 Weyl-invariance exception for the string among branes

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.