دانلود کتاب هندسه با ابرتقارن تلاقی می‌کند

In 1982, Edward Witten published his seminal paper [2] where he showed that some classical problems of differential geometry and differential topology such as the de Rham complex and Morse theory can be described in a very simple and transparent way using the language of supersymmetric quantum mechanics. However, according to what I could judge from my conversations with mathematicians, the language of supersymmetry has not become yet a common working tool in this branch of mathematics. The experts working in this field prefer to use more traditional methods. That is a pity because supersymmetry not only allows one to describe in a simple way the results known before, but it also allows one to derive many new mathematical results. And this is the raison d’ˆetre for this book. I try to explain the supersymmetry formalism in a way understandable both for physicists and mathematicians and use it to derive both old and new mathematical results for differential geometry of manifolds. One can make here a general remark. The relationship of mathematics and theoretical physics is obvious. One can recall that geometry was first developed in ancient Babylon as an applied discipline to serve the needs of farmers and tax assessors. Only later people realized that geometry is not only useful but also beautiful and began to play with geometric problems for their own sake. And still later they were surprised to realize that purely abstract, as it seemed, multidimensional geometric constructions turned out to be useful for description of real physical world. The dialog between pure maths and theoretical physics is mutually benefitial — there is no need to list here a vast number of examples justifying this point. Unfortunately, this dialog is now often hampered by a difference of scientific languages used by the two communities. It is difficult for a physicist to understand a paper written by a pure mathematician, even if it is written on a subject that they know fairly well, because it is expressed in a different form, to which they are not accustomed. On the other hand, it is difficult for a mathematician to understand papers written by physicists. At this junction, I would like to make an important comment. I used above the words physics and physicist. But what do they precisely mean? Well, physics is understood in a traditional sense: it is the science studying the fundamental properties of the world we live in. And a physicist has been traditionally understood as a scientist doing physics and studying these laws. Physicists did so in the time of Galileo and Newton and still did a half-a-century ago. That concerned both theorists and experimentalists. A theorist could use complicated mathemical tools, but the essense of his work was to explain the experimental results and make predictions, which his colleage experimentalist could verify. The supreme judge was the experiment, and all the rest, including the logical rigour of the predictions, was not so important. And now the situation is very much different. We practically reached the technological limits in building new accelerators, and high-energy physics faces a serious crisis: there are practically no new interesting experiments and the living link between theory and experiment is practically broken. Under these circumstances, many theorists ceased to study nature, hence ceased to be physicists in a traditional sense of the word and are now studying a multitude of many imaginal worlds using methods that were developed in the last century to study the physical universe. In the absense of experiment, the logical consistency of the derivations rests the only criterium of truth. In fact, a new branch of mathematics has been created, but it is being developed by people who got their education at the physical departments of universities and are often having an experience of working as “physical physicists”. And the mutual Babylonian misunderstanding mentioned above is now, in a considerable extent, the misunderstanding between the scientists working in this new branch of science, one may call it physical mathematics, and the mathematicians working in traditional fields. Such a rift did not exist 200 years ago, while 100 years ago, it was already there but not so deep. It deepened in the middle of the last century after Nicolas Bourbaki published an influential series of monographs in different branches of mathematics written in an extremely austere formal manner. Like it or not, but this rift is now a fact of life that one cannot ignore. Going back to our book, we have written it in a slang representing a mixture of the two languages. We did so in a hope that this slang will be understandable to both communities.