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Moduli spaces of pseudoholomorphic curves do not carry the structure of a (compact) differentiable manifold in general (due to transversality issues). Nevertheless one would like to at least associate a fundamental class to the moduli space in question.
It looks like two approaches dominate: Kuranishi structures and polyfolds.
Both seem to be mammoth projects. Before diving seriously into one of them I may ask:
What are the advantages/drawbacks of these approaches? What is their motivation? Are they rigourosly proved? Are there reasonable alternatives?

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3 Answers 3

up vote 13 down vote accepted

Kuranishi models are a traditional - and beautiful - technique for describing the local structure of moduli spaces cut out by non-linear equations whose linearization is Fredholm. A more elaborate version, "Kuranishi structures", are used by Fukaya-Oh-Ohta-Ono (FOOO) and Akaho-Joyce to handle transversality for moduli spaces of pseudo-holomorphic polygons with Lagrangian boundary conditions, and the compactifications of these spaces. FOOO's book is the result of a decade of dedicated thought by a superb team, but few have assimilated it (I certainly haven't).

Polyfolds, Hofer's "infrastructure project", are designed with the severe demands of symplectic field theory (SFT) foremost in mind. This is a more radical rethink of how to handle transversality, whose aims include absorbing the difficulties of the lack of canonical coordinates when gluing Morse-Floer trajectories. (What difficulties? Try to prove that the moduli space of unparametrized broken gradient flow-lines of an ordinary Morse function is a smooth manifold with corners, and you'll find out.) Several papers into the project, it's still not completely clear how efficiently it will work in applications, especially those outside SFT. I'd hope that polyfolds will help us set up Cohen-Jones-Segal Floer homotopy-types, for instance - but there may still be severe difficulties. I've also never heard a compelling argument that Kuranishi structures are insufficient for SFT.

This paper of Cieliebak-Mohnke suggests an intriguing alternative approach.

My view would be that these mammoths are worth chasing only if you have very clear aims in mind. There are many excellent problems in symplectic topology that don't need such gigantic foundations. If you're interested in Fukaya categories, there's lots to be proved using the definition from Seidel's book, which deals with exact symplectic manifolds. If you want to prove things about contact manifolds, try using symplectic cohomology, a close cousin of SFT requiring less formidable analysis.

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Perhaps you would like to take a look at a recent preprint by McDuff and Wehrheim

http://arxiv.org/abs/1208.1340

This rather long paper seeks to explain the topological and analytic issues involved and provide the beginnings of a framework for resolving them.

These authors are well known for their attention to detail and expository ability, so it might be worthwhile to take a look.

I've heard rumors that an eventual Polyfold approach and the one taken by McDuff and Wehrheim share quite a bit of overlap in the end, but am not enlightened enough to say more on this matter.

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In another relevant recent preprint, Wehrheim constructs smooth structures with corners on moduli spaces of Morse trajectories: arxiv.org/abs/1205.0713 - this is a "baby case", but it still looks pretty technical! –  Jonny Evans Aug 20 '12 at 20:53

You might also enjoy this video of a talk by Katrin Wehrheim which compares the approaches a bit. It's actually also quite entertaining. http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/4474/show_video

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