nLab conformal anomaly




In quantum field theory a conformal anomaly is a quantum anomaly that breaks conformal invariance.


Weyl anomaly of relativistic string

Discussion of the conformal anomaly (Weyl anomaly) of the relativistic string 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.

QCD trace anomaly

In quantum chromodynamics: See at QCD trace anomaly.



  • Daniel Freed, Determinants, torsion, and strings, Comm. Math. Phys. Volume 107, Number 3 (1986), 483-513. (Euclid)

  • Nicolas Boulanger, Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions, Phys. Rev. Lett.98:261302, 2007 (arXiv:0706.0340)

See also:


Discussion via AdS/CFT:

  • Mans Henningson, Kostas Skenderis, The Holographic Weyl anomaly, JHEP 9807 (1998) 023 (arXiv:hep-th/9806087)

  • Mozhgan Mir, On Holographic Weyl Anomaly, JHEP 1310:084, 2013 (arXiv:1307.5514)

For more see at QCD trace anomaly the references there.

Last revised on September 11, 2019 at 06:14:13. See the history of this page for a list of all contributions to it.