 Auxiliary-metric brane action

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 $p=1$.