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Mathematical Aspects of String Theory (Advanced Series in Mathematical Physics)

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Although the mathematical constructions used in string theory have changed considerably since this book was published in 1988, it could still serve as an introduction to some of the mathematics of string theory, although the subject currently is dominated by M-theory. In this volume, composed of papers presented at a conference in the summer of 1986, one can see origins of the interplay between the theory of partial differential equations, non-Abelian gauge theories, fiber bundles, algebraic geometry, number theory, representation theory, operator theory, and supersymmetry. All of these developmemts, both in this volume and in current research, involve some of the most fascinating constructions in all of mathematics, and future historians of physics and mathematics will no doubt view this period as one which marked the beginning of an era where pure mathematics dominated the field of elementary particle physics.

Physicists deciding to specialize in string/M-theory will face a mountain of mathematics that must be mastered before entering the frontiers of the subject. Textbooks and monographs on the mathematics though are typically written from a formal point of view, and therefore to get more an understanding of the intuition behind this mathematics, it is important to get involved in the "oral tradition", and attend conferences like the one represented in this book. The questions asked and the motivations presented by the speakers will allow greater insight into the ideas behind these esoteric constructions, and which do not usually find their way in the printed summaries. This book does include a few informal presentations that are helpful, and that therefore give the reader the much-needed motivation.
For the beginning string/M-theorist, the article by Polchinski on relating the Polyakov functional integral to an integral over moduli space of Riemann surfaces would be one of these, as would the article by D'Hoker and Phong on the geometry of quantized string theory. Polchinksi's article is mathematically non-rigorous but helpful from a physics vantage point. The roles of the Fadeev-Popov procedure, zero modes, and the determinants of differential operators in calculating the path integral are made very clear. The D'Hoker/Phong article is very illuminating in that it gives more details on what Polchinski did, and extends it to higher loop amplitudes.

Mathematicians might be interested in the article by Gerd Faltings on arithmetic intersections for surfaces.The article is very short however, quickly overviewing what is being done in Arakelov theory, and because of its level not really suitable for beginners. His goal is to prove a Riemann-Roch theorem for arithmetic surfaces, but of course does not do so in this article because of its length. The article on SU(3) holonomy by Edward Witten might also be of interest, if taken in context of modern developments.