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 K¨ 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 K¨ 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 K¨ 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 K¨ 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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