# Which QFTs were rigorously constructed?

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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G. Scharf thinks he constructed QED rigorously in his "Finite Quantum Electrodynamics" (a realistic QFT) ;-) –  Vladimir Kalitvianski Oct 17 '11 at 13:30
@Vladimir: Scharf provides a perturbative construction only, using the method of Epstein and Glaser. I think the question pertains to rigorous nonpertubative constructions. –  Abdelmalek Abdesselam Oct 17 '11 at 16:25
@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? –  Abdelmalek Abdesselam Oct 17 '11 at 16:31
@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 –  Squark Oct 17 '11 at 18:18
Gents, so far all answers concern the 2D case only. I suppose there are some interacting examples in 3D at least, no? –  Squark Oct 17 '11 at 18:29

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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OK, these are nice examples. But have anyone compiled a complete list of rigorous 2d QFTs? Or at least CFTs? –  Squark Oct 17 '11 at 18:20
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... –  Marcel Oct 17 '11 at 19:53
Well, I don't need a classification, a mere list of known constructions will suffice –  Squark Oct 17 '11 at 21:07
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 –  Marcel Oct 19 '11 at 21:25

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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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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OK, but I was really thinking about QFT in Minkowski spacetime (or at least Euclidean spacetime) –  Squark Oct 17 '11 at 18:25
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? –  Marcel Oct 17 '11 at 19:45
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. –  Urs Schreiber Oct 17 '11 at 20:49
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 ncatlab.org/nlab/show/perturbation%20theory#ReferencesInAQFT –  Urs Schreiber Oct 17 '11 at 20:50

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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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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All QFT's on lattice are well defined. It may be true that all well defined QFT's are either lattice theory or the low energy limit of a lattice theory. See a related post Rigor in quantum field theory

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Cranky nitpick: All QFTs on a finite lattice are well-defined. An infinite lattice can present real challenges. –  user1504 Dec 16 '12 at 4:08
We never have infinite lattice. Even observable universe has a finite volume. –  Xiao-Gang Wen Dec 16 '12 at 14:04