Stability andwall-crossing

Tom Bridgeland

University of Sheffield

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1. Hearts and tilting

Definition of a Torsion pair

Let A be an abelian category.

A torsion pair (T ,F) ⊂ A is a pair of full subcategories such that:

(a) Hom(T ,F ) = 0 for T ∈ T and F ∈ F .

(b) for every object E ∈ A there is a short exact sequence

0 −→ T −→ E −→ F −→ 0

for some pair of objects T ∈ T and F ∈ F .

T F

A

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Definition of a heart

Let D be a triangulated category.

A heart A ⊂ D is a full subcategory such that:

(a) Hom(A[j ],B[k]) = 0 for all A,B ∈ A and j > k .

(b) for every object E ∈ D there is a finite filtration

0 = Em → Em+1 → · · · → En−1 → En = E

with factors Fj = Cone(Ej−1 → Ej) ∈ A[−j ].

· · · · · ·A[1] A A[−1]

D

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Properties of hearts

(i) It would be more standard to say that A ⊂ D is the heart of abounded t-structure on D. But any such t-structure isdetermined by its heart.

(ii) The basic example is A ⊂ Db(A).

(iii) In analogy with that case we define H jA(E ) := Fj [j ] ∈ A.

(iv) A is automatically an abelian category.

(v) The short exact sequences in A are precisely the triangles in Dall of whose terms lie in A.

(vi) The inclusion functor gives an identification K0(A) ∼= K0(D).

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The tilt of a heart at a torsion pair

Suppose A ⊂ D is a heart, and (T ,F) ⊂ A a torsion pair.

We can define a new, tilted heart A] ⊂ D as in the picture.

F [1] T FT [1] T [−1] · · ·· · ·

A

A]

An object E ∈ D lies in A] ⊂ D precisely if

H−1A (E ) ∈ F , H0A(E ) ∈ T , H iA(E ) = 0 otherwise.

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Example of tilting: threefold flop

F+[1] T+ F+ · · ·· · · Db(X+)

Coh(X+)

Per+(X+/Y )

F−[1] T− F−

∼=

· · ·· · · Db(X−)

∼=Per−(X−/Y )

Coh(X−)

X−

Y

X+

F+ = 〈OC+(−i)〉i>1, F− = 〈OC−(−i)〉i>2.

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Stable pairs as quotients in a tilt

Consider tilting A = Coh(X ) ⊂ D(X ) with respect to the torsion pair

T = {E ∈ Coh(X ) : dimC supp(E ) = 0},

F = {E ∈ Coh(X ) : HomX (Ox ,E ) = 0 for all x ∈ X}.

T F T [−1] · · ·· · ·

A]

A

Note that OX ∈ F ⊂ A]. We claim that

Pairs(β, n) =

{quotients OX � E in A]with ch(E ) = (0, 0, β, n)

}.

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Proof of the claim about stable pairs

Given a short exact sequence in the category A]

0 −→ J −→ OXf−−→ E −→ 0,

we take cohomology with respect to the standard heart A ⊂ D.

0→ H0A(J)→ OXf−−→ H0A(E )→ H1A(J)→ 0→ H1A(E )→ 0.

T F T [−1] · · ·· · ·

A]

A

It follows that E ∈ A ∩A] = F and coker(f ) = H1A(J) ∈ T .

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Last time ...

(a) Hall algebras: Hallfty(C), Hallmot(C).

0→ A→ B → C → 0

&&ww

(A,C ) B

M×M (a,c)←−−− M(2) b−−−→ M

(b) Character map ch : K0(C)→ N ∼= Z⊕n.(c) Quantum torus: Cq[N] =

⊕α∈N C(t) · xα with

xα ∗ xγ = q−12

(γ,α) · xα+γ.

(d) Integration map: I : Hall(C)→ Cq[N].10 / 31

Positive cones and completions

Choosing a basis (e1, · · · , en) for the group N gives an identification

C[N] = C[x±11 , · · · , x±1n ].

We often need to use the positive cone

N+ ={ n∑

i=1

λiei : λi > 0}⊂ N ,

and the associated completion

C[[N+]] ∼= C[[x1, · · · , xn]].

We can similarly define the completed quantum torus Cq[[N+]].

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Sketch proof of the DT/PT identity

(i) Reineke’s identity: δOA = QuotOA ∗δA and δOA] = Quot

OA] ∗δA] .

(ii) Torsion pair identities: δA = δT ∗ δF and δA] = δF ∗ δT [−1].

T F T [−1] · · ·· · ·

A]

A

(iii) Torsion pair identities with sections:

δOA = δOT ∗ δOF and δOA] = δ

OF ∗ δOT [−1].

(iv) All maps OX → T [−1] are zero, so δOT [−1] = δT [−1].

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Conclusion of the sketch proof

(v) Reineke’s identity again: δOT = QuotOT ∗δT .

(vi) Putting it all together: QuotOA ∗δT = QuotOT ∗δT ∗ QuotOA] .

(vii) Restrict to sheaves supported in dimension 6 1. The Euler formis then trivial so the quantum torus is commutative. Thus

I(QuotOA) = I(QuotOT ) ∗ I(QuotOA]).

(viii) Setting t = ±1 then gives the required identity∑β,n

DT(β, n)xβyn =∑n

DT(0, n)yn ·∑β,n

PT(β, n)xβyn.

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2. Generalized DT invariants

Moduli spaces of framed sheaves

Let X be a Calabi-Yau threefold.

So far we have been discussing moduli spaces of objects in thecategory Db Coh(X ) equipped with a kind of framing.

ExampleThe Hilbert scheme parameterizes sheaves E ∈ Coh(X ) equippedwith a surjective map f : OX � E .

(i) This framing data eliminates all stabilizer groups, so the modulispace is a scheme, and therefore has a well-defined Eulercharacteristic.

(ii) In this context wall-crossing can be achieved by varying thet-structure on the derived category Db Coh(X ).

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What about unframed DT invariants?

Fix a polarization of X and a class α ∈ N , and consider the stack

Mss(α) ={E ∈ Coh(X ) : E is semistable with ch(E ) = α

}.

(a) In the case when α is primitive, and the polarization is general,this stack is a C∗-gerbe over its coarse moduli space M ss(α),and we set

DTnaive(α) = e(M ss(α)) ∈ Z.Genuine DT invariants are defined using virtual cycles or by aweighted Euler characteristic as before.

(b) In the general case, Joyce figured out how to define invariants

DTnaive(α) ∈ Q

with good properties, and showed that they satisfy wall-crossingformulae as the polarization is varied.

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Quantum and classical DT invariants

(a) The generating function for the quantum DT invariants is

q-DTµ = I([Mss(µ) ⊂M]

)∈ Cq[[N+]].

(b) The generating function for the classical DT invariants is

DTµ = limq→1

(q − 1) · log q-DTµ ∈ C[[N+]].

A difficult result of Joyce shows that this limit exists in general.

(c) The DT invariants are also encoded by the Poissonautomorphism

Sµ = exp{

DTµ,−}∈ AutC[[N+]].

This coincides with the q = 1 limit of conjugation by q-DTµ.

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Example: a single rigid stable bundle

Suppose there is a single rigid stable bundle E of slope µ. Then

Mss(µ) = {E⊕n : n > 0} =⊔n>0

BGL(n,C).

Set α = ch(E ) ∈ N . Applying the integration map we calculate

(a) The quantum DT generating function is

q-DTµ =∑n>0

xnα

(qn − 1) · · · (q − 1)∈ Cq[[N+]].

We recognise the quantum dilogarithm Φq(xα).

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A single stable bundle continued

(b) The classical DT generating function is

DTµ = limq→1

(q − 1) · log Φq(xα) =∑n>1

xnα

n2

and we conclude that DT(nα) = 1/n2.

(c) The Poisson automorphism Sµ ∈ AutC[[N+]] is

Sµ(xβ) = exp{∑

n>1

xnα

n2,−}

(xβ) = xβ · (1 + xα)〈α,β〉

where the RHS should be expanded as a power series.

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3. Stability conditions

Stability conditions

Let A be an abelian category.

DefinitionA stability condition on A is a map of groups Z : K0(A)→ C suchthat

0 6= E ∈ A =⇒ Z (E ) ∈ H̄,

where H̄ = H ∪ R

Phases and stability

Definitions(a) The phase of a nonzero object E ∈ A is

φ(E ) =1

πarg Z (E ) ∈ (0, 1],

(b) An object E ∈ A is Z -semistable if

0 6= A ⊂ E =⇒ φ(A) 6 φ(E ).

Z (A)

Z (E )

φ(A)φ(E )

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Harder-Narasimhan filtrations

DefinitionA stability condition Z has the Harder-Narasimhan property if everyobject E ∈ A has a filtration

0 = E0 ⊂ E1 ⊂ · · · ⊂ En ⊂ E

such that each factor Fi = Ei/Ei−1 is Z -semistable and

φ(F1) > · · · > φ(Fn).

(i) If A has finite length this condition is automatic.(ii) When they exist, HN filtrations are necessarily unique, because

the usual argument shows that if F1,F2 are Z -semistable then

φ(F1) > φ(F2) =⇒ Hom(F1,F2) = 0.

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Another Reineke identity

Let C be a finitary abelian category equipped with a stabilitycondition Z having the Harder-Narasimhan property. Let

δss(φ) ∈ Ĥallfty(A)

be the characteristic function of Z -semistable objects of phase φ ∈ R.

Lemma (Reineke)

There is an identity δC =∏→

φ∈R δss(φ).

Proof.The product is taken in descending order of phase. The result followsfrom existence and uniqueness of the HN filtration.

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Wall-crossing formula

(a) The LHS of the above identity is independent of Z so given twostability conditions we get a wall-crossing formula

−→∏φ∈R

δss(φ,Z1) =−→∏φ∈R

δss(φ,Z2).

(b) If C has global dimension 6 1 we can apply the integration mapI to get an identity in the ring Cq[[N+]].

(c) We can then take the q = 1 limit and obtain an identity in thegroup of automorphisms of the Poisson algebra C[[N+]].

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Example: the A2 quiver

Let C be the abelian category of representations of the A2 quiver. Ithas 3 indecomposable representations:

0 −→ S2 −→ E −→ S1 −→ 0.

We have N = K0(A) = Z⊕2 = Z[S1]⊕ Z[S2],

〈(m1, n1), (m2, n2)〉 = m2n1 −m1n2,

and there are isomorphisms

Cq[[N+]] = C〈〈x1, x2〉〉/(x2 ∗ x1 − q · x1 ∗ x2)

C[[N+]] = C[[x1, x2]], {x1, x2} = x1 · x2.

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Quantum pentagon identity

The space Stab(A) is isomorphic to H̄2 and there is a single wall

W = {Z ∈ Stab(A) : ImZ (S2)/Z (S1) ∈ R>0}

where the object E is strictly semistable.

Z(S1)Z(S2)

Z(E)W

E unstable E stable

Z(S2)Z(S1)

Z(E)

The wall-crossing formula in Cq[[N+]] becomes the pentagon identity

Φq(x2) ∗ Φq(x1) = Φq(x1) ∗ Φq(√q · x1 ∗ x2) ∗ Φq(x2).

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Semi-classical version

Z(S1)Z(S2)

Z(E)W

E unstable E stable

Z(S2)Z(S1)

Z(E)

The semi-classical version of the wall-crossing formula is the clusteridentity

C(0,1) ◦ C(1,0) = C(1,0) ◦ C(1,1) ◦ C(0,1).

Cα : xβ 7→ xβ · (1 + xα)〈α,β〉 ∈ AutC[[x1, x2]].

It can be viewed in the group of birational automorphisms of (C∗)2.

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4. Stability in triangulatedcategories

Stability in triangulated categories

Let D be a triangulated category.

DefinitionA stability condition on D is a pair (Z ,A) where(i) A ⊂ D is a heart,(ii) Z : K0(A)→ C is a group homomorphism,such that Z defines a stability condition on A with the HN property.

An object E ∈ D is defined to be semistable if E = A[n] for someZ -semistable A ∈ A. The phase of E is then φ(E ) := φ(A) + n.

A[1] A A[−1] · · ·· · · D

R

φ=0 φ=−1φ=1φ=2

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Space of stability conditions

We consider only stability conditions satisfying the extra conditions

(a) The central charge Z : K0(D)→ C factors via our fixed map

ch : K0(D) −→ N ∼= Z⊕n.

(b) There is a K > 0 such that for any semistable object E ∈ D

Z (E ) > K · ‖ch(E )‖.

The set Stab(D) of such stability conditions has a natural topology.

TheoremSending a stability condition to its central charge defines a localhomeomorphism

Stab(D) −→ HomZ(N ,C) ∼= Cn.

In particular, Stab(D) is a complex manifold.

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