SPACES OF STABILITY CONDITIONS 5
charge Z(E) is always nonzero, and there is a distinguished choice of phase
φ(E) =
1
π
arg Z(E) R.
Axiomatising the properties of the subcategories P(φ) D of BPS branes of phase
φ leads to the definition of a stability condition given in the next section.
A good example to bear in mind for heuristic purposes is the case when the
SCFT is a sigma model on a Calabi-Yau threefold (X, I, β + iω), and
D =
Db
Fuk(X, β + iω)
is the A-model topological twist. Then the leaf L N is expected to be the space of
complex structures on X considered up to diffeomorphisms isotopic to the identity.
To each point of a bundle with fibre
C∗
over L there is a well-defined holomorphic
three-form on X. Given this choice of Ω, one can associate to a Lagrangian
submanifold L X the complex number
Z(L) =
L
C.
Recall that a Lagrangian submanifold L X is said to be special of phase φ R/2Z
if
Ω|L = exp(iπφ) volL,
where volL is the volume form on L. Note that one then has
Z(L) =
L
eiπφ
volL R
0
exp(iπφ).
To be slightly more precise, the objects of the Fukaya category are graded La-
grangians. This means that the phases φ of the special Lagrangians can be lifted
to elements of R up to an overall integer indeterminacy which can be eliminated
by passing to a Z cover. Thus corresponding to each point of a bundle with fibre C
over L and each real number φ R there is a full subcategory P(φ) D of special
Lagrangians of phase φ.
If one varies the complex structure on X and hence the holomorphic three-form
Ω, the numbers Z(L) and the subcategories of special Lagrangians will change.
This process was studied by Joyce [35]. The analogy with variations of stability in
algebraic geometry was explained by Thomas [47]. These ideas provided the basis
for Douglas’ work on Π-stability for D-branes in a general SCFT context.
3. Basic definitions
This section is a summary of the essential definitions and results from [12].
Proofs and further details can be found there.
3.1. Stability conditions. For the rest of the paper D will denote a trian-
gulated category and K(D) its Grothendieck group. In the previous section we
considered A∞ categories whose cohomology categories were triangulated; no such
enhancements will be necessary for what follows.
Definition 3.1. A stability condition σ = (Z, P) on D consists of a group
homomorphism Z : K(D) C called the central charge, and full additive subcate-
gories P(φ) D for each φ R, satisfying the following axioms:
(a) if 0 = E P(φ) then Z(E) = m(E) exp(iπφ) for some m(E) R 0,
(b) for all φ R, P(φ + 1) = P(φ)[1],
5
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