Auxiliary-metric brane action ($\int \sqrt{-\gamma }({\gamma }^{\mu \nu }{h}_{\mu \nu }-(p-1))$)

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 $p\ne 1$, 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 $p=1$.

Under ${\gamma }_{\mu \nu }\mapsto e^{2\omega }{\gamma }_{\mu \nu }$, the quadratic kinetic density of a $p$-brane scales as $e^{(p-1)\omega }$ and its constant density scales as $e^{(p+1)\omega }$. At $p=1$ the first factor is one and the constant term in the auxiliary-metric brane action vanishes. The traced metric equation is then an identity, leaving a Weyl transformation freedom. For $p\ne 1$, the metric equation instead identifies the auxiliary and induced worldvolume metrics.