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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,

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?

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In works on SU(2) anomaly people have used definitions of the fermion measure by expanding the fields via eigenfucntions of a Dirac operator. Not sure it applies here – Peter Kravchuk Apr 13 '13 at 19:59
It would be a lot more sensible if there were a derivative of $\zeta$. This is what one usually encounters when dealing with $\zeta$-regularization.. But $\zeta(0)$, I dont get it. Its like summ of 1 for every eigenvalue of the operator.. – Peter Kravchuk Apr 13 '13 at 20:57
Okay, there is this paper, it uses $\zeta$-regularization for chiral Jacobian, which is in curved spacetime and indeed has this $\zeta(0)$. Looks like 4.8' has something to do with conformal anomaly, but I just don't know much about this. – Peter Kravchuk Apr 13 '13 at 21:08

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Finally, I got it. The sketch is as following:

In order to understand the fermionic measure, suppose that we did something suitable so that the spectrum of the appropriate Dirac operator is discrete (for exapmle, take the volume to be finite, work on compact manifold). Let $\psi_n(x)$ be the eigenbasis for the operator. Then we use $\det C = \exp(Tr \ln C)$ for $C$ the transition operator. In our case it multiplies by $\exp(\phi)$, so $\ln C$ is multiplication by $\phi$. Trace $\sum_n\int d\mu(x)\phi(x)\bar{\psi}_n(x)\psi_n(x)$ is regularised either by $\zeta$-regularization or by introducing $\exp(-D^2/M^2)$ with $D$ the Dirac operator. Basically the same as the chiral anomaly in, but we take the finite transformation instead of infinitesimal one.

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Feels awkward, but never mind – Peter Kravchuk Apr 13 '13 at 21:42

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