4 TOM BRIDGELAND

The sigma models on X all lie in one connected component MK3 of the moduli

space M. The corresponding connected component N

K3

of the Teichm¨ uller space

N referred to above is the set of pairs

(Ω, ) ∈ P(Λ ⊗ C)

satisfying the following relations in terms of the pairing (−, −)

(Ω, Ω) = ( , ) = (Ω, ) = (Ω,

¯)

= 0,

(Ω,

¯

Ω) 0, ( ,

¯)

0,

with the extra condition

(∗) there is no δ ∈ ∆(Λ) such that (Ω, δ) = 0 = ( , δ).

The moduli space MK3 is the discrete group quotient N

K3

/O+(Λ).

Note that

N

K3

does indeed have a local product structure in this case. The mirror map

preserves the connected component MK3 (this is the statement that K3 surfaces

are self-mirror) and simply exchanges Ω and .

In fact Aspinwall and Morrison do not impose the last condition (∗). In the

physics there is indeed some sort of theory existing at the points where (∗) fails, but

the brane with the corresponding charge δ has become massless, so that the strict

SCFT description breaks down, and non-perturbative corrections have to be taken

into account. In any case, it is clear that for our purposes it is important to leave

out the hyperplanes where (∗) fails, since one obtains interesting transformations

by taking monodromy around them.

The sigma model deﬁned by a complex structure I and a complexiﬁed K¨ahler

class β+iω on X corresponds to the pair (Ω, ) where Ω is the class of a holomorphic

two-form on (X, I) and

=

[eβ+iω]

= [(1, β + iω,

1

2

(β +

iω)2)]

∈ P(Λ ⊗ C).

It is important to note that in the K3 case, in contrast to the case of a simply-

connected threefold, the set of choices of this data does not deﬁne an open subset

of the moduli space MK3. In fact the point of MK3 deﬁned by a pair (Ω, ) can

only come from a sigma model of the usual sort if there is a hyperbolic plane in Λ

orthogonal to Ω. To obtain an open set in MK3 one must also include sigma models

deﬁned using Hitchin’s generalised complex structures [32, 38]. As explained by

Huybrechts [33] one then obtains the entire space MK3 as the moduli space of

generalised K3 structures on X.

2.4. Stability for D-branes. Suppose that one is given an SCFT together

with one of its topological twists. As above, this twist corresponds mathematically

to a Calabi-Yau A∞ category D. Some deformations of the SCFT will induce

deformations of D, but there is a leaf L in the Teichm¨ uller space N whose points

all have the same topological twist. The notion of a stability condition comes about

by asking what the signiﬁcance of these extra parameters is for the category D.

Douglas argued [23, 24] that at each point on the leaf L ⊂ N there is a full

subcategory of semistable objects or BPS branes in the category D. At each point

of a bundle with ﬁbre C over L one can also assign complex numbers Z(E) called

central charges to all objects E ∈ D. Moreover, for a BPS brane E the central

4