I came across a new type of gluing in the FOOO paper http://arxiv.org/abs/1002.1660 (maybe not so new to experts). The situation is as follows: one considers a relative (to lagrangian) class and want to understand the evaluation maps of the moduli space to the lagrangian.
An option is to do the original FOOO thing--Kuranishi structure, multisection perturbation and evaluation; however, if there is a separating contact type hypersurface in the manifold, where the lagrangian entirely lies on one of the pieces, there's another option. I can first do neck-stretching in SFT, then evaluate the fiber product of moduli space involving punctures on the two sides, fiber product taken over evaluation on the punctures where the moduli spaces can be glued. My question is:
When do the two options give the same chain (in an appropriate sense that I don't know how to formulate)$/$class?
I suppose this has been used for a while in GW theory, but I am not sure what can be claimed "standard" in this case of Floer homology. According to the above paper,
When the puncture is simple and of minimal period, this conclusion is standard.
I suspect there's another implicit assumption they used but suppressed here, that the evaluation always gives a cycle due to torus action. I do not have a clear idea why the assumption on simple puncture plays a role. The Li-Ruan's paper and settings therein do not seem to be responsible for this assumption. Notice that, in the setting of the above paper, the evaluation should involve a virtual perturbation since the moduli space in question is (a maslov=$2$ disk)+($k$ copies of maslov=$0$ spheres).
The following special case should help concretize my question: consider still a toric manifold, where the Kuranishi structure and perturbation seem much easier to construct according to toric FOOO I&II. If one considers a certain contact type hypersurface which is Morse-Bott and invariant under torus action, and an arbitrary classes with maslov=$2$, can one still claim such a gluing result "standard"?
