that the dependence of the time evolution of $h_{t,x}(A^\pm)$ consists just in a prefactor \[3.25\] \_t h\_[t,x]{}(A\^)=(vu)\_x h\_[t,x]{}(A\^) and the second factor in (\[3.25\]) is independent of $u,v$. More precisely we have \[3.26\] {H(u,v),h\_[t,x]{}(A\^)} &=& \_[S\^1]{} dy{H(u,v),Y\^\_(t,y)}\
&=:& \_[S\^1]{} dy\
&=& \_t h\_[t,x]{}(A\^) One can of course verify by direct computation from the expression (\[3.18\]) that the $T^n(A^\pm)$ are Dirac observables for the string.

Since we can trade the power $n\le N$ for considering increasing rank $N$ of the matrices $\tau_I$ the only interesting invariant is the generating functional \[3.27\] Z\^T\_:=T\^1(A\^)=N+\_[n=1]{}\^ Z\_\^[\_1..\_N]{} where $T^\mu=T^\mu_I \tau^I$ and we have used the expansion (\[3.18\]). It is understood that $N$ can be arbitrarily large and that the complex matrices $T^\mu\in GL(N,\Cl)$ are freely specificable while $Z^T_\pm$ is an invariant. Thus we conclude that the expansion coefficients are themselves invariants. These are the [**Pohlmeyer Charges**]{} \[3.28\] Z\_\^[\_1..\_N]{} &=& \[R\^[\_1..\_N]{}(x)+R\^[\_2..\_N \_1]{}(x)+.. +R\^[\_N \_1..\_[N-1]{}]{}(x)\]\
R\_\^[\_1..\_N]{}(x) &=& \_x\^[x+2]{} dx\_1\_[x\_1]{}\^[x+2]{} dx\_2..\_[x\_[N-1]{}]{}\^[2+2]{} dx\_N Y\^[\_1]{}\_(x\_1).. Y\^[\_N]{}\_(x\_N)\
&=:& \_[xx\_1 ..x\_N x+2]{} d\^Nx Y\^[\_1]{}\_(x\_1)..Y\^[\_N]{}\_(x\_N) The functionals $R^{(N)}_\pm(x)$ depend explicitly on $x$, they are not invariants. It is only after cyclic summation $C_N\cdot R^{(N)}_\pm=
Z^{(N)}_\pm$ that they become invariants.

The [**Pohlmeyer Charges**]{} exepectedly form a complicated, closed subalgebra of the full Poisson algebra 