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So I have what might be a fairly basic question, but my understanding that in the quantization of the the string, or the 1-brane, there are conditions on the number of spacetime dimensions to ensure that the Lorentz algebra still holds. For the bosonic string we have D=26 and for the superstring we have D=10.

Now I know that an equivalent condition on the critical dimension comes up by requiring that the conformal anomaly vanish. This is the prelude to my question: Are there any similar or equivalent restrictions on the number of spacetime dimensions are come up in the quantizing more general branes? Is it only the string because only the worldsheet theory has conformal invariance that we want to ensure holds in the quantized theory? If this statement is equivalent to the Lorentz algebra closing, why for a general p-brane do we not need to impose any conditions on the dimensions for this to hold?

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Related: – Qmechanic May 9 '13 at 7:58
up vote 2 down vote accepted

From what we understand today, p-branes are honest degrees of freedom, on equal footing with strings. Shop you have a good question. But I don't think anyone had so far managed to consistently quantize a p-brane. Loosely, a brane has much more degrees of frerdom than a string and it's difficult to get them under control. So quantizing it is a technical challenge. Most of the brane related computations we do are (semi)classical.

EDIT: Another special property working in favour of the string/1-brane is the fact that conformal symmetry in 2 dimensions is infinite dimensional. So it's sufficiently constraining to render the theory well-behaved. That's how we come to the condition of the vanishing conformal anomaly in the critical dimension. It's not at all clear (to me) how something like that could be implemented for higher dimensional branes.

That's the sketchy picture I have. Maybe someone who undersands the technical details could comment.

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Ah, I see @Qmechanic's answer at the linked post does just that. – Siva May 9 '13 at 16:33

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