SPACES OF STABILITY CONDITIONS 3
2.2. Sigma models. Suppose X is a Calabi-Yau manifold, and for simplicity
assume that X is simply-connected and has complex dimension three. The non-
linear sigma model on X defines an SCFT depending on a complex structure I on
X together with a complexified ahler class β +
H2(X,
C). In fact, while it is
expected that the integrals defining the sigma model will converge providing that
the ahler class ω is sufficiently positive, this has not been proved even to physicists’
satisfaction. Ignoring this problem, the sigma model construction defines an open
subset of the moduli space M. More precisely, a point in M is defined by data
(X, I, β + iω) considered up to some discrete group action. For example changing
the B-field β by an integral cohomology class does not effect the isomorphism class
of the associated SCFT. Note also that many points of M will not be defined by
any sigma model, and that points in a single connected component of M may be
defined by sigma models on topologically distinct Calabi-Yau manifolds.
The A and B twists of the sigma model define TCFTs associated to the Calabi-
Yau manifold X. The corresponding A∞ categories are expected to be the derived
Fukaya category of the symplectic manifold (X, ω), suitably twisted by the B-
field β, and an enhanced version of the derived category of coherent sheaves on
the complex manifold (X, I) respectively. By enhanced we mean that one should
take an A∞ category whose underlying cohomology category is the usual derived
category of coherent sheaves. If two sigma models corresponding to geometric data
(Xj , Ij , βj + iωj ) are exchanged by the mirror map one therefore has equivalences
Db
Coh(X1, I1)

=
Db
Fuk(X2, β2 + iω2)
Db
Fuk(X1, β1 + iω1)

=
Db
Coh(X2, I2).
This is Kontsevich’s homological mirror symmetry proposal [39].
In the neighbourhood of a sigma model defined by data (X, I, β + iω) the two
foliations of N correspond to varying the complex structure I or the complexified
ahler class β + iω. In the simply-connected threefold case it is expected that the
corresponding leaves in M are the spaces MC(X) of complex structures on X up
to diffeomorphism, and the so-called stringy ahler moduli space MK(X). This
latter space has no mathematical definition at present; this is one of the motivations
for introducing stability conditions. The mirror map induces isomorphisms
MC(X1)

=
MK(X2), MK(X1)

=
MC(X2),
and these are often taken as the definition of the ahler moduli spaces.
2.3. K3 surface case. The component of the moduli space M containing
sigma models on a K3 surface X was described explicitly by Aspinwall and Morrison
[4]. It thus gives a good example to focus on, although in some ways it is confusingly
different from the picture expected for a simply-connected threefold.
The total integral cohomology Λ =
H∗(X,
Z) has a natural integral symmetric
form (−, −) first introduced by Mukai defined by
(
(r1, D1, s1), (r2, D2, s2)
)
= D1 · D2 r1s2 r2s1.
The resulting lattice Λ is even and non-degenerate and has signature (4, 20). Define
∆(Λ) = Λ : (δ, δ) = −2}.
The group O(Λ) of isometries of Λ has an index two subgroup
O+(Λ)
consisting of
isometries which preserve the orientation of positive definite four-planes in Λ R.
3
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