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**Topological quantum field theories.**
*(English)*
Zbl 0692.53053

The starting point to topological quantum field theory was given by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)] where he explained the geometric meaning of super-symmetry, pointing out that for super-symmetric quantum mechanics the Hamiltonian is just the Hodge- Laplacian. He then outlined the corresponding ideas for super-symmetric quantum field theories viewed as differential geometry of certain infinite dimensional manifolds, including the associated analysis and topology. This shows that there may be interesting topological aspects of quantum field theory and that these should be important for physics. On the other hand one can use these quantum field theories as a conceptual tool to suggest new mathematical results. Indeed this reversed process led to spectacular progress in the understanding of geometry in low dimensions.

The author starts with presenting a set of axioms for topological quantum field theories following G. B. Segal [The definition of conformal field theory, Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165-171 (1988; Zbl 0657.53060)], then lists examples of theories (known to exist) satisfying such axioms. These include in \(d=1\) the Floer/Gromov theory and holomorphic conformal field theories (Segal); in \(d=2\) the Jones/Witten theory, Casson theory, Johnson theory and Thursten theory and in \(d=3\) the Floer/Donaldson theory.

The author starts with presenting a set of axioms for topological quantum field theories following G. B. Segal [The definition of conformal field theory, Differential geometrical methods in theoretical physics, Proc. 16th Int. Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 165-171 (1988; Zbl 0657.53060)], then lists examples of theories (known to exist) satisfying such axioms. These include in \(d=1\) the Floer/Gromov theory and holomorphic conformal field theories (Segal); in \(d=2\) the Jones/Witten theory, Casson theory, Johnson theory and Thursten theory and in \(d=3\) the Floer/Donaldson theory.

Reviewer: R.Schmid

### MSC:

53C80 | Applications of global differential geometry to the sciences |

57N99 | Topological manifolds |

81T99 | Quantum field theory; related classical field theories |

### Keywords:

topological quantum field theories; Floer/Gromov theory; holomorphic conformal field theories; Jones/Witten theory; Casson theory; Johnson theory; Thursten theory; Floer/Donaldson theory
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\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)

### References:

[1] | M. F. Atiyah, New invariants of three and four dimensional manifolds, inThe Mathematical Heritage of Herman Weyl, Proc. Symp. Pure Math.,48, American Math. Soc. (1988), 285–299. |

[2] | S. K. Donaldson, Polynomial invariants for smooth four-manifolds, to appear inTopology. |

[3] | A. Floer, Morse theory for fixed points of symplectic diffeomorphisms,Bull. A.M.S.,16 (1987), 279–281. · Zbl 0617.53042 |

[4] | A. Floer,An instanton invariant for three manifolds, Courant Institute preprint, to appear. · Zbl 0684.53027 |

[5] | M. Gromov, Pseudo-holomorphic curves in symplectic manifolds,Invent. Math.,82 (1985), 307–347. · Zbl 0592.53025 |

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[11] | E. Witten, Super-symmetry and Morse theory,J. Diff. Geom.,17 (4) (1982), 661–692. · Zbl 0499.53056 |

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[13] | E. Witten, Topological quantum field theory,Comm. Math. Phys.,117 (1988), 353–386. · Zbl 0656.53078 |

[14] | E. Witten, Topological sigma models,Comm. Math. Phys.,118 (1988), 411–449. · Zbl 0674.58047 |

[15] | E. Witten, 2 + 1 dimensional gravity as an exactly soluble system,Nucl. Phys. B,311 (1988/89), 46–78. · Zbl 1258.83032 |

[16] | E. Witten, Topology changing amplitudes in 2 + 1 dimensional gravity,Nucl. Phys. B (to appear). |

[17] | E. Witten, Elliptic genera and quantum field theory,Comm. Math. Phys.,109 (1987), 525–536. · Zbl 0625.57008 |

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