Quantization of the Nonlinear Sigma Model Revisited

Timothy Nguyen

Published 2014-08-19Version 1

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 nonlinear sigma model; (ii) a transitive group of isometries in the special case when the target space is a homogeneous space. We show that there are no anomalies in case (i) and if in addition H_1(X) = 0 then (ii) is also anomaly-free, 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 theory. Our approach follows the rigorous formulation of perturbative quantum field theory in the Batalin-Vilkovisky formalism due to K. Costello.