Sigma model renormalization group flow, “central charge” action, and Perelman’s entropy

Zamolodchikov’s c-theorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be a gradient one to all loop orders. However, the monotonicity of the flow of the target-space metric is not obvious since the metric on the space of metric-dilaton couplings is indefinite. To leading (one-loop) order when the RG flow is simply the Ricci flow the monotonicity was proved by Perelman [G. Perelman, math.dg/0211159.] by constructing an “entropy” functional which is essentially the metric-dilaton action extremized with respect to the dilaton with a condition that the target-space volume is fixed. We discuss how to generalize the Perelman’s construction to all loop orders (i.e. all orders in ${\alpha }^{\prime }$). The resulting entropy is equal to minus the central charge at the fixed points, in agreement with the general claim of the c-theorem.