An Introduction to String Theory and D-Brane Dynamics

Rating:
Summary: A little short but does its job.
Review: From a mathematical perspective, string theory, and its modern metamorphosis, M-theory, is the most complex physical theory ever constructed. From a physical and experimental perspective, these theories completely lack any support. Mathematical elegance and the need for a consistent physical theory have driven the research in string theory, and to this day it remains one of the hottest, if not the most esoteric, topics in theoretical physics. Those physicists with a penchant towards mathematics have joined the ranks of those doing research in string theory. Mathematicians are also benefiting immensely from the insights that string theory offers to myriads of concepts and results in mathematics.

This book gives a very quick overview of the main results in string and M-theory, and would be suitable only for those readers who have had a lot of prior exposure to the subject. There are no in-depth explanations given for the physical and mathematical concepts needed in string and M-theories in the book, and therefore it might be difficult for the newcomer curious about these theories to gain an appreciation of them. The mathematics behind these theories is formidable, requiring years of study to digest, and the insight and motivation behind this mathematics is usually not given in the literature, unfortunately.

After a brief discussion of the history of string theory in chapter 1, and also a very brief discussion of the classical dynamics of strings in chapter 2, the author begins a study of how to quantize the bosonic string in chapter 3. This is done using the familiar canonical quantization of quantum field theory but here applied to the (1+1)-dimensional worldsheet field theory. The mass-shell constraints of classical string dynamics appearing as Virasoro operators are subjected to normal ordering in the quantization procedure. The origin of the bosonic critical dimension of spacetime as 26 is not explained in enough detail that will allow the reader to appreciate it. Also discussed, but only briefly, are the `vertex operators', which have become very important recently, especially in mathematics. In this book vertex operators are introduced as an analogy to the operator-state correspondence that is found in ordinary quantum field theory.

Superstring theory is studied in chapter 4, motivated by the need for eliminating tachyonic states and for including fermions in the spectrum. The Ramond-Neveu-Schwarz (RNS) and light-cone Green-Schwarz formalisms are mentioned as two techniques for dealing with superstrings, but the author only uses the RNS formalism in the book. The role of boundary conditions as the origin of the Ramond (R) and Neveu-Schwartz (NS) sectors is explained very well, but the author leaves to the reader (as an exercise) the canonical quantization of the superstring. The origin of the superstring critical dimension as 10 is thus delegated to the reader. The ubiquitous `GSO projection' is introduced as a device for making the theory of interacting strings consistent, one example being the elimination of tachyonic states. The GSO projection is discussed in both the NS and R sectors. The modular invariance of the bosonic string partition function is left as an exercise for the reader. The author does explain well the origin of `spin structures', i.e. their connection with the introduction of fermions, and the consequent use in the superstring theory to get rid of the tachyonic instability. He gives brief discussions of the five different types of string theories, but restricts himself to only Type II superstrings in the remainder of the book. The origin of the famous `T-duality' for closed strings and its relation to the existence of D-branes in superstring theory is explained very well. The author assigns a very interesting exercise for the reader on showing that T-duality interchanges the definition of normal and tangential derivatives, and therefore exchanges Neumann and Dirichlet boundary conditions. This exercise, in this reviewer's opinion, should eliminate sometimes held view of T-duality as being somewhat mysterious. D-branes are explained as being essential for superstring theory, in that there are missing R-R charges in the perturbative string states, i.e. the vertex operators for the R-R states only involve the fields. The D-branes are thus nonperturbative states that carry the R-R charges.

D-branes and their (fascinating) relation to gauge theory are discussed in detail in chapter 6, their dynamics in chapter 7, and their R-R couplings in the last chapter. D-branes are described nonperturbatively, with the massless modes of open strings equated to the fluctuations of D-branes. Massless fields are interpreted as a 10-dimensional gauge theory on the D-brane worldvolume. The guage fields have components as U(1) gauge fields on the D-brane as well as scalar field components that describe the fluctuations of the D-brane. The gauge theory is actually, and most interestingly, a dimensional reduction to the D-brane of supersymmetric Yang-Mills theory. Via a consideration of Wilson lines of the gauge fields, the author shows how T-duality maps gauge fields in open string theory to positions on the D-branes. The dynamics of D-branes is further described in terms of (supersymmetric Yang-Mills) gauge theory, giving the famous AdS/CFT correspondence. This correspondence is quite exciting if one views it from the standpoint of how difficult it is to do nonperturbative calculations in gauge theories. Interactions between D-branes are studied, and the author describes the coincidence (resulting from spacetime supersymmetry) between the D-brane tension and the R-R charge, i.e. that the R-R repulsion between parallel D-branes cancels their gravitational and dilaton attraction. A brief discussion is given of `BPS states' and their relation to D-branes, i.e. that a D-brane is a state that preserves only half of the spacetime supersymmetries. In addition, and similar to the case in gauge field theories where chirality is present, anomalies can arise in D-branes. These arise, as the author shows, on the chiral worldvolume field theory on the intersection of two or more D-branes. Requiring anomaly cancellation will determine completely the coupling between the D-brane and the fields of the R-R sector.