16 TOM BRIDGELAND
The web of categories indexed by elements of B3 described in Theorem 6.3
is very similar to the picture obtained by physicists studying cascades of quiver
gauge theories on X = ωP2 [28]. To obtain an exact match one should consider the
subcategories A(g) D up to the action of the group Aut(D). In the physicists’
pictures the operation corresponding to tilting is called Seiberg duality [8, 31] and
the resulting webs are called duality trees. Physicists, particularly Hanany and
collaborators, have computed many more examples (see for example [7, 28, 29]).
6.3. Resolutions of Kleinian singularities. There are a couple of interest-
ing examples where the theory described above enables one to completely describe
a connected component of the space of stability conditions.
Let G SL(2, C) be a finite group and let f : X Y be the minimal resolution
of the corresponding Kleinian singularity Y =
C2/G.
Define a full subcategory
D = {E
Db
Coh(X) : Rf∗(E) = 0}
Db
Coh(X).
The groups G have an ADE classification so we may also consider the associated
complex semi-simple Lie algebra g = gC with its Cartan subalgebra h g and
root system Λ
h∗.
The Grothendieck group K(D) with the Euler form can be
identified with the root lattice
h∗
equipped with the Killing form.
It was proved in [15] (see also [48]) that a connected component
Stab†(D)

Stab(D) is a covering space of
hreg
= h : θ(α) = 0 for all α Λ}.
The regions corresponding to stability conditions with a fixed heart are precisely the
connected components of the inverse images of the complexified Weyl chambers. If
we set
Aut†(D)
to be the subgroup of Aut(D) preserving this connected component
one obtains
Stab†(D)
Aut†(D)

=
hreg
W e
,
where W
e
= W Aut(Γ) is the semi-direct product of the Weyl group of g with
the finite group of automorphisms of the corresponding Dynkin graph.
In the same geometric situation one can instead consider the full subcategory
ˆ
D
Db
Coh(X) consisting of objects supported on the exceptional locus of f. A
connected component of the space of stability conditions is then a covering space
of the regular part of the affine Cartan algebra
ˆ
h and
Stab†(
ˆ
D )
Aut†(
ˆ
D )

=
ˆreg
h
ˆ
W e
,
where now
ˆ
W e =
ˆ
W
Aut(ˆ
Γ) is the semi-direct product of the affine Weyl group of
g with the finite group of automorphisms of the corresponding affine Dynkin graph.
For more details see [15].
In the An case the spaces Stab(D) and Stab(
ˆ
D ) are known to be connected
and simply-connected [34]. A similar but more difficult example involving crepant
resolutions of three-dimensional singularities has been considered by Toda [49] (see
also [17]). He has also considered three-dimensional Calabi-Yau categories defined
by considering a formal neighbourhood of a fibre of a K3 or elliptic fibration [50].
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