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 Ω on X. Given this choice of Ω, one can associate to a Lagrangian

submanifold L ⊂ X the complex number

Z(L) =

L

Ω ∈ 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