# M2-brane wrapped twice around $S^1$

M-theory wrapped around $S^1$ is type IIA, especially in the limit of zero radius. An M2-brane wrapped around $S^1$ once is a type IIA string. The zero mode anyway. In the limit of very small radius, if we project to states with tiny energy, we get a collection of type IIA strings only.

What does an M2-brane wrapped around the circle twice correspond to? Its energy is also low. It has to be. Pretend it is an M-theory compactification with twice the radius. Orbifold it over half a periodic translation. It's back to the original. So, is it made up of type IIA strings? Is it two type IIA strings? Or a single type IIA string? Surely it's not a different kind of type IIA string with $1/\sqrt{2}$ times the string scale and $\sqrt{8}$ times the string coupling?

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If this object were the only one - and the original singly wrapped membrane wouldn't exist - then your double wrapped object would really be the fundamental string with twice as high string tension and a different string coupling.

However, the singly wrapped membrane states also exist in your theory. And they make a lot of difference. In particular, your doubly wrapped membrane - which clearly carries the same winding charge as two ordinary type IIA strings - may decay into the pair of type IIA strings. At least, the charge conservation laws don't prohibit such a decay.

Imagine that your membrane is wrapped on the compact dimension $X^{10}$ and extended in another dimension $X^{9}$. Take a slice for a fixed value of $X^9$. Then the cross section of the membrane looks like a doubly wound string that has been called "state II" in this question:

Statistics and macrolocality in string theory

As discussed in the answer to the question above, this double wound string may self-intersect and a crossing-over interaction may occur at the intersection. This will turn the string into a pair of strings, and the point where this transition occurs may spread in both directions along the $X^9$ axis I neglected, eventually turning the double wrapped membrane into a pair of singly wrapped membranes.

For some non-minimal values of charges, one must always be careful and check whether a stable state exists, and whether it is a bound state with a nonzero binding energy or a threshold bound state. In this case, I would say it doesn't exist.

A way to see that it doesn't exist is matrix string theory,

Good introductory text for matrix string theory

the full light-cone gauge model for M-theory on a circle. In the spectrum of this 1+1-dimensional maximally supersymmetric gauge theory, you find the ordinary strings with an arbitrary $P^+$ but you won't find any "doubly wrapped membranes" at all. They would correspond to some block-diagonal matrices that depend on $\sigma$ but those states may be manifestly seen to be equivalent to pairs of ordinary strings.

The matrix model can't even recognize which strand of the membrane is which. This fact is related to another fact that even in the BFSS matrix model where membranes are expressed by non-commutative geometry, the symmetry under the volume-preserving diffeomorphisms on the M2-brane is actually extended to a full $U(N)$ gauge symmetry that may permute - and mix - arbitrary "cells" of the membrane. So the only physical information is hiding in the approximate positions of the "cells" of the M2-brane and they don't have any other IDs.

It should be emphasized that this conclusion - that no other stable independent states with charges greater than one exist - isn't universal at all. In particular, the threshold bound states of $N$ D0-branes of type IIA string theory do exist: they're D0-branes with a proportionally higher charge and mass. By T-duality, this is true for any fully toroidally wrapped D-branes.

One could think that there is a contradiction because by this procedure, one may uncover genuine and independent new threshold bound states of two (or more) D2-branes (wrapped on a two-torus) of type IIA string theory - and D2-branes are the same objects as M2-branes in the corresponding M-theoretical description, aren't they?

However, this argument is flawed because of one difference: in the type IIA description, the D2-branes correspond to M-theory M2-branes that are localized in the infinitesimal 10th direction of M-theory (that produces type IIA). It's the differences in the location in this tiny new circular direction of space that may distinguish a pair of D2-branes (wrapped on another 2-torus) from a doubly wrapped D2-brane. However, in your example, there's no other "tiny direction" in which the M2-branes are localized, so this extra information doesn't exist. Consequently, the doubly wrapped M2-branes don't produce any new independent objects.

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