hall algebras techniques for dt theorypeople.mpim-bonn.mpg.de/gborot/files/dt3-dtintro.pdf · s...

Hall algebras techniques for DT theory

Severin Charbonnier

April 24, 2020

Severin Charbonnier (∴) Hall algebra – DT April 24, 2020 1 / 40

Overview

1 Physics of DT invariantsD-branesHilbert schemes and DT invariantsCorrespondences with GW and PT

2 Toy model: finitary Hall algebrasFinitary Hall algebras: Quotient identity and Wall-crossing formulaIntegration map

3 Motivic Hall algebrasStacks of flagsMotivic Hall algebraIntegration map


Table of Contents





D-branes

Space-time in string theory: R1,3 × X , where X is a Calabi-Yau 3-fold(CY3) with Kahler form J.There exists 5 consistent versions of superstring theory:

type I; type IIA; type IIB; E8 × E8 heterotic; SO(32) heterotic︸︷︷︸limiting cases of theory M

Particles are closed and open strings propagating in space-time. At theend points of open strings: one must specify the boundary conditions.

D-branes (for “Dirichlet boundary conditions”) are submanifolds ofR1,3 × X on which end points can live.

Dp-branes have real dimension p + 1: the p spatial dimensions lay in X ,the left one is time dimension of R1,3. p is even for type IIA, odd for typeIIB.


In the large volume limit: Vol(X ) =∫X J3 →∞, the aim is to engineer a

Bogomol’nyi-Prasad-Sommerfield (BPS) state using D-branes.

Definition (BPS D-brane for type IIA)

A BPS D-brane of B-type is the data of a pair (Y ,E ) where:

Y ⊆ X is a holomorphic submanifold

E −→ Y is a polystable holomorphic vector bundle

More generally, away from the large volume limit, a BPS D-brane is aBridgeland stable object E ∈ Db(Coh(X )).

BPS states correspond to black holes, and the condition Y holomorphiccorresponds to the minimization of the mass (=volume) of the brane. Thestability condition ensures stability of the blackhole (see Campbell’s talk,p.27 and discussion on nLab).

Aim of DT invariants in string theory: count the number of BPS states.


https://ncatlab.org/nlab/show/black+holes+in+string+theory

Hilbert schemes and DT invariants

Let β ∈ H2(X ,Z), n ∈ Z. The Hilbert scheme of points and curves in Xof type β, n is:

Hilbβ,n(X ) = {Y ⊂ X , 1− dim subscheme | [Y ] = β, χ(X ,OY ) = n}

Examples

β = 0: Hilb0,n(X ) is the Hilbert scheme of points (subschemes oflength n).

If Y is non-singular projective surface, let nβ := −12β · (KY + β):

Hilbβ,n(Y ) ∼= Divβ(Y )︸︷︷︸1−dim

× Hilb0,n−nβ (Y )︸︷︷︸0−dim

Goal: count the elements in Hilbβ,n(X ).


Theorem

There exists an isomorphism of projective schemes:

Hilbβ,n(X ) ∼= Mβ,n(X )

Mβ,n(X ) = {F −→ X | det(F ) ∼= OX , ch(F) = (1, 0,−β,−n)}/ ∼

Hilbβ,n(X ) and Mβ,n(X ) carry different obstruction theories. On Mβ,n(X ),it is symmetric, perfect, and has expected dimension 0. ApplyingBehrend-Fantechi machinery:

Definition (DT invariants)

DTβ,n(X ) =

∫[Mβ,n]vir

1 ∈ Z

Example: if Y ⊂ P4 is the quintic threefold, there are 2875 lines, soDT1,1(Y ) = 2875.


Correspondences with GW and PTX is a CY3, β ∈ H2(X ,Z), n ∈ Z, g ∈ Z≥0. Gromov-Witten invariants aregiven by GWg ,β(X ) =

∫[Mg,0(X ,β)]vir 1.

Pandharipande-Thomas pairs:

Pairβ,n(X ) =

{OX

f−→ E |(

dimE = 1dim supp coker(f ) = 0

), ch(E ) = (0, 0, β, n)

}Naive PT invariants: PTβ,n = e(Pairβ,n(X )) ∈ Z

GW/DT and PT/DT correspondence: conjecture and theorem

∑β∈H2(X ,Z)

n∈ZDTβ,n(X )(−e iλ)nxβ∑

n∈Z DT0,n(X )(−e iλ)n=

exp

(∑β∈H2(X ,Z)g∈Z≥0

GWg ,β(X )λ2g−2xβ

)exp

(∑g∈Z≥0

GWg ,0(X )λ2g−2)

∑n∈Z DTβ,n(X )qn∑n∈Z DT0,n(X )qn

=∑n∈Z

PTβ,n(X )qn ∀β ∈ H2(X ,Z)


Table of Contents





Strategy

1 Describe the relevant phenomenons about DT invariants (e.g.Wall-Crossing) as statements on a category A (e.g. existence anduniqueness of Harder-Narasimhan filtrations for different stabilityconditions): see Campbell’s (tremendous) talk;

2 Translate those statements in algebraic relations in a Hall algebraH(A);

3 Use a ring homomorphism (the integration map) I : H(A) −→ Cq[N]to get tractable identities on generating functions.


Finitary Hall algebras

Let A be an essentially small abelian category, i.e.

Every E ∈ A has finitely many subobjects;

Every group Ext iA(E ,F ) is finite.

Example: A = VectFq the category of finite dim. vector spaces over Fq.

Definition (Finitary Hall Algebra)

The finitary Hall algebra of A is the associative algebraH∧fty (A) = {f : (Ob(A)/ ∼=)→ C} endowed with the convolution product:

(f1 ∗ f2)(B) =∑A⊂B

f1(A) · f2(B/A)

Hfty (A) ⊂ H∧fty (A) is the subalgebra of functions with finite support.


For E ∈ A, let δE , κE ∈ Hfty (A) and δA ∈ H∧fty (A):

δE (B) = δE ,B ; κE = |AutE | · δE ; δA(B) = 1.

H∧fty (A) is an associative algebra with unit δ0.

Example: A = VectFq : δn(E ) = δn,dimE

δn ∗ δm = |Grn,n+m(Fq)|δn+m =(qn+m − 1) . . . (qm+1 − 1)

(qn − 1) . . . (q − 1)δn+m


Quotient identity

Fix P ∈ A, and define δPA,QuotPA ∈ H∧fty (A):

δPA(E ) = |HomA(P,E )| ; QuotPA(E ) = |Hom�A (P,E )|

Example: A = VectFq , dimP = n:

δPA =+∞∑m=0

qnmδm

QuotPA = δ0 +n∑

m=1

(nm

)(qm − 1) . . . (q − 1)q

m(m−1)2

+m(n−m)δm


Lemma (Quotient identity)

δPA = QuotPA ∗ δA

Proof: On E ∈ A, f ∈ Hom(P,E ) factors uniquely via its image:

imf

P Ef

so |HomA(P,E )| =∑A⊂E|Hom�

A (P,A)| · 1 �

Example: A = VectFq , dimP = n:

+∞∑m=0

qnmδm =+∞∑m=0

n∑p=0

(np

)(qm+p − 1) . . . (qm+1 − 1)q

p(p−1)2

+p(n−p)δm+p


Wall-crossing formula

Assume that A is equipped with the stability condition – Campbell p.30 –(Z ,P) and define δss(φ) ∈ H∧fty (A) the characteristic function of P(φ).

Lemma (Reineke)

δA =→∏φ∈R

δss(φ) where→∏

stands for the product on descending order of

phases.

Proof: Since ∀φ ∈ R, 0 ∈ P(φ), δss(φ) = δ0 + δss6=0(φ) where δss6=0(φ) is thecharacteristic function of non zero semi-stable elements.

→∏φ∈R

δss(φ) =→∏φ∈R

(δ0 + δss6=0(φ))

= δ0 +∑k≥1

∑φ1>···>φk

δss6=0(φ1) ∗ · · · ∗ δss6=0(φk)


Evaluated at 0, the r.h.s. yields 1, and at E ∈ A with E 6= 0, it gives:∑k≥1

∑φ1>···>φk

∑0=M0⊂M1⊂···⊂Mk=E

δss6=0(φ1)(M1/M0) . . . δss6=0(φk)(Mk/Mk−1)

Each term of the sum gives 1 if 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mk = E is aHN-filtration. Since HN filtration exists and is unique, the sum gives 1. �

δA does not depend on the stability condition, so we get

Corollary (Wall-crossing formula)→∏φ∈R

δss(φ; (Z1,P1)) =→∏φ∈R

δss(φ; (Z2,P2))


Integration map

Assume that A is linear over k, and Ext-finite:∑i∈Z

dimExt iA(E ,F ) <∞.

Denote by K (A), χ, N = K (A)/K (A)⊥ the Grothendieck group, theEuler form and the numerical Grothendieck group. N is equipped with abilinear form (·, ·).Define a group homomorphism (Chern character): ch : K (A)→ N, locallyconstant in families.Example: A = VectFq , ch(E ) = dimE .

Definition (Quantum torus)

The quantum torus associated to N is the non-commutative algebraCt [N] =

⊕α∈N C(t)xα with the product:

xα ∗ xβ = t−(α,β)xα+β


In general, the existence of an integration map I : H(A) requires:

either (finitary case) A is hereditary: Ext iA(E ,F ) = 0 for i > 1;

or (motivic case) A satisfies the CY3 condition:Ext iA(E ,F ) ∼= Ext3−i

A (E ,F )∗

Lemma

If A is hereditary, there exists an algebra homomorphism:

I : H∧fty (A) −→ Ct [N]|t=√q

f 7−→∑E∈A

f (E )

|AutE |xch(E)

The codomain is the quantum torus for the form (·, ·) = 2χ(·, ·)


Proof: for A,B,C ∈ A, Ext1A(C ,A)B are the extensions

0→ A→ B ′ → C → 0 where B ′ ∼= B.

κA ∗ κC =∑B∈A

|Ext1A(C ,A)B |

|HomA(C ,A)|κB =

∑B∈A

qdimExt1A(C ,A)B

qdim HomA(C ,A)κB

since A is Fq-linear. Therefore:

I(κA ∗ κC ) =1

qdim HomA(C ,A)

∑B∈A

qdimExt1A(C ,A)B ·

xch(A)+ch(C)︷︸︸︷xch(B)

= qdimExt1A(C ,A)−dim HomA(C ,A)xch(A)+ch(C)

= t−2χ(C ,A)xch(A)+ch(C)|t=√q = I(κA) ∗ I(κC )

which gives the claim with (·, ·) = 2χ(·, ·) �.


https://www.youtube.com/watch?v=Rh6xOtrDeRQ

A word on κA ∗ κC (with a pinch of salt):

κA ∗ κC (E ) =∑F⊂E|AutA| · δA(F ) · |AutC | · δC (E/F )

= |{F ⊂ E |F ∼= A, E/F ∼= C}|

=∑B∈A|{F ⊂ B |F ∼= A, B/F ∼= C}|δB(E )

=∑B∈A

|{0→ A→ B ′ → C → 0 |B ′ ∼= B}||HomA(C ,A)|

κB(E )

=∑B∈A

|Ext1A(C ,A)B |

|HomA(C ,A)|κB(E )


Example: A = VectFq , ch(E ) = dimE :

I(δn) =q

n2 xn

(qn − 1) . . . (q − 1)

I(δn ∗ δm) =q

n+m2 xn+m

(qn − 1) . . . (q − 1)(qm − 1) . . . (q − 1)

I(δA) =+∞∑n=0

qn2 xn

(qn − 1)(q − 1)=: Eq(x)

Eq(x) is the quantum dilogarithm. In this context, the quotient identitygives, after applying I:

+∞∑m=0

qm(n+ 12

)xm

(qm − 1) . . . (q − 1)=

+∞∑m=0

n∑p=0

(np

)qp(n+ p

2)xp

qm2 xm

(qm − 1) . . . (q − 1)


Table of Contents





A word on “Motivic”

It refers to invariants ψ such that, if Y ⊂ X is a closed subvariety:

ψ(X ) = ψ(Y ) + ψ(X\Y )

e.g. the Euler characteristic e(X ) =∑

i∈Z(−1)i dimCH i (Xan,C); the DTinvariants (expressed as a weighted Euler characteristic).

In this part: X is a CY 3-fold along with H1(X ,OX ) = 0, A = Coh(X ), Sis the category of schemes.


Preliminary on stacks

Definition (Category over S)

A category over S is a category C with a covariant functor π : C → S.E ∈ C is a lifting of S ∈ S if π(E ) = S .

Definition (Groupoid fibration)

C over S is a groupoid fibration (GF) over S if for all f ∈ HomS(T ,S)and every E lifting of S , there exists a unique (up to isomorphism) liftingof f to C: f : F → E .

For {Si → S} an open covering of S and for Ei a lifting of Si to C, denoteSij the intersection Si ×S Sj and Ei |Sij the pullback of Ei to Sij .


Definition (Descent datum)

Let C be GF over S. A descent datum for C is

an open covering {Si → S};for all i , a lifting Ei of Si ;

for all i , j , an isomorphism αij : Ei |Sij → Ej |Sij satisfying the cocyclecondition αik = αjk ◦ αij on Sijk .

A descent datum is effective if there exists a lifting E of S withisomorphisms αi : E | Si → Ei such that αij = αj | Sij ◦ (αi |Sij)−1.

Definition (Isomorphisms are a sheaf)

Let C be GF over S. Isomorphisms are a sheaf for C if for any E , E ′

liftings of S , for all covering {Si → S} and every collection ofisomorphisms αi : E | Si → E ′ | Si satisfying αi | Sij = αj |Sij , there existsa unique isomorphism α : E → E ′ such that α | Si = αi .


Definition (Stack)

A stack is a GF over S such that isomorphisms are a sheaf and everydescent datum is effective.

Definition (S-flat sheaf)

A quasi-coherent sheaf F on S is S-flat if for all p ∈ S , Fp is a flatOS ,p-module, i.e. for all exact sequence of OS,p-modules:

0→ A→ B → C → 0

the tensor product is exact:

0→ A⊗ Fp → B ⊗ Fp → C ⊗ Fp → 0


Stacks of flags

Definition (stack of flags)

The moduli stack of n-flags of coherent sheaves on X is the categoryM(n) where:

the objects over S × X (S ∈ S) are flags of coherent sheaves0 = E0 ↪→ E1 ↪→ · · · ↪→ En = E such that for all i , Fi := Ei/Ei−1 isS-flat.

the morphisms over a morphism of schemes Sf→ T is the collection

of isomorphisms θi : f ∗(Ei )→ E ′i and the cartesian diagram:

(0 = f ∗(E0) f ∗(E1) . . . f ∗(En) = f ∗(E )) S

( 0 = E ′0 E ′1 . . . E ′n = E ′ ) T

θ0 θ1

π

θn f

π

M :=M(1).


As a consequence of the definition, the Ei ’s are S-flat.

The stacks M(n) are algebraic.For all i ∈ {1, . . . , n}, we have morphisms of stacks ai and b:

ai : M(n) →M b : M(n) →M(E1 ↪→ · · · ↪→ En) 7→ Fi (E1 ↪→ · · · ↪→ En) 7→ En

M(2) is the stack of short exact sequences in A: 0 ↪→ E1 ↪→ E2 → F2 → 0and we have the diagram:

M(2) M

M×M

b

(a1,a2)


Motivic Hall algebra

The motivic Hall algebra H(A) is the relative Grothendieck group of

stacks over M, K (St/M). The objects of H(A) are classes [Af→M].

They satisfy the motivic property:

[Af→ E ] = [B

f |B→ E ] + [A\Bf |A\B→ E ]

Pushforward and pullback of a morphism of stack a : E → F .

a∗ : K (St/E )→ K (St/F ) a∗ : K (St/F )→ K (St/E )

[Af→ E ] 7→ [A

f ◦a→ F ] [Bg→ F ] 7→ [A

f→ E ]

(for the pullback, f is deduced from the cartesian square

A B

E F

f g

a

)


Definition (Convolution product)

The convolution product on H(A) ism = b∗ ◦ (a1, a2)∗ : H(A)⊗H(A)→ H(A).

(a1, a2)∗ : H(A)⊗H(A)→ K (St/M(2))

[A1 × A2f1×f2→ M×M] 7→ [P

h→M(2)]

b∗ : K (St/M(2))→ H(A)

[Ph→M(2)] 7→ [P

b◦h→ M]

Summed up in:

T M(2) M

A1 × A2 M×M

h b

(a1,a2)

f1×f2


Theorem (Joyce)

(H(A),m) is an associative unital algebra.

The unit is given by [M0 ↪→M]. M0: substack of zero objects in A.

For a subcategory C of A, MC , MOC and HilbC ⊂MOC are the stacksparametrizing respectively sheaves E ∈ C ⊂ A, sheaves with a section

OXf→ E , and sheaves with a surjective section OX

f� E .

Elements of H(A) Definition Analogous in H∧fty (A)

δOC = [MOCf→M] forgets the section δPA

QuotOC = [HilbCf |Hilb→ M] forgets the section QuotPA

δC = [MCid→M] identity δA


Quotient identity

Theorem (Quotient identity)

δOA = QuotOA ∗ δA

Proof: on the r.h.s. the product is given by:

T M(2) M

Hilb×M M×M

h b

(a1,a2)

f×id

The points of the stack T over S are diagrams:

OS×X

D = 0 E1 E2 F2 0

γ δ

α


ϕ : T →MO

D 7→ (OS×Xδ→ E2)

is not an isomorphism. However, if S = Spec(C), δ factors uniquely via itsimage:

OSpec(C)×X

E1 E2

γ δ

α

therefore MO can be stratified by locally closed substacks on which ϕ is

an isomorphism on each piece, so [T b◦h→ M] = [MO f→M] = δOA �


Motivic wall-crossing

Let A ⊂ Db(Coh(X )). Reinier’s and Campbell’s talk: t-structure, heartsand tilting on A. Define the torsion pair:{

T = {E ∈ Coh(X ) | dim suppE = 0}F = {E ∈ Coh(X ) |HomX (Ox ,E ) = 0 ∀x ∈ X}

and A] := Tilt(T ,F)(A) the tilt associated to (T ,F).

Principle of wall-crossing in the motivic context: varying the t-structure onDb(A) through this tilting amounts to get various interpretations ofsurjective sections.

Theorem (motivic wall-crossing)

QuotOA ∗ δT = QuotOT ∗ δT ∗ QuotOA]

Proof: appendix.


Integration mapA is C-linear and Ext-finite. We denote:

χ the Euler form;

N = K (A)/K (A)⊥ the numerical Grothendieck group, χ descends onN × N (Riemann-Roch);

ch : K (A)→ N the Chern character;

ν : M→ Z the constructible function of Behrend.

Since the Chern character is locally constant in families, there is adecomposition of M into closed and open substacks:

M =⊔α∈NMα ; H(A) =

⊕α∈N

K (St/Mα)

Remark

X is a CY3, so A satisfies the CY3 condition:Ext iA(E ,F ) ∼= Ext3−i

A (E ,F )∗.


Restrictions

Let σ ∈ {−1,+1}Instead of the quantum torus Ct [N], we rather consider itssemi-classical limit as t → σ: Cσ[N] =

⊕α∈N

C · xα is the commutative

Poisson algebra with:

xα · xγ = limt→σ

xα ∗ xγ = σ2χ(α,γ)xα+γ

{xα, xγ} = limt→σ

xα ∗ xγ − xγ ∗ xα

t2 − 1= 2χ(α, γ)xα · xγ

The algebra of regular elements Hreg (A) ⊂ H(A) is the span of

[Sf→M] where S is a variety. The semi-classical limit is the

commutative algebra Hsc(A) = Hreg (A)/(L− 1)Hreg (A).


Theorem (Integration map)

There exists a Poisson and homomorphism:

Iσ : Hsc(A)→ Cσ[N]

[Sf→Mα] 7→

{e(S)xα, ifσ = +1

e(S , f ∗(ν))xα, ifσ = −1

Example: application to the wall-crossing identity.

Iσ(QuotOA ∗ δT ) = Iσ(QuotOT ∗ δT ∗ QuotOA])

Iσ(QuotOA) = Iσ(QuotOT ) ∗ Iσ(QuotOA])∑β∈H2(X ,Z)

n∈Z

DTβ,n(X )ynxβ =∑n≥0

DT0,n(X )yn∑

β∈H2(X ,Z)n∈Z

PTβ,n(X )ynxβ

We recover DT/PT correspondence (Bridgeland, Toda).


Thank you!

Physical insights on DT invariants: master thesis of Stephen PietromonacoAn Introduction to Modern Enumerative Geometry with Applications tothe Banana Manifold

Survey on Hall algebras and DT of Tom Bridgeland (again, very clear):Hall algebras and Donaldson-Thomas invariants

Details for the motivic Hall algebra: Bridgeland’s article An introductionto motivic Hall algebras

Stacks: Barbara Fantechi’s survey Stacks for everybody

Next time: representations of quivers, stay safe and tuned...


https://arxiv.org/pdf/1905.07085.pdf

https://arxiv.org/pdf/1905.07085.pdf

https://claymath.org/sites/default/files/bridgeland.pdf

http://www.tom-bridgeland.staff.shef.ac.uk/publications/pub14.pdf


https://www.researchgate.net/publication/228581690_Stacks_for_Everybody

Appendix : proof of the motivic wall-crossing formula

See Bridgeland: Hall algebras and curve-counting invariants.

(T ,F) is a torsion pair for A, so every E ∈ A sits in a unique exactsequence 0→ T → E → F → 0, where T ∈ T , F ∈ F . In the samemanner for E ∈ A] with the torsion pair (F , T [−1]):0→ F → E → T [−1]→ 0. In the Hall algebra, those exactsequences give (lemma 4.1):

δA = δT ∗ δF (1) δA] = δF ∗ δT [−1] (2)

With similar techniques as for the quotient identity, and applyingH0(X , ·) to the previous exact sequences (lemma 4.2):

δOA = δOT ∗ δOF (3) δOA] = δOF ∗ δOT [−1] (4)

Quotient identities:

δOA = QuotOA ∗ δA (5) δOA] = QuotOA] ∗ δA] (6) δOT = QuotOT ∗ δT (7)

H0(X ,T [−1]) = 0 so: δOT [−1] = δT [−1] (8)



Above the equality signs are the equations of previous slide used at eachstep:

(QuotOT ∗ δT ) ∗ (QuotOA] ∗ δA])(7)∗(6)

= δOT ∗ δOA](4)= δOT ∗ δOF ∗ δOT [−1]

(3)∗(8)= δOA ∗ δT [−1]

(5)= QuotOA ∗ δA ∗ δT [−1]

(1)= QuotOA ∗ δT ∗ δF ∗ δT [−1]

(2)= QuotOA ∗ δT ∗ δA]

In the end:QuotOT ∗ δT ∗ QuotOA] = QuotOA ∗ δT


hall algebras techniques for dt theorypeople.mpim-bonn.mpg.de/gborot/files/dt3-dtintro.pdf · s...

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