Generalizes Ricci flow to brane flows with n-forms, proves monotonicity for fixed field-dependent volume flows and that steady solitons are gradient solitons, including a new functional for Chern-Simons cases.
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.
verdicts
UNVERDICTED 3representative citing papers
Functional renormalization group flow is recast as a potential-modified Ricci flow on the Fisher information metric of coupling space, with an RG-flow entropy serving as the infinite-dimensional analog of Perelman's F-entropy and fixed points appearing as Ricci solitons.
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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Brane flows
Generalizes Ricci flow to brane flows with n-forms, proves monotonicity for fixed field-dependent volume flows and that steady solitons are gradient solitons, including a new functional for Chern-Simons cases.
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Functional Renormalization Group as a Ricci Flow: An \(\mathcal{F}\)-Entropy Perspective on Information Metric Dynamics
Functional renormalization group flow is recast as a potential-modified Ricci flow on the Fisher information metric of coupling space, with an RG-flow entropy serving as the infinite-dimensional analog of Perelman's F-entropy and fixed points appearing as Ricci solitons.
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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.