Supergravity.

*(English)*Zbl 1245.83001
Cambridge: Cambridge University Press (ISBN 978-0-521-19401-3/hbk; 978-1-139-36806-3/ebook). xiii, 607 p. (2012).

This book on supergravity starts by reviewing aspects of relativistic field theory in Minkowski spacetime. After introducing the relevant ingredients of differential geometry and gravity, some basic supergravity theories for \(D=4\) and \(D=11\), \(D\) being the dimension of space-time, and the main gauge theory tools are explained. In the second half of the book, complex geometry and \(N=1\) and \(N=2\) supergravity theories are covered. Classical solutions and a chapter on the anti de Sitter/conformal field theory (AdS/CFT) duality complete the book.

What is not covered here: theories in \(D < 4\), higher derivative actions, and fully quantized theories. In this sense, supergravity is interpreted quite classically in this book.

The 23 chapters and 2 appendices are structured as follows: Chapter 1 deals with the scalar field, its action and the corresponding wave equation. Chapter 2 introduces the Dirac field and the related spinor equations. Chapter 3 presents the Clifford algebras and their relation to spinors. Chapter 4 is on the Maxwell and the Yang-Mills equations. Chapter 5 presents the Rarita-Schwinger field and the several variants for massive gravitinos. Chapter 6 is on basic supersymmetric (SUSY) field theory. These six chapters refer solely to field theories in the Minkowski space-time, i.e., to special relativity as background theory.

Chapter 7 is a short compendium on differential geometry and ends with a section on the non-linear \(\sigma\)-model. Chapter 8 introduces the Einstein field equation both in the first order and in the second order formalism.

With these 8 introductory chapters, the reader is prepared to get to the core of the book: Chapters 9 to 12 present the several variants of supergravity, its algebras, its local and global symmetries, and the dependences of its properties on the dimension of space-time.

Chapters 13 and 14 deal with the same items, but here based on complex geometry. Chapters 15 to 17 concentrate on conformal symmetries related to supergravity. Chapters 18 to Chapter 21 present several applications, and Chapter 22 gives classical solutions of gravity and supergravity. The final chapter 23 deals with the anti de Sitter/conformal field theory (AdS/CFT) duality.

Appendix A fixes the notation for space-time, spinors, and differential forms, whereas Appendix B summarizes Lie algebras and superalgebras and their representations. The reference list contains 365 items. Finally, a subject index closes this very carefully presented monograph.

A website featuring solutions to some exercises and additional reading material can be found at http://www.cambridge.org/supergravity.

In [the authors, “Ingredients of supergravity”, Fortschr. Phys. 59, No. 11-12, 1118–1126 (2011; Zbl 1243.83004)], one can read that the book project has been presented in lectures given in Corfu in 2010, and that feedback from the audience has been incorporated into the present book.

What is not covered here: theories in \(D < 4\), higher derivative actions, and fully quantized theories. In this sense, supergravity is interpreted quite classically in this book.

The 23 chapters and 2 appendices are structured as follows: Chapter 1 deals with the scalar field, its action and the corresponding wave equation. Chapter 2 introduces the Dirac field and the related spinor equations. Chapter 3 presents the Clifford algebras and their relation to spinors. Chapter 4 is on the Maxwell and the Yang-Mills equations. Chapter 5 presents the Rarita-Schwinger field and the several variants for massive gravitinos. Chapter 6 is on basic supersymmetric (SUSY) field theory. These six chapters refer solely to field theories in the Minkowski space-time, i.e., to special relativity as background theory.

Chapter 7 is a short compendium on differential geometry and ends with a section on the non-linear \(\sigma\)-model. Chapter 8 introduces the Einstein field equation both in the first order and in the second order formalism.

With these 8 introductory chapters, the reader is prepared to get to the core of the book: Chapters 9 to 12 present the several variants of supergravity, its algebras, its local and global symmetries, and the dependences of its properties on the dimension of space-time.

Chapters 13 and 14 deal with the same items, but here based on complex geometry. Chapters 15 to 17 concentrate on conformal symmetries related to supergravity. Chapters 18 to Chapter 21 present several applications, and Chapter 22 gives classical solutions of gravity and supergravity. The final chapter 23 deals with the anti de Sitter/conformal field theory (AdS/CFT) duality.

Appendix A fixes the notation for space-time, spinors, and differential forms, whereas Appendix B summarizes Lie algebras and superalgebras and their representations. The reference list contains 365 items. Finally, a subject index closes this very carefully presented monograph.

A website featuring solutions to some exercises and additional reading material can be found at http://www.cambridge.org/supergravity.

In [the authors, “Ingredients of supergravity”, Fortschr. Phys. 59, No. 11-12, 1118–1126 (2011; Zbl 1243.83004)], one can read that the book project has been presented in lectures given in Corfu in 2010, and that feedback from the audience has been incorporated into the present book.

Reviewer: Hans-Jürgen Schmidt (Potsdam)

##### MSC:

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

83C45 | Quantization of the gravitational field |

83E50 | Supergravity |

83A05 | Special relativity |

83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |

83E30 | String and superstring theories in gravitational theory |

81T20 | Quantum field theory on curved space or space-time backgrounds |

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

15A66 | Clifford algebras, spinors |

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

83C22 | Einstein-Maxwell equations |

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

53Z05 | Applications of differential geometry to physics |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

53B35 | Local differential geometry of Hermitian and Kählerian structures |

83F05 | Relativistic cosmology |