= Dirac membrane action
{c}
{title2=$S=-T\int d^3\xi\sqrt{-\det h}$}
A relativistic membrane sweeps out a three-dimensional <worldvolume> with <induced worldvolume metric> $h_{ij}=\partial_iX^m\partial_jX^n\eta_{mn}$. Its leading geometric <action> is tension times its invariant volume. In static gauge in four-dimensional spacetime, one transverse field $Y$ gives $h_{ij}=\eta_{ij}+\partial_iY\partial_jY$, and the <matrix determinant lemma> gives $S=-T\int\sqrt{1+\partial_iY\partial^iY}$. Expanding supplies the translation-zero-mode kinetic term. A <domain wall> of finite thickness also has higher-curvature corrections outside the long-wavelength limit.
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