# Gauge fixing and residual symmetries

1. Jan 28, 2008

### StatusX

This question comes from reading Schwarz' string theory book, which is why I put it in this section. But it seems like a general QFT question, so maybe this isn't the right forum for it.

Starting with the sigma model action, reparametrization and Weyl invariance allow us to "gauge fix" the auxilliary world sheet metric $h_{\alpha \beta}$ so that $h_{\alpha \beta}=\eta_{\alpha \beta}$, the 2D minkowski metric. This requires retaining the equation of motion of $h_{\alpha \beta}$ 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 $L_m$ (which, classically, are functions of the coefficients of the mode expansion of $X^\mu$), this requires each $L_m$ to vanish.

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

$$\{L_m, L_n\} = i(m-n) L_{m+n}$$

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?