Not even wrong. The failure of string theory and the search for unity in physical law.

*(English)*Zbl 1128.81025
New York, NY: Basic Books (ISBN 0-465-09275-6/hbk; 978-0-465-09276-5/pbk). xix, 291 p. 26.95/hbk; $ 16.95/pbk (2006).

This book is interesting and useful in getting to know the active interplay
between the frontiers of physics and those of mathematics. The description is
highly condensed and modest. The first nine chapters deal with a quick review
of modern physics from the birth of quantum theory around the turn between the
19th century and the 20th century to the establishment of the standard model
in the early 1970s. The tenth chapter is devoted to the dazzling interactions
between quantum field theory and mathematics. This chapter is most interesting
to the reviewer. It deals with various topics such as:

- 1.
- Instantons in Yang-Mills theory and in mathematics such as seen in Donaldson theory;
- 2.
- Lattice gauge theory;
- 3.
- t’Hooft’s interesting idea of generalizing quantum chromodynamics from an \(SU(3)\) gauge theory involving three colors to one where the number of colors is some arbitrary large number \(N\) and the corresponding symmetry group is \(SU(N)\);
- 4.
- Two-dimensional quantum field theories including conformal field theories, which culminated in the Wess-Zumino-Witten model in 1983;
- 5.
- Very little being known about the theory of representations of groups of gauge symmetries in four dimensions, and renormalization being much tricker in four dimensions than in two dimensions, while a great deal being known about the two-dimensional case, because things in two dimensions are determined by certain Kac-Moody groups whose representations are well understood;
- 6.
- Witten’s topological quantum field theory in four dimensions whose Hilbert space is the Floer homology of the boundary three-dimensional space, and whose observable quantities are Donaldson’s topological invariants;
- 7.
- The Chern-Simons-Witten theory, which bestowed a Fields medal on Edward Witten in Kyoto in 1990;
- 8.
- A topological sigma model, which has a great deal to do with algebraic geometry
(in particular, the Candelas group’s success in getting a stunning formula,
which gives the number of analytic fields for all degrees at once.

Reviewer: Hirokazu Nishimura (Tsukuba)

##### MSC:

83E30 | String and superstring theories in gravitational theory |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

00A08 | Recreational mathematics |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

00A79 | Physics |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

83-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to relativity and gravitational theory |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

81T45 | Topological field theories in quantum mechanics |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

14J80 | Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |

81T25 | Quantum field theory on lattices |

81V05 | Strong interaction, including quantum chromodynamics |