The impact of physics on geometry.

*(English)*Zbl 0657.53054
Differential geometrical methods in theoretical physics, Proc. 16th Int Conf., NATO Adv. Res. Workshop, Como/Italy 1987, NATO ASI Ser., Ser. C 250, 1-9 (1988).

[For the entire collection see Zbl 0646.00009.]

The underlying theme of this lecture is the intimate interaction between geometry and physics, with special emphasis on the benefits bestowed upon the former by ideas and concepts that originated in the latter. At the outset special mention is made of a paper by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)] in which the basic relationship between supersymmetric quantum field theories and topology is described. Many ideas alluded to later in this lecture have their origin in Witten’s paper. A brief review is given of Yang-Mills theory with special reference to the theorems obtained by S. K. Donaldson on the geometry of 4-manifolds and the more recent results on instanton invariants for 3- manifolds due to A. Floer (to appear). This is followed by a discussion of manifolds and modular forms, and the paper concludes with a brief description of new results concerning polynomial invariants for knots obtained by V. F. R. Jones [Bull. Am. Math. Soc., New Ser. 12, 103- 111 (1985; Zbl 0564.57006)].

The underlying theme of this lecture is the intimate interaction between geometry and physics, with special emphasis on the benefits bestowed upon the former by ideas and concepts that originated in the latter. At the outset special mention is made of a paper by E. Witten [J. Differ. Geom. 17, 661-692 (1982; Zbl 0499.53056)] in which the basic relationship between supersymmetric quantum field theories and topology is described. Many ideas alluded to later in this lecture have their origin in Witten’s paper. A brief review is given of Yang-Mills theory with special reference to the theorems obtained by S. K. Donaldson on the geometry of 4-manifolds and the more recent results on instanton invariants for 3- manifolds due to A. Floer (to appear). This is followed by a discussion of manifolds and modular forms, and the paper concludes with a brief description of new results concerning polynomial invariants for knots obtained by V. F. R. Jones [Bull. Am. Math. Soc., New Ser. 12, 103- 111 (1985; Zbl 0564.57006)].

Reviewer: H.Rund

##### MSC:

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

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

81T08 | Constructive quantum field theory |