I have to remark that with this different definition, while it is obviously gauge invariant, the gauge fixing does not work out due to anomalies of the Polyakov action and measure which only vanish in 0 dimensions.

I don't think the determinant should be anything else, it is just the general rule that $\delta(f(g(x)))=\sum\frac{\delta(g(x)-g_0)}{|\det df(g_0)|}$.

@mthibodeau Yeah, normally it is defined in that way and then it is trivially gauge invariant, but then the argument Tong wants to make with it becomes impossible due to other anomalies which vanish only in 0 dimensions.

The thing is, when writing the Fadeev-Popov determinant as the partition function of the $c=-26$ CFT in the general background it clearly is not gauge invariant: $g^{ab}\frac{\delta}{\delta g^{ab}}\ln Z[g]\sim cR$, where $R$ is the ricci scalar.

Hello goldmans, for a 'larger' gauge transformation, why can't the leading term be non unity and thus have a non unit determinant? Especially: if the $\epsilon$ is a pure Weyl transformation, then $h^\epsilon=\phi*\epsilon$, and multiplying by phi has certainly non unit determinant.

But this assumption is just plain wrong, isn't it? And that renders the also the whole discussion in 5.3 meaningless since they use a wrong expression for the gauge fixed partition function. Am I misunderstanding something?