SPACES OF STABILITY CONDITIONS 19

7.3. An example. Consider again the example of Section 6.3 relating to the

resolution of the Kleinian singularity Y =

C2/G.

Consider the function on Stab(D)

deﬁned by

(∗) F (Z) =

α∈Λ+

Z(α)2

log Z(α)

Here the sum is over all positive roots and Z denotes the central charge of a given

point of Stab(D). Of course, unless Stab(D) is the universal cover of hreg the

function F may be many-valued. This function F satisﬁes the WDVV equation

(see for example [51]) and thus deﬁnes an associative multiplication on the tangent

bundle to Stab(D). Explicitly, one can deﬁne double and triple point functions

θ1, θ2 =

α∈Λ+

θ1(α)θ2(α)

θ1, θ2, θ3 =

α∈Λ+

θ1(α)θ2(α)θ3(α)

Z(α)

where θi : K(D) → C are tangent vectors to Stab(D) at a point σ = (Z, P), and

then deﬁne a multiplication on tangent vectors by

θ1

∗ θ2, θ3 = θ1, θ2, θ3 = θ1, θ2 ∗ θ3 .

This gives an associative multiplication on the tangent bundle to Stab(D) whose

identity is the vector ﬁeld Z. Since this identity is not flat this does not quite deﬁne

a Frobenius manifold, but rather forms what Dubrovin calls an almost Frobenius

manifold [27]. In fact this structure is the almost-dual of the Frobenius manifold

of Saito type on the unfolding space of the surface singularity X (see [27, Sections

5.1, 5.2]).

It would be nice to generalise the function F to some other examples. In [22]

there are formulae for prepotentials of gauge theories which look like (∗) (see for

example Equation (3.1)) but with correction terms involving sums over graphs.

One of these was checked to satisfy the WDVV equation in [20]. This connection

looks worthy of further investigation.

As a ﬁnal remark, very recently Joyce [36] has constructed a flat connection on

the space of stability conditions on certain abelian categories satisfying the Calabi-

Yau condition. Extending this work to derived categories seems to be problematic

for several reasons, not least because of questions of convergence. Nonetheless,

at present Joyce’s approach seems to be our best hope for deﬁning interesting

structures on spaces of stability conditions.

Acknowledgements. The author is supported by a Royal Society University

Research Fellowship.

References

[1] P. Aspinwall and M. Douglas, D-Brane stability and monodromy, J. High Energy Phys.

2002, no. 5, no. 31.

[2] P. Aspinwall, R. Horja and R. Karp, Massless D-Branes on Calabi-Yau threefolds and mon-

odromy, Comm. Math. Phys. 259 (2005), no. 1, 45–69.

[3] P. Aspinwall and I. Melnikov, D-Branes on vanishing del Pezzo surfaces, J. High Energy

Phys. 2004, no. 12, 042 (2005).

[4] P. Aspinwall and D. Morrison, String theory on K3 surfaces, Mirror symmetry, II, 703–716,

AMS/IP Stud. Adv. Math., 1, Amer. Math. Soc., Providence, RI, (1997).

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