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conformal anomaly

Contents

Idea

The standard quantum anomaly arising in a 2-dimensional sigma-model, also called the Weyl anomaly

Details

Discussion as an anomalous action functional is in (Freed 86, 2.). The following summary of this is taken from this MO answer by Pavel Safranov.

Let Σ\Sigma be a compact surface (worldsheet) and MM a Riemannian manifold (spacetime). The string partition function looks like

Z string= gMet(Σ)dg σMap(Σ,M)dσexp(iS(g,σ)).Z_{string}=\int_{g\in Met(\Sigma)}dg\int_{\sigma\in Map(\Sigma,M)}d\sigma\exp(iS(g,\sigma)).

Here Met(Σ)Met(\Sigma) is the space of Riemannian metrics on Σ\Sigma and S(g,σ)S(g,\sigma) is the standard σ\sigma-model action S(g,σ)= Σdvol Σdσ,dσS(g,\sigma)=\int_{\Sigma} dvol_\Sigma \langle d\sigma,d\sigma\rangle. In particular, SS is quadratic in σ\sigma, so the second integral Z matterZ_{matter} does not pose any difficulty and one can write it in terms of the determinant of the Laplace operator on Σ\Sigma. Note that the determinant of the Laplace operator is a section of the determinant line bundle L detMet(Σ)L_{det}\rightarrow Met(\Sigma). The measure dgdg is a ‘section’ of the bundle of top forms L gMet(Σ)L_g\rightarrow Met(\Sigma). Both line bundles carry natural connections.

However, the space Met(Σ)Met(\Sigma) is enormous: for example, it has a free action by the group of rescalings Weyl(Σ)Weyl(\Sigma) (gϕgg\mapsto \phi g for ϕWeyl(Σ)\phi\in Weyl(\Sigma) a positive function). It also carries an action of the diffeomorphism group. The quotient \mathcal{M} of Met(Σ)Met(\Sigma) by the action of both groups is finite-dimensional, it is the moduli space of conformal (or complex) structures, so you would like to rewrite Z stringZ_{string} as an integral over \mathcal{M}.

Everything in sight is diffeomorphism-invariant, so the only question is how does the integrand change under Weyl(Σ)Weyl(\Sigma). To descend the integral from Met(Σ)Met(\Sigma) to Met(Σ)/Weyl(Σ)Met(\Sigma)/Weyl(\Sigma) you need to trivialize the bundle L detL gL_{det}\otimes L_g along the orbits of Weyl(Σ)Weyl(\Sigma). This is where the critical dimension comes in: the curvature of the natural connection on L detL gL_{det}\otimes L_g (local anomaly) vanishes precisely when d=26d=26. After that one also needs to check that the connection is actually flat along the orbits, so that you can indeed trivialize it.

References

Last revised on February 2, 2014 at 10:43:01. See the history of this page for a list of all contributions to it.