Can Lattice Formulations Accommodate Modern Quantum Field Theories?

[Gvido_Anselmi]

Hello everybody,

I have three quite mathematical questions in modern QFT.

1) Why it's supposed that N=2 SUSY Yang-Mills probably cannot be put on a lattice?

2) What is the recent status of lattice approach to conformal quantum field theories?
This question is motivated by the following reason. There are some interesting mathematical structures (such as discrete holomorphicity and random walks) which arise in lattice approach to 2D statistical physics and help one to derive rigorous results (ctitical percolation, conformal invariance of scaling limit etc). Have these structures anything to say about 2D quantum CFT (Liouville theory for example).

3) E.Witten has shown in his work "Gauge theories and integrable lattice models" (1989) that calculation of expectation values of Wilson loops in 3D Chern-Simons theory with compact gauge group can be done with the help of partition functions of some integrable discrete statistical models. However, I've seen no explanation why these models need to be integrable (is they are at all). So what is known nowadays about the connection between 3D Chern-Simons theory and integrability in 2D? What is known about Chern-Simons theory with non-compact gauge group?

I would be gratefull for anybody who answer me.

 

[Demystifier]

Putting a scale invariant theory on the lattice necessarily breaks the scale invariance. I think that answers your 1).

 

[Jimster41]

Putting a scale invariant theory on the lattice necessarily breaks the scale invariance. I think that answers your 1).

But lattice models can produce scale invariance?

http://arxiv.org/pdf/cond-mat/0503462v1.pdf
Log-Poisson Statistics and Extended Self-Similarity in Driven Dissipative Systems
Kan Chen, C. Jayaprakash
(Submitted on 18 Mar 2005)
The Bak-Chen-Tang forest fire model was proposed as a toy model of turbulent systems, where energy (in the form of trees) is injected uniformly and globally, but is dissipated (burns) locally. We review our previous results on the model and present our new results on the statistics of the higher-order moments for the spatial distribution of fires. We show numerically that the spatial distribution of dissipation can be described by Log-Poisson statistics which leads to extended self-similarity (ESS). Similar behavior is also found in models based on directed percolation; this suggests that the concept of Log-Poisson statistics of (appropriately normalized) variables can be used to describe scaling not only in turbulence but also in a wide range of driven dissipative systems.http://arxiv.org/pdf/1509.08858v1.pdf
Conformal invariance of boundary touching loops of FK Ising model
Antti Kemppainen, Stanislav Smirnov
(Submitted on 29 Sep 2015)
In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces.
Some other results in this article are convergence of the exploration process of the loop ensemble (or the branch of the exploration tree) to SLE(κ,κ−6), κ=16/3, and convergence of a generalization of this process for 4 marked points to SLE[κ,Z], κ=16/3, where Z refers to a partition function. The latter SLE process is a process that can't be written as a SLE(κ,ρ1,ρ2,…) process, which are the most commonly considered generalizations of SLEs.

 

[Physics Monkey]

Regarding 1, I would argue that any theory which is not anomalous can be put on a lattice. In practice the lattice model may need to be heavily fine tuned to reach to the target theory in the IR. I'm curious in what context you've heard it claimed that N=2 SUSY YM cannot be put on a lattice?

 

[Demystifier]

I would argue that any theory which is not anomalous can be put on a lattice.

Does it mean that anomalous theory can not be put on a lattice? If this is what it means, then why?