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The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds
By John W. Morgan PRINCETON UNIVERSITY PRESS
Copyright © 1996 Princeton University Press
All rights reserved.
ISBN: 978-0-691-02597-1
CHAPTER 1
Introduction
Beginning with the groundbreaking work of Donaldson in about 1980 it became clear that gauge-theoretic invariants of principal bundles and connections were an important tool in the study of smooth four-dimensional manifolds. Donaldson showed the importance of the moduli space of anti-self-dual connections. The next fifteen years saw an explosion of work in this area leading to computations of Donaldson polynomial invariants for a wide class of four-dimensional manifolds, especially algebraic surfaces. These computations yielded many powerful topological consequences including, for example, the diffeomorphism classification of elliptic surfaces. For some of the results obtained by using these techniques see [1], [6], and [2].
Last fall, motivated by new work in quantum field theory, Seiberg and Witten [9], introduced a different gauge-theoretic invariant which they claimed should be closely related to Donaldson's invariants. Indeed they gave an explicit formula for the relationship of their invariant to Donaldson's. While the link claimed by Seiberg-Witten between their invariants and Donaldson's has not yet been established mathematically, it is clearly true and can be shown to hold in all computed examples. Nevertheless, one can forget this supposed link and work directly with the new invariants as a substitute for the anti-self-dual invariants. This has been the approach during the last year or so.
It was clear from the beginning that the new invariants would be easier to work with since they involved principal bundles with structure group the circle instead of the non-abelian groups such as SU(2) which arise in Donaldson theory. The surprise was that such simple invariants could capture the subtlety that Donaldson's invariants revealed. But in short order, Witten [17] and then Taubes, Kronheimer, and Mrowka showed that indeed these new invariants did capture these subtleties and that they were easier to compute, at least in many cases. They did this by explicitly solving the Seiberg-Witten equations over Kähler surfaces. (See [3] for one account of the results for Kähler surfaces.) There followed in quick succession a series of remarkable theorems, each extending in a different way partial results from Donaldson's anti-self-dual theory. In fact, conjectures which seemed reasonable from the perspective of Donaldson theory but technically difficult, if not unreachable within that theory, suddenly became the standard test cases for the power of the new invariants. One by one these conjectures were established – leaving only one classical conjecture outstanding. The remaining one, called the 11/8ths-Conjecture, deals with the quadratic forms that arise as intersection forms of simply connected spin four-manifolds.
It is the purpose of these notes to lay the groundwork for the Seiberg-Witten theory and then to show how one computes these invariants for most Kähler surfaces. We begin with the basics of Clifford algebras, spin structures and their cousins Spinc-structures, spin representations, and spinor bundles. We then consider the Dirac operator on the spin bundles over a riemannian four-manifold. The connection with Kähler geometry is facilitated because of the close connection on a Kähler manifold between the Dirac operator and [bar.[partial derivative]]. All of this is examined in detail. The book of Lawson-Michelson gives a complete introduction to this material as well as an elaborate treatment of many more topics in the spinor geometry.
We then exhibit the Seiberg-Witten equations and show how to use these equations to produce a finite dimensional manifold, the moduli space of solutions to these equations modulo changes of gauge, inside an infinite dimensional configuration space. The homology class of this moduli space is then the value of the Seiberg-Witten invariant. We establish analogues of familiar theorems from anti-self-dual theory. The equations defining the moduli space are elliptic modulo the group of changes of gauge. A generic perturbation of the equations leads to a smooth orientable moduli space whose dimension can be computed by the Atiyah-Singer index theorem. To orient this moduli space it suffices to choose an orientation of H+2 and H1 of the underlying four-manifold. As long as b+2 > 1 the moduli space varies smoothly as we vary the metric and perturbation. When b+2 = 1 the generic perturbation yields a smooth moduli space, but there will be singularities when we vary the metric and perturbation. These singularities are reducible solutions to the equations. This leads to a chamber structure for the Seiberg-Witten invariants in case b+2 = 1, similar to what happens in the anti-self-dual theory.
We go on to establish one special property of this theory, namely the compactness of the moduli space of solutions. This is a consequence of a priori bounds for the pointwise norms of solutions to the equations. This result has no analogue for the anti-self-dual equations. Much of the geometric richness and much of the technical complexity of the anti-self-dual theory is directly related to the non-compactness of the moduli space. For both better and worse, that is all missing here. The Seiberg-Witten moduli spaces are compact and vary by a compact bordism as we vary the metric.
Having presented all these technical results, it is now clear that the homology class of the moduli space in the ambient space of configurations is an invariant of the underlying smooth four-manifold and the isomorphism class of the Spinc structure on that manifold. By definition, the Seiberg-Witten invariant of the Spinc structure is this homology class. We finish by explicitly computing the moduli spaces of solutions on 'most' algebraic surfaces, leading immediately to a computation of the Seiberg-Witten invariant for any Spinc structure over the surface. There are some special cases that we do not treat. These can, however, be treated by an elaboration of the techniques that we introduce. A complete discussion is contained in.
There is much more to be said about the Seiberg-Witten equations and the invariants that are defined from these equations. There are analogues of the theorems for Kähler manifolds which hold for symplectic manifolds, see [12, 13, 14]. There are also gluing theorems which lead to MeyerVietoris principals for solutions to the equations, see [8]. These then lead to product formulas for the invariants, which is one approach to gaining a more topological understanding of the invariants. In spite of the striking progress over the last year, much remains to be done. It is my hope that these notes will serve as an introduction making it possible for more mathematicians to contribute to this progress.
These notes are a written version of lecture series given at Columbia University and Princeton University during the past year. I wish to thank the graduate students at both Universities who attended these lectures and who, by their comments and questions, helped shape these notes.
CHAPTER 2
Clifford Algebras and Spin Groups
For any n > 2 the orthogonal group SO(n) has fundamental group Z/2Z and hence has a universal covering group called Spin(n) which is a non-trivial double covering. Thete is a beautiful algebraic construction which yields the Spin groups (and much more as well). This is the subject of Clifford algebras.
2.1 The Clifford Algebras
An example. Befote delving into the complexities of Clifford algebras in general, let us consider a simple example. Consider the unit sphere S3 inside the quaternion algebra H. Multiplication of quaternions induces a group structure on S3. Let us consider the action of this group on H by conjugation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This action preserves the norm, i.e., is an orthogonal action. It also leaves invariant the center of H which is R [subset] H, and hence it leaves invariant the perpendicular complement to R which is the three-dimensional space Im H of purely imaginary quaternions. Of course, Im H is naturally identified with the Lie algebra of S3 and the action we are considering is the adjoint action of S3 on its Lie algebra.
It is an easy geometric exercise to see that a unit quaternion a, different from ±1, acts by conjugation on H preserving the complex plane Cα spanned by α and its perpendicular Cαj. On the first complex plane conjugation by α acts trivially and on the second it acts by rotation through twice the angle θ between α and 1. It follows that the conjugation action of α on 1m H leaves invariant the line tangent to the circle generated by α and acts by rotation through 2θ on the perpendicular complement and thus that every rotation of Im H is in the image of the representation
S3 -> SO(Im H) = SO(3).
It is also clear that the kernel of this representation is the intersection of S3 with the center of H, which is the center of S3 and is {±1}.
In this way we construct the double covering group of SO(3) and identify it with the group of quaternions of length one under multiplication.
The definition of the Clifford algebra associated to an positivedefinite inner product space. Now let us turn to the general case. Let V be a finite dimensional vector space over R with a positive definite inner product <·, ·> leading to a norm denoted [parallel] · [parallel]. We consider the tensor algebra
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
generated by V. It is an associative algebra with unit 1. The Clifford algebra Cl(V) generated by V is the quotient of T(V) by the two-sided ideal generated by all elements of the form
υ [direct sum] υ + [parallel] υ [parallel]21
for υ [member of] V. Notice that the grading on T(V) descends to a Z/2Z grading on Cl(V), giving a decomposition
Cl(V) = Cl0(V) [direct sum] Cl1 (V)
where Cl0(V) is a subalgebra and Cl1(V) is a module over this subalgebra. Corresponding to this splitting, we can write any υ [member of] Cl(V) as υ0 + υ1. We denote by
ε: Cl(V) -> Cl(V)
the algebra homomorphism that is multiplication by +1 on Cl0(V) and is multiplication by -1 on Cl1(V); i.e., ε(υ0 + υ1) = υ0 - υ1.
Choosing an orthonormal basis {e1, ..., en} for V we can write Cl(V) in terms of generators and relations. Namely, Cl(V) is the algebra over R generated by {e1, ..., en} subject to the relations e2i = -1 for all i ≤ n and ei x ej = -ej x ei for all i ≠ j. In particular, it follows that every element of CI(V) can be written uniquely as a sum of products of the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where i1 ... it. Thus, the dimension of Cl(V) as a real vector space is 2d where d is the dimension of V over R.
Examples. (i) Let Rn denote the usual Euclidean space of dimension n. Then Cl(R1) = R]x]/(x2 + 1) [congruent to] C. The subalgebra Cl0(R1) is identified with the reals and Cl1(R1) with the purely imaginary complex numbers.
(ii) Similarly, Cl(R2) is the algebra generated by x, y subject to the relations
x2 = -1; y2 = -1; xy = -yx.
Hence, Cl(R2) is isomorphic to the quaternion algebra H. The subalgebra Cl0(R2) is generated by xy and can be identified with C [subset] H.
(iii) Cl(R3) is of dimension 8 over R. It is generated by x, y, z with x2 = y2 = z2 = -1 and xy = -yx, xz = -zx, yz = -zy. This algebra is isomorphic to H [direct sum] H. The isomorphism between H and the first factor, resp., the second factor, is given by sending 1, i, j, k to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
resp., to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The subalgebra Cl0(R3) is identified with the diagonal copy of H in this decomposition.
(iv) For any inner product space V, we have an isomorphism of algebras Cl(V) [congruent to] Cl0(V [direct sum] R). Letting e be a unit vector in R, the isomorphism is given by
υ0 + υ1 -> υ0 + υ1e.
It is an easy exercise to show that this map is an isomorphism of algebras. In particular, Cl0(R4) is isomorphic to H [direct sum] H.
Comparison with the Exterior Algebra. The grading on T(V) induces an increasing filtration
0 [subset] F1 [subset] F2 [subset] ...
of Cl(V) by linear subspaces; namely, we set Ft eqQal to the image in Cl(V) of
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Clearly, the multiplication in Cl(V) preserves this filtration in the sense that it induces maps
Fi [cross product] Fj -> Fi+j.
Thus, there is an associated graded algebra
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
with the induced multiplication.
Claim 2.1.1 GrF. (Cl(V)) is naturally isomorphic to the exterior algebm Λ*(V).
Proof. Let [??] be the increasing filtration of T(V) coming from the grading. Clearly, there is a natural algebra isomorphism between T(V) and Gr[??]. (T(V)). Consequently, we have a commutative diagram
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Clearly, the quotient of T(V) by the two-sided ideal generated by υ [cross product] υ for υ [member of] V is exactly the exterior algebra Λ*(V). Thus, this diagram induces the claimed isomorphism.
There is a natural splitting σ of the map Cl(V) -> Λ*(V). The map σis linear but not multiplicative. It is defined as follows. Consider an elementary element in ΛkV (; i.e., one contained in Λk(W) [subset] Λk(V) for some k-dimensional subspace W [subset] V. Such an element can be written as re1 [conjunction] ... [conjunction] ek where the ei form an orthonormal basis for W and where r > 0. We define
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It is an easy exercise to show that this determines a well-defined mapping which splits the natural projection.
Using this isomorphism, we can view Cl(V) as being given by a new multiplication on Λ*(V). This new multiplication is generated by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [angle] is the contraction:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
2.2 The groupsPin(V)andSpin(V)
Let Clx(V) denote the multiplicative group of units of the algebra Cl(V). We define the group Pin(V) as the subgroup of Clx(V) generated by elements υ [member of] V with [parallel] υ [parallel]2 = 1. Notice that the given generators of Pin(V) are units since the square ohny one of them is -1. We let Spin(V) be the intersection of Pin(V) with Cl0(V), i.e., the kernel of the group homomorphism Pin(V) -> Z/2Z induced by the splitting of Cl(V) = Cl0(V) [direct sum] Cl1(V). Since the generators of Pin(V) are contained in Cl1(V), Spin(V) is the subgroup of index two consisting of all elements in Pin(V) which can be written as a product on an even number of the given generators for Pin(V). Let us compute these groups in the first three examples.
(i) The group Pin(1) is the subgroup of C generated by ±i. Hence it is a cyclic group of order 4. The subgroup Spin(1) is the group of order two ±l inside R.
(ii) The group Pin(2) is the subgroup of H generated by the circle through j and k, i.e., by all elements of the form cos(θ)j + sin(θ)k for θ [member of] S1. It is easy to see that this group is the union of two circles - the usual unit circle in the complex plane and j times it. Hence, the group Spin(2) is isomorphic to S1.
(Continues...)
Excerpted from The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds by John W. Morgan. Copyright © 1996 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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