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By Michael Atiyah

Professor Atiyah is among the maximum dwelling mathematicians and is celebrated in the course of the mathematical global. he's a recipient of the Fields Medal, the mathematical identical of the Nobel Prize, and continues to be on the top of his occupation. His large variety of released papers, targeting the parts of algebraic geometry and topology, have right here been accrued into six volumes, divided thematically for simple reference by means of participants drawn to a selected topic. Volumes III and IV disguise papers written in 1963-84 and are the results of a protracted collaboration with I. M. Singer at the Index concept of elliptic operators.

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E. flat connection). The results are not used in any important way in the book (only in the Appendix to Chapter 4). The discussion is not really specific to the Chern–Simons case. Two general contexts in which the ideas are developed are on the one hand the theory of ‘centre manifolds’ in PDE theory and on the other hand in the Kodaira–Spencer–Kuranishi Basic material 28 deformation theory of geometric structures; the structures in our case being flat connections over 3-manifolds. We will first fix some notation.

To obtain the Euclidean Yang–Mills solutions in this framework we merely have to change the sign of the potential, so that for example the trivial flat solution now becomes highly unstable. 3 into the general framework in mechanics reviewed above. Locally in BP we can regard the Chern– Simons function ϑ as a real-valued function and we have | grad ϑ|2 = F 2 = V. e. the Yang–Mills instantons) give solutions to the Euclidean Yang–Mills equations becomes a special case of the general principle above.

We have seen that the flat connections over Y can be viewed as the zeros of the vector field V, the critical points of the Chern–Simons function, and the instantons on the tube are gradient flow lines of this vector field. This is of course reminiscent of a familiar picture in the calculus of variations where one characterises geometric objects as the critical points of a Lagrangian on some infinite-dimensional function space, for example geodesics and the energy function on the space of loops. In such a case also one has an associated evolution equation representing, at least formally, the gradient flow of the Lagrangian.

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