The Embedded Contact Homology (ECH), introduced by M. Hutchings, is an invariant of (contact) three-manifolds. Since its introduction, well-known conjectures in symplectic/contact topology in dimension $3$, including the Weinstein conjecture and its gen erali zations, the Arnold chord conjecture, are proved based on this invariant. Furthermore, the ECH capacities are used to establish a Weyl law, which is the foundation of the equidistribution result for generic contact forms. There are also nice applications of ECH to symplectic embedding problems in dimension $4$.
The question is: why is ECH so powerful?
To illustrate the question, the Symplectic Field Theory (SFT) should capture more information from pseudo-holomorphic curves than ECH but the above results were not (yet) proved using SFT. One quick answer to this question would be the celebrated ECH=SWF theorem, which allows one to use powerful results/computations from monopole Floer homology. But eventually, one might hope to stay in the world of pseudo-holomorphic curves without appealing to gauge-theoretic invariants, in order to go to higher dimensions. So the question might be phrased as:
Is ECH a low-dimensional miracle? If not, what lessons should we learn from the selections of pseudo-holomorphic curves which define ECH?
