a priori just "wedge-local" nets but he managed to show for a class that to show the existence of local observables.

• Lechner. Construction of Quantum Field Theories with Factorizing S-Matrices. Commun.Math.Phys. 277, 821-860 (2008)
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answered Oct 17, 2011 by (300 points)
OK, these are nice examples. But have anyone compiled a complete list of rigorous 2d QFTs? Or at least CFTs?

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You can make a list of constructions, but for example every even lattice gives a conformal net or Vertex Operator Algebra (VOA). Even the classification of even selfdual lattices seems to be hopeless task...

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Well, I don't need a classification, a mere list of known constructions will suffice

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BTW there is als a work in progress by Carpi, Kawahigashi, Longo, Weiner how to go from a unitary VOA (+ som technical ass.) to a conformal net

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The list would be a bit too long here. It also depends on how demanding you are on the notion of "being constructed". If you take a rather restrictive definition as: all the Wightman axioms have been established then that excludes Yang-Mills even though important work has been done by Balaban as mentioned by Jose and also other authors: Federbush, Magnen, Rivasseau, Seneor. Examples of theories where all the Wightman axioms have been checked:

• Massive 2d scalar theories with polynomial interactions, see this article by Glimm, Jaffe and Spencer.

• Massive $\phi^4$ in 3d, see this article by Feldman and Osterwalder as well as this one by Magnen and Seneor.

• Massive Gross-Neveu in 2d see this article by Gawedzki and Kupiainen and this one by Feldman, Magnen, Rivasseau and Seneor.

• Massive Thirring model, see this article by Frohlich and Seiler and this more recent one by Benfatto, Falco and Mastropietro.

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answered Oct 20, 2011 by (640 points)
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Notice that a conformal AQFT net as in the replies of Marcel and Pieter only gives the "chiral data" of a CFT, not a full CFT defined on all genera. For the rational case the full 2d CFTs have been constructed and classified by FFRS. Also Liang Kong has developed notions that promote a chiral CFT to a full CFT (rigorously), see this review.

Beyond that, of course topological QFTs have been rigorously constructed, including topological sigma-models on nontrivial targets. Via "TCFT" this includes the A-model and the B-model in 2d.

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answered Oct 17, 2011 by (6,025 points)
OK, but I was really thinking about QFT in Minkowski spacetime (or at least Euclidean spacetime)

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I did not make the comment yet in my list. One can always take a product of two chiral parts $\mathcal A_+ \otimes \mathcal A_-$ to obtain a model on 2D Minkowski space and further extensins of this. I guess FFRS is about the construction of a CFT on a space with non-trivial topology?

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Yes, for nontrivial topology. That's what I mean by "for all genera". Kong's construction also deals with that case, though less explicitly, I think.

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By the way, just for the record: while maybe it does not count as a "full construction", there has recently been quite some work on how to turn the usual tools of perturbative QFT into rigorous constructions of "perturbative AQFT nets". Some references are here http://ncatlab.org/nlab/show/perturbation%20theory#ReferencesInAQFT

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I assume you know that free field theories can be constructed (in arbitrary dimension of spacetime, I believe).

In algebraic quantum field theory (a la Haag), there is for example the conformal Ising model. You can find more about this in these references:

• Mack, G., & Schomerus, V. (1990). Conformal field algebras with quantum symmetry from the theory of superselection sectors. Communications in Mathematical Physics, 134(1), 139–196.
• Böckenhauer, J. (1996). Localized endomorphisms of the chiral Ising model. Communications in Mathematical Physics, 177(2), 265–304.

In the latter "localized endomorphisms" as in the Doplicher-Haag-Roberts programme on superselection sectors. See for example this paper on the arXiv.

There are surely more examples, also in the Wightman setting, but I'm not too familiar with them.

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answered Oct 17, 2011 by (550 points)
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An approach to the rigorous construction of gauge theories is via the lattice. There were some papers in the 1980s -- I remember those of Tadeusz Bałaban (MathSciNet) (inSPIRE) in Communications -- on this topic.

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answered Oct 19, 2011 by (2,315 points)

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