This is a comment to Lubos' article
on deconstruction. As it got quite long and I would really like to have an answer, I post it here as well:
Maybe this is the right place to ask a question about deconstruction that I have had for quite some time: IIRC, deconstruction (at least for the 6D (2,0) theory relevant for M5 and NS5 branes) starts by looking at D3 branes in a A_N orbifold
which can be written as C^2/Z_N. This gives you the A_N quiver theory.
Then you take a limit in which you take N to infinity while moving away from the orbifold sigularity. The resulting formulas look to me pretty much like you modded out Z from C^2 to end up with a cylinder R^3 x S^1.
Furthermore, what used to be the quiver theory looks like Wati Taylor's version of T-duality in M(atrix) (or any other D-brane gauge theory. In the classic
hep-th/9611042
he describes how you grow an extra dimension from a "SU(N x inifinity)" theory. This should give you a D4 in IIA and you
can apply the usual M-Magic to turn it into a M5 or NS5. So what is more in deconstruction that M(atrix)-T-duality?
Some people I asked this have suggested that the advantage is that you express everything in terms of a renormalizable 4D theory. But this is strictly true only for finite N.
If that were the case you could look at any 6D theory and compactify it on a 2 torus of finite size. You fourier decompose all fields in the compact direction. The fourier components are formally fields in 4D and thus have renormalizable couplings (for a gauge theory say). Of course, non-renormalizabiity comes back when you realize that you have an infinite number of component fields and the sum over all the components will diverge.
Note that I am not trying to argue that all large N limits are ill defined (that would be stupid), I just say that the argument as I understand it sounds too simple to me.
PS: blogspot.com is sicker than ever. How long will it take me to get this posted?
Thursday, March 10, 2005
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2 comments:
Hi Robert,
My understanding of deconstruction is the following: We start from a 4d quiver gauge theory which is derived as the world volume theory of D3-branes located transverse to a C^2/Z_N orbifold. The exact map between the parameters of the string theory l_s, g_s, N, d (=distance from the orb. singularity) and the field theory parameters R_5, R_6 (=radii of extra dim.), g_YM, v (vev), are known in this
background. The question is: Does there exist a limit N->infinity of the 4d field theory and which theory does it describe in this limit?
The answer is given by considering the D-brane set-up in the N->infinity limit: By T-duality and lift to M-theory, the D3-branes map to M5-branes wrapped around a 2-torus. Since there is a precise map to the field theory parameters, string theory predicts the N->infinity limit of the field theory: It is the 6d (2,0) SCFT living on the M5 brane. Without string theory we could not tell which theory the 4d quiver theory approaches on the Higgs branch in the N->infinity limit. We couldn't even tell whether there exists a well-defined limit. String theory tells us how to take the limit in the field theory to get the 6d SCFT. So for this particular quiver theory, the description of the higher-dimensional theory in terms of a renormalizable 4d field theory is true for infinite N!
This demonstrates nicely how we can gain knowledge about certain field theories from string theory (Of course, there are many other examples like AdS/CFT etc). I hope my comment was useful.
Best, Ingo
Let me add the following clarifying comment: Deconstruction is not simply the inverse of discretization. Of course, you can consider the quiver theory also as the discretization of the (weakly coupled) 5d YM theory living on D4 branes. The crucial point is that the limit N->infinity is taken in such a way that the quiver theory becomes strongly coupled on the Higgs branch. In other words, the theory approaches the D4-brane theory at STRONG (!) coupling which is equivalent to the M5-brane theory. So you deconstruct the UV completion of the 5d YM theory which by string theory is shown to be the 6d (2,0) theory.
In summary, the quiver theory on the Higgs branch in the N->infinity limit describes more than just the higher-dimensional (nonrenormalizable) YM theory; since it is strongly coupled, it describes the UV completion (=6d (2,0) here) of the YM theory.
Remark: The strong coupling is generally considered as a disadvantage of deconstruction and probably impedes further progress towards the understanding of the M5-brane theory.
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