The functional renormalization group is equivalent to a diffeomorphism-modified Ricci flow on the information metric of coupling space, with the log effective action generating diffeomorphisms and an F-entropy serving as the driving Lyapunov functional toward a Ricci soliton.
On sigma model RG flow, "central charge" action and Perelman's entropy
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abstract
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.
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UNVERDICTED 2representative citing papers
Riemannian manifolds with a closed parallel torsion 3-form are locally N × G (G semisimple), enabling simplified proofs and explicit classification of strong G2, Spin(7), and certain 8D HKT manifolds.
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Functional Renormalization Group as a Ricci Flow: An $\mathcal{F}$-Entropy Perspective on Information Metric Dynamics
The functional renormalization group is equivalent to a diffeomorphism-modified Ricci flow on the information metric of coupling space, with the log effective action generating diffeomorphisms and an F-entropy serving as the driving Lyapunov functional toward a Ricci soliton.
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On the rigidity of special and exceptional geometries with torsion a closed $3$-form
Riemannian manifolds with a closed parallel torsion 3-form are locally N × G (G semisimple), enabling simplified proofs and explicit classification of strong G2, Spin(7), and certain 8D HKT manifolds.