The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions by Shing-Tung Yau (PDF)

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String theory says we live in a ten-dimensional universe, but that only four are accessible to our everyday senses. According to theorists, the missing six are curled up in bizarre structures known as Calabi-Yau manifolds. In The Shape of Inner Space, Shing-Tung Yau, the man who mathematically proved that these manifolds exist, argues that not only is geometry fundamental to string theory, it is also fundamental to the very nature of our universe. Time and again, where Yau has gone, physics has followed. Now for the first time, readers will follow Yau’s penetrating thinking on where we’ve been, and where mathematics will take us next. A fascinating exploration of a world we are only just beginning to grasp, The Shape of Inner Space will change the way we consider the universe on both its grandest and smallest scales.

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⭐In 1953, mathematician Calabi raised a question: Suppose a compact Kahler manifold has the vanishing first Chern class, can the manifold admit Ricci-flat metric? In 1977, Yau proved its existence for the first time (Calabi-Yau theorem), and mainly by that finding, Yau won the Fields Medal in 1982. Later, it was found that the manifolds are extremely important in string theory. In 1985, physicists coined the spaces as Calabi-Yau spaces (or manifolds). That is, a Calabi-Yau space is topologically a compact Kahler manifold with the vanishing first Chern class, and geometrically Ricci-flat.In this book, “The Shape of Inner Space” by Shing-Tung Yau and Steve Nadis, the authors explain in detail what a Calabi-Yau space is. In my opinion, if you are at the level of a senior math-undergraduate student interested in physics, you can enjoy the book without much difficulty. I truly recommend the book to such level of readers because, first of all, Yau himself explains Calabi-Yau space from scratch.Before reading the book, by reading Brian Green’s Elegant Universe, I knew that Calabi-Yau manifolds are very important in string theory. The theory claims that our apparently four dimensional space-time is in fact a ten dimensional space-time; one time dimension and nine space dimension. The remaining six spatial dimensions are extremely small so that we cannot see them. Some prominent string theorists including Brian Green proposed that the hidden space is a Calabi-Yau manifold, and the Calabi-Yau space determines the particles in our universe and all of their properties including mass. But up to date, we don’t know exactly what the Calabi-Yau space that constitutes our universe is.To understand what a Calabi-Yau space is, you need to know the notions such as those of compact space, Kahler manifolds, Chern classes, and Ricci-flatness. In the book, the authors explain these notions from scratch. Moreover, the authors describe in detail the following.1. Why did Chern study Chern class?1. Why did Calabi pose the conjecture?2. Before proving the Calabi-Yau theorem, what had Yau been studying?3. How did Yau become interested in Calabi’s conjecture?4. His many unsuccessful attempts to disprove its existence5. His idea to prove its existence6. Many mathematical offshoots of the theorem7. Its unexpected applications to physics. Yau proved the purely mathematical theorem in 1977. Only after 1985, the spaces came to be a focus point in string theory.Here are my comments about detailed points of the book.1. Before reading the book, I didn’t know the following.a. What the Calabi conjecture was.b. What a Calabi-Yau space is.c. What the positive mass conjecture is (now a theorem).d. What geometric analysis is.e. What mirror symmetry is.f. What super-symmetry is, in particular, its position in or relation to string theory.After reading the book, I can say that I understand these things enough to be satisfied. I thank the authors. Moreover, there are many interesting topics for physics manias including Hawking’s Information Paradox, Maldacena’s duality conjecture, and Witten’s M-theory as well as the topics for geometry manias including Donaldson’s work and Seiberg-Witten theory and Schubert’s problems. For an instance, Schubert’s problem was a century-old puzzle in algebraic geometry. It is a problem of counting the number of rational curves (complex 1-dimensional objects) in a quintic threefold which is a Calabi-Yau space (a complex 3-dimensional space). Mirror symmetry is the phenomenon such that two different Calabi-Yau spaces give rise to the same physical world in string theory. The good thing about mirror symmetry is that something is difficult to be handled in one Calabi-Yau space can be easy to be handled in its mirror partner Calabi-Yau space. By applying mirror symmetry, physicists corrected some errors of mathematical experts in the Schubert problem, and furthermore, suggested good directions to research.2. In the book, the authors deal with not only Calabi-Yau spaces, but also various related topics. The main tool for the Yau’s solution was geometric analysis.”The goal of this approach, broadly stated, is to exploit the powerful methods of analysis, an advanced form of differential calculus, to understand geometric phenomena and, conversely, to use geometric intuition to understand analysis.””Approximating the nonlinear world with linear mathematics is a common practice, but, of course, it does nothing to change the fact that the universe is, at its heart, nonlinear. To truly make sense of it, we need techniques that merge geometry with nonlinear differential equations. That’s what we mean by geometric analysis…”I come to know that geometric analysis is the passion of Yau and he is one of the top experts in this area. They explain geometric analysis in a very comprehensive way so that readers can get a deep understanding of its relation to various areas of mathematics.For example, if you are interested in mathematics, you may heard of Poincare’s conjecture and Grigori Perelman’s settlement. The authors introduce the readers to the story. Perelman proved the Poincare conjecture by proving a more general conjecture, that is, Thurston’s Geometrization conjecture. Geometrization conjecture is that any topological compact 3 dimensional manifold without boundary can be cut into pieces such that each piece can admit one of eight geometries suggested by Thurston. Here we see that Calabi conjecture and Thurston’s Geometrization conjecture ask something in common. Given a topological space, does the space admit a particular geometry? Not only the two conjectures are similar in the sense, but also their resolution use geometric analysis in common. In the book, the authors provide a gentle introduction to geometric analysis over several chapters.The application of geometric analysis to 4-manifolds is another example. Around 1980, M. Freedman made a great contribution to the classification of topological classification of 4-manifolds. On the other hand, there are little known facts about geometric structures on 4-manifolds. In 1982, S. Donaldson used nonlinear partial differential equations called Yang-Mills equations to study geometric structures of 4-manifolds.”The Yang-Mills equations operate within the context of four-dimensional space. Rather than just trying to solve the equations, which one would normally attempt by drawing on the geometric and topological features of the underlying space, Donaldson turned the problem on its head: The solution to those equations, he reasoned, should yield information about the four-dimensional space in which they operate. More specifically, the solutions should point to key identifying features — what mathematicians call invariants — that can be used to determine whether four dimensional shapes are different or the same.”The solution of the positive mass conjecture is another example also. It asks if for an any isolated system, its mass (or equivalently, energy due to Einstein) is always positive. Almost physicists believed that it should be. But many mathematicians including M. Gromov believed that it would be not true. Schoen and Yau proved that the answer is yes, using minimal surface theory which is absolutely a branch of geometric analysis. The book provides a nice introduction to minimal surface theory as well.3. The Glossary of the book was helpful and easy to understand. In fact, itself was a good material of reading for me.4. If it is possible to give six stars in this review at Amazon, I could. But the book is also not perfect. At some places, the explanations are vague. In the book, the authors explain what the vacuum Einstein equation is at the beginning part of the book, and after that, they just call it the Einstein equation in many places. For the readers, it would better call its full name. For another instance, they explained the concept of internal symmetry of Kahler manifolds over several pages, but it was hard to understand. I was able to follow their arguments line by line, but fail to understand what the internal symmetry is. For another instance, they introduce John Nash’s embedding theorem as follows.”In the 1950s, John Nash had proved that if you put a Riemannian manifold in a space of high enough dimension, you can get any induced metrics that you want.”I think that although the authors explained about induced metric, if a reader is a real beginner, he would have difficulty in understanding this sentence. Nash’s embedding theorem is that for any Riemannian manifold, you can find a Euclidean space of high enough dimension such that the Riemannian manifold can be embedded as a (geometric) subspace of the Euclidean space.For another instance, when they say “a non-Kahler manifold”, it seems that it does not mean the general sense, that is, a manifold that is not Kahler. It seems to be used in a special sense, that is, a Hermitian manifold that is not Kahler.5. The authors say that Calabi-Yau theorem implies that there is a non-trivial solution to the vacuum Einstein equation. Mathematically, the solutions of the vacuum Einstein equations are metrics whose Ricci curvature vanishes at all points. Physically, the solution represents a universe without matter and energy. There can be trivial solutions and non-trivial solutions. Although the authors omitted the definition, I think that the trivial solution of the equation is the spaces whose Riemann curvature tensor is identically zero. This means that the space is really flat, not curved. If a space has Identically zero Riemann curvature tensor, then the space is Ricci-flat. The converse need not be true in general.But why is that implication true? Why is it true that if there are compact Kahler manifolds with the vanishing Chern class that admit the Ricci-flat curvature, then even if there were no matter and energy in our universe, it is possible that our universe has non-vanishing curvature (non-vanishing Riemann curvature tensor)? The authors seem to think that it is obvious, and so they devote their efforts only in explaining what a Calabi-Yau space is and what the vacuum Einstein equation is.But I am somewhat confused about why that implication is true. Let me see. The trivial solutions seems to be manifolds with the vanishing first Chern class since a space has the zero first Chern class means that the space has no singularity. Any trivial space will have no singularity. Thus, for their implication to hold, the “trivial” solutions of the vacuum Einstein equation must be not Kahler or non-compact. That is, if the Calabi conjecture is true, that is, if there is a “compact” “Kahler” manifolds with the vanishing first Chern class that admit the Ricci-flat metric, we can say that there can be a “non-trivial” solution of the vacuum Einstein equation, so the implication holds. Watch the compactness and being Kahler to be non-trivial.Good. Simple. Now I seem to understand the implication. But in the book, the authors say as follows.”Before my proof, the only known compact spaces that satisfied the requirements set down by Einstein were locally homogeneous, meaning that any two points near each other will look the same. But the spaces I identified were both inhomogeneous and asymmetrical — or least lacking in a sweeping global symmetry, though they were endowed with the less visible internal symmetry we discussed in the last chapter.”In the paragraph, the space described as “both inhomogeneous and asymmetrical or least lacking in a sweeping global symmetry” seems to mean the Kahler manifolds. This paragraph means that before the finding of Yau, there were already known non-trivial solutions of the Einstein vacuum equation. Watch the compactness. But except this phrase, I surely got the impression that Yau was the first one who found the non-trivial solution. This is somewhat confusing. I am afraid that I misunderstood some points.6. There is an extremely interesting chapter in the book. Its title is “The End of Geometry”.”A lake’s surface may look perfectly smooth on a calm, windless day, but that is an illusion. When we examine the surface at higher resolution, it appears jagged rather than smooth. We see that it’s actually composed of individual water molecules that are constantly jiggling around… “”Classical geometry is like this… it only provides an approximate description of nature rather than an exact, or a fundamental, one.””To probe the universe at the level of the hidden dimensions or individual strings, we’re going to need a new kind of geometry — sometimes referred to as quantum geometry…””Geometry, as it appears in physics might be a phenomenon that’s “emergent” rather than fundamental.””You cannot look at waves on the surface of a lake and, from that, deduce anything about the molecular structure… On the other hand, if you know where all the molecules are and how they’re moving, you can in principle deduce everything about the body of water and its surface features. The microscopic description, in other words, contains the macroscopic information. That’s why we consider the microscopic description to be more fundamental, and the macroscopic properties emerge from it.”For example, from the temperature and pressure of a gas, we cannot deduce anything about the gas molecules. But if you know the positions and velocities of all the molecules of the gas, then you can deduce the temperature and pressure of the gas as Boltzmann really did that about one hundred years ago. In some sense, temperature and pressure are illusions. Some physicists seem to believe that time and gravity are illusions also. Very interesting. Then what is it from which geometry is emergent? According to the authors, we don’t know yet, but studying Calabi-Yau spaces may give us several hints to the right road.””What we learned is that when the classical geometry of the Calabi-Yau appears to be singular, the four-dimensional physics looks smooth,” explains Aspinwall. “The masses of particles do not go to infinity, and nothing bad happens.” So the quantum geometry of string theory must somehow have a “smoothing effect” taking something that classically looks singular and making it nonsingular.”7. In the book, the authors don’t pretend to be modest. They try to show full range of the meanings and possibilities in the applications of Calabi-Yau space to mathematics and physics. But as much as that amount, their position seems to be open to criticism, and so healthy. For an instance, up to now, we don’t have the Calabi-Yau space where we actually live in. There have been a lot of criticisms about that. Some physicists claim that the hidden space may be not a Calabi-Yau space. Instead, they say we have to investigate other spaces similar to Calabi-Yau spaces, but they need not be exactly Calabi-Yau. About this, the authors quote Strominger’s remark.”Although Calabi-Yau spaces may not be the ultimate destination, they may well be stepping stones to the next level of understanding”I believe that most of the readers would absolutely agree with Strominger’s remark.

⭐This is one of the best pieces of popular science writing I’ve ever run across — and I’ve read hundreds of them. It is superbly, clearly and engagingly written, but it is also one of the most conceptually sophisticated and difficult popularizations I’ve read.In many ways, it’s an autobiography of S-T Yau, a Harvard mathematician, but it’s as much a biography of Calabi-Yau spaces and their role in String Theory. I give Yau and his co-author enormous credit for making a very complex subject understandable (albeit only dimly — I’m not that good a mathematician!) and even entertaining.I’d be overjoyed to find other books that go into scientific material to the same depth while also being so readable.

⭐I truly appreciated the authors’ efforts to describe their contribution to the unification effort in physics in a way accessible to a determined reader. Thanks! There is lots of enjoyable material in this book.Years ago I’d read in popular science magazines about the successes that physicists had had in explaining and unifying the basic forces of nature other than gravity. So much imagination and intellect! To bring in gravity the space that we live in had to be highly curved but it wasn’t so that appeared to be that. The authors here have addressed this. Also way back I’d taken a class in the math area differential geometry, which did not go well, but it did give me a new understanding about geodesics and parallel transport.The first chapters in the book referenced almost every concept I’d seen in that course. If you are up for a review of math and physics topics that you saw years ago in school, then not again. then go for it. If it is going to be new material then still go for it, it’ll be worth it.

⭐What a privilege to have one of the leading experts of higher-dimensional geometry (the Yau of Calabi-Yau spaces) guide us along the boundary between String theory and mathematics. It’s a personal journey and history of String theory as well, with clear explanations of the concepts introduced from math and physics along the way.String theory has been criticized as not making testable predictions, but it has had a huge impact in revitalizing several branches of mathematics, solving longstanding problems, and providing mathematicians with powerful new tools.The book is up to date, and includes current problems, speculations, and research directions.

⭐I had read this material before under another cover; it may contain updates on dark energy and the LHC. Shing-Tung Yau has been immersed in Geometrical Physics a long time and it’s simply amazing all he has been involved with and I enjoyed reading about it all again. If you have not read other contemporary books for laymen on string theory and Topology etc., you may have difficulty treading through some of the weirdness, one thing after another to learn, remember and swallow as good science; not pure fiction. Some familiarity with what the Calculus is about (not actually required to do any) as well as how algebraic equations relate to geometry is very helpful. No actual mathematical manipulations or proofs appear to worry about. Weird ideas and theories are another matter. Start reading this stuff, it grows on you. This material has been out there for awhile and it’s still fascinating, especially as told by Yau. It would be hard to find another as authoritative, yet accessible to the general reader.

⭐As someone who did an astrophysics degree in the late seventies, I’ve tried over the years to stay in touch with developments in cosmology by reading the odd popular science book now and again. Inevitably, since the string theory revolution of the eighties, that has involved digesting a few of the more or less well known lay-expositions of string theory, and its associated ideas. With each such book I have been left with a dissatisfied feeling that I would like to have been given just a bit more of the relevant conceptual mathematical framework. Well, with this book I got my wish, and its fair to say that I’ve come away understanding about 10% of what I’ve read. It is apparent that my rusty undergraduate physics level maths doesn’t even get you to the front door of where string theory picks up from.Yet, despite all this, I found the book to be compelling, and I found a way of reading it that allowed me to take much of what passed on faith, and just enjoy the handful of concepts and images that I did manage to abstract from the flow. Somewhat like watching for patterns in clouds or fire. There is a surface level story of extraordinarily gifted people, mathematicians and physicists, all attending conferences, then beavering away at impossibly difficult proofs and calculations. The history of who proved or calculated what, when, thus enabling whoever else, to prove or calculate whatever came next is, to someone like me, a plausible human interest story with a certain level of excitement. But with regard to the maths, then, if I am honest, it was really just a question of hanging on by the coattails. I may vaguely be able to remember the significance of a complex Riemann surface, and I may be able to follow the simple 3D-or-less analogies that help me to conceptualise compact Kähler metrics with vanishing first Chern class. But I have no choice but to accept on faith the significance of whether an object combining these properties has a Ricci-flat metric, the Calabi conjecture, along with all the mathematical paraphernalia that the author draws upon to prove it. Likewise the ultimate utility to physics and string theory of the class of objects, the Calabi-Yau manifolds, that his proof bought to light. I cannot visualise these things, and indeed no one can. At this level there are just the equations, and the rules of their transformation into other equations, but there are no equations in this book, and even if there were, I would not be able to understand them. There is just too much background required. Still the grist of the book sparked a rich flow of insights, however trivial, to engage and, at times, delight my humble brain. Of particular interest was the description of how the controversial missing entropy might be `encoded’ into black holes, through configurations of higher dimensional geometries.So, that’s what I think awaits the prospective reader. I can’t conceive of having any kind of traction on this book without some level of maths background, but I don’t know just how much maths you would need in order to be able to claim that you had understood everything you read. I suspect as much as would be needed to start delving into the actual equations. But, on the other hand, surprisingly, one doesn’t need a complete grasp of everything in order to get a quite rewarding experience out of the book.Finally, there is the question of what, having lifted the bonnet an inch on the mathematical nuts and bolts of string theory, do I think of string theory as a basis for progress in cosmology? Well, I think I’m slightly appalled actually. When I was studying cosmology, a theory of everything seemed just around the corner, and notions of aesthetic mathematical beauty still proved a useful guide that ultimately led to the building of the Standard Model we have today. It was thought for a long while that the final theory, when it came, would be brief and breathtakingly elegant. I’m sure that to mathematicians Calabi-Yau manifolds and their geometric ilk are beautiful things. But, that such a huge amount of mathematical apparatus should be required to describe the geometry of reality seems rather unpalatable and intuitively unlikely. The hallowed relation between truth and beauty seems to have left by the back door. Still, who am I to judge such esoteric matters?P.S. Congratulations to reviewer Nigel Seel for his superb outline summary of the book’s main argument.

⭐Having read a couple of String Theory pro and con books by physicists, I thought this might provide me with a less partisan perspective, though I was a bit worried I might have to give up on the maths, as I did with the (admittedly excellent) ‘The Road To Reality’.But no need to worry! Rather than reams of equations and ‘homework’ problems this was much more of a narrative account of this mathematician’s exploration of the (his) geometrical base for the theory with virtually no maths. Excellent analogies were widely used, which I found unusually helpful too. Humour, historical background, and relationships with other researchers were also liberally used, which made it a very entertaining and informative book. Although narrative, it did cover quite a lot of the more abstruse aspects, but in an approachable way.As a collaborative effort of two writers it was seamless too, and not disjointed at all.So it seemed a pretty unbiased account by a mathematician who had only a limited investment (though a considerable and genuine one as he one a principal developer of the maths involved) in whether the maths fitted a particular physical reality. Quite happy that the maths field had received tremendous support from string theory work, yet not heavily committed to it except as a mathematical structure. If it didn’t work out there, well it might somewhere else, and in any case the maths was the main thing – no research grant dependency here!Provided an interesting perspective on the perennial question of ‘why does mathematics describe the physical world so well?’ Doesn’t actually answer it of course, but does give a bit of different food for thought.I found it quite a ‘foil’ to the previous books I had read, and would recommend it to anyone who has a passing sympathy with the mathematical enterprise and string theory. A longish book but VERY easy to get through.

⭐Although this book emanates from a mathematical context, it has been co-authored by Shing-Tung Yau and Steve Nadis and impeccably offers two complementary perspectives.In doing so it provides a fascinating and unusual account not normally described in many general physics books extolling the viewpoint of the mathematician over that of the physicist, highlighting the fundamental differences in approach each takes to similar physical problems and how mathematics manages to overcome many of the obstacles faced by the physicist in explaining Quantum reality. This is incredibly interesting to discover and not something I have seen elsewhere.However, the main premise of the book leaves you utterly breathless as both Steve Nadis and Shing-Tung Yau take us on a journey never before presented in a way that opens up the mysteries of Calabi-Yau manifolds and ten and eleven dimensional spacetime. Shing-Tung Yau describes the heartache as well as the ecstasy of discovering the intricacies of multi dimensional spaces.There is little in the way of mathematics other than fairly straight-forward schoolboy algebra and precious little of that. However, the physical concepts he describes, textually, whilst written in plain English, are truly astounding and well worth taking the ride to discover.This is a truly worthwhile read. Please do not be put off by others attempts to place this book in an elite league that can only be understood by physics professors. Nothing could be further from the truth. The only perquisite to enjoyment of this book is an open mind and a healthy desire to learn. Please enjoy this book for what it is, an outstanding view of quantum reality.

⭐This book, from a mathematician, covers the period from the first proof that Calabi-Yau spaces actually might exist to their current central place as a preferred model for String Theory’s extra dimensions. Shing-Tung Yau is the Fields Medallist godfather of the eponymous manifolds and Steve Nadis had the unenviable task of writing it all down so that the rest of us could have a prayer of understanding it. He also did the interviews and fleshed out the physics side. The best way to review this book is just to explain what it says chapter by chapter.Chapter 1: The universe is a big place, maybe infinite. Even if its overall curvature suffices to close it, observations suggest that its volume may be more than a million times the spherical volume of radius 13.7 billion light year we actually see. The unification programme of theoretical physics doesn’t really work, however, if it’s confined simply to three large spatial dimensions plus time. It turns out that replacing the point-like objects of particle physics with tiny one-dimensional objects called strings, moving in a 10 dimensional spacetime may permit the unification of the electromagnetic, weak and strong forces plus gravity. Well, today it almost works.We see only four space-time dimensions. Where are the other six? The suggestion is that they are compactified: rolled up to be very small. But that’s not all, to make the equations of string theory valid, the compactified six dimensional surface must conform to a very special geometry. That is the subject of the rest of the book.Chapter 2: Yau was born in mainland China in 1949. His father was a university professor but the pay was poor and he had a wife and eight children to support. When Yau was 14 his father died leaving the family destitute: Yau’s destiny seemed to be to leave school and become a duck farmer to pay his way but in a flash of inspiration he decided instead to become a paid maths tutor, teaching as he was learning. Yau’s astounding talent led him from this humble background to the University of California at Berkeley by the time he was 20. As well as autobiographical details, this chapter also outlines the idea of a metric on curved spaces, introducing Einstein’s theory of gravity.Chapter 3: Yau’s early work at Berkeley was in the area of geometric analysis, used in the proof of the Poincare conjecture (1904). This conjecture states that a compact three dimensional space is topologically equivalent to a sphere if every possible loop which can be drawn in that space can be shrunk to a point without tearing. The conjecture was proved in 2002 by the controversial Russian mathematician Grisha Perelman. Work in this area set the scene for Yau’s celebrated proof of the Calabi conjecture: that what subsequently became known as `Calabi-Yau’ (CY) spaces actually exist.Chapter 4: The Calabi conjecture is simple to state if not to understand: it asks whether a complex Riemann surface (conformal, orientable) which is compact (finite in extent) and Kähler (the metric is Euclidean to second order) with vanishing first Chern class has a Ricci-flat metric. All these concepts are explained in this chapter. One of the more interesting features of a space satisfying Calabi’s conjecture (if it existed) was that it would satisfy Einstein’s vacuum field equations automatically.Chapter 5. Yau initially didn’t believe the Calabi conjecture and at a conference held at Stanford in 1973 went so far as to give a seminar “disproving” it. Calabi contacted Yau a few months later asking for details and Yau set to furious work, the argument slipping out of his hands the harder he tried to make it rigorous. Yau concluded that in fact the conjecture must be correct and spent the next three years working on the problem. In 1976 he got married and on his honeymoon the last piece of the puzzle dropped into place. The conjecture was proved correct.Chapter 6. What Yau had proved was a piece of mathematics but he was sure there must be applications in theoretical physics. However, nothing happened until 1984. Parallel developments in string theory (ST) had determined that ten dimensions were needed to allow sufficiently diverse string vibrations to occur to capture the four fundamental forces and to induce `anomaly cancellation’. The search was on for a six dimensional compactified space to complement four dimensional space-time. The chapter describes how physicists came to CY spaces via supersymmetry and holonomy.CY manifolds within ST are very small (a quadrillion times smaller than an electron) and are riddled with multidimensional holes (up to perhaps 500). The way strings wrap around the CY surface, threading through holes, is intended to reproduce observed particles and their masses. This has proven a fraught task as it requires a very special CY manifold to even get close. Yau has estimated there might be 10,000 different manifolds but no-one really knows.The chapter closes with a discussion of M-theory, Edward Witten’s framework for uniting the five different string theories developed in the 1990s. M-theory is defined in 11 dimensions and includes `branes’ of anything from 0-9 dimensions. Apparently the universe could have 10 and 11 dimensions simultaneously but the mathematics (via CY spaces) works better in 10.Chapter 7 discusses a challenge to the applicability of CY spaces due to the quantum field theory requirement for conformal and scale invariance. The CY metric doesn’t (without tweaking) allow for this. This research led to a concept called mirror symmetry which associates CY manifolds with distinct topologies with the same Conformal Field Theory (CFT). This proved important for calculation.Chapter 8 talks about the success of ST in deriving the Bekenstein-Hawking formula for (supersymmetric) black hole entropy. The very large number of required black hole microstates are constituted by wrapping branes around sub-surfaces of a CY manifold to build the black hole. The chapter ends by extending these ideas to the celebrated AdS/CFT correspondence.Chapter 9 notes that ST has yet to reproduce the Standard Model (SM) and recounts some of the attempts being made. Yau’s favourite is E8 x E8 heterotic ST and the technique is to break the many symmetries of E8 down to the 12 required by the SM [SU(3) with 8D symmetry, 8 gluons; SU(2) with 3D symmetry, W+, W-, Z; U(1) with 1D symmetry, photon]. We are not there yet.Chapter 10 talks about mechanisms to keep the compactified dimensions small when energetically they would prefer to be large. The CY manifolds are stabilised by quantised fluxes. Suppose there are 10 values (0-9) for a flux loop and 500 holes in a CY manifold then there are 10 ** 500 different stable states. This extraordinary crude estimate has been widely publicised as “The Landscape Problem” for those who were hoping that there would be exactly one CY model for the universe. Yau is unimpressed, never having believed in such uniqueness in the first place. Chapter 11 continues the theme of `explosive decompactification’ and recommends not being around if and when it happens.Chapter 12 surveys the search for hidden dimensions. They may be visible `out there’ for telescopes to pick up. Alternatively there’s the LHC. Chapter 13 is an essay on truth and beauty in mathematics.The final chapter raises a deep question. CY manifolds are solutions to Einstein’s gravitational field equations in a vacuum. But Einstein’s theory is classical – smooth all the way down (except for rare singularities). However, the QM view of space-time at the Planck scale is anything but smooth: the term `quantum foam’ has been coined. What kind of geometry – quantum geometry – could model this?Yau’s view is that at present no-one has much of a clue although he describes some ideas exploring CY topology changes via singularity introduction – the flop transition -which could shed some light on what quantum geometry could look like.In summary this is not a book for the faint-hearted. It gives a mountain-top view of the research area which is Calabi-Yau theory and its application to String Theory. One never forgets however how much inaccessible mathematics and physics lies behind Steve Nadis’s persuasive and fluent writing.

⭐the cover looks nice

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