The Seiberg-Witten equations and 4-manifold topology.

*(English)*Zbl 0872.57023Since 1982 the use of gauge theory, in the shape of the Yang-Mills instanton equations, has permeated research in 4-manifold theory. In the last part of the year 1994 this research area was turned on its head by the introduction of a new kind of differential-geometric equation by Seiberg and Witten. In the space of a few weeks long-standing problems were solved, new and unexpected results were found, along with simpler new proofs of extending ones, and new vistas for research opened up.

The paper under review is a report on some of these developments specially due to P. B. Kronheimer and T. S. Mrowka [Bull. Am. Math. Soc., New Ser. 30, 215-221 (1994; Zbl 0815.57010); Math. Res. Lett. 1, 797-808 (1994; Zbl 0851.57023)] and C. H. Taubes [Math. Res. Lett. 1, 809-822 (1994; Zbl 0853.57019); 2, 9-13 (1995; Zbl 0854.57019)] building on the seminal work of N. Seiberg [Phys. Lett. B 318, 469 (1993)] and N. Seiberg and E. Witten [Nucl. Phys. B 431, 581-640 (1994)]. It is written as an attempt to take stock of the progress stemming from this initial period of intense activity.

In conclusion a wonderful work which will be useful to all those interested in 4-manifold theory.

The paper under review is a report on some of these developments specially due to P. B. Kronheimer and T. S. Mrowka [Bull. Am. Math. Soc., New Ser. 30, 215-221 (1994; Zbl 0815.57010); Math. Res. Lett. 1, 797-808 (1994; Zbl 0851.57023)] and C. H. Taubes [Math. Res. Lett. 1, 809-822 (1994; Zbl 0853.57019); 2, 9-13 (1995; Zbl 0854.57019)] building on the seminal work of N. Seiberg [Phys. Lett. B 318, 469 (1993)] and N. Seiberg and E. Witten [Nucl. Phys. B 431, 581-640 (1994)]. It is written as an attempt to take stock of the progress stemming from this initial period of intense activity.

In conclusion a wonderful work which will be useful to all those interested in 4-manifold theory.

Reviewer: M.Puta (Timişoara)

##### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

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

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\textit{S. K. Donaldson}, Bull. Am. Math. Soc., New Ser. 33, No. 1, 45--70 (1996; Zbl 0872.57023)

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