 Solution

The real canonical variables $x^m,p_m$ describe the center-of-mass position and total momentum of the open string. The nonzero string oscillators are complex Fourier modes with ${\alpha }_{-k}^m=({\alpha }_k^m{)}^{\ast }$; the Lagrange multipliers obey ${\lambda }_{-j}={\lambda }_j^{\ast }$. The Virasoro constraints satisfy $L_{-j}=L_j^{\ast }$, so the multiplier term is real.

For each $k>0$, the oscillator kinetic term differs from the manifestly real expressionby $\frac{i}{2k}\frac{d}{dt}({\alpha }_k\cdot {\alpha }_{-k})$. Thus the written action is real up to a boundary term, which does not change its symplectic form or bulk dynamics. Adding the corresponding endpoint term makes reality exact.

The independent nonzero Poisson brackets areAll brackets between these independent center-of-mass and oscillator variables vanish. The dependent zero mode is ${\alpha }_0^m=p^m/\sqrt{\pi T}$, so, if it is used, $\{x^m,{\alpha }_0^n\}={\eta }^{mn}/\sqrt{\pi T}$.

The Fourier coefficients of the Virasoro constraints areTheir Poisson brackets areThey form the classical Witt algebra, with no Virasoro central extension. In particular they are first-class constraints, closing on the constraint surface, and generate the remaining worldsheet diffeomorphisms rather than independent physical excitations.