Mathematics – Logic
Scientific paper
Oct 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.6085p&link_type=abstract
Classical and Quantum Gravity, Volume 23, Issue 20, pp. 6085-6086 (2006).
Mathematics
Logic
Scientific paper
The early 1980s, when I first learned theory, were desperate times for graduate students. We searched frantically for coherent introductions, passing tattered copies of review articles around like samizdat, struggling over obscure references to ancient models of strong interactions, and flocking to lectures—not least those by Joe Polchinski—that promised to really explain what was going on. If only this book had been around, it would have saved much grief.
Volume I, The Bosonic String, offers a clear and well organized introduction to bosonic string theory. Topics range from the 'classical' (spectra, vertex operators, consistency conditions, etc.) to the 'modern' (D-branes first appear in an exercise at the end of chapter 1, noncommutative geometry shows up in chapter 8). Polchinski does not hesitate to discuss sophisticated matters—path integral measures, BRST symmetries, etc.—but his approach is pedagogical, and his writing is lucid, if sometimes a bit terse. Chapters end with problems that are sometimes difficult but never impossible. A very useful annotated bibliography directs readers to resources for further study, and a nearly 30-page glossary provides short but clear definitions of key terms.
There is much here that will appeal to relativists. Polchinski uses the covariant Polyakov path integral approach to quantization from early on; he clearly distinguishes Weyl invariance from conformal invariance; he is appropriately careful about using complex coordinates on topologically nontrivial manifolds; he keeps the string world sheet metric explicit at the start instead of immediately hiding it by a gauge choice. Volume II includes an elegant introduction to anticommuting coordinates and superconformal transformations. A few conventions may cause confusion—%, Polchinski's stress energy tensor, for instance, differs from the standard general relativistic definition by a factor of -2π, and while this is briefly mentioned in the text, it could easily be missed—but these are minor drawbacks.
Readers will find clear answers to many 'frequently asked questions.' Are D-branes really necessary? Polchinski begins with T-duality for the closed string, and shows that the extension to open strings requires the existence of D-branes. How does string theory incorporate gravity? The two standard answers are that string theory contains a massless spin two 'graviton' and that consistent string propagation in a curved background requires that the background metric satisfy the Einstein field equations; Polchinski links the two, showing that the background metric can be viewed as a coherent state of the spin two excitations.
Volume II, Superstring Theory and Beyond, extends Volume I to superstring theory, and then proceeds to treat a range of more advanced subjects: effective actions for branes, dualities and equivalences among string theories, M theory, stringy black holes, compactifications and four-dimensional field theories, and the like. The tone of this volume changes a bit—it is not as self-contained, and reads less like a textbook and more like an extended review article. I suspect, for example, that few students without a strong background in field theory will follow the discussion of anomalies in chapter 12. The change can be largely attributed to the content: the superstring is inherently more difficult than the bosonic string, and the newer material is not as deeply understood. But there are a few weaknesses in presentation as well: for instance, a discussion in chapter 11 of the relationship between symmetries and constraints omits any explanation of how one decides whether a transformation generates a symmetry or a constraint.
Any two-volume book on string theory is necessarily incomplete. In his introduction, Polchinski cites the lack of a more thorough treatment of compactifications on curved manifolds. I would personally have liked to see more about noncritical strings and Liouville theory and about the Green Schwarz superstring, but I will not argue with Polchinski's choices. Indeed, even as it is, this book contains enough material for several courses. (The preface suggests a variety of abbreviated routes through the book for readers with particular interests.)
Given the pace of the subject, any book on string theory is also necessarily outdated from the moment it appears. Polchinski has only a single paragraph on the AdS/CFT correspondence, and of course does not discuss such current hot topics as flux compactifications and the string 'landscape'. But a person who learns string theory from this book will have an excellent background for launching into the more recent developments.
For a more elementary introduction to string theory, a student might want to look at Zwiebach's A First Course in String Theory, and for a treatment of somewhat more recent topics, Johnson's D-Branes is a useful text. But for a clear, comprehensive introduction to string theory, Polchinski's book has no equals.
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