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As a generalization of point particle dynamics, one can conceive of a theory of $n-$dimensional objects with 'world-manifold' action given by

$$ S[X] = -\frac{T}{2} \int d^{n+1}\sigma \sqrt{h} h^{\alpha\beta}(\sigma) g_{\mu\nu}(X) \partial_\alpha X^\mu \partial_\beta X^\nu. $$

In page 60, Volume 1 of Superstring theory by Green, Schwarz, and Witten, it is claimed that the action $S[X]$ enjoys $n+1$ independent reparametrization gauge invariances. Among $(n+1)(n+2)/2$ independent components of $h_{\alpha\beta}$, remaining $n(n+1)/2$ will then survive, resulting in a theory whose quantization is untractable. The only exception arises when $n=1$, where the additional Weyl invariance comes to rescue for the theory of 1-manifolds, i.e. strings.

To me, however, it is unclear why the authors conclude there are $n+1$ independent reparametrizations of the world manifold. The number $n+1$ seems to count the reparametrizations $\delta\sigma^i = \lambda \sigma^i\; (n=0,1,2,\dots,n)$, but how could one exclude other possibilities?

Qmechanic
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Hyeongmuk LIM
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GSW is merely stating the fact that an $(n+1)$-dimensional world volume has $(n+1)$-dimensional local coordinate systems, or equivalently, is parametrized by $n+1$ independent parameters $\sigma^{\alpha}$. A local reparametrization $\sigma^{\prime\alpha}=f^{\alpha}(\sigma)$ involves $n+1$ independent component functions $f^{\alpha}$, which in principle can be used to gauge-fix $n+1$ components of the $(n+1)(n+2)/2$ independent components $h_{\alpha\beta}$ of the world-volume metric. Weyl symmetry kills 1 more component, i.e. there are $$ \frac{(n+1)(n+2)}{2} -(n+1) -1 ~=~ \frac{n(n+1)}{2} -1 ~=~\frac{(n-1)(n+2)}{2} $$ DOF left, which is only a non-positive number for $n=1$ dimensional strings and $n=0$ dimensional point particles, but not for $n\geq 2$ dimensional membranes.

Related Phys.SE post: Why one-dimensional strings, but not higher-dimensional shells/membranes?

Qmechanic
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