Introduction to Superstrings and M-Theory (Graduate Texts in Contemporary Physics)

Rating:
Summary: Decent but...
Review: I enjoyed this book because it simply portrays how the univeral workings of the universe function. It makes you think just how many forms string can take and in this case, it's one large piece of string. Perhaps, we can use these pieces of string as a source of fuel to help us survive so that more theories of strings can be created.

Rating:
Summary: Helpful in some places
Review: Superstring theory has come a long way since its beginnings in the theory of the strong interaction. The mathematical preparation needed back then was no where near as formidable as it is today, but the experimental motivation then greatly exceeded what is available today in superstrings. Students have to face a mountain of mathematics in order to enter research into superstring theory, and most of this is not explained satisfactorily in the mathematics textbooks and monographs. Therefore, students need to embed themselves in the "oral tradition" of mathematics in order to understand it and gain the insight needed to make original contributions to string theory. This book is somewhat helpful in explaining the mathematics behind string and M-theories, and so the places in which it is will be highlighted in this review.

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.

Rating:
Summary: Decent but...
Review: This is a well written book, but I think it lacks the depth necessary to actually learn string theory from it. I do recommend getting it, but get it along with Polchinski's book and use it as a supplement, something to read to reinforce the main ideas.

Rating:
Summary: a "schaum's outline" of string theory
Review: well, it doesn't exactly have those solved examples as in Schaum's Outline books, but the analogy is close enough for the notes. That means this is a terrible book to learn the subject from if you just barely know quantum field theory, but if you've already been exposed to quite a bit of current research topics, even superficially, here is a very neat set of notes/summaries of some core elements. Recommended for intermediate graduate students as a quick reference.