reparameterization invariance of strings
I'm working through <PERSON>'s First Course In String Theory.
I'm stuck on a problem (16.1) where I need to show that the coupling Kalb-Ramond field to a worldsheet have the following transformation:
$$\int d\tau' d\sigma' \frac{\partial X^\mu}{\partial \tau'} \frac{\partial X^\nu}{\partial \sigma'} B_{\mu \nu}(X) = sgn(\gamma) \int d\tau d\sigma \frac{\partial X^\mu}{\partial \tau} \frac{\partial X^\nu}{\partial \sigma} B_{\mu \nu}(X)$$
where sgn(x) is the sign of x and $\gamma = \frac{\partial \tau}{\partial \tau'} \frac{\partial \sigma}{\partial \sigma'} - \frac{\partial\tau}{\partial \sigma '}\frac{\partial \sigma}{\partial \tau '}$ and the reparameterization of the world sheet coordinates go from $(\tau, \sigma)$ to $(\tau'(\tau,\sigma),\sigma'(\tau,\sigma)))$.
I don't have any particular conceptual questions, but I do have a lot of things I've tried. I'm confused about when things are vectors and when they are not in this problem.
1. gamma is the same as the Jacobian in a change of variables, which is similar (maybe inverse?) as the square root of the determinant of the metric (of what?).
2. 818
$\gamma$ and what I've been calling$\gamma^{-1}$ where $\gamma^{-1} = \frac{\partial \tau'}{\partial \tau} \frac{\partial \sigma'}{\partial \sigma} - \frac{\partial\tau'}{\partial \sigma }\frac{\partial \sigma'}{\partial \tau }$ are not quite multiplicative inverses of each other unless the cross terms are zero (Or if the gammas were vectors and I took the dot product).
3. I can find the total derivatives for $d\tau'$ , $d\sigma'$, and the velocities, and multiply them together, but then I end up with something like $\gamma\gamma^{-1}$ (the unprimed coupling), plus a number of extra terms which all go to zero if $\frac{\partial \tau'}{\partial \tau} \frac{\partial \tau'}{\partial \sigma}=0$. This seems reasonable since the vectors along tau and sigma are perpendicular.
4. It seems like, algebraically, the only way for sgn(gamma) to appear in an equation would be to have$(\sqrt{sgn(\gamma)\gamma})^2\gamma^{-1}$ where the sign was installed to ensure that the sign under the square root was positive.
Could someone help me piece these observations together? 818