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).
19
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