From Polyakov to Perelman: History of the Ricci Flow (and a Millennium Prize)

It has just been announced by the Clay Mathematics Institute, that G. Perelman has won the Millennium prize for “resolution of the Poincare conjecture”.

Previously I’ve written about the Poincare conjecture and Perelman’s celebrated proof of it, which was based on the so-called Ricci flow equation. Now let me talk more about how this story is related to physics, and where this equation first appeared in history.

The Ricci flow is an analog of the heat equation for geometry – it is a diffusive process acting on the metric of a Riemannian manifold. It is a special case of so-called geometric evolution equations, that describe the deformation of the metric of Riemannian manifolds driven by their curvature.
Actually, what is usually called the Ricci flow equation, constitutes a very simple equation. Let M be a n-dimensional compact manifold with positive definite metric g_{ij}. The Ricci flow equation is

dg_{ij}/dt = -2 R_{ij}(g)

where t is an external parameter defined on some interval and i,j =1, 2, …, n; the Ricci tensor with components R_{ij} is calculated from the Riemann tensor as the trace R_{ij}=R^k_{ikj}.

In physics it was introduced by Friedan in the 1980’s in the renormalization-group (RG) flow of two-dimensional sigma models. Such sigma models describe the propagation of strings in curved backgrounds. A variant of the Ricci flow appeared already in the 1970’s in the renormalization group studies of non-linear sigma models; the original computation was performed by Polyakov who considered the class of O(n) sigma models with target space metric that of the round sphere on S^{n-1} for n>2. Polyakov (1975) originally considered the O(n) nonlinear sigma models. Here the radius of the sphere is the only parameter of the theory, and as such, it serves as the inverse of its coupling constant, i.e. R ~1/g. These two-dimensional models are not conformally invariant in the quantum regime, as it was found that the coupling constant runs with respect to the world-sheet renormalization scale parameter \Lambda to lowest order in perturbation theory.

The computation of the perturbative beta function was later extended by Friedan (1980) to encompass two-dimensional non-linear sigma models with generalized coupling given by the target space metric g_{ij} of arbitrary Riemannian manifolds. Here

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}-\frac{\alpha'^2}{2}R_i^{\, klm}R_{jklm}+...

where the dots stands for higher order curvature corrections, and the RHS is a perturbative expansion in \alpha'. 
The first-order term in \alpha' is

\Lambda^{-1}\frac{\partial g_{ij}}{\partial \Lambda^{-1}} = -\beta(g_{ij}) =-\alpha' R_{ij}.

Now, to see that the Ricci flow is essentially the same as RG flow, we identify the “time” t with the energy scale \Lambda through t=\log(\Lambda^{-1}), then \Lambda \sim e^{-t} and so large positive t corresponds to small energy \Lambda: flow in t is therefore a flow from the ultra-violet (UV) regime to the infra-red (IR) regime, or from small to large distance scales. The calculation above of Friedan led to the development of the sigma model approach to string theory, since the requirement of conformal invariance for the world-sheet quantum field theory gives rise to the vacuum Einstein equations in target space, R_{ij}=0 plus higher order corrections. Further generalizations can be considered by including other massless modes of the string, such as the dilaton \phi and the anti-symmetric torsion field B_{ij} to the action of the world-sheet sigma model.

Perelman’s proof (2003) of the Poincare  conjecture is related in an interesting way to this physical model. Perelman’s proof uses an “entropy” functional {\cal F}. Let {\cal M} be the space of smooth Riemann metrics on M. Define {\cal F}:{\cal M}\times C^\infty (M)\rightarrow R by

{\cal F}(g,f) =\int_M\left(R+|\nabla f|^2\right)e^{-f}dV

this is basically the Einstein-Hilbert action in general relativity, where f is a “dilaton” field; roughly speaking we go from Perelman to Polyakov by setting f=\phi. To my knowledge, the full extent of the connection between string theory and Perelman’s proof has not really been understood yet.

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One Response to From Polyakov to Perelman: History of the Ricci Flow (and a Millennium Prize)

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