In quantum field theory a conformal anomaly is a quantum anomaly that breaks conformal invariance.
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 be a compact surface (worldsheet) and a Riemannian manifold (spacetime). The string partition function looks like
Here is the space of Riemannian metrics on and is the standard -model action . In particular, is quadratic in , so the second integral does not pose any difficulty and one can write it in terms of the determinant of the Laplace operator on . Note that the determinant of the Laplace operator is a section of the determinant line bundle . The measure is a ‘section’ of the bundle of top forms . Both line bundles carry natural connections.
However, the space is enormous: for example, it has a free action by the group of rescalings ( for a positive function). It also carries an action of the diffeomorphism group. The quotient of 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 as an integral over .
Everything in sight is diffeomorphism-invariant, so the only question is how does the integrand change under . To descend the integral from to you need to trivialize the bundle along the orbits of . This is where the critical dimension comes in: the curvature of the natural connection on (local anomaly) vanishes precisely when . After that one also needs to check that the connection is actually flat along the orbits, so that you can indeed trivialize it.
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.