The Shape of Inner Space

Besides the Hawking book, which was a disappointment in many ways, I recently also finished reading a much better and more interesting book which deals with some of the same topics, but in a dramatically more substantive and intelligent manner. The Shape of Inner Space is a joint effort of geometer Shing-Tung Yau and science writer Steve Nadis. Yau is one of the great figures in modern geometry, a Fields medalist and current chair of the Harvard math department. He has been responsible for training many of the best young geometers working today, as well as encouraging a wide range of joint efforts between mathematicians and physicists in various areas of string theory and geometry.

Yau begins with his remarkable personal story, starting out with a childhood of difficult circumstances in Hong Kong. He gives a wonderful description of the new world that opened up to him when he came to the US as a graduate student in Berkeley, where he joyfully immersed himself in the library and a wide range of courses. Particularly influential for his later career was a course by Charles Morrey on non-linear PDEs, which he describes as losing all of its students except for him, many off to protest the bombing of Cambodia.

He then goes on to tell some of the story of his early career, culminating for him in his proof of the Calabi conjecture. This conjecture basically says that if a compact Kahler manifold has vanishing first Chern class (a topological condition), then it carries a unique Kahler metric satisfying the condition of vanishing Ricci curvature. It’s a kind of uniformisation theorem, saying that these manifolds come with a “best” metric. Such manifolds are now called “Calabi-Yau manifolds”, and while the ones of interest in string theory unification have six dimensions, they exist in all even dimensions, in some sense generalizing the special case of an elliptic curve (torus) among two-dimensional surfaces.

Much of the early part of the book is concerned not directly with physics, but with explaining the significance of the mathematical subject known as “geometric analysis”. Besides the Calabi conjecture, Yau also explains some of the other highlights of the subject, which include the positive-mass conjecture in general relativity, Donaldson and Seiberg-Witten theory, and the relatively recent proof of the Poincare conjecture. Some readers may find parts of this heavy-going, since Yau is ambitiously trying to explain some quite difficult mathematics (for instance, trying to explain what a Kahler manifold is). Having tried to do some of this kind of thing in my own book, I’m very sympathetic to how difficult it is, but also very much in favor of authors giving it a try. One may end up with a few sections of a book that only a small fraction of its intended audience can really appreciate, but that’s not necessarily a bad thing, and arguably much better than having content-free books that don’t even try to explain to a non-expert audience what a subject is really about.

A lot of the book is oriented towards explaining a speculative idea that I’m on record as describing as a failure. This is the idea that string theory in ten-dimensions can give one a viable unified theory, by compactification of six of its dimensions. When you do this and look for a compact six-dimensional manifold that will preserve N=1 supersymmetry, what you find yourself looking for is a Calabi-Yau manifold. Undoubtedly one reason for Yau’s enthusiasm for this idea is his personal history and having his name attached to these. Unlike other authors though, Yau goes into the question in depth, explaining many of the subtleties of the subject, as well as outlining some of the serious problems with the idea.

I’ve written elsewhere that string theory has had a huge positive effect on mathematics, and one source of this is the array of questions and new ideas about Calabi-Yau manifolds that it has led to. Yau describes a lot of this in detail, including the beginnings of what has become an important new idea in mathematics, that of mirror symmetry, as well as speculation (“Reid’s fantasy”) relating the all-too-large number of classes of Calabi-Yaus. He also explains something that he has been working on recently, pursuing an idea that goes back to Strominger in the eighties of looking at an even larger class of possible compactifications that involve non-Kahler examples. One fundamental problem for string theorists is that of too many Calabi-Yaus already, so they’re not necessarily enthusiastic about hearing about more possibilities:

University of Pennsylvania physicist Burt Ovrut, who’s trying to realize the Standard Model through Calabi-Yau compactifications, has said he’s not ready to take the “radical step” of working on non-Kahler manifolds, about which our mathematical knowledge is presently quite thin: “That will entail a gigantic leap into the unknown, because we don’t understand what these alternative configurations really are.”

Even in the simpler case of Calabi-Yaus, a fundamental problem is that these manifolds don’t have a lot of symmetry that can be exploited. As a result, while Yau’s theorem says a Ricci-flat metric exists, one doesn’t have an explicit description of the metric. If one wants to get beyond calculations of crude features of the physics coming out of such compactifications (such as the number of generations), one needs to be able to do things like calculate integrals over the Calabi-Yau and this requires knowing the metric. Yau explains this problem, and how it has hung up any hopes of calculating things like fermion masses in these models. He gives a general summary of the low-level of success that this program has so far achieved, and quotes various string theorists on the subject:

But there is considerable debate regarding how close these groups have actually come to the Standard Model… Physicists I’ve heard from are of mixed opinion on this subject, and I’m not yet sold on this work or, frankly, on any of the attempts to realize the Standard Model to date. Michael Douglas… agrees: “All of these models are kind of first cuts; no one has yet satisfied all the consistency checks of the real world”…

So far, no one has been able to work out the coupling constants or mass…. Not every physicist considers that goal achievable, and Ovrut admits that “the devil is in the details. We have to compute the Yukawa couplings and the masses, and that could turn out completely wrong.”

Yau explains the whole landscape story and the heated debate about it, for instance quoting Burt Richter about landscape-ologists (he says they have “given up.. Since that is what they believe, I can’t understand why they don’t take up something else – macrame for example.”) He describes the landscape as a “speculative subject” about which he’s glad to, as a mathematician, not have to take a position:

It’s fair to say that things have gotten a little heated. I haven’t really participated in this debate, which may be one of the luxuries of being a mathematician. I don’t have to get torn up about the stuff that threatens to tear up the physics community. Instead, I get to sit on the sidelines and ask my usual sorts of questions – how can mathematicians shed light on this situation?

So, while I’m still of the opinion that much of this book is describing a failed project, on the whole it does so in an intellectually serious and honest way, so that anyone who reads it is likely to learn something and to get a reasonable, if perhaps overly optimistic summary of what is going on in the subject. Only at a few points do I think the book goes a bit too far, largely in two chapters near the end. One of these purports to cover the possible fate of the universe (“the fate of the false vacuum”) and the book wouldn’t lose anything by dropping it. The next chapter deals with string cosmology, a subject that’s hard to say much positive about without going over the edge into hype.

Towards the end of the book, Yau makes a point that I very much agree with: fundamental physics may get (or have already gotten..) to the point where it can no longer rely upon frequent inspiration from unexpected experimental results, and when that happens one avenue left to try is to get inspiration from mathematics:

So that’s where we stand today, with various leads being chased down – only a handful of which have been discussed here – and no sensational results yet. Looking ahead, Shamit Kachru, for one, is hopeful that the range of experiments under way, planned, or yet to be devised will afford many opportunities to see new things. Nevertheless, he admits that a less rosy scenario is always possible, in the even that we live in a frustrating universe that affords little, if anything in the way of empirical clues…

What we do next, after coming up empty-handed in every avenue we set out, will be an even bigger test than looking for gravitational waves in the CMB or infinitesimal twists in torsion-balance measurements. For that would be a test of our intellectual mettle. When that happens, when every idea goes south and every road leads to a dead end, you either give up or try to think of another question you can ask – questions for which there might be some answers.

Edward Witten, who, if anything, tends to be conservative in his pronouncements, is optimistic in the long run, feeling that string theory is too good not to be true. Though, in the short run, he admits, it’s going to be difficult to know exactly where we stand. “To test string theory, we will probably have to be lucky,” he says. That might sound like a slender thread upon which to pin one’s dreams for a theory of everything – almost as slender as a cosmic string itself. But fortunately, says Witten, “in physics there are many ways of being lucky.”

I have no quarrel with that statement and more often than not, tend to agree with Witten, as I’ve generally found this to be a wise policy. But if the physicists find their luck running dry, they might want to turn to their mathematical colleagues, who have enjoyed their fair share of that commodity as well.

Update: I should have mentioned that the book has a web-site here, and there’s a very good interview with Yau at Discover that covers many of the topics of the book.

Update: There’s more about the book and an interview with Yau here.

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16 Responses to The Shape of Inner Space

  1. Anon2 says:

    Minor typo corrections:

    “…Particularly influential for his later career was a course by Charles Morrey on non-linear PDEs, which he describes losing all of it students except for him…” {its}

    “…anyone who reads it is likely to learn something and to get a reasonable, if perhaps overly optimistic summary of what it going on in the subject. …” {is}

    “… Looking ahead, Shamit Kachru, for one, is hopefuls that the range of experiments under way …” {hopeful}

    Just a comment about the claim Yau makes (which you agree with) that mathematics can provide inspiration when experimental data runs out:

    “But if the physicists find their luck running dry, they might want to turn to their mathematical colleagues, who have enjoyed their fair share of that commodity as well.” – Yau

    To the naive public, the problems in string theory are mathematical. Einstein added a time dimension to Euclidean space account for gravity, Kaluza and Klein added a spatial dimension for electromagnetism, and string theory adds further spatial dimensions to build in the rest of particle physics. Then the mathematics is so complex due to the need to hide the extra dimensions in a Planck scale sized Calabi-Yau manifold, it becomes an uncheckable and ugly mathematical guess. Similarly, Lisi’s E8 Lie algebra or older SU(5) GUT ideas have apparently come from mathematicians, and don’t really lead anywhere.

    In what way should physicists turn to mathematicians?

  2. Peter Woit says:


    Thanks, typos fixed.

    The problems with string theory are not mathematical, but physical. Adding an extra dimension is a physical idea. Yes, the mathematics string theorists end up using is not only sophisticated, but quite complicated and rather ugly. This is because they’re pursuing a wrong physical idea.

    My own ideas about where inspiration can be found in mathematics are more in representation theory than in geometry these days (although the two subjects are closely related). I’ve written about them here often.

    Please though, comments should be about Yau’s book, generic “math sucks, no it doesn’t” arguments are tedious and will be deleted.

  3. PM says:

    Peter: “Towards the end of the book, Yau makes a point that I very much agree with: fundamental physics may get (or have already gotten..) to the point where it can no longer rely upon frequent inspiration from unexpected experimental results”

    It always puzzles me when I hear that it’s the lack of experimental data that is the problem in physics. There is plenty left to explain about the data we already have: masses and coupling constants – still unexplained, structure of nuclei – still unexplained, origin of symmetries – still unexplained, dark matter and dark energy – still unexplained, and so on…

    As far as I see it the lack of experimental data is not the main problem, sure more data would make things easier but it’s not like we have nothing left to explain.

    I would say current problems are mostly:

    1. Cultural
    First many people manage to convince themselves that there are no explanations to be found – QM is complete and final. Take the decay of unstable nuclei for example it is thought to be acausal and random – an idea which I find absurd, but if you are OK with it you won’t look for a better explanation even though there may very well be one just waiting to be found.

    This is often striking in retrospect when fairly straightforward generalizations or concepts took decades simply because people weren’t looking for them.

    Second the existence of certain candidate theories which cannot be tested makes many simply stop looking for alternatives until these candidates can be confirmed or ruled out, but it may take decades or centuries to do so so we need to keep looking for alternatives.

    2. Foundational
    I think physics is currently trapped in a deep local minimum so to speak and the road in every direction will be up a steep hill for a while. To me we have simply reached the limits of the current conceptual framework and further progress will require a radical rethinking of the very foundations.

    This is problematic however as such alterations to foundations threaten to ruin all the intricate modern physics which rests upon them and since it works so well many feel the foundations should not be altered. But I think it’s a necessary step and I expect that once the better foundations are found all the current results will be rederived from them and many more.

    Two examples of such radical thinking from the past which I particularly like even though they failed are the geometrodynamics of Wheeler and Dirac attempt to reformulate classical theory without point charges. Those are the kind of imaginative and far reaching ideas which I think are needed to move us forward.

    Those were also true unification attempts since they tried to modify two distinct pieces of physics to unify them unlike certain modern approaches which just try to come up with some “glue” to stick them together.

    Finally I also like that they tried to improve the classical theory as I think this is where the problems are. I see quantum theory as simply emerging from an underlying classical theory and I believe it is this classical theory has to be improved to see progress..

  4. Peter Woit says:


    My point is not that there’s no experimental data still unexplained, but that for many years now we’ve had very little new such data. We have hints from experiment about what a better theory should look like, but the problem is that we haven’t been getting new hints, so are stuck facing the same intractable problems. Something physicists should be thinking about is how to make progress under these circumstances, but instead the tendency seems to be to just keep doing the same thing, no matter how long goes by without it working.

    But, again, I encourage people to discuss Yau’s book, not other unconnected speculative ideas about physics.

  5. Wouarnud says:


    Thanks for that. I may actually read this book, since, as you rightly say, it is rare to find a book that tries to explain complex ideas to people who don’t have all the necessary math background without using false and crude metaphors (in my case the math was somewhat there too long ago, those particular gray cells have been reallocated to fascinating tasks like management plans and shopping lists).

    I was wondering first if Yau addresses a question I often think about, regarding the part about new directions in physics. One rather glaring experimental fact is that of dark matter and dark energy. As far as I understand, it is lacking from standard models and theoretical explanations of it are, to say the least, incomplete. As far as I’ve seen as a member of the general public, mathematical theories about these facts are fairly basic, rather non-specific and unimaginative (modified gravity et al). I don’t really understand why this isn’t the hottest of the hot areas in fundamental physics and mathematics today rather than String theory or any competing alternative. Does Yau mention that in his book?

    As an aside, working in a Space Agency, I once had a scientist making the point to me that since dark energy is 90% of the actual energy in the universe, it should receive 90% of the funding (including of course the mission he was promoting, a far-UV telescope if I remember well). It’s hard to argue with that…


  6. Peter Woit says:


    Dark matter and dark energy are perhaps the hottest topics in fundamental physics. The problem is that no one has a really good idea about their origin, using geometry or any other argument. I forget whether Yau and Niadis say much about the problem in the book, but if so there’s just not much that the study of Calabi-Yaus adds to the question.

  7. Giotis says:

    I’ve read only half of the book and I like it so far. Yau is focusing mainly on the mathematical aspects which was expected in some extend and indeed he makes an ambitious honest attempt to explain these difficult issues. On the other hand he is rather sketchy regarding the physical concepts and String theory itself; at least so far. The autobiographic parts are embedded in a natural way and they serve the overall scope of the book.

    BTW I’ve noticed that on page 125 he attributes the 10 dimensions of the theory to the Green–Schwarz mechanism although it is a direct consequence of superconformal invariance which is of central importance. He mentions conformal variance on page 152 but he doesn’t relate it to the critical dimension.

  8. neo says:

    I thought it interesting that Yau, in his Discover interview, says that he ended up proving the Calabi conjecture because he began by thinking it was wrong, and that he had proven it so. Perhaps this is an approach that should be actively adopted more often.

  9. Coin says:

    Thanks for this review, this sounds like the kind of string theory book I have been wanting to read. I will try to give this one a shot. (And I’d actually be curious if there are any books which treat AdS/CFT in a similar non-physics-math-person friendly manner!)

    I do have a question, something that has perplexed me for awhile and I think I should try to get a better understanding of before I read Yau’s book– just so that I know how to interpret what I’m reading:

    When I look at people talking about string theory, I see two different notions of how space is treated in string theory– how we get from ten or eleven dimensions down to the perceived four. Sometimes people talking about string theory seem to be using one of these notions, sometimes they seem to be using the other, sometimes I can’t tell.

    The Calabi-Yau Manifold notion is the first. When string theory is described this way, “the universe” is in fact a ten dimensional manifold, it’s just that six of the dimensions are compactified. This manifold isn’t embedded in anything else, it stands alone as a background space by itself. Calabi-Yau theory describes this manifold’s structure.

    The other notion of space I sometimes see is this “M-Theory” picture, where there’s an open ten-eleven-ish dimensional “bulk” filled with arbitrary N-dimensional “brane” objects. In this picture “the universe” is described as something “stuck to” (or similar vague phrasing) the outside of one of the branes; the structure of the manifold our universe seems to be located in winds up being dictated by how some less-than-ten-dimensional brane restricts our universes’s strings.

    How do I reconcile these two pictures? Are the Calabi-Yau manifolds of the first picture actually embedded in the “bulk” of the second? Are the manifolds of the first picture somehow dual to specific configurations of branes in the second? Or are these two different incompatible versions of string theory altogether? In short, when I’m reading about how the Calabi-Yau manifolds work, how (if at all) do I connect that to things people say about M-Theory-alikes?

  10. Robert says:

    Hi Peter,

    a small correction:

    by Yau’s solution of the Calabi conjecture there exists a unique Ricci-flat Kahler metric in each Kahler class.

    If you don’t fix the Kaehler class (and the complex structure), the uniqueness statement is obviously wrong, e.g. if you take a Ricci-flat Kahler metric and rescale it you get another one.

  11. Chris Austin says:


    In brane compactifications of the 5 ten-dimensional superstring theories, the 6-dimensional Calabi-Yau space is the bulk, and the brane, called a Dirichlet brane or D-brane, is the Cartesian product of the 3 space dimensions we live in, and a closed surface embedded in a topologically non-trivial fashion in the Calabi-Yau. For example, if the closed surface is a 1-dimensional closed loop, embedded in the Calabi-Yau so that it can’t be contracted to a point without breaking it, then the brane is a 4-brane, meaning it has 4 spatial dimensions. Open strings are associated with matter fields and often have both ends attached to the brane, but free to move along it, while closed strings are associated with gravitational fields and are free to move through the bulk.

    M-theory usually refers either to the strong coupling limit of type IIA string theory, which is d = 11 supergravity compactified on a circle from 11 to 10 dimensions, or the strong coupling limit of the E8 x E8 heterotic string, which is d = 11 supergravity compactified to 10 dimensions on a finite 1-dimensional interval, with a d = 10 supersymmetric E8 Yang-Mills multiplet on the 10 dimensional spacetime at each end of the interval. This last version is also known as Horava-Witten theory, and the 9 space dimensions at each end of the Horava-Witten interval are sometimes regarded as 9-branes, with the Cartesian product of the interior of the Horava-Witten interval, and the 9 space dimensions perpendicular to it, constituting the bulk.

    Burt Ovrut, who Yau quotes twice in the extracts above, works on Horava-Witten theory, with 6 of the 9 space dimensions perpendicular to the Horava-Witten interval compactified on a 6-dimensional Calabi-Yau space. We live on the boundary at one end of the Horava-Witten interval, and the remaining 3 space dimensions perpendicular to the Horava-Witten interval are the 3 extended space dimensions that we see.

  12. Coin says:

    Chris Austin: Oh dear. Thank you for your helpful post, I am still a little confused. I think this is where I am getting stuck. You say: “In brane compactifications of the 5 ten-dimensional superstring theories, the 6-dimensional Calabi-Yau space is the bulk”. Should one expect the six dimensions of the Calabi-Yau space in this case to be compactified?

  13. Chris Austin says:

    I should more accurately have said that the the bulk is the Cartesian product of the 3 extended space dimensions and the 6-dimensional Calabi-Yau space, and the brane is the Cartesian product of the 3 extended space dimensions and a closed surface embedded in a topologically non-trivial fashion in the Calabi-Yau space.

    The six dimensions of the Calabi-Yau space are always compactified, but depending on the model, they can be as small as the Planck length around 10^{-35} metres, or as large as around 10^{-14} metres. In the latter case the closed surface embedded in a topologically non-trivial fashion in the Calabi-Yau space can be no larger than around 10^{-19} metres, so the Calabi-Yau space has to be able to support topologically non-trivial structures much smaller than itself. Calabi-Yau spaces with this property are sometimes called “swiss cheese” Calabi-Yaus.

    The latter case might result in superstrings and/or quantum gravitational effects eventually being observed at the LHC, but I don’t know whether Yau discusses this in the book.

  14. Nono says:

    Oh boy, Chris! Now I know what people mean when they talk about Rube Goldberg constructions. And you didn’t even mention fluxes…

  15. Chris W. says:

    Someday this work will be looked upon—by physicists—in more or less the same way that some of the ideas in Kelvin’s Baltimore Lectures are looked upon now. (Its mathematical interest is beside the point.)

  16. Geoff says:

    Hi Peter –

    Yau mentions a general relativity class at Berkeley that had a profound influence on him. Would that have been R.K. Sachs’ class?

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