Nambu–Goto phase-space constraints ($P^2+T^2{X}^{\prime 2}=0,P\cdot X^{\prime }=0$)

The two constraints are $P^2+T^2{X}^{\prime 2}=0$ and $P\cdot X^{\prime }=0$. They are first-class constraints associated with worldsheet diffeomorphisms. Their Lagrange multipliers are arbitrary gauge functions rather than propagating fields.

For a closed string, the currents $J_{\pm }=P\pm T{X}^{\prime }$ have vanishing mixed Poisson brackets. The densities ${\mathcal{H}}_{\pm }=J_{\pm }^2/(4T)$ therefore generate commuting constraint algebras. With opposite Fourier orientations their modes satisfy $\{L_m,L_n\}=-i(m-n)L_{m+n}$ and the identical relation for ${\tilde{L}}_n$. This is a direct sum of Lie algebras consisting of two Witt algebras.