Quantization of the Nonlinear Sigma Model Revisited
Abstract
We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization. The symmetries we consider are twofold: (i) diffeomorphism covariance for a general target manifold; (ii) a transitive group of isometries when the target manifold is a homogeneous space. We show that there are no anomalies in case (i) and that (ii) is also anomalyfree under additional assumptions on the target homogeneous space, in agreement with the work of Friedan. We carry out some explicit computations for the $O(N)$model. Finally, we show how a suitable notion of the renormalization group establishes the Ricci flow as the one loop renormalization group flow of the nonlinear sigma model.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1408.4466
 Bibcode:
 2014arXiv1408.4466N
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Differential Geometry;
 Mathematics  Quantum Algebra;
 81T70;
 81T15
 EPrint:
 51 pages