Zamolodchikov's c-theorem type argument (and also string theory effective action constructions) imply that the RG flow in 2d sigma model should be 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 (math.dg/0211159) by constructing an "entropy"

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... which is essentially the metric-dilaton action extremised 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'). The resulting "entropy" is equal to minus the central charge at the fixed points, in agreement with the general claim of the c-theorem.