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Introduction to Superstrings and M-Theory (Graduate Texts in Contemporary Physics) Hardcover – 1 Aug 1999
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From the reviews
Foundations of Physics, on the first edition:
"... the dedicated reader...will be well versed in this fascinating area of theoretical physics."
Physics Today, on the first edition:
"...presents a pedagogical survey on string theory. It covers material from early developments to present-day research ... divided into three parts ... results of quantization, string field theory, and phenomenology ... an impressive effort..."
FOUNDATIONS OF PHYSICS
"Kaku’s book, at 568 pages, is a comprehensive, self-contained text on string theory…[It] contains useful summaries of mathematical topics such as index theory, cohomology, and Kahler manifolds. This is a book for the really serious student of string theory; the dedicated reader who emerges after page 568 will be well versed in this fascinating area of theoretical physics.”
About the Author
Dr. Michio Kaku is a theoretical physicist and the cofounder of string field theory (a branch of string theory), and he continues Einstein s search to unite the four fundamental forces of nature into one unified theory. He is also the New York Times bestselling author of seven books, including his most recent work, Physics of the Future: How Science Will Shape Human Destiny and Our Daily Lives by the Year 2100. --This text refers to the Paperback edition.
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One of the places which it does this is in chapter 5 on multiloops and Teichmuller spaces. The author discusses the Schottky groups, the constant curvature metric formalism, theta functions, and the light cone formalism, the latter of which is dealt with in the context of string field theories in later chapters. The author points out the Schottky problem as one that has been solved and its connection to the parametrizing moduli space by the period matrix for the calculation of loop amplitudes beyond three loops. He does a good job of explaining how to calculate the multiloop amplitude using these different formalisms, particularly the origin of the "period matrix". An explicit formula is given for the multiloop amplitude in terms of the Schottky groups using the Nambu-Goto formalism. The functional integral does not fix uniquely the region of integration in this formalism, and so this region must be carefully truncated to avoid overcounting. This motivates the author to introduce the Polyakov formalism, which, interestingly, makes heavy use of the research of the 19th century on Riemann surfaces. Thus, string theory should not be thought of as a purely 21st century theory that found its way into the 20th, as some have described it. Much of the mathematics it uses comes from the latter half of the 19th century. The author shows how the singularity structure of the multiloop diagram can be expressed in terms of a Selberg zeta function. The redundancy in the path measure under conformal transformations is removed by gauge fixing, Weyl rescalings, and reparametrizations. All of this leads to the moduli space of constant curvature metrics so as to alleviate the problem of overcounting from reparametrization invariance. The moduli space, as usual, is written as Teichmuller space modulo the mapping class group, and the author shows how to relate the variation of the metric tensor to the quadratic differentials. All of these considerations are then generalized to superstrings, with the author showing how the presence of spinors complicates things to a certain extent. The author does mention the supermoduli space in connection with Grassmannians, but unfortunately refers the reader to the literature for further details. He justifies his avoidance of the Grassmannian approach by purusing a field theory of strings. The latter however is just as complicated, although for different reasons.
Another helpful discussion in the book is the one on Kac-Moody algebras and E8. The author motivates well the need for Kac-Moody algebras, namely that of making sense of the complicated spectrum of the heterotic string. The Kac-Moody algebras are first developed in the book in the context of conformal field theory wherein the author introduces the famous vertex operators. In the case of heterotic strings, the author uses the vertex operators to construct a representation of a Kac-Moody algebra that utilizes the Chevalley basis.
The discussion on F-theory, although very short, is also very interesting and helpful considering that most of the mathematical literature on this subject might be too difficult for newcomers to the subject. The author motivates well the need for F-theory, being that of a theory with twelve-dimensional symmetry that is compactified on the torus. F-theories are thus a Type IIB theory with SL(2,Z) modular symmetry. Elliptic fibrations, of much recent interest in the mathematics community, are shown to originate in the (non-perturbative) compactification of a Type IIB theory on a manifold B, via F-theory compactified on an elliptic fibration of the manifold B.
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