Which QFTs were rigorously constructed?

Question

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.

Answer 1

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.

Answer 2

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)

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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