In this question, Orbicular made the following comment to Feb7 and my own answers;
Please keep in mind that – even though it is stated very often – noncommutative geometry does not give "real" insight to physics. The reason is that they only have toy models, all of which are unphysical (in the sense that they predict things which differ from real world measurements). Furthermore even the toy models are usually extremely complicated, killing most expectations to get a "real" model (which is not toyish).
First, I want to thank Orbicular for pointing this out, as it is something that I 'kinda' knew, but often forget. The purpose of this question, is to ask for a deeper explanation, either from Orbicular or someone else. In particular
to what degree does Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry influence/assist 'real' models and actual physics related to the physical world?
I don't wish for this question to turn into a debate about whether or not these maths will later be applied in some beautiful stringy-quantum-symmetry theory; I would much rather it be some explanation of the real use of these things. Specifically, I am interested in hearing about the use of Quantum Groups and their representations to Physicists along with some thoughts on the actual usefulness of the results in NC Algebraic Geometry of those articles I posted over here. Another particularly interesting subject I would like to hear about is the usefulness of commutative algebraic geometry in physics.
Some things I have found
Just two references that I have found that at least address these things to some degree are Peter Woit's lecture notes on Representation Theory, and in Shawn Majid's book on Quantum Groups he discusses some definite physical motivation for studying quantum groups.
Thanks!
Best Answer
Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware of the extent of its pervasiveness. Towards the bottom of the answer I mention also the use of representation theory of vertex algebras in condensed matter physics.
The representation theory of the Poincaré group (work of Wigner and Bargmann) underpins relativistic quantum field theory, which is the current formulation for elementary particle theories like the ones our experimental friends test at the LHC.
The quark model, which explains the observed spectrum of baryons and mesons, is essentially an application of the representation theory of SU(3). This resulted in the Nobel to Murray Gell-Mann.
The standard model of particle physics, for which Nobel prizes have also been awarded, is also heavily based on representation theory. In fact, there is a very influential Physics Report by Slansky called Group theory for unified model building, which for years was the representation theory bible for particle physicists.
More generally, many of the more speculative grand unified theories are based on fitting the observed spectrum in unitary irreps of simple Lie algebras, such as $\mathfrak{so}(10)$ or $\mathfrak{su}(5)$. Not to mention the supersymmetric theories like the minimal supersymmetric standard model.
Algebraic Geometry plays a huge rôle in String Theory: not just in the more formal aspects of the theory (understanding D-branes in terms of derived categories, stability conditions,...) but also in the attempts to find phenomenologically realistic compactifications. See, for example, this paper and others by various subsets of the same authors.
Perturbative string theory is essentially a two-dimensional (super)conformal field theory and such theories are largely governed by the representation theory of infinite-dimensional Lie (super)algebras or, more generally, vertex operator algebras. You might not think of this as "real", but in fact two-dimensional conformal field theory describes many statistical mechanical systems at criticality, some of which can be measured in the lab. In fact, the first (and only?) manifestation of supersymmetry in Nature is the Josephson junction at criticality, which is described by a superconformal field theory. (By the way, the "super" in "superconductivity" and the one in "supersymmetry" are not the same!)