Book review: “The Shape of Inner Space” by Yau and Nadis

The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
Shing-Tung Yau and Steve Nadis


Yes, I said I have no intentions reading the book. But then I was offered a copy for free. And, since I had it anyway, I could as well read it, no?

“The Shape of Inner Space” is a curious mixture of Yau’s autobiography, a crash-course in differential geometry, and physics-themed popular science, sandwiched between an introduction to the history of geometry and philosophical considerations about the beauty of mathematical truth. The string that runs through the book and weaves it together are Calabi-Yau manifolds. Shing-Tung Yau, the “Yau” in “Calabi-Yau,” has spent pretty much his whole life on these manifolds and won the Fields Medal in 1982, among other achievements, for his proof of the Calabi conjecture. So the reader learns first hand from the world expert. Steve Nadis is a popular science writer, and the two have joined forces to produce the book.

The result is interesting and also courageous.

After the introduction, it follows a brief history of geometry. From Pythagoras and Plato over Euclid, Descartes, Gauss and Euler to Minkowski, Riemann, Einstein, Kaluza, Klein and, of course, Calabi. As we come closer to the 21st century, we learn about the geometrization of physics and its successes. To move on beyond Platonic solids, the reader is introduced to mathematical lingo in a rapid fire treatment. It starts with the innocent concept of derivative and integrals. From there it goes on to partial derivatives, curve integrals, non-linear partial differential equations, manifolds (differentiable, compact, orientable, product of), complex numbers, metric (in n dimensions, hermitian), parallel transport, geodesics, curvature and Ricci curvature, groups, tangent spaces, fibre bundles, exotic spheres, homeomorphic diffeomorphisms, harmonic equations, Betti numbers, Chern classes, holonomy and cohomology, Ricci flow, Riemann surfaces, Kähler manifolds and of course Calabi-Yau spaces. Just to mention a few. If you're afraid of math, this book is not for you.

In the later chapters follow the contemporary topics, and the connection to string theory is established. The reader learns about the Dirac equation, Yang-Mills theory, mirror symmetry and the Seiberg-Witten equations. We come across Yukawa-couplings, correlation functions, black hole information loss, moduli and the landscape problem. We meet familiar names like Hawking, Penrose, Guth, Strominger, Kachru, Witten, Greene, Gross, Susskind, Vafa, Giddings and more. Nadis has interviewed many researchers in the field and the text is frequently supplemented by quotations from these interviews (and other sources). One might find it an expression of laziness (or maybe cowardice) to export explanations and opinions into quotations from other people. But I found it very readable and interesting to hear the researchers’ comments and explanations of their work, and that of others, in their own words. I liked that a lot.

The mathematical and physical explanations are accomplished basically without equations (though there are a few examples) and without formal definitions. Sometimes the text is accompanied by figures that I found very helpful and well done, but figures only get you so far to understanding six dimensional spaces. Now all the used concepts are explained somewhere, and I was familiar with most of the terminology before reading the book anyway. But I suspect if you don’t know anything about field theory, differential geometry, and topology, “The Shape of Inner Space” is a very heavy read.

With use of the introduced mathematical concepts the reader then learns what Yau proved, what his colleagues proved and how the field has evolved within the last some decades. Then the authors explain how the connection to string theory came about and how this intersection of physics and math has been fruitful for both sides. That I found indeed the most interesting aspect of the book: The interrelation between mathematics and physics and the mutual benefit for both sides. Yau writes:

“[I] like to position myself at the interface between these two fields, math and physics, where a lot of interesting cross-pollination occurs. I’ve hovered around that fertile zone since the 1970s and have managed to get wind of many intriguing developments as a result.”

However, the book is very focused specifically on the cross-pollination between differential and algebraic geometry and string theory that has sprung from Calabi-Yau spaces. It is a pity there was not more about the recent and not-so-recent history of the math-physics exchange in a broader sense.

Towards the end of the book, after a somewhat bizarre interlude about the way you would die through false vacuum decay, we then find a chapter on experimental tests of string theory. Yau is a mathematician and takes the point of view of an interested outsider. His main interest is mathematical truth, and if physicists with their methods can help mathematicians discover previously unknown relationships, then what does it matter if the physics eventually turns out to be a description of reality? But one or the other reader might care.

“At the end of Dorothy’s adventures in the Land of Oz, she learned that she had the powers to get back home all along. After some decades of exploring the Land of Calabi-Yau, string theorists and their math colleagues (even those equipped with the penetrating powers of geometric analysis) are finding it hard to get back home – to the realm of everyday physics (aka the Standard Model) – and, from there, to the physics that we know must lie beyond. If only it were as easy as closing our eyes, tapping our heels together, and saying “There’s no place like home.” But then we’d miss out on all the fun.”

Thus, in the chapter “Back to the real world” we learn about possibilities to test string theory in the early universe, by bubble collisions and their relics, by cosmic strings or – in the case of large extra dimensions – at the LHC. (I guess this is pretty much the last time a popular science book will talk about the latter possibility.)

Unfortunately, it is not very clearly pointed out that all these tests are tests not of string theory itself but of string theory inspired phenomenological models. Finding such evidence would certainly be a boost for string theorists, but not finding it doesn’t need to bother them either. A quotation by McAllister states it very carefully correct: “It’s possible that string theory will predict a finite class of models, none of which are consistent with the observed properties of the early universe, in which case we could say the theory is excluded by observation.” Yes, it is possible. But at the moment it seems like there’s a string theory motivated model to explain whatever the data will be.

Yau and Nadis avoid commenting on the controversy about the usefulness of string theory as a description of reality. On the landscape problem Yau writes “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.”

“Critical treatments of [string theory], such as The Trouble with Physics and Not Even Wrong” are mentioned in the passing, decorated with quotations from Henry Tye saying “string theory is too beautiful, rich, creative, and subtle not to be used by nature,” and Michael Atiyah letting us know that “even if we can’t measure it experimentally, [string theory] appears to have a very rich… mathematical structure. [String theorists] are onto something, obviously. Whether that something is what God’s created for the universe remains to be seen. But if He didn’t do it for the universe, it must have been for something.” (Like, maybe the multiverse?)

It then follows some elaboration on beauty and mathematical truth, and its relevance for physics:

“Of course, if beauty is going to guide us in any way […] that leaves the problem of trying to define it […] There’s no doubt that a blind adherence to mathematical beauty could lead us astray, and even when it does point us in the right direction, beauty alone can never carry us all the way to the goal line. Eventually, it has to be backed up by something […] more substantial, or our theories will never go beyond the level of informed speculation, no matter how well motivated and plausible that speculation may be.”

But Yau and Nadis remove themselves from the debate about physical relevance by focusing on the mathematics:

“Whereas the final proof in physics is in experiment, that is not the case in math… If the mathematics associated with string theory is solid and has been rigorously proven, then it will stand regardless of whether we live in a ten-dimensional universe made of strings or branes.”

And that is what the book is about – it’s a book about the mathematics of Calabi-Yau spaces, not more and not less. Just so you know what to expect should you consider buying “The Shape of Inner Space:” It’s not, in the first line, a book about string theory and certainly not about quantum gravity*. It is a book about a special kind of manifold and the interaction between physicists and mathematicians it has brought.

The book is generally well written, though I found the writing style over long stretches somewhat uninspired. Many pages it goes along the lines that soandso wrote this paper on this, and then soandso wrote a paper on that, and then a student of soandso wrote a paper on this and that, and so on. Also, I found it somewhat disturbing that in several places technical terms are used that are only introduced in later chapters, sometimes with, sometimes without, mentioning of the later explanation (metric and entropy for example). The book has a glossary, but if hadn’t known anyway what they were talking about I’d have found it a quite annoying break in the reading flow.

The book is also discontinuous in the level of explanation. Over many pages it reads almost like a review paper on Calabi-Yau spaces, summarizing who proved what when by which method. And then there comes the occasional pop-sci explanation. Just to give you an impression, here’s a quotation from a randomly chosen page (133):

“The presence of those [covariantly constant] spinors helps ensure the supersymmetry of the manifolds in question, and the demand for supersymmetry of the right sort is what pointed Strominger and Candelas to SU(3) holonomy in the first place. SU(3), in turn, is the holonomy group associated with compact, Kähler manifolds with a vanishing first Chern class and zero Ricci curvature.”

(That supersymmetry partners bosons and fermions is btw explained only some pages later.) The level of the pop sci explanations are for example that of an exchange particle mediating an interaction by the common analogy to a ball being thrown, or for quantum foam by analogy to the British railway. (“The geometry, in other words, would be undergoing shifts so violently it hardly makes sense to call it geometry. It would be like a rail system where the tracks shrink, lengthen, and curve at will –a system that would never deliver you to the right destination and, even worse, would get you there at the wrong time.”).

The impression I had was that Yau wrote a draft, and Nadis then sprinkled pop sci explanations and quotations on it.

Taken together, I enjoyed reading the book more than expected. It is a very comprehensive summary of research I have a peripheral interest in, and Yau and Nadis have presented it very nicely, so I learned some relations that previously hadn't been clear to me. I was surprised though that the AdS/CFT correspondence is only briefly mentioned and its recent applications are not discussed at all. I'd have found it relevant to the question of what string theory is a theory of. And, there's no explanation of what is actually plotted in the omnipresent pictures of Calabi-Yau spaces you find for illustration all over the place.

Reading the book I couldn't help wondering what audience it is aimed at.
Readers should at the very least have read a fair share of popular physics books because they will not get an introduction to general relativity and quantum mechanics, not to mention quantum field theory, though these are essential to understanding big parts of the book. Black holes, entropy, the standard model, dark matter, inflation etc are explained with only a few sentences each. This, I will admit, was a great relieve to me because I’ve read more than enough stories about quantum pets and suicidal astronauts plunging into black holes. I’m just saying you better bring that knowledge along because otherwise you’ll miss big parts of the story. And, given the mathematical rapid fire treatment, the reader should at the very least have a high school exam, preferably a few semesters math in addition.

In summary, the book might be interesting for you if you have some, though not necessarily expert knowledge in math and physics. “The Shape of Inner Space” will give you a good impression about the state of the art, the history, and a glimpse on the possible future of research on Calabi-Yau spaces. You will learn about the interaction between math and physics it has inspired, and it will give you opportunity to ponder eternal truth and beauty in mathematics, and its relevance for Nature.

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* In the introduction it is made clear that “Because of our focus on so-called Calabi-Yau manifolds and their potential role in providing the geometry for the universe’s hidden dimensions – assuming such dimensions exist – this book will not explore loop quantum gravity, an alternative to string theory that does not involve extra dimensions […]” And that's the first and last time alternative approaches to quantum gravity are mentioned.