= Chiral constraint algebra of a closed string
{title2=$\mathrm{Diff}_1\oplus\mathrm{Diff}_1$}
For a <closed string>, the currents $J_\pm=P\pm TX'$ 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 $\widetilde L_n$. This is a <direct sum of Lie algebras> consisting of two <Witt algebras>.
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