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 K¨ ahler class β + iω ∈

H2(X,

C). In fact, while it is

expected that the integrals deﬁning the sigma model will converge providing that

the K¨ 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

K¨ 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 K¨ 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 K¨ 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) = {δ ∈ 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.

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