Nambu-Goto phase-space action

The first-order string action isThe multipliers impose the two Virasoro constraints. The canonical equations give a spatial momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$.

Translations and Lorentz transformations give Noether charges ${\mathcal{P}}_m=\int P_{m}d\sigma$ and ${\mathcal{J}}_{mn}=\int (X_m{P}_n-X_n{P}_m)d\sigma$. Their time derivatives are boundary fluxes, which vanish for a closed string or a free-ended open string.

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.