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The Theory of Gauge Fields in Four Dimensions (Cbms Regional Conference Series in Mathematics)

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Lawson's "The Theory of Gauge Fields in Four Dimensions" is somewhat of a forgotten book. Though it treats almost exactly the same material as Freed & Uhlenbeck's Instantons and Four-Manifolds (F&U), is almost as easy to understand, and is still in print, it no longer is cited much in the gauge theory literature. But considering that F&U is OOP and consequently sometimes expensive, it is a worthwile substitute with even a few advantages over its more famous rival.

You can see an introduction to Donaldson gauge theory under my review of F&U, but essentially, Donaldson's theorem was the first major topological result in gauge theory that really put the subject on the (mathematical) map and won Simon Donaldson a Fields Medal. The theorem is based on the result that characteristic-1 solutions modulo gauge equivalence to the self-dual Yang-Mills equation over a closed smooth 4-manifold themselves form a manifold with various properties (nonempty, 5-dimensional, orientable, with a noncompact end that is diffeomorphic to the underlying 4-manifold, smooth except at singularities that are homeomorphic to cones on complex projective space). The proof, as is typical in gauge theory, involves a mixture of hard nonlinear analysis, differential geometry, index theory, and spin geometry, and is very technical, long (practically the whole 100-page book is a single proof, and incomplete at that), and difficult for the beginner to wade through. Lawson breaks it down stepwise, introducing the necessary geometry and analysis (quarternion line bundles, connections, curvature, Sobolev spaces, Fredholm operators, the Sard-Smale theorem, the Atiyah-Singer index theorem, etc.) along the way. The proof is largely complete, with a proof of Uhlenbeck's gauge fixing lemma and some concluding details from one of Donaldson's papers being the most notable omissions.

The easiest way to evaluate this book is to compare it to F&U. First of all, it is written at a little higher level, assuming the reader to be more confortable with, e.g., principal bundles or K-theory. The Sobolev inequalities and multiplication theorems are given, but not much time is spent explaining them. Lawson's proof of orientability is a bit different, more abstract but maybe a little easier, while the Sard-Smale theorem is here applied to perturbations of the equations rather than the metrics, as Uhlenbeck's generic metrics theorem had only just been announced when this book was written. (The generic metrics theorem of Uhlenbeck is one of the unique features in the SU(2) YM theory that is lacking in, say, Seiberg-Witten theory, but Lawson's perturbation method is more generally applicable.) The proofs of Taubes's existence theorem are similar, but Lawson makes fewer mistakes, although he messes up the inequalities and choices of constants sometimes, too. (Everyone who writes these sorts of proofs seems to do this - F&U, Lawson, Donaldson, even Taubes himself; they're very tedious.)

Both books begin with the classification of simply connected 4-manifolds and symmetric bilinear forms, as well as a construction of a fake R^4, i.e., a manifold that is homeomorphic but not diffeomorphic to R^4, but I prefer Lawson's treatment, which for the classification emphasizes the similarities with the 2-dimensional case. F&U, however, also include a chapter on Fintushel & Stern's SO(3) theory, which Lawson doesn't, but which is found in Petrie & Randall's Connections, Definite Forms, and Four-manifolds. Neither Lawson nor F&U (or P&R) treat Donaldson invariants, since they were discovered several years later, and both still work with self-dual rather than anti-self dual connections, but in the 2nd edition of F&U, released in 1991, these developments were at least mentioned.

There are a number of mistakes with signs, factors of 2, missing or reversed subscripts and superscripts, C instead of H in eqn. II.8.6, "=" instead of "+" in eqn. III.2.3, etc., which seems about par for gauge theory books. The most serious errors are in the description of instantons on the 4-sphere on pp. 13 and 45 (cf. AHS's paper), the omission of the word "reducible" from Prop. IV.3.3, the word "minimum" instead of "maximum" twice on p. 70, and, as already noted, a number of errors in the choice of constants, order of fixing parameters, and inequalities in Chapter VI, especially the statement of Lemma 1.8. Fortunately, all of these problems are surmountable, and in fact F&U is probably worse in this regard.

All in all, this book could serve as a good introduction to Donaldson theory if F&U is unavailable or to be read concurrently and if the reader has a good background in spin geometry and nonlinear analysis. But it is only an introduction, since it doesn't cover the invariants themselves, probably the most important part of the theory (cf. Donaldson & Kronheimer's The Geometry of Four-Manifolds, the seminal work in the field). Moreover, Donaldson theory itself has fallen out of favor since Seiberg-Witten gauge theory can reproduce the same results much more easily (cf. Morgan's The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. or Nicolaescu's Notes on Seiberg-Witten Theory).