Soon I am supposed to give two mini-courses on ECH. I am also supposed to write some accompanying lecture notes. As preparation for both, I am going to start writing some rough notes here. If you have any questions about these notes, please feel free to post a comment and I will try to answer. Note that my second mini-course will cover a lot more material than the first, because the lectures will be longer and there will also be a number of related lectures by others. So, while I have not yet decided exactly what I will cover in the first mini-course, it will probably skip some of the following material (in which case the skipped material explained here might be a useful supplement).
The first goal of these lecture notes is to explain something about where ECH comes from. This material is more difficult than the definition of ECH itself, and I don’t want to scare anyone away from my minicourses! Fortunately this material is dispensable if one is willing to simply accept the definition of ECH as it is and start from there. But the definition of ECH may seem somewhat mysterious, and if you want some more motivation, this might help.
The starting point for the definition of ECH is Taubes’s “SW=Gr” theorem relating Seiberg-Witten invariants of symplectic four-manifolds to counting holomorphic curves. To describe this, we first need to briefly review the Seiberg-Witten invariants of four-manifolds.
Seiberg-Witten invariants of four-manifolds
The Seiberg-Witten invariants of a four-manifold (all manifolds in these notes are smooth) count solutions to the Seiberg-Witten equations on . This material is somewhat difficult, and I do not have time to give a fully satisfactory explanation here. However it is not really necessary to understand the details of the Seiberg-Witten equations in order to get the basic idea of where ECH comes from, so I will just briefly review the equations and encourage you to briefly skim them. I will explain things more slowly and carefully (and the material will become more elementary) when we get to the ECH material that is my focus.
Let be a closed oriented connected four-manifold with a Riemannian metric. Recall that a spin-c structure on is a lift of the frame bundle from a principal -bundle to a principal -bundle, where . The set of (isomorphism classes of) spin-c structures on is an affine space over which does not depend on the choice of Riemannian metric. A spin-c structure determines a spin bundle . Here is a rank 2 complex vector bundle with a Hermitian metric. A section of is called a spinor. There is also a Clifford action which satisfies the relation for . The Clifford action extends using the metric to an action of on , and then by antisymmetry to an action of on . A spin connection on is a connection on such that
for every vector field and spinor . Here on the right hand side denotes the Levi-Civita connection. It turns out that a spin connection is equivalent to a connection on the line bundle , and henceforth we will regard a spin connection as a connection on . (This convention matters in that adding an imaginary-valued 1-form to the connection on adds to the connection on .) A spin-c connection determines a Dirac operator defined by first applying the spin covariant derivative and then contracting using Clifford multiplication.
Fix a spin-c structure on . The Seiberg-Witten equations concern a pair where is a connection on and is a section of . The equations are
Here denotes the self-dual part of the curvature of (regarded as an imaginary-valued 2-form), and is a quadratic bundle map defined by
Finally, is a self-dual 2-form which is a parameter in the equations. The gauge group acts on the set of solutions to the Seiberg-Witten equations by the rule . Two solutions are called gauge equivalent if they differ by the action of the gauge group, and we usually mod out by this equivalence. A solution is called reducible if , and irreducible otherwise. Note that the gauge group acts freely on the set of irreducible solutions, but not on the set of reducible solutions.
We now want to count solutions to the Seiberg-Witten equations modulo gauge equivalence. To do so, we need the following analytic facts. First, the set of solutions to the Seiberg-Witten equations modulo gauge equivalence is always compact. (This is the “miracle” of the equations and depends crucially on the sign in the second equation.) Furthermore, the linearized equations are elliptic, and so a standard transversality argument shows that for generic , the set of irreducible solutions modulo gauge equivalence is a smooth manifold. (Smoothness does not work for the reducibles because of the failure of the gauge group to act freely.) A calculation using the Atiyah-Singer index theorem (I told you this material was not so elementary) shows that for generic , the dimension of the space of irreducible solutions modulo gauge equivalence is
Here denotes the signature of .
If there are no reducible solutions, then the set of all solutions modulo gauge equivalence is (for generic ) a compact smooth manifold, which we can use to define counting invariants. So when are there reducible solutions? A reducible solves the Seiberg-Witten equations if and only if . The Hodge decomposition can be used to show that the space of all self-dual 2-forms on can be decomposed as
Here denotes the space of harmonic self-dual 2-forms, and denotes the composition of with the projection . Furthermore, represents the class . It follows that there is a reducible solution if and only if the harmonic part of agrees with the self-dual part of the harmonic representative of . This is a codimension condition on , where denotes the dimension of a maximal positive definite subspace of . Conclusion: if then we can choose so that there are no reducibles, and if then any two such are connected by a path of forms for which there are no reducibles.
Now if is a closed oriented connected four-manifold with and is a spin-c structure on , we define the Seiberg-Witten invariant as follows. First, to fix signs in the counting (more formally to orient the moduli spaces of solutions to the Seiberg-Witten equations modulo gauge equivalence), we need to fix a homology orientation of , which is an orientation of the vector space , where denotes any maximal positive definite subsapce of . Switching the homology orientation will switch the sign of the Seiberg-Witten invariant. Now if , we define . Otherwise choose a generic for which there are no reducibles. If , then the moduli space of Seiberg-Witten solutions modulo gauge equivalence is a finite set of points, and counts them, with signs determined by the homology orientation in a manner which I will not explain. When one first cuts down the moduli space to be zero dimensional using certain natural cohomology classes on the configuration space of pairs modulo gauge equivalence and then counts. (The “simple type conjecture” asserts that whenever and and ).
If one can define the Seiberg-Witten invariant similarly. However now the set of for which there are no reducibles has two components, called “chambers”, and the Seiberg-Witten invariant depends on the choice of a chamber.
Whew! That completes our sketch of the definition of the Seiberg-Witten invariants of four-manifolds. If you find this definition rather impenetrable, you are not alone. In addition, computing examples of Seiberg-Witten invariants directly from the definition is almost impossible except in some extremely simple examples (although there are axiomatic properties which one can use to compute interesting examples). However, because of the deep applications which the Seiberg-Witten invariants have to the topology of smooth four-manifolds (which I did not discuss here), we would like to understand them as much as we can. Fortunately, Taubes’s “SW=Gr” theorem, which I will explain in the next installment of notes, makes the Seiberg-Witten invariants MUCH easier to understand for symplectic four-manifolds.