Review of: The Shape of Inner Space – Shing-Tung Yau and Steve Nadis.
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.