I am reading the paper "Bosonization in a Two-Dimensional Riemann-Cartan Geometry", Il Nuovo Cimento B Series 11 11 Marzo 1987, Volume 98, Issue 1, pp 25-36, http://link.springer.com/article/10.1007%2FBF02721455
Equation 4.8' on p. 34 suggests a particular transformation law for the measure under the Weyl scaling 4.8. I am concerned with the fact that this law is somehow dependent on the Dirac operator in given metric. Certainly this general form is expectable (it involves a product of $\exp(\sigma)$ over all points), but I want to understand where the particular factors come from. Zeta-function of Dirac operator suggests that it is obtained via some zeta-regularization of something.
Does anybody know what is this all about (may be a sketch of derivation), or have a reference?