# Relation between solutions to Yang-Baxter equations, integrability and exact solvability?

Wikipedia mentions that there is an implication: Yang-Baxter solutions yield integrable models, what 1D systems concerns.

In arbitrary dimensions, what is the relation, if any, between solutions to Yang-Baxter equations, integrability and exact solvability?

If somebody could provide a no-go theorem or cases, where integrability was already ruled out, it would be great.

This is a very difficult question. In one dimensions a two body scattering is the minimum interaction you can have. In multiple dimensions you can have more bodies scattering as the minimum interaction.

The Yang Baxter equation is formally, $$(R\otimes \mathbf{1})(\mathbf{1}\otimes R)(R\otimes \mathbf{1}) =(\mathbf{1}\otimes R)(R\otimes \mathbf{1})(\mathbf{1}\otimes R)$$ This is graphically represented as,

It basically says that three particle scattering can be reduced to a two body problem. So, no matter how large the system under study is we can always reduce the corresponding scattering matrices to two body problems.

This is no longer the case in higher dimensions and we need to generalize the Yang Baxter equation. An example of this is the so-called "Tetrahedron equation" in 2+1 (or 3) dimensions. It's called that because the above graph now looks like a tetrahedron. See, figure 1 of this paper: http://arxiv.org/pdf/hep-th/9401076.pdf

However, this equation is very difficult to work with and is poorly understood.

First of all, there is no generally accepted definition for the notion "quantum integrability". It is used in different, though closely related, contexts and with definitions that are most appropriate for the case at hand. In analogy with the classical mechanical case the underlying idea is that the model should admit so many 'compatible', i.e. commuting, and independent symmetries that it can be solved exactly, at least in principle.

The discrete context deals with statistical-physical models on 2(+0)d lattices or quantum spin chains in 1+1 dimension; note that it's really 2d that is special: don't forget about time in the quantum setting. Here quantum integrability is often taken to mean the presence of an underlying Yang-Baxter equation, which via the construction of transfer matrices lead to many symmetries.

The other context is that of (quantum) field theories in 1+1d. Here the YBE for the S-matrix expresses factorized scattering, which is a compatibility condition for the presence of many symmetries. I have tried to explain this in Section 5.2 of these notes, following a nice exposition of Witten. This time the relation between commuting independent symmetries (which one might call 'quantum integrability') and the YBE is more clearly associated with a no-go theorem. Indeed, loosely speaking, the presence of many spacetime symmetries of the S-matrix leads to a trivial (non-interacting) theory by Coleman-Mandula. In 2d the argument does not hold, but leads to factorized scattering instead.