SPACES OF STABILITY CONDITIONS 17
7. Geometric structures on spaces of stability conditions
This section will be of a more speculative nature than the previous ones. I
shall try to use ideas from mirror symmetry to make a few remarks about what
geometric structures the space of stability conditions should carry.
7.1. Stability conditions and the stringy ahler moduli space. Let X
be a simply-connected Calabi-Yau threefold and set D =
Db
Coh(X). In Remark
3.9 it was argued that one should expect an embedding of the stringy ahler moduli
space MK(X) in the double quotient
Aut(D)\ Stab(X)/C.
It is tempting to suggest that these two spaces should be identified. In fact, as we
now explain, it is easy to see using Theorem 3.5 that this could never be the case.
Put simply, the space Stab(X) is too big and too flat.
For concreteness let us take X to be the quintic threefold. The stringy K¨ahler
moduli space MK(X) is, more or less by definition, the complex moduli space of
the mirror threefold Y . As is well-known this is a twice-punctured two-sphere with
a special point. The punctures are called the large volume limit point and the
conifold point, and the special point is called the Gepner point. The periods of
the mirror Y define holomorphic functions on MC(Y ) which satisfy a third order
Picard-Fuchs equation which has regular singular points at these three points.
Under mirror symmetry the periods of Lagrangian submanifolds of Y corre-
spond to central charges of objects of D. Thus we see that the possible maps
Z : K(D) C occurring as central charges of stability conditions coming from
points of MK(X) satisfy the Picard-Fuchs equation for Y . Since these satisfy no
linear relation, comparing with Theorem 3.5 we see that the space Stab(X) must
be four-dimensional and the double quotient above is a three-dimensional space
containing MK(X) as a one-dimensional submanifold. The embedding of this sub-
manifold in Stab(X) is highly transcendental.
More generally, for a simply-connected Calabi-Yau threefold X we would guess
that the space Stab(X) is not the stringy ahler moduli space, whose tangent
space can be identified with
H1,1(X),
but rather some extended version of it, whose
tangent space is
p
Hp,p(X).
To pick out the ahler moduli space as a submanifold of
Aut(D)\ Stab(X)/C,
we would need to define some extra structure on the space of stability conditions.
7.2. Related moduli spaces. There are at least three other types of moduli
spaces occurring in the mirror symmetry story which have a similar flavour to
spaces of stability conditions: universal unfolding spaces, big quantum cohomology
and the extended moduli spaces of Barannikov-Kontsevich. All these spaces carry
rich geometric structures closely related to Frobenius structures and all of them
are closely related to moduli spaces of SCFTs. In each case one can make links
with spaces of stability conditions, although none of these are close to being made
precise. We content ourselves with giving the briefest outlines of the connections
together with some references.
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