= Auxiliary-metric brane action
{title2=$\int\sqrt{-\gamma}(\gamma^{\mu\nu}h_{\mu\nu}-(p-1))$}
= Polyakov-type brane action
{c}
{synonym}
The auxiliary-metric brane action is a quadratic embedding action with an independent <worldvolume metric>. Its embedding equation is $\partial_\mu(\sqrt{-\gamma}\gamma^{\mu\nu}\partial_\nu X^a)=0$. Its metric equation is $h_{\mu\nu}-\gamma_{\mu\nu}(\gamma^{\rho\sigma}h_{\rho\sigma}-(p-1))/2=0$. For $p\ne1$, tracing determines $\gamma^{\rho\sigma}h_{\rho\sigma}=p+1$ and hence $\gamma=h$; eliminating the auxiliary field gives twice the invariant <worldvolume> volume, before including the overall <brane tension>. The <Weyl-invariance exception for the string among branes> prevents unique elimination at $p=1$.
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