Lectures on Seiberg-Witten Invariants (Lecture Notes in Mathematics, 1629) 2nd Edition by John D. Moore (PDF)

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Riemannian, symplectic and complex geometry are often studied by means ofsolutions to systems ofnonlinear differential equations, such as the equa­ tions of geodesics, minimal surfaces, pseudoholomorphic curves and Yang­ Mills connections. For studying such equations, a new unified technology has been developed, involving analysis on infinite-dimensional manifolds. A striking applications of the new technology is Donaldson’s theory of “anti-self-dual” connections on SU(2)-bundles over four-manifolds, which applies the Yang-Mills equations from mathematical physics to shed light on the relationship between the classification of topological and smooth four-manifolds. This reverses the expected direction of application from topology to differential equations to mathematical physics. Even though the Yang-Mills equations are only mildly nonlinear, a prodigious amount of nonlinear analysis is necessary to fully understand the properties of the space of solutions. . At our present state of knowledge, understanding smooth structures on topological four-manifolds seems to require nonlinear as opposed to linear PDE’s. It is therefore quite surprising that there is a set of PDE’s which are even less nonlinear than the Yang-Mills equation, but can yield many of the most important results from Donaldson’s theory. These are the Seiberg-Witte~ equations. These lecture notes stem from a graduate course given at the University of California in Santa Barbara during the spring quarter of 1995. The objective was to make the Seiberg-Witten approach to Donaldson theory accessible to second-year graduate students who had already taken basic courses in differential geometry and algebraic topology.

User’s Reviews

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⭐First of all, this review is for the first edition only. Depending on whether the front matter is being included in the page count, the second edition appears to have either 4 or 15 more pages.This was the second book published on Seiberg-WItten gauge theory, just after John Morgan’s

⭐. Since then, 2 more books devoted to the subject also have been published: Nicolaescu’s

⭐and Marcolli’s

⭐. As Moore states in his preface, the purpose of this book was to make the subject of gauge theory accessible to second-year graduate students who have studied differential geometry and algebraic topology and to prepare them for more advanced treatments, such as that of Morgan. Thus 2/3 of the book is devoted to preliminary material on vector bundles, connections, characteristic classes, hodge theory, spinors, clifford algebras, Dirac operators, and the Atiyah-Singer index theorem, although if a student really has studied differential geometry already (and you really shouldn’t be learning it from this book), vector bundles and connections should be familiar. The third chapter introduces the Seiberg-Witten equations and establishes standard results about their moduli spaces (see my review of John Morgan’s book for a brief explanation of what gauge theory is if you are unfamiliar with this terminology), such as compactness, generic smoothness, and orientability. This then allows the SW invariants to be defined, which are subsequently computed for Kaehler surfaces. Finally, a few topological results on 4-manifolds are deduced, much more easily than with the older Donaldson gauge theory.The preliminary material that is covered in the first 2 chapters is done rather well, with an explicit representation frequently used for the Clifford algebra to make the calculations more clear (in contrast to the more formal presentation of Morgan). The introduction to the Atiyah-Singer index theorem in particular is good, with it being applied to give easy proofs of Rohlin’s theorem, Lichnerowicz theorem, and (a sketch of) the Hirzebruch signature theorem.His coverage of the properties of the SW moduli space is not as thorough as that of Morgan, but he does give a very compact proof of compactness (albeit in the simply connected case only). The treatment of Sobolev spaces and elliptic estimates is rather inadequate, which is an unexpected shortcoming in a book that aims to be an introduction to gauge theory – the reader needs to follow up with his references (such as Freed and Uhlenbeck’s

⭐; Donaldson and Kronheimer’s

⭐is far, far beyond the level of this book). His explanation of how to apply the Sard-Smale theorem to deduce the smoothness of the moduli space for generic perturbations is excellent, something that is sorely lacking in Morgan (and moreover Moore defines the SW equations with the perturbations already included, which avoids repetition). Moore also does a good job of explaining the mechanics of proofs of orientability in gauge theory, probably the most uninteresting part of the theory. He does, however, leave out such important steps as demonstrating that the quotient space in which the moduli space is defined is a smooth Hausdorff manifold (except at reducible points), or proving (rather than just stating) that the invariants are independent of the Riemannian metric on the underlying manifold. His derivation of the homotopy type of the quotient space is clearer than Morgan’s, although he only states it for the simply connected case, which makes a big difference. There is also no discussion at all of wall-crossing formulas, as b+ is assumed to be > 1 in the definition of the invariants, which limits the applicability of the results a little.For the applications of SW invariants, more is covered than in Morgan (but for more restrictive cases), such as a simple proof of (part of) Donaldson’s Theorem that the only negative definite unimodular form represented by a compact smooth simply connected 4-manifold is -I (this proof occupies virtually the entire book of Freed and Uhlenbeck), which shows that not all topological 4-manifolds carry a smooth structure. Finally some invariants for some Kaehler surfaces (in much less generality than in Morgan) are calculated, following a trick that Witten used in his original paper, and as a corollary, an example is found (relying heavily upon other algebraic-geometric references) of a compact 4-manifold with an infinite number of smooth structures.There aren’t an excessive number of typos/errors in the book, but the ones that are present tend to be more apt to confuse. For example, on pg. 57, in the equation between equations 2.10 and 2.11, the letter e appears twice where a gradient sign (a “nabla”) with an e subscript was intended. On pg. 64, in the 4th equation from the bottom, the first term on the RHS should be 2, not 1. On pg. 76, in the third line from the top, the subscript on W should be -, not +. Also on pg. 76, 2 paragraphs up from the Transversality theorem, it should read “codimension being = dim(Ker(D…,” not <= as is stated. Near the bottom of pg. 77, the words "for F=0" should be added after "dF is surjective." On pg. 79, in the line above the last displayed equation, the V should have a subscript 1. And on pg. 80, the word injective should be replaced with surjective (which of course is a big difference).Overall, this is probably the best introduction to Seiberg-Witten gauge theory for those who are not familiar with Yang-Mills/Donaldson theory. It constitutes good preparation for being able to move on to more advanced works, such as Morgan, Marcolli, or the many reasearch papers in the field. On the other hand, for anyone with a stronger background in differential geometry and analysis to begin with, you should be able to breeze through this book very quickly. Nicolaescu's newer and larger book is far more comprehensive and even more oriented toward novices, but it is a bit overly large and difficult to follow, so for a first taste of the subject Moore is probably superior. ⭐This book is a short and elementary introduction to the Seiberg-Witten equations, which created quite a stir back in 1994 when they were first proposed. The book is elementary enough that it could be read by someone without a background in the intricacies of the geometry and topology of 4-dimensional topological and smooth manifolds, but the results can be better appreciated if one already has such a background. A background in quantum field theory, specifically the guage theory of the strong interaction, called quantum chromodynamics, will also help in the appreciation of the book. A lot of work has been done in elucidating the properties of the Seiberg-Witten equations since this book was written, but the book could still serve as an introduction to these developments. The author gives a brief introduction to the use of Seiberg-Witten equations in chapter 1, along with a review of the background needed from the theory of vector bundles, differential geometry, and algebraic topology needed to read the book. All of this background is pretty standard, although the appearance of spin structures may not be as familiar to the mathematician-reader, but completely familiar to the physicist reader. Detailed proofs of the main results are not given, but reference to these are quoted. Also, the theory of characteristic classes is outlined only briefly so no insight is given as to why they work so well. In particular, the reason for the vanishing of the second Stiefel-Whitney class as a precondition for the manifold having a spin structure is not given. In chapter 2, the author goes into the spin geometry of 4-manifolds in more detail. After discussing the role of quaternions in this regard, spin structures are defined. A spin structure on a manifold M, via its cocycle condition, give two complex vector bundles of rank two over M. The complexified tangent bundle can thus be represented in terms of these vector bundles, which are themselves quaternionic line bundles over M. The author also defines spin(c) structures, and shows how, using an almost complex structure, to obtain a canonical spin(c) structure on a complex manifold of complex dimension two. The spin(c) structure also allows a construction of the "virtual vector bundles" W+, W-, and L, for manifolds that do not have a spin structure. These bundles play a central role in the book. Clifford algebra becomes meaningful on the direct sum W of W+ and W-, and spin connections can be defined on W. In particular given a unitary connection on a complex line bundle L over a spin manifold M, one can obtain a connection on the tensor product of W and L. When M is not a spin manifold, this is still possible but one must use the "square" L^2 of L. One can then define the Dirac operator over the sections of this tensor product, which the author does and extends it to one with coefficients in a general vector bundle. The author then discusses, but does not prove, the Atiyah-Singer index theorem and the Hirzebruch signature theorem. These theorems, the author emphasizes, are proved in the context of linear partial differential equations, and give invariants of 4-manifolds. This sets up the discussion in chapter 3, which deals with the problem of how to find invariants of 4-manifolds if one works in the context of nonlinear partial differential equations. Those familiar with the Donaldson theory, which was done using the (nonlinear!) Yang-Mills equations, will understand the difficulties of this approach. The strategy of the nonlinear approach as outlined by the author is to show that the solution set of a nonlinear PDE is compact and a finite-dimensional compact manifold. The solution set depends on the Riemannian metric, but its cobordism class does not, and this may give a topological invariant. The fact that it is defined in terms of a PDE might give a way of distinguishing smooth structures. The Seiberg-Witten theory is one method for doing this. The Seiberg-Witten equations are nonlinear, but the nonlinearity is "soft" enough that it can be dealt with. They arise in the context of oriented 4-dimensional Riemannian manifolds with a spin(c) structure and a positive spinor bundle W+ tensored with L. A connection on L^2 and a section of this spinor bundle are chosen to satisfy these equations, which involve the self-dual part of the connection. One also needs to work with the "perturbed" Seiberg Witten equations, where a self-dual two-form is added. The moduli space of the solutions to the perturbed Seiberg-Witten equations is shown to form a compact finite-dimensional manifold. The proof follows essentially from the Weitzenbock formula, the Sobolev embedding theorem, and Rellich's theorem. Sard's theorem shows that the moduli space is smooth and the Fredholm theory shows it is oriented. The Seiberg-Witten invariants are associated to virtual complex line bundles over the 4-manifold, and when the dimension of the self-dual harmonic two-forms is greater than or equal to 2, and the dimension of the moduli space is even. Their definition does involve the Riemannian metric, but changing this metric only alters the moduli space by a cobordism. It is proved that oriented Riemannian manifolds with positive scalar curvature have vanishing Seiberg-Witten invariants. Kahler surfaces are shown to have positive Seiberg-Witten invariants, and the author proves that there is a compact topological manifold with infinitely many distinct smooth structures. Unfortunately though, an explicit example of one of these is not given. Such an example may be very important from the standpoint of physics, for the behavior of dynamical systems or quantum field theories might be very different for different smooth structures.

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