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We know that 11D M-theory is described by BFSS matrix model and for M-theory on a torus $T^p$ (at least for small $p$s) the description is given by SYM theory in $p+1$ dimensions by using the dualities.

My question is: Does the correspondence holds only for torus with Planck size? I didn't think so. If not, is there anything special about the Planck size torus? E.g. in arXiv:hep-th/9911068, Banks have discussed this whole issue seemingly from the point of view of the Plack size torus. Please explain. Thanks a ton!

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First of all, regardless of speculative comments in the original BFSS paper, different superselection sectors – different parts of the Hilbert space specified by the "background" (at least at infinity) – require different matrix model descriptions.

The matrix model is known for $T^p$ with any radii; for $p\leq 3$, the theory is a super Yang-Mills theory on the dual torus, for $p=4$, the 4+1-dimensional Yang-Mills on $T^4$ is non-renormalizable but automatically completed to the $d=6$ (2,0) superconformal field theory, and for $p=5$, one gets little string theory on a $T^5$. There exists no non-gravitational matrix model description for $p\geq 6$ although such models would be interesting as they would exhibit exceptional Lie symmetries.

The radii may be anything; but the 11-dimensional Planck scale is the only natural "unit of length" in the 11-dimensional M-theory, so whatever the radii are, they are naturally expressed as multiples of the Planck length. The numerical multiples are pure numbers; if you invert them, you get the radii of the dual torus in the Yang-Mills matrix model expressed in its natural length scale given by the Yang-Mills scale. (For $p=3$, the theory is conformal, so some parameters get translated to the dimensionless gauge coupling. For $p=4$, the theory in $d=6$ is also conformal but there's one extra dimension whose radius defines a preferred scale. The little string theory for $p=5$ has its own "string scale".)

But the matrix model works even if the radius is much longer or much shorter than the 11-dimensional Planck scale. For example, M-theory on a parameterically short $S^1$ circle produces weakly coupled type IIA string theory whose matrix model, Yang-Mills on $S^1\times R$, is also known and may be shown to describe the IIA strings via the so-called matrix string theory, including all the degrees of freedom on the strings of any length, their interactions, and D-branes.

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I see.. Thanks a lot again! So I guess when Banks said Planck size torus, he probably just meant Planck size from the point of view you mentioned, namely because it's the only natural scale in M-theory. –  user1349 Oct 17 '12 at 21:18
    
Hi, I am confident that when Tom - my (former) adviser - says Planck size torus, he really means a torus whose radii are of order the Planck length. Why don't you believe he says what he apparently does? ;-) When the radii are much larger, one may get simplifications and a different, easier matrix model ("decompactification"), so the matrix model is "really" one for $T^p$ if the torus is of order Planck length in size. –  Luboš Motl Oct 18 '12 at 4:50
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