# In what conceivable way can supersymmetric Yang-Mills theory help us understand traditional Yang-Mills?

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Supersymmetric Yang-Mills theory seems to be an active area of research. I'm wondering what its value is beyond just being a toy model. For example in what conceivable way can supersymmetric Yang-Mills theory help us understand traditional Yang-Mills? On the surface it looks qualitatively different and seems to need a big stretch before it can shed light on traditional QCD, but attempts clearly exist. I'd also be interested in the status quo of this research direction.

PS. I only know supersymmetric Yang-Mills very superficially, so feel free to throw in some education if necessary.

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Let us make some very basic considerations. On one hand side we have the "normal" QCD we know (quarks, gluons etc) whith gauge groups $SU(3)_c$ where c stands for color. On the other hand we have a theory called super-Yang-Mills (SYM) theory with gauge group $SU(N)$which can be interpeted as a theory that has N colors. Note that there might be other symmetries as well, like flavor ones for both theories. QCD has a specific flavor while for a generic SYM we can have various options for flavor symmetries. Now, we know that in the "real" world the very short distance physics, inside the hadrons, is described by QCD, the interactions between quarks and gluons. While QCD in very high energies is asymptotically free, in small energies it is confining and strongly coupled. This makes it impossible to study QCD in this regime since, being strongly coupled, we cannot use our best tool for centuries: perturbation theory. One way to shed light into the dynamics of QCD is to look for cousin theories of it. This is why people are interested in SYM. QCD is a subset of the general class of SYM that does not contain supersymmetry and has some fixed gauge group. Now, let us consider the normal QCD you are familiar with but let us take the rank of the group go to infinity, $N \to \infty$. It turns out that once you do this the coupling of the theory $g$the goes to zero, $g \to 0$ while their product remains finite. It turns out that the essential perturbative and non-perturbative features you wanted to study in the normal QCD have remained in this "planar" theory. Despite this  However, an analytical description of the planar limit has remained elusive and people had to use Lattice methods. One big breakthrough came with the inclusion of supersymmetry. Then, supersymmetry proved to be an extremely useful tool for obtaining analytical results through the dualities. Various results have been obtained, for example, the gluino condensate and its $\beta$functions, to the astonishing Seiberg dualities that relate two different SQCD's in the UV to the same theory in some fixed point of the renormalization group flow in the IR (the theories being interpreted as electric and magnetic), the exact solution of the $\mathcal{N}=2$ Seiberg-Witten  theory whith the elliptic curves (which gave new insight in some deep mathematical problems, see Donaldson theory), which upon to some small perturbation can be broken to an  $\mathcal{N}=1$ theory and gives us the color confinement we so much want to understand in normal QCD. More recently people started suspecting that some non-supersymmetric YM theories, cousins of QCD, were equivalent to gluondynamics in their bosonic sector in the large N limit. If true, this statement has really far-reaching consequences because a large number of prediction we have made using supersymmetry , such as spectral degeneracies, low-energy theorems, and so on, hold (to leading order in 1/N) in strongly coupled non-supersymmetric gauge theories. around 2004 some people came up with something called  "orientifold field theory" and found out that if you only take its bosonic sector you get a full equivalence (perturbative and non-perturbative) with SYM at large N limit. In this example, when taking the N=3 limit you get QCD (but with only 1 flavor)!

Now, some other ideas come from the gauge/gravity correspondence which offers a new framework to tackle strongly coupled problems (with many advantages compared to lattice QCD such as allowing real time and non-chemical potential). Now, even we still do not have the full picture of AdS/CFT we can still find out many interesting qualitative results and various universal results about QCD through the correspondence. While there are many pathologies (QCD confining but SYM not at low energies, QCD has no SUSY but SYM has maximal susy, QCD is not conformal  invariant, QCD has operators in the fundamentals and anti-fundamentals, SYM only in the adjoint) and so on. By the way in this paragraph SYM is the maximal supersymmetric one. Still, Witten was the first to attempt to find a QCD-like dual assuming aType IIA string theory with N D4-branes. The gauge theory on the branes is of course a five  dimensional supersymmetric $SU(N_c)$ gauge theory with fermions and scalars in the adjoint representation. The he compactified one spacelike dimension on a circle of radius 1/M, with anti-periodic boundary conditions for the fermions and the supersymmetry gets broken, the fermions acquire mass at tree-level and the scalars at 1-loop. Only the gauge fields zero modes around the circle remain massless due to gauge invariance and finally prevail at energies lower than the energy scale M. Therefore for energies much smaller than M the theory is a pure four-dimensional $SU(N_c)$ theory. Also, the background is sourced by D4-branes and this has this exhibits a "cigar" like geometry which encodes confinement! This construction manages to reproduce features of the QCD IR dynamics (though it fails at higher energies because of the infinite towers of K-K modes). If we restrict to the low-energy regime and in order to find the mesonic spectrum we have to use some other techniques that go by the name Sakai-Sugimoto model and involve $D8-\bar{D}8$ brane systems embedded in D4 backgrounds at two different points in the circle.

Similar constructions have been made since then but let me just mention something that for me is very very interesting and this is just D-branes wrapping Calabi-Yau singularities which reproduce $\mathcal{N}=1$ SYM theories (in e.g. Klebanov-Witten background or Klebanov-Strassler background). Now this is very promising because we get away with only 4 supersymmetries (the maximum number that can be realized, one day, in the real world) and shows also a deep connection with string theory. I am not going to say much more for these because they require an entire course.

So to summarize, people have been trying to construct various models that catch the basic qualitative and universal properties of QCD, recently based on gauge/gravity dualities. Now, as you see there are many lines of research so once you have a better idea maybe you can ask for something more specific. E.g. the recent progress on improved holographic QCD and so on. A status-quo of the current research on SYM theories requires an at least 1000 page review.

Still, I hope my answer has indeed given you some directions to look further.

answered Nov 7, 2014 by (3,535 points)
edited Nov 7, 2014

This is pretty informative, thank you. By the way, what do you mean by

"take $\mathcal{N}=3$ limit"? How is it related to $N=3$?

Typo, I will fix it. the curly N is always supersymmetries while the normal N is always the rank of the group.

Ok, then how can one extrapolate from large-N to N=3?

In the orientifold field theory you do this by some deformation of the orientifold. I guess that it corresponds to some kind of a reverse 't Hooft process but I really do not know about it! I think Shifman has worked on it btw.

If you find something interesting maybe you should mention it here.

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