Proof. First note that the group
GL +(2, R) can be thought of as the set of
pairs (T, f) where f : R R is an increasing map with f(φ + 1) = f(φ) + 1, and
T :

is an orientation-preserving linear isomorphism, such that the induced
maps on
= R/2Z =
\ {0})/R
are the same.
Given a stability condition σ = (Z, P) Stab(D), and a pair (T, f)
GL +(2, R),
define a new stability condition σ = (Z , P ) by setting Z = T
Z and P (φ) =
P(f(φ)). Note that the semistable objects of the stability conditions σ and σ are
the same, but the phases have been relabeled.
For the second action, note that an element Φ Aut(D) induces an automor-
phism φ of K(D). If σ = (Z, P) is a stability condition on D define Φ(σ) to be the
stability condition (Z
P ), where P (t)=Φ(P(t)).
Neither of the two group actions of Lemma 3.8 will be free in general. In
particular, if σ = (Z, P) is a stability condition in which the image of the central
charge Z : K(D) C lies on a real line in C then σ will be fixed by some subgroup
GL +(2, R). However there is a subgroup C
GL +(2, R) which does act freely.
If λ C then λ sends a stability condition σ = (Z, P) to the stability condition
λ(σ) = (Z , P ) where Z (E) =
and P (φ) = P(φ + Re(λ)). Note that
for any integer n the action of the shift functor [n] on Stab(D) coincides with the
action of n C.
Remark 3.9. Return for a moment to the discussion of Section 2.4 in which
D =
Fuk(X, β + iω).
The action of C on Stab(D) clearly corresponds to rotating the holomorphic three-
form Ω. It also seems reasonable to guess that the action of Aut(D) on Stab(D)
corresponds to the discrete group quotient N M. Thus we might expect an
embedding of the complex moduli space MC(X) in the double quotient
Aut(D)\ Stab(D)/C.
The mirror statement is that if X is a Calabi-Yau with a given complex structure
and D =
Coh(X) then the above quotient contains the stringy ahler moduli
space MK(X). In the next section we will examine this suggestion in some simple
4. Compact examples
In this section I review some examples of stability conditions on smooth projec-
tive varieties. The only Calabi-Yau examples are elliptic curves and K3 and abelian
4.1. Elliptic curves. Let X be a complex projective curve of genus one. It
was shown in [12] that the action of
GL +(2, R) on Stab(X) is free and transitive.

GL +(2, R)

C × H
where H C is the upper half-plane. Quotienting by the group of autoequivalences
of D =
Coh(X) gives

SL(2, Z)
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