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When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic discs with incidence conditions. To acheive a virtual fundemental class, a Kuranishi structure is built on these moduli spaces and then a virtual class is obtained by a (multi) section perturbation.

I understand that each Kuranishi chart corresponds do the picture of the del-bar operator as a section in the corresponding banach bundle. But now (multi)section perturbations don't necessarily correspond to perturbing J or adding a Hamiltonian term. So in concrete terms what actually happens?

It seems to me that after the perturbation what I am counting are no longer solutions to the $\bar\partial_J$ but something else. Can the equation they satisfy be described in some concrete way? Which propeties of J-holomorphic curves these solutions still satisfy?

Thanks

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  • $\begingroup$ "the book by FOOO"??? $\endgroup$
    – Gerry Myerson
    Commented Jun 29, 2017 at 23:47
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    $\begingroup$ @GerryMyerson FOOO is a standard abbreviation in the field for Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. $\endgroup$
    – j.c.
    Commented Jun 30, 2017 at 3:55
  • $\begingroup$ bookstore.ams.org/amsip-46 or bookstore.ams.org/ast-376? $\endgroup$
    – Gerry Myerson
    Commented Jun 30, 2017 at 5:20
  • $\begingroup$ I was referring to the first one (it should have two parts, though, or maybe now they are sold together), where everything is constructed. They later apply the theory to the torix case in a series of papers to get remarkable results on lagrangian fibers $\endgroup$
    – Yaniv Ganor
    Commented Jun 30, 2017 at 7:06

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You are right that the solutions of the perturbed equation do not satisfy the $\bar\partial_J$ equation for any $J$ anymore. Please note that this is a feature: if they would still be properly $J$-holomorphic, you could not in general achieve transversality. Not even with multisections.

If you don't know such examples it is fun to construct one: Take a holomorphic sphere that's cut out transversally and calculate the Fredholm indices of branched coverings. Moving around the branch points will allow you to construct solution spaces that have a dimension that's higher than the virtual dimension predicts. Finally note that you can't perturb them away by changing $J$.

When it comes to solutions of an abstractly perturbed equation, they are a priori meaningless. I.e. you forget that you're talking about spaces of maps and just do differential geometry in infinite dimensions. They in general do not even satisfy a PDE anymore, whatever that might mean for you. You can however always choose abstract perturbations that retain some meaning. For that, note that to perturb you need to choose a section of the cokernel, and the value of a section of the cokernel can be identified with a vector field along a map (the base point). A solution u will then satisfy $\bar\partial_J u = p(u)$. Here $p(u)$ denotes a vector field along $u$.

What kind of geometric meaning the solutions of the perturbed equation have is then really up to you. If you choose the set of allowed perturbations cleverly, you can for example ensure that the solutions will still be J-curves on large parts of the domain.

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