M-theory dynamics on a manifold of \(G_2\) holonomy.

*(English)*Zbl 1033.81065The motivation is given by the very fact that \(G_2\) holonomy is the condition for unbroken supersymmetry in four dimensions, on the one hand, and that the behavior at a singularity is always one of the crucial problems in studying supersymmetric compactifications of string theories, on the other hand.

Basically, the authors study the behavior of M-theory on the following three simply-connected types of manifolds: a cone on \(\mathbb{C}\mathbb{P}^3\), \(SU(3)/U(1)\times U(1)\), or \(S^3\times S^3\), as each of these cones can be deformed into a smooth, complete and asymptotically conical manifold \(X\) with a metric of \(G_2\) holonomy. Besides these types of manifolds, also additional examples obtained by taking quotients modulo a finite group are discussed with regard to the behavior of M-theory on them.

As to the utilized techniques for constructing compact manifolds of \(G_2\) holonomy, the reader should consult the recent, very detailed monograph by D. D. Joyce [ Compact manifolds with special holonomy, Oxford: Oxford University Press (2000; Zbl 1027.53052)].

The present treatise is subdivided into six sections, including a brief introduction as Section 1.

Section 2 is entitled “Known examples and their basic properties”. The authors introduce the examples mentioned above, describe their basic properties, and make a proposal for the M-theory dynamics on the first two of them.

Section 3 provides additional evidence for this proposal by relating the inspected manifolds of \(G_2\) holonomy to certain configurations of branes in \(\mathbb{C}^3\) that have been studied by D. Joyce as examples of singularities of special Lagrangian manifolds. Extending their arguments, the authors also give examples of four-dimensional chiral fermions arising from models of \(G_2\) holonomy.

Section 4 gives a subtle analysis of the more complicated example of a cone on \(S^3 \times S^3\). In this example, the authors argue that there is a certain moduli space of theories of complex dimension one which interpolates smoothly, without a phase transition, between three different classical space-time models. This interpolation is precisely described by introducing some natural physical observables, within that concrete example, and the outcome is a just as precise description of the proposed moduli space mentioned above.

Section 5 comes with the title “Role of a fermion anomaly”. Here the authors compare their results derived in Section 4 to some topological subtleties occurring in the membrane effective action within fermion theory.

Finally, Section 6 is devoted to further examples obtained by dividing the previously analyzed manifolds by a finite group. The authors again give a precise description of the moduli spaces in these examples. All together, this both mathematically and physically highly advanced and topical treatise is overflowing with ingenious ideas, stunning approaches, deep insights, guiding initiations, and systematizing orientations in M-theory. Apparently addressed to experts in the field of M-theory, its reading requires a broad knowledge in both complex geometry and quantum field theory. However, the exposition is very lucid and sufficiently detailed, and indeed a highly valuable connection link to the vast recent literature on the subject of M-theory.

Basically, the authors study the behavior of M-theory on the following three simply-connected types of manifolds: a cone on \(\mathbb{C}\mathbb{P}^3\), \(SU(3)/U(1)\times U(1)\), or \(S^3\times S^3\), as each of these cones can be deformed into a smooth, complete and asymptotically conical manifold \(X\) with a metric of \(G_2\) holonomy. Besides these types of manifolds, also additional examples obtained by taking quotients modulo a finite group are discussed with regard to the behavior of M-theory on them.

As to the utilized techniques for constructing compact manifolds of \(G_2\) holonomy, the reader should consult the recent, very detailed monograph by D. D. Joyce [ Compact manifolds with special holonomy, Oxford: Oxford University Press (2000; Zbl 1027.53052)].

The present treatise is subdivided into six sections, including a brief introduction as Section 1.

Section 2 is entitled “Known examples and their basic properties”. The authors introduce the examples mentioned above, describe their basic properties, and make a proposal for the M-theory dynamics on the first two of them.

Section 3 provides additional evidence for this proposal by relating the inspected manifolds of \(G_2\) holonomy to certain configurations of branes in \(\mathbb{C}^3\) that have been studied by D. Joyce as examples of singularities of special Lagrangian manifolds. Extending their arguments, the authors also give examples of four-dimensional chiral fermions arising from models of \(G_2\) holonomy.

Section 4 gives a subtle analysis of the more complicated example of a cone on \(S^3 \times S^3\). In this example, the authors argue that there is a certain moduli space of theories of complex dimension one which interpolates smoothly, without a phase transition, between three different classical space-time models. This interpolation is precisely described by introducing some natural physical observables, within that concrete example, and the outcome is a just as precise description of the proposed moduli space mentioned above.

Section 5 comes with the title “Role of a fermion anomaly”. Here the authors compare their results derived in Section 4 to some topological subtleties occurring in the membrane effective action within fermion theory.

Finally, Section 6 is devoted to further examples obtained by dividing the previously analyzed manifolds by a finite group. The authors again give a precise description of the moduli spaces in these examples. All together, this both mathematically and physically highly advanced and topical treatise is overflowing with ingenious ideas, stunning approaches, deep insights, guiding initiations, and systematizing orientations in M-theory. Apparently addressed to experts in the field of M-theory, its reading requires a broad knowledge in both complex geometry and quantum field theory. However, the exposition is very lucid and sufficiently detailed, and indeed a highly valuable connection link to the vast recent literature on the subject of M-theory.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

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

81T60 | Supersymmetric field theories in quantum mechanics |

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

83E15 | Kaluza-Klein and other higher-dimensional theories |

53C29 | Issues of holonomy in differential geometry |

53C56 | Other complex differential geometry |