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

March 19, 2010

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.


String Theory: Crash Course

April 21, 2007

A Tool for Living in the 21st Century:

The excellent Seed magazine offers a cribsheet [PDF] which includes a basic introduction to string theory.

Prof. Clifford Johnson over at Asymptotia.com was an adviser.

Other interesting cribsheets are one on stem cells, and one on climate change.


Deconstructing Strings?

November 14, 2006

“Deconstruction” is a method of critical analysis of philosophical and literary language that emphasizes the internal workings of language and conceptual systems, the relational quality of meaning, and the assumptions implicit in forms of expression.

Todays most fascinating paper is without doubt the one by Bert Schroer entitled:

String theory deconstructed (a detailed critique of the content of ST from an advanced QFT viewpoint)

In this paper Bert (BS) supposedly gives “a detailed and comprehensive critique of claims and methods of string theory from an advanced quantum field theoretical viewpoint.” BS starts out by listing nine claims of string theory “which afterwards will be shown to be fundamentally flawed”. These nine claims are:

1 ) The Kaluza-Klein argument can be used in QFT (or ST) to encode compactified spatial coordinates into inner symmetries
2 ) In ST supersymmetry is spontaneously broken
3 ) Holography is a construct which needs quantum gravity as a prerequisite
4 ) The Maldacena conjecture is about a AdS—CFT holography
5 ) The counting zero mode degree of freedom estimate about the cosmological
constant is consistent with the principle of local covariance
6 ) String theory solves the “information paradox”
7 ) Strings are quantum objects with a localization in spacetime which is
string- instead of point-like
8 ) It has been shown that ST contains QFT in the limit of low energies.
9) The S-matrix of ST has the properties of a particle physics S-matrix

For one thing, BS does not like KK compactifications (claim #1), since “I recently red that already Pauli had shown that this is impossible, but there was no reference given.” And concerning claim 3), BS states that “I think that anybody who knows the framework of particle physics (say beyond the level of recent QFT texts which where written by string theorists) would agree that holography from d+1 to d dimension and its possible inversion cannot be anything else than a radical change of the spatial encoding of a specified algebraic substrate; using this word for anything else would be a misuse and lead to misunderstandings.” This seems to debunk the idea that holography should be related to quantum gravity.

But BS’s arguments against “the Maldacena conjecture” are even stronger. For example, he says that “I do not know any competent quantum field theorist who does not accept Rehren’s work as the correct formulation of AdS—CFT holography (Hollands, Wald, Brunetti, Fredenhagen, Verch, Buchholz, …)”. It is – at least to me – unclear what the …’s stand for here; but even more staggering are BS’s adventures into advanced psychoanalysis: “For psychologically understandable reasons it was this metaphoric QG connection which attracted the attention of string theorists (QG is the raison d’etre for string theory) and which led Maldacena to formulate a conjecture involving a vague idea of supersymmetric string under the KK curling (with its even more vague idea of its QG content) on the dual AdS side in case one starts from a (supersymmetric) conformal field theory”.

For some reason also, quantum mechanics seems to be enough to understand black hole physics: “Of course one can use Bekenstein’s classical formula and equate it with this microscopically computed entropy to determine epsilon (I have not done this, but there can be no doubt that at this point the Planck length enters and determines the size of the vacuum polarization cloud). The calculations are in two papers […]”.

With Maldacena (and … and ….) literally on his knees, an alternative resolution of the apparent clash between quantum mechanic and general relativity was put forward by Wald: “His proposed solution was the start of the modern theory of QFT in CST in which the Lagrangian formalism is abandoned in favor of the adoption of the dichotomy of AQFT between the algebraic structure of QFT and the admissible states on such algebras.”

Numerous other advanced arguments seem to kill the claims 5) and 6) above. And for 7) I learned, that: “The localized algebras are monades with very different properties from algebras one meets in QM. There can be no doubt that the understanding of their positioning in a common Hilbert space will be an important step on the long way towards QG.”

But of course the “monad” (or, in biological terms, flagella) point of view also call into question whether string theory contains quantum field theory in its low-energy limit (8): “The message from this last case is that metaphoric arguments (e.g. looking at functional representations without actually doing the functional integrals) may turn out to lead to wrong results. Take for example the case of 2+1 dimensional QFT which have braid-group statistics. If the spin is anyonic (i.e. not semi-integer) the statistics is plektonic and the upholding of the spin-statistics theorem in such a case prevents the nonrelativistic limit to be a (second quantized) QM; it remains a nonrelativistic QFT. Only if one relinquishes the plektonic commutation relations, but preserves the anyonic spin one finds Wilczek’s anyons in the form of quantum mechanical Aharonov-Bohm dyons […]”, and then “The message from this illustration is that a theory can only be asymptotically (e.g. for long distances) contained in a more fundamental one if their structures harmonize.”

But the flagella (monads) also kill the S-matrix arguments 9): “A much more detailed correspondence of Leibniz’s image of reality in terms of indivisible monades to the conceptual structure of particle physics is provided by the algebraic setting of QFT (AQFT). If one identifies Leibniz’s monades with copies of the unique hyperfinite type III_1 factor algebras then it can be shown that any QFT permits a faithful encoding into the relative positions of a finite number of monades”.

At this point I started thinking: is this all a joke? Was I being fooled? Staring a the screen I was wondering whether or not I had really been fooled. On the one hand, if I wasn’t fooled, then this paper was serious, hence I was fooled by my understanding of physics. But if I was fooled, then I did get what I expected from BS, so in what sense was I fooled?.

Then suddenly I realized, that I had seen this text before, but just in another (isomorphic) disguise. It was the famous “Sokal hoax”, a hoax paper published by physicist Alan D. Sokal in 1994, entitled “Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity”. (Sokal’s hoax served a public purpose, that of attracting attention to what Sokal saw as a decline of standards of rigor in the academic community; for this reason, Sokal’s text was “liberally salted with nonsense”). For example, quoting from Sokal:

“In mathematical terms, Derrida’s observation relates to the invariance of the Einstein field equation […] under nonlinear space-time diffeomorphisms (self-mappings of the space-time manifold which are infinitely differentiable but not necessarily analytic). The key point is that this invariance group “acts transitively”: this means that any space-time point, if it exists at all, can be transformed into any other. In this way the infinite-dimensional invariance group erodes the distinction between observer and observed; the [pi] of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity; and the putative observer becomes fatally de-centered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone.

and

“It is still too soon to say whether string theory, the space-time weave or morphogenetic fields will be confirmed in the laboratory: the experiments are not easy to perform. But it is intriguing that all three theories have similar conceptual characteristics: strong nonlinearity, subjective space-time, inexorable flux, and a stress on the topology of interconnectedness.”

.

So, was I fooled or not?


A Chat with Prof. Lisa Randall

November 9, 2006

Discover magazine is conducting a chat with Prof. Lisa Randall of Harvard University today at 2:00 PM EST:

www.discover.com

I have reviewed Lisa’s excellent book, “Warped Passages”, here.


Perelman, the Poincare Conjecture and a Fields Medal

August 22, 2006

Poincare’s conjecture (1904) is related to the nature of three-dimensional manifolds, or spaces. It is in many ways deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.

(The conjecture is fundamental to topology, and incidentally one of the $1 million Millenium Prize Problems awarded by The Clay Mathematics Institute of Cambridge, Massachusetts).

In three dimensions, it is hard to discern the overall shape of something; so for example, when we envision the surface of a sphere, we are really seeing a two-dimensional object embedded in three dimensions. To understand the problem it is therefore easier to start with two dimensions.

The topology of two-dimensional manifolds or surfaces has been well understood since the 19th century. In fact, there is a simple list of all possible smooth compact orientable surfaces. Any such surface has a well-defined genus g ≥ 0, which can be described intuitively as the number of holes; and two such surfaces can be put into a smooth one-to-one correspondence with each other if and only if they have
the same genus.

This means that all such two-dimensional surfaces are classified by a single integer, g. A surface of genus zero is topologically a sphere (an apple), a surface of genus one is a torus (a doughnut) and a surface of genus two is, well a dougnut with two handles (or whatever you want to call it), and so on.

In the case of two dimensions, it is easy to see how the conjecture should be understood: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.

The corresponding question in higher dimensions is much more difficult. The most basic example of such a manifold is the three-dimensional unit sphere, that is, the set of all points (x,y,z,w) in four-dimensional Euclidean space which have distance exactly 1 from the origin: x^2 + y^2 + z^2 + w^2 = 1.

A distinguishing feature of the two-dimensional sphere is that every simple closed curve in the sphere can be deformed continuously to a point without leaving the sphere, or that “you can’t lasso a two-sphere”.

In mathematical terms, the Poincare Conjecture can be phrased as follows: If a compact three-dimensional manifold M has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M is homeomorphic to the sphere S^3 ?

From the first, the apparently extremely simple nature of this statement has led mathematicians to overdo it. For about 100 years, several mathematicians have tried to prove the conjecture. But they all failed. Well, except from one, it seems.

Grigori Perelman’s proof is based on a study of the geometry of the infinite dimensional space consisting of all Riemannian metrics on a given smooth three-dimensional manifold.

The metric g_ij on a Riemannian manifold determines the length between points according to ds^2 = gij dx^i dx^j. (From the first and second derivatives of this metric tensor, one can compute the Ricci curvature tensor R_ij , and the scalar curvature R; for example, Euclidean flat space has R_ij = R = 0, while a usual three-dimensional sphere of radius r, has Ricci curvature R_ij = 2g_ij /r^2 and scalar curvature R = 6/r^2).

Perelman’s proof is based on the study of the Ricci flow, i.e. the solutions to the differential equation:

dg_ij/dt = −2R_ij ,

where t is like an external time-parameter. In other words, the metric is required to change with time so that distances decrease in directions of positive curvature.

Perelman’s succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process; however, the fates of more complicated objects were previously thought to be problematic. As the Ricci flow progressed, kinks and neck pinches – or singularities – could appear, pinch off and even shrink away. Perelman has showed that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started (his papers at arXiv.org).

For his work, Dr. Perelman has just been awarded the Fields Medal (an analog of the Nobel Prize) and as said before, also the Millenium Prize; but Perelman seems to have refused to accept the award (news.google.com). What a coolheaded guy 😉

The other winners of the Fields Medal are, Terence Tao, Wendelin Werner, and Andrei Okounkov.

In the context of string theory, the Ricci flow represent the world-sheet renormalization group (RG) flow; Perelman’s proof might have important implications for understanding space-time singularities.

To be continued…


Not Even Wrestling?

June 16, 2006

Peter Woit’s book, “Not Even Wrong: The Failure of String Theory & the Continuing Challenge to Unify the Laws of Physics”, is out. The book description at Amazon.com says:

How does the world work and what is mathematics’ role in its description? An authoritative and well-reasoned account of string theory’s fashionable status among today’s theoretical physicists, and promising new directions, including the role of beauty in mathematics and physics.

As of yet, the book has not been reviewed in any major physics journals (and might never be?). There is a rather positive (but also rather uninformed) piece on the book in the Times; there is a positive review in the Sunday Times; there is also a comment on the book in the Financial Times (registration required); and finally, Lubos Motl has a very critical review at his blog.

There is also a recent review by John Horgan in the August issue of the British magazine Prospect.

One of Woit’s main objections to string theory seem to be, that it offers no possibility of producing experimental evidence. And that even the proposed Superconducting Super Collider would have failed to provide any clue as to whether the theory had merit.

The problem with this conclusion is that Woit seem to be implying that you need to reach something like the order of the Planck energy to “confirm” string theory.

In spite of all this, the book should be interesting to read; but I have to add that most likely, will we never see universes wrestling, not even wrestling (which Peter Woit might be disappointed about…).

PS1: I coined the term “Not Even Wrestling”, and then Lubos commentet that: “we might never observe Universes wrestling: not even wrestling”;

PS2: This post DOES NOT refer to the book: “Not Even Wrong: Adventures in Autism”, but for some, I guess, this is more or less the same.

Update: Lubos has written a 17-page-long list of comments to the book. Peter Woit has written an errata page for the book.


CSL-1: The End of Cosmic Strings?

January 17, 2006

In the last couple of years there has been some enthusiasm both in the string theory and cosmology communities regarding the possible observation of an astronomical effect of a cosmic string. A pair of images – named CSL-1 – has been observed with the right properties such that they are candidates to arise from a single object lensed by a cosmic string such that it actually appears doubled. See for example the following paper.

The recent Hubble Space Telescope observations of CSL-1 clearly shows that it is not a cosmic string, but just a pair of interacting elliptical galaxies. If there were a string lensing a single object, you would have to see a discontinuity of the picture along the string – very different from lensing by a point mass for example. And even though the shapes are quite similar, the actual shapes would have to have been much more similar.

If CSL-1 could have been interpreted as a cosmic string, surely it would have had a very important impact on our understanding of fundamental physical laws – basically, it could either indicate a field theory phase transition at high energies or that it is possible for some superstrings from the early universe to remain macroscopic and still stable. At any rate, this particular case has not falsified the idea of cosmic strings in general. This should answer the question posed above.

The hypothesis that we may observe cosmic strings has been around for long – and long before it was shown that these cosmic strings could be identified with strings from string theory (the other possibility being that cosmic strings might arise as gauge theory solitons). The idea, that a cosmic string could be a fundamental string (F-string) is actually rather new. Before 1995 it was believed that F-strings have a tension close to the Planck scale and observational data precludes such heavy strings – cosmic strings are bound to have a tension at least two orders lower. Furthermore, Witten showed that long BPS F-strings (in the heterotic string theory) are unstable and hence would never be seen.

There are of course a number of instabilities that would prevent superstrings from growing to cosmic size (but it should be remembered that there is no fundamental principle preventing strings from growing to cosmic sizes – it’s basically just a matter of how much energy the string carries).

One is related to gauge strings. For gauge strings, the U(1) symmetry is exact with a magnetic flux running along the core of the string. However, one expects that should be electric and magnetic sources for every flux, so that the string can break by creation of a monopole/anti-monopole pair.

Another instability is related to global strings: that there are no global exact symmetries in string theory (this is because black holes can destroy global charges). This implies that a domain wall will force the string to collapse. The heterotic and Type II strings are effectively global strings, because they couple to the two-form field B. The Type I string couples to no form field and it can potentially break. The Type I and Type II cosmic superstrings have appeared in brane inflation models. However, it now seems that long superstrings can be stable – they generate networks (generally of (p,q) strings), radiate and lens distant objects (this was exactly what many hoped to see in CSL-1). More precisely, in Type II and Type I string theories it has been realized that it is possible – with the advent of D-branes – to construct long superstrings which are not BPS but are nonetheless stable and potentially observable. And in warped compactifications – as in the Randall-Sundrum model – superstring tensions can be reduced by several orders of magnitude.

I should be said, that a problem might be, however, that string theory generally produces too many and too many kinds of cosmic superstrings. For example, a cosmic superstring may arise as a Dp brane wrapped on a compact (p-1) dimensional cycle and general Calabi-Yau compactifications often have a huge number of S^2 and S^3 cycles.

After the observation of CSL-1, the existence of cosmic strings might be considered less likely for some people. But as such, it was not really a blow to string theory (as Peter Woit seems to be implying), or to the possibility of observing cosmic string, or to new physics in general. Sure, it was really bad luck – think about this: how many years did it take before black holes were “observed”? Does anybody believe that black holes do not exist? Or Dark Matter, for that matter 😉 ?

The CLS-1 has also been discussed by Lubos Motl, by Peter Woit (as mentioned above) and at Cosmic Variance.