String theory has been criticized since it was first invented but not to the degree that it has now, this criticism mostly focusing on its failure to connect with observation. The criticism has increased dramatically in recent years however, and some of this has been too vituperative to be useful to those curious about string theory as a viable physical theory. But criticism, however harsh, can be healthy, since it motivates the proponents of a theory to more carefully elucidate its foundations and content. This is usually not the case when a theory is popular, as researchers are in a competitive spirit and are hesitant to share the knowledge to possible competitors. At this stage in the game however, string theorists it seems are now on the defensive, and have thus taken the time to discuss in-depth what this reviewer still believes is the most complex and beautiful theory ever constructed in mathematical physics. String theory still has a long way to go before it gains status as being a physical theory, but hopefully by the end of the next few decades one will see the appearance of charts, graphs, and numerical calculations in books on string theory, much like one finds in the most successful of all physical theories to date: relativistic quantum field theory.<br><br>Some highlights in the book that are particularly insightful include:<br><br>1. The observation that Dirichlet boundary conditions (for the open string) break Poincare invariance, but that this leads to the introduction of Dp-branes as positions of the endpoints of the open string. Poincare invariance is recovered as long as Dp-brane is space filling, i.e. has a dimension one less than the background spacetime.<br><br>2. The view that the BRST quantization of the path integral is really a conformal field theory. This is interesting in that BRST analysis is typically thought of as a procedure for quantizing constrained systems (gauge theories being predominant examples).<br><br>3. The `Myers effect'. Sometimes referred to as the `D-brane dielectric effect', it is part of an attempt to understand the physics of non-Abelian D-branes for strong fields. One of the challenges in this understanding involves the validity of the Dirac-Born-Infeld action in these kinds of circumstances, which as the authors remark is designed for situations where the background fields and world-volume gauge fields do not vary appreciably over the distances on the order of the string scale.<br><br>4. The origin of the (classical) Virasoro algebra as the freedom of choice of gauge in the reparametrization symmetry. And along these same lines, the quantization of the Virasoro algebra is defined to the normal ordering of the Virasoro generators, and their commutators give an expression consisting of the ordinary classical term plus a "quantum" correction, the famous central extension. Thus the quantum Virasoro algebra can be viewed as a "quantum deformation" of the classical Virasoro algebra, with the central parameter as being the deformation parameter. This philosophy of deformation has found generalization in what are now called `quantum groups' (even though strictly speaking they are much more complicated objects than ordinary groups).<br><br>5. The connection of the dilaton to the Euler characteristic.<br><br>6. The role of the GSO projection in insuring consistency in the state spectrum.<br><br>7. The use of (vector bundle) K-theory to classify D-brane charges. This use arises when it is realized that the conserved R-R charges cannot be identified with cohomology classes of gauge field configurations. Instead, the D-branes are classified by K-theory classes.<br><br>8. The discussion on `primitive cohomology' and its relation to de Rham cohomology and Hodge theory.<br><br>9. The role of the Born-Infeld structure in ensuring Lorentz invariance of the T-dual description. The Born-Infeld action was once viewed as a mere historical curiosity, namely as a nonlinear generalization of the Maxwell theory, with no experimental backing. That it finds such a natural place in string theory is very interesting (but still of course lacking in experimental support).<br><br>10. The derivation of a lower bound for Newton's constant from heterotic M-theory, which is close to the observed value.<br><br>11. The argument, beautifully elucidated in this book, that type IIA supergravity may be obtained from 11-dimensional supergravity by dimensional reduction.<br><br>12. The discussion on warped space-times and the gauge hierarchy. The authors cleverly motivate this subject by asking why Newtonian gravity follows an inverse-square law rather than an inverse-cube law.<br><br>13. An entire chapter is devoted to "stringy" geometry, which is a fascinating subject given that it touches so many areas of modern mathematics.<br><br>14. The discussion of the `hidden sector' and its conjectured relation to dark matter and supersymmetry breaking.<br><br>15. The author's treatment of the AdS/CFT conjecture is superb and is by far the most interesting part of the book. The dualities shown to exists between gauge theory and string theory are a possible route to a full understanding of nonperturbative quantum chromodynamics, which to this date has defied resolution.<br><br>Some major omissions or discussions that need more elaboration include:<br><br>1. The difficulties that are actually involved in quantizing the Nambu-Goto action. The authors remark that this is due to the presence of the square root, but it would have been interesting if they would have indicated just where the trouble rises explicitly when a quantization procedure is attempted with the Nambu-Goto action. In ordinary quantum field theory, the presence of the square root is interpreted as a "nonlocal" problem, but even there this issue is not usually dealt with in a manner that is very transparent.<br><br>2. A more detailed treatment of string field theory for those readers who want to compare it to what is done in second quantization in ordinary quantum field theory.<br><br>3. The role of the Beltrami differentials in the attaining of a measure for moduli space that is invariant under reparametrizations of the moduli space.<br><br>4. No in-depth discussion of characteristic classes over and above the algebra involved in their manipulation (i.e. the wedge products). An understanding of characteristic classes is crucial to understanding superstring and brane theory, but the pages of this book mislead the unsuspecting reader that there is nothing to characteristic classes except algebraic manipulation of the differential forms. But characteristic classes have a deep geometrical meaning, and obtaining insight into this meaning has been proven to be difficult for students of string theory. This book does not provide any of this insight, nor do any of the other books currently in print on string theory.<br><br>5. Is supersymmetry absolutely necessary for the incorporation of fermions into string theory? The authors seem to argue that it is, but an explicit proof is lacking.<br><br>6. The proof that `threshold bound states' are stable is omitted, disappointing the more mathematically sophisticated reader. As the authors remark, the proof involves a special type of index theory involving non-Fredholm operators, and where one must deal with a continuous spectrum. The usual index theory breaks down since one is only dealing with elliptic operators, and contributions to the index from bosons and fermions do not necessarily have to be integers.<br><br>7. The authors should have included more discussion on mirror symmetry, beautiful subject that it is.<br><br>8. Dp-branes are asserted to be useful in incorporating non-Abelian gauge symmetries in string theory, in that they appear "naturally" as confined to world volumes of multiply-coincident Dp-branes. But is this the best way to introduce these symmetries? Is there a method, other than this one and `compactification', that is just as "natural" and does not have the contrived element that the introduction of Dp-branes sometimes has?<br><br>9. The authors need to elaborate in more detail on the definition of "stable" and "unstable" D-brane.<br><br>10. The omitting of the proof that string theories are ultraviolet finite theories of quantum gravity. This is by far the most serious omission in the book. This reviewer does not know of a reference that proves this assertion, and many in the physics community have pointed to this omission as being a sign that the string theory research community has been misled by false assertions of proof.