Dirac membrane action ($S=-T\int d^3\xi \sqrt{-\det h}$)

A relativistic membrane sweeps out a three-dimensional worldvolume with induced worldvolume metric $h_{ij}={\partial }_i{X}^m{\partial }_j{X}^n{\eta }_{mn}$. Its leading geometric action is tension times its invariant volume. In static gauge in four-dimensional spacetime, one transverse field $Y$ gives $h_{ij}={\eta }_{ij}+{\partial }_{i}Y{\partial }_{j}Y$, and the matrix determinant lemma gives $S=-T\int \sqrt{1+{\partial }_{i}Y{\partial }^{i}Y}$. Expanding supplies the translation-zero-mode kinetic term. A domain wall of finite thickness also has higher-curvature corrections outside the long-wavelength limit.