WZW model

1. Aug 27, 2009

Jim Kata

Hi I'm trying to understand very basic aspects of the WZW model, and would appreciate any help. The main thing i don't get is how you replace

$$I = \frac{1} {2}\int\limits_0^1 {\gamma _{\alpha \beta } } \frac{{dx^\alpha }} {{d\tau }}\frac{{dx^\beta }} {{d\tau }}d\tau$$

with
$$I = \frac{{ - k}} {4}\int\limits_0^1 {tr} (g^{ - 1} \frac{d} {{d\tau }}g)^2 d\tau$$

for some group g

My thoughts are I don't know how the metric enters and the connection doesn't. For example consider

$${\mathbf{g}} = P\exp \int\limits_0^1 {\frac{{dx^\mu }} {{d\tau }}} {\mathbf{A}}_\mu d\tau$$

then

$${\mathbf{g}}^{ - {\mathbf{1}}} \frac{d} {{d\tau }}{\mathbf{g}} = - \frac{{dx^\mu }} {{d\tau }}{\mathbf{A}}_\mu$$

so the connection enters but not sure how the metric does cos I'm not familiar with any identity that says

$$tr\left( {{\mathbf{A}}_\mu {\mathbf{A}}_\tau } \right) = \gamma _{\mu \tau }$$