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 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 is the class of a holomorphic
two-form on (X, I) and
=
[eβ+iω]
= [(1, β + iω,
1
2
+
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
Previous Page Next Page