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 ZΛ ⊂

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

= {θ ∈ 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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