  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'' functional 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.

High Energy Physics - Theory;Differential Geometry