Starting with the sigma model action, reparametrization and Weyl invariance allow us to "gauge fix" the auxilliary world sheet metric [itex]h_{\alpha \beta}[/itex] so that [itex]h_{\alpha \beta}=\eta_{\alpha \beta}[/itex], the 2D minkowski metric. This requires retaining the equation of motion of [itex]h_{\alpha \beta}[/itex] as a constraint, which amounts to requiring the world sheet energy momentum tensor to vanish. If we expand the energy momentum tensor in modes with coefficients [itex]L_m[/itex] (which, classically, are functions of the coefficients of the mode expansion of [itex]X^\mu[/itex]), this requires each [itex]L_m[/itex] to vanish.

Here's my question. In section 2.4, it is said that these mode expansion coefficients satisfy the algebra:

[tex] \{L_m, L_n\} = i(m-n) L_{m+n} [/tex]

where the bracket is the poisson bracket (and translates to the commutator after quantization). It is then said this is a result of the fact that the gauge fixing leaves a residual group of reparametrization symmetries whose lie algebra satisfy the same relations. I'm having a difficult time seeing how these two algebras are related. Can someone help me out?