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Since nobody else answered this question, I'll give it a try from the point of view of mirror symmetry. This is probably a very backwards way of telling the story and there are maybe better answers from, say, the point of view of calibrated geometry. A Bridgeland stability condition on a triangulated category $D$ consists of a group homomorphism:

$$Z : K(D) → C \mathbb{C}$$

and a "slicing" $P$ of $D$ such that if $E \neq 0$ in $P(\phi)$ then Z(E) = $m(E)e^{iπφ}$ for some $m(E) ∈ R>0$. See Definition 5.1. of the paper which began this subject: http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf

Just as for ordinary stability conditions in geometric invariant theory, there are notions of stability and semi-stability which give us a notion of moduli spaces of objects. There is a conjecture that the derived Fukaya category of the symplectic manifold X carries a Bridgeland stability condition such that the semi-stable objects are Floer theoretically unobstructed special Lagrangians.

Edit: Another important theorem that I totally forgot to mention is that which is due to Thomas and Yau http://arxiv.org/pdf/math/0104197.pdf and which says that there can be at most one special Lagrangian in a Hamiltonian deformation class. We can therefore say that, roughly speaking, the special Lagrangian condition picks out a nice geometric representative in a semi-stable isomorphism class in the Fukaya category. Unfortunately, this is probably only a partial truth since there will probably be semi-stable objects which do not correspond to smooth Lagrangians, but singular ones.

3 added 285 characters in body

Since nobody else answered this question, I'll give it a try from the point of view of mirror symmetry. This is probably a very backwards way of telling the story and there are maybe better answers from, say, the point of view of calibrated geometry. A Bridgeland stability condition consists of a group homomorphism:

$$Z : K(D) → C$$

and a "slicing" $P$ of $D$ such that if $E \neq 0$ in $P(\phi)$ then Z(E) = $m(E)e^{iπφ}$ for some $m(E) ∈ R>0$. See Definition 5.1. of the paper which began this subject: http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf

Just as for ordinary stability conditions in geometric invariant theory, there are notions of stability and semi-stability which give us a notion of moduli spaces of objects. There is a conjecture that the derived Fukaya category of the symplectic manifold X carries a Bridgeland stability condition such that the semi-stable objects are Floer theoretically unobstructed special Lagrangians.

Edit: Another important theorem that I totally forgot to mention is that which is due to Thomas and Yau http://arxiv.org/pdf/math/0104197.pdf and which says that there can be at most one special Lagrangian in a Hamiltonian deformation class. We can therefore say that, roughly speaking, the special Lagrangian condition picks out a nice geometric representative in an a semi-stable isomorphism class in the Fukaya category. Unfortunately, this is probably only a partial truth since there will probably be semi-stable objects which do not correspond to smooth Lagrangians, but singular ones.

2 deleted 10 characters in body

Since nobody else answered this question, I'll give it a try from the point of view of mirror symmetry. This is probably a very backwards way of telling the story and there are maybe better answers from, say, the point of view of calibrated geometry. A Bridgeland stability condition consists of a group homomorphism:

$$Z : K(D) → C$$

and a "slicing" $P$ of $D$ such that if $E \neq 0$ in $P(\phi)$ then Z(E) = $m(E)e^{iπφ}$ for some $m(E) ∈ R>0$. See Definition 5.1. of the paper which began this subject: http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf

Just as for ordinary stability conditions in geometric invariant theory, there are notions of stability and semi-stability which give us a notion of moduli spaces of objects. There is a conjecture that the derived Fukaya category of the symplectic manifold X carries a Bridgeland stability condition such that the semi-stable objects are Floer theoretically unobstructed special Lagrangianscorrespond to the semi-stable objects.

We can say that the special Lagrangian condition picks out a nice geometric representative in an isomorphism class in the Fukaya category. Unfortunately, this is probably only a partial truth since there will probably be semi-stable objects which do not correspond to smooth Lagrangians, but singular ones.

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