TITLE: Obstruction bundle for spaces with Kuranishi structure
QUESTION [13 upvotes]: In the symplectic topology view on Gromov-Witten-Invariants some authors use what they call a Kuranishi structure on the moduli of stable maps. These were introduced by Fukaya and Ono and are also used in their big book on Fukaya categories.They are also used a lot in recent papers by Joyce. 
The key feature of the Kuranishi structure is that there locally exists a homeomorphism or a diffeomorphism, depending on whether you follow Fukaya or Joyce, to a zero set of a section of the "obstruction bundle".
On the algebraic geometry side there also is something involving the term obstruction, namely the perfect obstruction theory on the moduli space of maps. This is a morphism [E-1 -> E0] -> LX, where L is the cotangent complex.
Here's my question: To what does the obstruction bundle on the symplectic topology side correspond on the algebraic side?
There are three candidates I can think of:

E-1 
the kernel of E-1 -> E0
the thing the kernel of E-1 -> E0 maps surjectively to, mostly called the T2

The definition of the obstruction bundle on the symplectic side is a finite dimensional supspace of the cokernel of the linearized operator of the pseudoholomorphic curve equation, p.978 of Fukaya-Ono:"Arnold Conjecture and Gromov Witten Invariant." As a die hard algebraic geometer, that's just to hard to digest...

REPLY [9 votes]: By the way, the notion of Kuranishi structure is an attempt to 
formalize ideas that had been used in the gauge theory literature somewhat eariler.
See for example Friedman and Morgan's book Smooth Four Manifolds and Complex Surfaces.
In that book the obstruction bundle idea is used in the definition of the Donaldson
invariants.
The obstruction bundle idea for studying  nonlinear PDE problems of course goes back 
at least to  Kuranshi.  Taubes used this idea to study solutions to the Yang-Mills equations
near bubbling in the 1982  paper "Self-dual Yang-Mills connections on non-self-dual 4-manifolds" in way that lead to the modern use.