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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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    $\begingroup$ G. Scharf thinks he constructed QED rigorously in his "Finite Quantum Electrodynamics" (a realistic QFT) ;-) $\endgroup$
    – Vladimir Kalitvianski
    Oct 17, 2011 at 13:30
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    $\begingroup$ @Vladimir: Scharf provides a perturbative construction only, using the method of Epstein and Glaser. I think the question pertains to rigorous nonpertubative constructions. $\endgroup$
    – Abdelmalek Abdesselam
    Oct 17, 2011 at 16:25
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    $\begingroup$ @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? $\endgroup$
    – Abdelmalek Abdesselam
    Oct 17, 2011 at 16:31
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    $\begingroup$ Gents, so far all answers concern the 2D case only. I suppose there are some interacting examples in 3D at least, no? $\endgroup$
    – Squark
    Oct 17, 2011 at 18:29
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    $\begingroup$ @Pieter: The Wightman axioms need to be extended to deal with theories like QCD & 2d sigma models where the algebra of observables is generated by local operators which are not themselves observables (because they do not them have well-defined correlation functions). In these theories, the state-operator correspondence which is baked into the Wightman axioms doesn't quite work. This isn't an insurmountable obstacle: something like the Holland-Wald axioms is probably fine. $\endgroup$
    – user1504
    Nov 4, 2012 at 23:11

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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 Bałaban as mentioned by José and also other authors: Federbush, Magnen, Rivasseau, Sénéor. 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 Sénéor.

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

  • Massive Thirring model, see this article by Fröhlich 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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  • $\begingroup$ OK, but I was really thinking about QFT in Minkowski spacetime (or at least Euclidean spacetime) $\endgroup$
    – Squark
    Oct 17, 2011 at 18:25
  • $\begingroup$ 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? $\endgroup$
    – Marcel
    Oct 17, 2011 at 19:45
  • $\begingroup$ 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. $\endgroup$
    – Urs Schreiber
    Oct 17, 2011 at 20:49
  • $\begingroup$ 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 $\endgroup$
    – Urs Schreiber
    Oct 17, 2011 at 20:50
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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.

  • Jürg Fröhlich and Fabrizio Gabbiani. Operator algebras and conformal field theory. Comm. Math. Phys. Volume 155, Number 3 (1993), 569-640. Link

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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  • $\begingroup$ OK, these are nice examples. But have anyone compiled a complete list of rigorous 2d QFTs? Or at least CFTs? $\endgroup$
    – Squark
    Oct 17, 2011 at 18:20
  • $\begingroup$ 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... $\endgroup$
    – Marcel
    Oct 17, 2011 at 19:53
  • $\begingroup$ Well, I don't need a classification, a mere list of known constructions will suffice $\endgroup$
    – Squark
    Oct 17, 2011 at 21:07
  • $\begingroup$ 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 $\endgroup$
    – Marcel
    Oct 19, 2011 at 21:25
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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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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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    $\begingroup$ Cranky nitpick: All QFTs on a finite lattice are well-defined. An infinite lattice can present real challenges. $\endgroup$
    – user1504
    Dec 16, 2012 at 4:08
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    $\begingroup$ We never have infinite lattice. Even observable universe has a finite volume. $\endgroup$
    – Xiao-Gang Wen
    Dec 16, 2012 at 14:04

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