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 =
Consider the function on Stab(D)
(∗) F (Z) =
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 ) and thus deﬁnes an associative multiplication on the tangent
bundle to Stab(D). Explicitly, one can deﬁne double and triple point functions
θ1, θ2 =
θ1, θ2, θ3 =
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
∗ 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 . 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
It would be nice to generalise the function F to some other examples. In 
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 . This connection
looks worthy of further investigation.
As a ﬁnal remark, very recently Joyce  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
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