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In M-theory, our universe is a brane. But this brane is 2-dimensional, so how does our universe fit in? Is our universe just a part of the brane?

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In M-theory, there are two kinds of fundamental branes, the 2-dimensional ones and the 5-dimensional ones (and recall that this counts only the spatial dimensions, and in each case you have to add one time-like dimension).

I don't know why the OP says that our universe is a brane. What could be true is that the space-time of our universe, with its $1+3$ dimensions, could be supported by the world-volume of a 3-brane (this kind of objects exists for instance in type $IIB$ superstring theory, where it is realized as a D-brane). The OP is right when saying that the $M2$ brane is not large enough to support our $1+3$-dimensional space-time. However, one possibility would be to use $M5$ branes. The world-volume is then $1+5$-dimensional, and with an appropriate compactification of two space-like dimensions, we obtain the wanted dimensionality.

Note however that realistic models of our universe using M-theory, with the correct field content, are quite difficult to engineer, and are presumably, if they exist, much more complicated than the basic construction alluded to here.

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