Question

Is there any source, different than BEHWZ paper, that one can learn about the geometric nature of bubbling-off process, high level building structures (and so on...) in the SFT setting?

I want to see how different scenarios of gradient blow up (like different convergence behaviours of sequence of points with diverging gradient norms) might lead to different types of breakings.

Thanks for your interest.

Answer 1

For non-specialist readers:

SFT = symplectic field theory

BEHWZ = Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder, the authors of the paper which establishes the basic compactness theorem for pseudo-holomorphic curves in symplectic manifolds with convex or concave ends.

There is an attractive alternative approach to SFT compactness due to Cieliebak and Mohnke:

Compactness for punctured holomorphic curves. J. Symplectic Geom. 3 (2005), no. 4, 589-654.

Their method is founded not on Deligne-Mumford-type degenerations of the source curves, but rather on the degenerations of their images in a symplectic manifold with a lengthening cylindrical neck. These degenerations are controlled by the Morse theory of the function on the source curve given by the holomorphic map followed projected to the cylindrical coordinate.