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 +
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
Coh(X1, I1)

Fuk(X2, β2 + iω2)
Fuk(X1, β1 + iω1)

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

MK(X2), MK(X1)

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 =
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
consisting of
isometries which preserve the orientation of positive definite four-planes in Λ R.
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