ADHM Sheaf Theory and Wallcrossing

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國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

www.math.ntu.edu.tw/ ~ mathlib/preprint/2011- 18.pdf

ADHM Sheaf Theory and Wallcrossing

Wu-yen Chuang

December 28, 2011

Wu-yen Chuang^♠ Abstract.

In this article we survey the recent developments in ADHM sheaf theory on a smooth projective variety X. When X is a curve the theory is an alternative construction of stable pair theory of Pandharipande and Thomas or Gromov-Witten theory on local curve geometries. The construction relies on relative Beilinson spectral sequence and Fourier- Mukai transformation. We will present some applications of the the- ory, including the derivations of the wallcrossing formulas, higher rank Donaldson-Thomas invariants on local curves, and the coholomogies of the moduli of stable Hitchin pairs.

§1. Introduction

Recently we have seen much progress in the study of curve enumera- tions on Calabi-Yau 3-folds. Three different types of theories have been proposed, including Gromov-Witten (GW) theory, Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) theory. They are conjectured to be equivalent at the level of generating functions after suitable changes of variables.

First let us briefly recall the definitions of these theories. Let X be smooth projective Calabi-Yau 3-fold over C. For g ≥ 0 and β ∈ H₂(X, Z), the GW invariant Ng,β is defined as the integration of the virtual class,

Ng,β= Z

[Mg(X,β)]^vir

1 ∈ Q ,

where Mg(X, β) is the Deligne-Mumford stack of the stable map f : C → Z with C a genus g curve and f_∗[C] = β. The reduced GW

Received Month Day, Year.

Revised Month Day, Year.

2000 Mathematics Subject Classification. Primary 14N35; Secondary 81T30.

♠Golden-Jade Fellow, Kenda foundation, Taiwan

generating function is

ZGW = exp X

g,β6=0

Ng,βλ^2g−2v^β ,

where we have omitted the contribution from the constant maps.

Now we define the DT theory. Consider the Hilbert scheme I_n(X, β) of 1-dimensional subschemes Z ⊂ X with χ(O_Z) = n and β ∈ H₂(X, Z).

We regards In(X, β) as the moduli parametrizing the ideal sheaves of 1-dimensional subschemes. The DT invariant In,β is given by

In,β= Z

[In(X,β)]^vir

1 ∈ Z .

The reduced DT generating function is given by Z_DT⁰ =X

n,β

In,βqⁿv^β/X

n

In,0qⁿ .

and the MNOP conjecture [23][24] states that ZGW = Z_DT⁰ after the change of variables q = −e^iλ.

The stable pair of Pandharipanda and Thomas [31] is, by definition, a pair (F, s), where s is a section of F ,

s : OX→ F

such that F is a pure sheaf of dimension 1 and s has zero dimensional cokernel. The moduli Pn(X, β) of PT stable pair (F, s) with [F ] = β and χ(F ) = n is constructed in and also equipped with a symmetric perfect obstruction theory. The PT invariants and the generating function are given by

P_n,β= Z

[P_n(X,β)]^vir

1 ∈ Z , ZP T =X

n,β

P_n,βqⁿv^β .

Pandharipande and Thomas conjectured that Z_DT⁰ = Z_{P T} as a wall- crossing formula where the denominator in Z_DT⁰ is the wallcrossing dif- ference. This conjecture has been proved in [34][33][2].

Motivated by string theoretical consideration, ADHM sheaf theory was first introduced by Diaconescu [7] and the theory has a natural variation of the stability conditions [6, 3]. In an asymptotic chamber of the stability condition space the ADHM sheaf theory on a projective curve X over C is equivalent to admissible pair theory on the projective plane bundle over X. When the twisting data (M1, M2) are chosen such

that M1⊗XM2' K_X⁻¹, this pair theory becomes a stable pair theory on the local Calabi-Yau 3-fold of the total space M₁⁻¹⊗XM₂⁻¹ over X.

The key ingredients of the construction consist of a relative version of Beilinson spectral sequence and Fourier-Mukai transformation.

Using Joyce-Song theory of generalized Donaldson-Thomas invari- ants [20], explicit wallcrossing formulas for ADHM invariants on curves have been derived [3], which also give rise to a further generalization of higher rank ADHM invariants [4]. This part of higher rank generaliza- tion is also motivated by the work of Toda [35] and Stoppa [32].

Another interesting application of ADHM sheaf theory on curves is the computation of Betti and Hodge number of moduli spaces of stable Hitchin pairs [5]. The application is based on refined generalizations of wallcrossing formulas, generalized Donaldson-Thomas invariants, and multicover formulas.

The purpose of the article is to survey the aforementioned devel- opments and related background material. The paper is organized as follows. In section 2 we review the definition of ADHM sheaf theory and prove a correspondence between stable pair theory and ADHM sheaf the- ory on curves. In section 3 we give a brief review of Joyce-Song theory of generalized Donaldson-Thomas invariants and present our results about wallcrossing formulas and higher rank invariants. A computational com- parison between Joyce-Song and Kontsevich-Soibelman formulas is also presented. In the final section we give our conjectural recursive relations for the Poincar´e polynomials of the Hitchin moduli space.

Acknowledgements. The author is grateful to the organizers of the conference ”Algebraic Geometry in East Asia” held in National Tai- wan University in November 2011 for giving him an opportunity to par- ticipate in such a fruitful event. He would also like to thank Duiliu- Emanuel Diaconescu and Guang Pan for the collaboration on the topics presented in this article. He is supported by NSC grant 99-2115-M-002- 013-MY2.

§2. ADHM Sheaf Theory

Let X be a smooth projective scheme over C equipped with a very ample line bundle O_X(1).

Definition 2.1. Let M1, M2 be fixed line bundles on X. Set M = M1⊗XM2. For fixed data X = (X, M1, M2), let Q_X denote the abelian category of (M1, M2)-twisted coherent ADHM quiver sheaves. An object of Q_X is given by a collection E = (E, E_∞, Φ1, Φ2, φ, ψ) where

• E, E∞ are coherent OX-modules

• Φi : E ⊗XMi → E, i = 1, 2 , φ : E ⊗XM1⊗XM2 → E∞, ψ : E_∞ → E are morphisms of OX-modules satisfying the ADHM relation

(2.1) Φ1◦ (Φ2⊗ 1M₁) − Φ2◦ (Φ1⊗ 1M₂) + ψ ◦ φ = 0.

The morphisms are natural morphisms of quiver sheaves i.e. col- lections (ξ, ξ_∞) : (E, E_∞) → (E⁰, E⁰_∞) of morphisms of OX-modules satisfying the obvious compatibility conditions with the ADHM data.

Let CX be the full abelian subcategory of QX consisting of objects with E∞ = V ⊗ OX, where V is a finite dimensional vector spaces over C. Note that given any two objects E, E⁰ of CX, the morphisms ξ_∞ : V ⊗ O_X → V⁰⊗ OX must be of the form ξ_∞ = f ⊗ 1_OX, where f : V → V⁰ is a linear map.

The numerical type of an object E of C_X is the collection (rank(E), deg(E), dim(V )) ∈ Z≥0× Z × Z≥0 .

An object of C_X is called an ADHM sheaf. An ADHM sheaf with ψ and φ identically 0 is called a Higgs sheaf.

Definition 2.2. Let δ ∈ R be a stability parameter. The δ-degree of an object E of C_X is deg_δ(E ) = d(E ) + δv(E ). If r(E ) 6= 0, the δ-slope of E is defined by µδ(E ) = deg_δ(E )/r(E ). A nontrivial object E of C_X is δ-(semi)stable if

(2.2) r(E) deg_δ(E⁰) (≤) r(E⁰) deg_δ(E ) for any proper nontrivial subobject 0 ⊂ E⁰⊂ E.

It was proved that the real parameter δ ∈ R gives a stability con- dition in the abelian category C_X of ADHM sheaves, ie. it has see-saw property and every object has a unique Harder-Narasimhan filtration.

For fixed numerical type (r, e, v) of an ADHM sheaf there are finitely many critical stability parameters dividing the real axis into chambers.

The set of δ-semistable ADHM sheaves is constant within each chamber.

When the numerical type is (r, e, 1), the strictly semistable objects may exist only if δ takes a critical value and the origin δ = 0 is a critical value for all (r, e, 1) ∈ Z≥1× Z × 1.

Definition 2.3. Let X be a projective curve and Y the total space of the projective bundle Proj(OX⊕ M1⊕ M2). Let d ∈ Z≥1, n ∈ Z. An admissible pair of type (d, n) on Y is a pair (Q, ρ), where Q is a coherent OY-module and ρ ∈ H⁰(Y, Q), such that ρ is not identically zero, Q is

flat over X, (ch0(Q), ch1(Q), ch2(Q)) = (0, 0, d[X]), χ(Q) = n and the cokernel of ρ : OY → Q is of pure dimension 0.

Let Π be the canonical projection Π : Y → X. Let z₀∈ H⁰(Y, O_Y(1)) corresponding to 1 ∈ H⁰(X, O_X) under the canonical isomorphism

H⁰(Y, O_Y(1)) ' H⁰(X, O_X) ⊕ H⁰(X, M₁) ⊕ H⁰(X, M₂) . It can be shown that if (Q, ρ) is an admissible pair on Y , then Q is of pure dimension one and supp(Q) is disjoint from D_∞= {z₀= 0}. Notice that Y \D_∞ is the total space of M₁⁻¹⊕ M₂⁻¹ over X. Therefore the admissible pair on Y is equivalent to the stable pair theory on Y \D_∞. Next is the theorem relating the ADHM sheaf theory on X with the admissible pair theory on Y [7].

Theorem 2.4. There exist a bijection between an S-family of ad- missible pairs on Y with certain support property and an S-family of (δ  1) stable ADHM sheaves with E_∞= O_X on X.

Sketch of the proof. First we have the resolution of the diagonal

∆ ∈ Y ×XY. The Koszul resolution K∆of ∆ is given by

(2.3) 0 → OY(−2)  Ω²Y /X(2) → OY(−1)  Ω¹Y /X(1) → OY  O^Y . Secondly we have the identity Fourier-Mukai functor

(2.4) Rp1∗(p^∗₂() ⊗ O^∆).

From (2.3) and (2.4) we construct a spectral sequence with (2.5) E₁^i,j= Rⁱp1∗(p^∗₂(Q) ⊗ K^j_∆) ,

converging to Q if i + j = 0 and 0 otherwise. In the end we obtain a three term complex centered at (−1) position with cohomology Q. We then take the cone construction for (O_Y → Q), which turns out to give^ρ a (δ  1) stable ADHM sheaf with E_∞= O_X on X. The generalization to an S-family is straightforward and details can be found in [7, sec 6,7]. Also note that this theorem does not need the condition on the deg(M₁) + deg(M₂).

 We also have the following results concerning moduli spaces of ADHM sheaf theory with E_∞= OX:

• For fixed (r, e) ∈ Z≥1 × Z and δ ∈ R there is an algebraic moduli stack M^ss_δ (X , r, e) of finite type over C of δ-semistable ADHM sheaves. If δ ∈ R is noncritical, M^ssδ (X , r, e) is a quasi- projectve scheme equipped with a perfect obstruction theory.

• For fixed (r, e) ∈ Z≥1×Z and δ ∈ R there is a natural algebraic torus S = C^× action on the moduli stack M^ss_δ (X , r, e) which acts on C-valued points by scaling the morphisms (Φ1, Φ₂) → (t⁻¹Φ₁, tΦ₂), t ∈ S. If δ is noncritical the stack theoretic fixed locus M^ss_δ (X , r, e)^S is proper over C. Therefore we define the ADHM invariants to be the residual ADHM invariants A^S_δ(r, e) by equivariant virtual integration in each stability chamber.

• (Theorem 2.4) For (r, e) ∈ Z≥1× Z there exists a critical value δ_M ∈ R>0so that for any δ > δ_M, M^ss_δ (X , r, e) is isomorphic to the moduli space of stable pairs of Pandharipande and Thomas on the total space of M₁⁻¹⊕M₂⁻¹over X. This identification in- cludes the equivariant perfect obstruction theories establishing an equivalence between local stable pair theory and asymptotic ADHM theory.

§3. Wallcrossing and Higher Rank ADHM Invariants

For completeness we include a very brief review of Joyce-Song the- ory. This review is far from self-contained and the interested readers are encouraged to refer to the original papers [20, 16, 17, 18, 19].

3.1. Joyce-Song Theory of Generalized Donaldson-Thomas Invariants

Joyce-Song Theory is a virtual counting theory on an algebraic mod- uli stack M_A locally of finite type over C, parametrizing all the objects in the abelian category A. The central element in Joyce-Song theory is the stack function algebra SF(M_A), which is Grothendieck group gen- erated over Q by isomorphism classes of pairs [(X, ρ)] where X is an algebraic stack of finite type over C and ρ : X → MA is a representable morphism of stacks.

Let Exact_Abe the moduli stack locally of finite type over C, parametriz- ing all the three term exact sequences 0 → E1 → E2 → E3 → 0 in A.

There are also three natural projections

p1, p2, p3: Exact_A→ MA.

We define a Q-bilinear operation ∗ : SF(MA) × SF(M_A) → SF(M_A) as follows. Given two stack functions [(X1, ρ1)] and [(X3, ρ3)], set the Ringel-Hall multiplication

(3.1) [(X1, ρ1)] ∗ [(X3, ρ3)] = [(p1, p3)^∗(X1× X3), p2◦ f)] , where f is determined by the following Cartesian diagram of stacks.

(3.2) (p1, p3)^∗(X1× X3)



f // Exact_A

(p₁,p₃)



p₂

// MA

X1× X3

f1×f3 // MA× M_A

According to Joyce-Song, (SF(M_A), ∗, δ_[0]) is an associative algebra with unity, where δ_[0] = [(Spec(C), 0)] is the stack function of the zero object in M_A. One then defines a Lie subalgebra SF^ind_alg(M_A) imposing certain conditions on the stabilizers of closed points x of the stacks X.

Namely, the subscript alg stands for ’algebra stabilizers’, which requires each such stabilizer Stab(x) to be identified with the group of invertible elements in a certain subring of the endomorphism ring EndA(ρ(x)). The upperscript ind stands for ’virtually indecomposable’ stack functions, which requires the closed points x to have virtual rank one stabilizers.

The definition of virtual rank is very technical and will not be reviewed here in detail. The important point here is the subspace SF^ind_alg(MA) is closed under the Lie bracket determined by the product ∗. Therefore it has a Lie algebra structure. Moreover SF^ind_alg(M_A) is spanned over Q by elements of the form [(U × [Spec(C)/C^×], ρ)], where U is a quasi- projective variety over C.

We could look at a simple example of the ∗-multiplication for char- acteristic delta functions of two object E1, E3 ∈ obj(A) to have a feel about it. We have

δ_Ei = [(Spec(C), ρi)] , ρ_i(Spec(C)) = Ei∈ M_A. Then δE₁∗ δE₂ is given by

δE₁∗ δE₂ =hhExt¹(E3, E1) Hom(E3, E1) i

, ρi ,

such that ρ is a 1-morphism sending the extension class u ∈ Ext¹(E3, E1) representing the exact sequence 0 → E1 → E2→ E3→ 0 to the object E2 ∈ obj(A) modulo the trivial action Hom(E3, E1). In the following theorem we consider the characteristic delta functions of τ -semistable objects in A.

Lemma 3.1. [18, Thm 8.7] Let (τ, T, ≤) be a stability condition on A. Define the stack function ¯δ_ss^α(τ ) in SFalg(M_A) for α ∈ K(A) to be the characteristic function of the moduli substack M^α_A(τ ) of τ -semistable

objects with α ∈ K(A) in MA. We then define ¯^α(τ ) in SFalg(MA) to be

(3.3) ¯^α(τ ) = X

n≥1,α₁,··· ,α_n∈C(A) α1+···+αn=α,τ (αi)=τ (α)

(−1)ⁿ n

δ¯^α_ss¹(τ ) ∗ ¯δ^α_ss²(τ ) ∗ · · · ∗ ¯δ_ss^αn(τ ),

where ∗ is the Ringel-Hall multiplication in SFalg(MA). Moreover, ¯^α(τ ) is in SF^ind_alg(M_A). We call ¯^α(τ ) a log stack function.

Theorem 3.2. [19, Sec 4, Sec 5] Under certain appropriate assump- tions let (τ, T, ≤), (˜τ , ˜T , ≤) and (τ⁰, T⁰, ≤) be three stability conditions on A with (τ⁰, T⁰, ≤) dominating (τ, T, ≤) and (τ, ˜T , ≤). (ie. (τ⁰, T⁰, ≤) is the critical stability condition on the wall.) Then for α ∈ K(A) we have the following wallcrossing formulas in terms of stack functions, (3.4)

¯δ_ss^α(˜τ ) = X

n≥1,α₁,··· ,α_n∈C(A) α₁+···+α_n=α

S(α₁, · · · , α_n; τ, ˜τ )¯δ_ss^α1(τ ) ∗ ¯δ^α_ss²(τ ) ∗ · · · ∗ ¯δ^α_ssⁿ(τ ) ,

¯

^α(˜τ ) = X

n≥1,α1,··· ,αn∈C(A) α1+···+αn=α

U (α1, · · · , αn; τ, ˜τ )¯^α1(τ ) ∗ ¯^α2(τ ) ∗ · · · ∗ ¯^αn(τ ) ,

where the coefficients S(∗; τ, ˜τ ) ∈ Z and U (∗; τ, ˜τ ) ∈ Q are certain com- binatorial coefficients which are complicated enough to be omitted here and there are only finitely many terms in the summation.

We now specialize to the case of the coherent sheaf A = coh(M ) on a Calabi-Yau 3-fold M or A is simply a CY3 algebra. Using the Serre duality on Calabi-Yau spaces or Calabi-Yau algebra we have an antisymmetric Euler form χ([E], [F ]) for E, F ∈ obj(A). We define a Lie algebra L(M ) to be the Q-vector space generated by the λ^αfor each α ∈ K(A), with the Lie algebra

(3.5) [λ^α, λ^β] = (−1)^χ(α,β)χ(α, β)λ^α+β .

Theorem 3.3. [20, Thm 5.14] There exist a canonical Lie algebra morphism Ψ : SF^ind_alg(M_A) → L(M ), which gives the Q-valued general- ized DT invariants DT^α(τ ). Namely, Ψ(¯^α(τ )) = DT^α(τ )λ^α. Moreover, conjecturally the multicover formula defines the Z-valued DT invariants dDT^α(τ ) for generic τ by

(3.6) DT^α(τ ) = X

m≥1,m|α

1

m²dDT^α/m(τ ) .

Remark 3.4. Recall that SF^ind_alg(MA) is spanned by elements of the form [(U × [Spec(C)/C^∗], ρ)] over Q, where U is a quasi-projective vari- ety. Then the canonical Lie algebra morphism will send this to

Ψ [(U × [Spec(C)/C^∗], ρ)]
= χ(U, ρ^∗(νMA)),

where ρ^∗(νMA) is the pullback of the Behrend function νMA to a con- structible function on U × [Spec(C)/C^∗].

3.2. Wallcrossing Formulas in ADHM Sheaf Theory We now consider ADHM sheaf theory on curves with line bundles M1and M2such that M1⊗XM2' K_X⁻¹. The following definition is used to take care of the summation over all the possible Harder-Narasimhan filtrations in the wallcrossing formulas.

Definition 3.5. Let α = (r, e) ∈ Z≥1× Z. For any integer l ≥ 1 and v = 1, 2 let HN₋(α, v, δ_c, l, l − 1) denote the set of order sequence (α_i)_1≤i≤l, α_i = (r_i, e_i) ∈ Z≥1 × Z satisfying α1+ · · · + α_l = α and e₁/r₁= · · · = e_l−1/r_l−1= (e_l+ vδ_c)/r_l= (e + vδ_c)/r.

For any integer l ≥ 2 let HN₋(α, 2, δ_c, l, l − 2) denote the set of order sequence (αi)1≤i≤l, αi = (ri, ei) ∈ Z≥1× Z satisfying α1+ · · · + αl= α and e1/r1 = · · · = el−2/rl−2 = (el−1+ δc)/rl−1 = (el+ δc)/rl = (e + 2δc)/r.

In the ADHM sheaf theory with E_∞ = O_X we have proved the following ADHM rank one wallcrossing formula [3].

Theorem 3.6. Let δc be a critical stability parameter and (α, 1) = (r, e, 1) be the numerical types of ADHM sheaves.

(i) The following wallcrossing formula holds for δc > 0

(3.7)

A+(α, 1) − A₋(α, 1) = X

l≥2

1 (l − 1)!

X

(α_i)∈HN−(α,1,δ_c,l,l−1)

A−(αl, 1)

l−1

Y

i=1

f1(αi)H(αi).

(ii) The following wallcrossing formula holds for δc = 0.

(3.8)

A₊(α, 1) − A₋(α, 1) = X

l≥2

1 (l − 1)!

X

(α_i)∈HN₋(α,1,δ_c,l,l−1)

A₋(αl, 1)

l−1

Y

i=1

f1(αi)H(αi)

+X

l≥1

1 l!

X

(αi)∈HN−(α,1,0,l,l−1) l

Y

i=1

f1(αi)H(αi) ,

where f1(α) = (−1)v(e−r(g−1))v(e − r(g − 1)) for α = (r, e) and A±(α, 1) and H(α) are generalized DT invariants with numerical invariants (r, e, 1) and (r, e, 0) respectively for α = (r, e).

Strategy of the proof. The first step to derive the wallcrossing formu- las is usually to write the formulas in terms of characteristic delta stack function. Take (i) for example. First we have a relation in SF(M_A)

d^α₊− d^α₋=X

l≥2

(−1)^l X

(α_i)∈HN₋(α,1,δ_c,l,l−1)

h^α1∗ h^α2∗ · · · ∗ [d^α₋^l, h^αl−1] ,

where d^α_±and h^αi are the characteristic stack functions for ADHM sheaf moduli and Higgs sheaf moduli.

The second step is to transform all the characteristic stack functions in the formula to the log stack function. Assume that the log stack functions for the stack functions h^αiis g^αi. After the transformation we have

(3.9)

d^α₊−d^α₋ =X

l≥2

(−1)^l (l − 1)!

X

(α_i)∈HN−(α,1,δ_c,l,l−1)

[g^α1, [g^α2, [· · · [g^α₋^l−1, d^αl] · · · ] .

Notice that the automorphism group of all ADHM sheaves with E∞ = OX is isomorphic to C^×. Therefore all the stack functions in (3.9) belong to the Lie algebra SF^ind_alg(M_A). We arrange the formula as a sequence of Lie brackets such that when applying the canonical Lie algebra homomorphism Ψ to (3.9) we extract the invariants directly.

 For fixed r ∈ Z^≥1, and fixed δ ∈ R^>0\ Q let

Zδ(q)r=X

e∈Z

q^e−r(g−1)Aδ(r, e), Z∞(q)r=X

e∈Z

q^e−r(g−1)A∞(r, e) .

Z_∞(q)_r is the generating function of degree r local stable pair in- variants of the data X = (X, M₁, M₂). Using the above theorem the following rationality result is proven, which is a consequence of the BPS expansion conjectured by Gopakumar-Vafa.

Theorem 3.7. For any r ∈ Z≥1, and any δ ∈ R>0 \ Q, Zδ(q)_r, Z_∞(q)rare Laurent expansions of rational functions of q. Moreover, the rational function corresponding to Z_∞(q)r is invariant under q ↔ q⁻¹. If g ≥ 1, Zδ(q)r = Z_∞(q)r is a polynomial in q, q⁻¹ invariant under q ↔ q⁻¹.

3.3. Higher Rank ADHM Invariants

In this section we consider ADHM sheaves with E∞ = V ⊗ OX, where V is a finite dimensional vector space over C. We proved the following rank 2 wallcrossing formulas for ADHM sheaf theory [4].

Theorem 3.8. The ADHM invariants with numerical types (α, 2) = (r, e, 2) satisfy the following wallcrossing formula

(3.10)

A₋(α, 2) − A₊(α, 2) = X

l≥2

1 (l − 1)!

X

(α_i)∈HN₋(α,2,δ_c,l,l−1)

A+(αl, 2)

l−1

Y

i=1

f2(αi)H(αi)

−1 2

X

l≥1

1 (l − 1)!

X

(α_i)∈HN₋(α,2,δ_c,l+1,l−1)

g(αl+1, αl)×

A₊(α_l, 1)A₊(α_l+1, 1)

l−1

Y

i=1

f₂(α_i)H(α_i)

+1 2

X

(α1,α2)∈HN−(α,2,δc,2,0)

X

l₁≥1

X

l₂≥1

1 (l₁− 1)!

1 (l₂− 1)!

X

(α1,i)∈HN−(α1,1,δc,l1,l1−1) (α2,i)∈HN−(α2,1,δc,l2,l2−1)

g(α₁, α₂)A₊(α_1,l1, 1)A₊(α_2,l2, 1)×

l1−1

Y

i=1

f₁(α_1,i)H(α_1,i)

l2−1

Y

i=1

f₁(α_2,i)H(α_2,i)

where A_±(α, 2) are the generalized DT invariants of the numerical in- variants (α, 2) and f_v(α) and g(α₁, α₂) are given by

fv(α) = (−1)v(e−r(g−1))v(e − r(g − 1)), v = 1, 2

g(α1, α2) = (−1)^{e1−e2−(r1−r2)(g−1)}(e1− e2− (r1− r2)(g − 1)) . Strategy of the proof. The proof strategy of this theorem is almost the same as v = 1 case. But since the numerical invariants are of the form (α, 2), we need to tranform the characteristic stack function d^(α,2)_± to its corresponding log stack functions e^(α,2)_± by (3.3) when α is divisible by 2.



An application of this theorem gives rank 2 genus zero invariants.

Consider the following generating function (3.11) Z_{X ,v}(u, q) =X

r≥1

X

n∈Z

u^rqⁿA_∞(r, n − r, v)

where v = 1, 2. Using the wallcrossing formula the following closed formulas are proven [4].

Corollary 3.9. Suppose X is a genus 0 curve and M1 ' OX(d1), M2' OX(d2) where (d1, d2) = (1, 1) or (0, 2). Then

(3.12)

Z_{X ,1}(u, q) =

∞

Y

n=1

(1 − u(−q)ⁿ)^{(−1)d1−1n}

Z_{X ,2}(u, q) =1 4

∞

Y

n=1

(1 − uqⁿ)^{2(−1)d1−1n}

−1 2

X

r1>r2≥1, n1,n2∈Z or r₁=r₂≥1, n2>n₁ or r1≥1, n1∈Z, r2=n2=0

(n1− n2)(−1)^(n1−n2)×

A_∞(r1, n1− r1, 1)A_∞(r2, n2− r2, 1)u^r1+r2qⁿ¹⁺ⁿ². Proof. Let C_X⁰ be the full abelian subcategory of C_X consisting of ADHM sheaves E with φ = 0. For any δ ∈ R, an object E of CX⁰ will be called δ-semistable if it is δ-semistable as an object of C_X. One can see that the properties of δ-stability and moduli stacks of semistable objects in C_X⁰ are analogous to those of C_X.

Given an ADHM sheaf E = (E, V, Φi, ψ) ∈ C_X⁰ of type (r, e, v), it cab be checked that for δ < 0 the proper nontrivial object (E, 0, Φi, 0) is always destabilizing. Therefore the main difference between C_X⁰ and CX

is that for any (r, e, v) ∈ Z^≥1× Z × Z≥1the moduli stack of δ-semistable objects of C_X⁰ of type (r, e, v) is empty if δ < 0.

Let E = (E, 0, Φ_i, 0) be a semistable Higgs sheaf of C_X⁰ of type (r, e, 0), (r, e) ∈ Z≥1× Z. If (d1, d₂) = (1, 1), E must be isomorphic to O_X(n)^⊕r for some n ∈ Z, and Φi= 0 for i = 1, 2. If (d₁, d₂) = (0, 2), E must be isomorphic to O_X(n)^⊕r for some n ∈ Z, and Φ2= 0.

This implies that for (d1, d2) = (1, 1) the moduli stack M^ss(r, e, 0) is isomorphic to the quotient stack [∗/GL(r)] if e = rn for some n ∈ Z, and empty otherwise. For (d1, d2) = (0, 2) the moduli stack M^ss(r, rn, 0), n ∈ Z, is isomorphic to the moduli stack of trivially semistable repre- sentations of dimension r of a quiver consisting of one vertex and one

arrow joining the unique vertex with itself. If e is not a multiple of r, the moduli stack M^ss(X , r, e, 0) is empty.

Performing a computation similar to [20, Sect. 7.5.1] we obtain the Higgs sheaf invariant H(r, e) and the only invariants in δ < 0 chamber of C_X⁰ are given by

(3.13) H(r, e) =







(−1)^d1−1

r² if e = rn, n ∈ Z 0 otherwise ,

(3.14) Aδ(0, 0, 1) = 1 , Aδ(0, 0, 2) = 1 4 .

Then we could apply (3.10) or the analogous wallcrossing formulas of Kontsevich and Soibelman to compute the invariants in the asymptotic δ  0 chamber. We leave out the remaning details and refer the inter-

ested readers to [4]. 

3.4. Comparison with Kontsevich-Soibelman Formula The goal of this section is to illustrate that the ADHM wallcrossing formulas (3.7)(3.10) are in agreement with the wallcrossing formulas of Kontsevich and Soibelman [21], which will be referred to as the KS formula in the following.

Numerical types of ADHM sheaves will be denoted by γ = (α, v), α = (r, e) ∈ Z≥1× Z, v ∈ Z≥0. In order to streamline the computations, let L(X )_≤2 denote the truncation of the Lie algebra L(X ) defined by (3.15)

[λ(α1, v1), λ(α2, v2)]_≤2=

 [λ(α1, v1), λ(α2, v2)] if v1+ v2≤ 2

0 otherwise.

Furthermore, it will be more convenient to use the alternative notation eα= λ(α, 0), fα= λ(α, 1), and gα= λ(α, 2).

Given a critical stability parameter δcof type (r, e, 2), (r, e) ∈ Z^≥1× Z, there exist two pairs α = (rα, e_α) and β = (r_β, e_β) with

eα+ δc

r_α = eβ

r_β = µ_δc(γ)

so that any η ∈ Z≥1× Z with µδ_c(η) = µδ_c(γ) can be uniquely written as η = (qβ, 0), (α + qβ, 1), or (2α + qβ, 2), with q ∈ Z≥0.

For any q ∈ Z^≥0 the following formal expressions will be needed in the KS formula,

(3.16)

Uα+qβ= exp(fα+qβ+1

4g2α+2qβ) , U2α+qβ= exp(g2α+qβ) , Uqβ = exp(X

m≥1

e_mqβ m² ) . Moreover, let

H = X

q≥0

H(qβ)eqβ,

where the invariants H(α) are the Higgs sheaf invariants. Then the wallcrossing formula of Kontsevich and Soibelman reads

(3.17)

exp(H) Y

q≥0, q↓

U_2α+qβ^{A+(2α+qβ,2)} Y

q≥0, q↓

U_α+qβ^A+(α+qβ,1)

= Y

q≥0, q↑

U_α+qβ^{A−(α+qβ,1)} Y

q≥0, q↑

U_2α+qβ^{A−(2α+qβ,2)}exp(H)

where an up, respectively down arrow means that the factors in the corresponding product are taken in increasing, respectively decreasing order of q. Note that the bar-ed invariants A_±(α, v) are the integral invariants corresponding to dDT^(α,v)(τ_±) in (3.6), while the unbar-ed in- variants A_±(α, v) correspond to rational invariants DT^(α,v)(τ_±). (Sorry about the notation.) In this case we have

A_±(2α + qβ, 2) = A_±(2α + qβ, 2) +1

4A_±(α + qβ/2, 1).

Expanding the right hand side, equation (3.17) yields (3.18)

exp(X

q≥0

A₋(2α + qβ, 2)g_2α+qβ+

X

q2>q1≥0

1

2g(q1β, q2β)A−(α + q1β, 1)A−(α + q2β, 1)g2α+(q₁+q₂)β) = exp(H) exp(X

q≥0

A+(2α + qβ, 2)g2α+qβ

+ X

q1>q2≥0

1

2g(q1β, q2β)A+(α + q1β, 1)A+(α + q2β, 1)g_{2α+(q1+q2)β}) exp(−H), modulo terms involving fγ. In fact the terms involving fγ precisely give us the v = 1 wallcrossing formula (3.7).

The BCH formula

(3.19)

exp(A)exp(B)exp(−A) = exp(X

n=0

1

n!(Ad(A))ⁿB)

= exp(B + [A, B] +1

2[A, [A, B]] + · · · ), yields

(3.20)

exp(H) exp(g2α+qβ) exp(−H) = exp(g2α+qβ+X

q1>0

f2(q1β)H(q1β)g_2α+(q+q1)β

+ 1 2!

X

q₁>0,q₂>0

f2(q1β)H(q1β)f2(q2β)H(q2β)g2α+(q+q₁+q₂)β+ · · · )

= exp X

l≥0,q_i>0

1 l!(

l

Y

i=1

f2(qiβ)H(qiβ))g_{2α+(q+q1+···+ql)β}

Substituting (3.20) in (3.18) results in (3.21)

exp X

q≥0

A−(2α + qβ, 2)g2α+qβ

+ X

q2>q1≥0

1

2g(q1β, q2β)A₋(α + q1β, 1)A₋(α + q2β, 1)g_{2α+(q1+q2)β}

= exp X

q≥0,l≥0 qi>0

A+(2α + qβ, 2)1 l!(

l

Y

i=1

f2(qiβ)H(qiβ))g2α+(q+q₁+···+q_l)β

+ X

q⁰₁>q₂⁰≥0 l≥0,qi>0

1

2g(q⁰₁β, q⁰₂β)A+(α + q₁⁰β, 1)A+(α + q₂⁰β, 1)×

1 l!(

l

Y

i=1

f2(qiβ)H(qiβ))g_2α+(q⁰

1+q⁰₂+q1+···+ql)β



In order to further simplify the notation, let

A_±(vα + qβ, v) ≡ A_±(q, v), g2α+qβ≡ gq .

Comparing the coefficients of gQ in (3.18), yields (3.22)

A₋(Q, 2) = X

q⁰≥0, l≥0, qi>0 q⁰+q1+···+ql=Q

A₊(q⁰, 2)1 l!(

l

Y

i=1

f₂(q_iβ)H(q_iβ))

+1 2

X

q⁰₁>q⁰₂≥0 l≥0, q_i>0 q⁰₁+q₂⁰+q₁+···+q_l=Q

g(q⁰₁β, q⁰₂β)A₊(q₁⁰, 1)A₊(q⁰₂, 1)1 l!(

l

Y

i=1

f₂(q_iβ)H(q_iβ))

−1 2

X

q⁰₂>q₁⁰≥0, q⁰₁+q⁰₂=Q

g(q₁⁰β, q₂⁰β)A−(q⁰₁, 1)A−(q₂⁰, 1) .

Using the v = 1 wallcrossing formula (3.7) to transform the last term in (3.22) we finally obtain the v = 2 wallcrossing formula.

(3.23)

A−(Q, 2) = X

q⁰≥0, l≥0, qi>0 q⁰+q₁+···+q_l=Q

A+(q⁰, 2)1 l!(

l

Y

i=1

f2(qiβ)H(qiβ))

+1 2

X

q⁰₁>q⁰₂≥0 l≥0, q_i>0 q⁰₁+q₂⁰+q1+···+q_l=Q

1

2g(q⁰₁β, q₂⁰β)A+(q₁⁰, 1)A+(q⁰₂, 1)1 l!(

l

Y

i=1

f2(qiβ)H(qiβ))

−1 2

X

q₂>q₁≥0 q₁+q₂=Q l≥0, ˜l≥0 q₁⁰≥0, q⁰₂≥0

ni>0,˜ni>0 q⁰₁+n1+···+nl=q1

q⁰₂+˜n1+···+˜n˜l=q2

g(q1β, q2β)A+(q⁰₁, 1)A+(q⁰₂, 1)×

1 l!(

l

Y

i=1

f₁(n_iβ)H(n_iβ))1

˜l!(

˜l

Y

i=1

f₁(˜n_iβ)H(˜n_iβ)) .

This formula agrees with (3.10).

§4. Cohomology of the Moduli Space of Hitchin Pairs

In this section we present a conjectural formalism to determine the Poincar´e (Hodge) polynomial of Hitchin moduli space [5]. More pre- cisely refined wallcrossing formulas and the refined ADHM invariants

with E∞ = OX in the asymptotic chamber are conjectured on local curves. It is shown that these formulas yield a recursive relation which correctly determines the Poincar´e (Hodge) polynomial of the moduli space of Hitchin pairs with coprime rank and degree on a smooth pro- jective curve of genus at least two.

We recall the defintion of a Hitchin pair. Let X be a smooth pro- jective curve and L is an invertible sheaf on X. A Hitchin pair is a pair (E, Φ) where E is a coherent sheaf on X and Φ : E → E ⊗XL a morphism of coherent sheaves. A Hitchin pair is (semi)stable if for any proper subsheaf 0 ⊂ E⁰⊂ E such that Φ(E⁰) ⊂ E⁰⊗XL, we have

(4.1) deg(E⁰)

rank(E⁰) < (≤)deg(E) rank(E) .

Note that if rank(E) > 0, semistability of the Hitchin pair implies that E is locally free. Assume that deg(L) ≥ 2g − 2. According to [12, 30, 1] we have an algebraic moduli stack Hit(X, L, r, e) locally of finite type parametrizing the semistable Hitchin pairs with numerical type (r, e). If (r, e) are coprime, this stack is a C^×-gerbes over a smooth quasi-projective variety Hit(X, L, r, e).

One of the major observations, which make the enumerations pos- sible, is the relation between semistable Higgs sheaves and semistable Hitchin pairs as follows.

• Suppose M1 = OX, M2 = K_X⁻¹ and let (r, e) ∈ Z^≥1× Z be coprime. Then there is an isomorphism

(4.2) Higgs(X , r, e) ' C × Hit(X, KX, r, e).

• Suppose M2is a line bundle of degree 2−2g−p, where p ∈ Z^>0. Then there is an isomorphism

(4.3) Higgs(X , r, e) ' Hit(X, M₂⁻¹, r, e).

Both statements rely on the fact that for coprime (r, e) slope semista- bility is equivalent to slope stability. Therefore the endomorphism ring of any semistable ADHM sheaf E is canonically isomorphic to C.

Then note that in the first case, given any semistable object E = (E, Φ₁, Φ₂) the relation Φ₁◦ (Φ₂⊗ 1_M1) − Φ₂◦ (Φ₁⊗ 1_M2) = 0 implies that Φ₁: E → E is an endomorphism of E since it obviously commutes with itself. Therefore it must be of the form Φ₁ = λ1_E for some λ ∈ C. In particular, it preserves any subsheaf E⁰ ⊂ E. Generalizing this observation to flat families it follows that there is an forgetful morphism

Higgs(X , r, e) → Hit(X, KX, r, e)

projecting (E, Φ1, Φ2) to (E, Φ2⊗ 1K_X). The isomorphism (4.2) then follows easily.

In the second case, note that given a semistable Higgs sheaf (E, Φ₁, Φ₂), of type (r, e), the data

E⁰=

E ⊗XM₁⁻¹, Φ1⊗ 1_M−1

1 , Φ2⊗ 1_M−1 1



determines a semistable Higgs sheaf of type (r, e − rdeg(M1)) = (r, e − rp). The relation

(Φ1⊗1_M⁻¹

1 )◦((Φ2⊗1_M⁻¹

1 )⊗1M₁)−(Φ2⊗1_M⁻¹

1 )◦((Φ1⊗1_M⁻¹

1 )⊗1M₂) = 0 implies that Φ1⊗ 1_M−1

1 is a morphism of (semistable) Higgs sheaves.

However µ(E ) > µ(E⁰) since p > 0, therefore any such morphism must vanish. Then (4.3) follows.

 Besides these observations we make the following conjectures.

Conjecture 4.1. Let α = (r, e) ∈ Z≥1× Z. Then for δ not crit- ical there exist refined equivariant ADHM invariants Aδ(r, e, 1)(y) ∈ Z[y, y⁻¹], for any δ ∈ R, and refined equivariant Higgs sheaf invari- ants H(r, e)(y) ∈ Q(y) so that A^δ(r, e, 1)(1) = Aδ(r, e, 1), H(r, e)(1) = H(r, e) and refined wallcrossing formulas hold. The conjectural refined wallcrossing formulas are obtained by the following direct substitution:

A_δ(r, e, 1) → A_δ(r, e, 1)(y), H(r, e) → H(r, e)(y), and (−1)^{(e−r(g−1))}v(e−

r(g − 1)) → (−1)^{(e−r(g−1))}[(e − r(g − 1))]y in the rank one wallcrossing formulas, where [n]y= ^y_y−y^n−y−1⁻ⁿ. Moreover H(r, e)(y) ∈ Z[y, y⁻¹] if (r, e) are coprime.

Conjecture 4.2. The following refined multicover relation holds for any (r, e) ∈ Z≥1× Z

(4.4) H(r, e)(y) = X

k∈Z, k≥1 k|r, k|e

1 k [k]y

H r k,e

k

 (y^k) ,

where H(r, e)(y) ∈ Z[y, y⁻¹] .

Remark 4.3. The invariants A_δ(r, e, 1)(y) ∈ Z[y, y⁻¹], H(r, e)(y) are refined generalizations of Joyce-Song invariants of ADHM sheaves.

They are conjecturally related to Kontsevich-Soibelman invariants Aδ(r, e, 1)(y) ∈ Z[y, y⁻¹], H(r, e)(y) ∈ Z[y, y⁻¹] by a refined multicover formula in Con-

jecture 4.2. For v = 1 invariants this formula states simply that A_δ(r, e, 1)(y) = Aδ(r, e, 1)(y).

The last essential piece in the construction is the Nekrasov partition function of instanton counting. Physically we can geometrically engineer a five dimensional SU (2) gauge theory with g adjoint hypermultiplets by putting M-theory on the total space Z of canonical bundle K_S on the ruled surface S = P(OX⊕ M1). Inside S there are two sections X₁ and X₂. The normal bundle N_X1|Z of X₁ inside Z is given by

NX1|Z ' M₁⁻¹⊕ M₂⁻¹ .

This construction involving ruled surfaces is needed because other con- structions are problematic physically, ie. other geometries do not admit geometric engineering of five dimensional gauge theory. However the subtlety in ruled surface construction is that we need to separate the contributions coming from the other section X2.

Given such a five dimensional theory, Nekrasov has constructed an equivariant instanton partition function Z_inst^(p) (Q, 1, 2, a1, a2, y), where

1, 2, a1, a2 are equivariant parameters for a natural torus action, Q is a formal variable counting instanton charges, and y is another formal variable.

It has been verify by string theorists in other local geometries that the instanton partition function could be identified with the (refined) topological string after suitable changes of variables [8, 29, 13, 14, 9, 15].

Mathematically Z_inst^(p) (Q, ₁, ₂, a₁, a₂, y) is the generating function for the χ_y-genus of a certain holomorphic bundle on a partial compact- ification of the instanton moduli space M (r, k) as described in the fol- lowing.

Hirzebruch genus Let M (r, k) denote the moduli space of rank r framed torsion-free sheaves (F, f ) on P² with second Chern class k ∈ Z≥0. The framing data is an isomorphism

(4.5) f : F |l∞ → O_l^⊕r

∞.

along the line l_{inf ty} at infinity defined by [0, z₁, z₂] in terms of the ho- mogeneous coordinates of P².

M (r, k) is a smooth quasi-projective fine moduli space i.e. there is an universal framed sheaf (F, f) on M (r, k) × P². Let V = R¹p_1∗F ⊗ p^∗₂O_P2(−1) where p₁, p₂: M (r, k) × P²→ M (r, k), P²denote the canoni- cal projections. It follows from [26] that V is a locally free sheaf of rank k on M (r, k).

There is a torus T = C^××C^××(C^×)^×raction on acting on M (r, k), where the action of the first two factors is induced by the canonical action on C^×× C^× on P², and the last r factors act linearly on the framing.

According to [27] the fixed points of the T-action on M (r, k) are isolated and classified by collections of Young diagrams Y = (Y1, . . . , Yr) so that the total number of boxes in all diagrams is |Y | = |Y₁| + · · · |Yr| = k. Let Yr,k denote the set of all such r-tuples of Young diagrams. Note also that both the holomorphic cotangent bundle T_{M (r,k)}^∨ and the bundle V constructed in the previous paragraph carry canonical equivariant structures.

The K-theoretic instanton partition function of an SU (2) theory with g adjoint hypermultiplets and a level p Chern-Simons term is given by the equivariant residual Hirzebruch genus of the holomorphic T- equivariant bundle

(T_{M (2,k)}^∨ )^⊕g⊗ (det V)^−p.

This is defined by equivariant localization as follows [28, 22]. Let (₁, ₂, a₁, a₂) be equivariant parameters associated to the torus T. Then the localiza- tion formula yields [28, 22]

(4.6) Z_inst^(g,p)(Q, ₁, ₂, a₁, a₂, y) =

∞

X

k=0

Q^kZ_k^(g,p)(₁, ₂, a₁, a₂; y)

where Z₀^(g,p)(₁, ₂, a₁, a₂; y) = 1 and (4.7)

Z_k^(g,p)(₁, ₂, a₁, a₂; y) = X

(Y₂,Y₂)∈Y_2,k

Z_(Y^(g,p)

2,Y₂)(₁, ₂, a₁, a₂; y)

= X

Y ∈Y_2,k 2

Y

α=1



e^−|Yα|aα Y

(i,j)∈Yα

e^{(i−1)1+(j−1)2}

^p

2

Y

α,β=1

Y

(i,j)∈Y_α



1 − ye^{(Yβ,jt −i)1−(Yα,i−j+1)2+aαβ}g



1 − e^{(Yβ,jt −i)1−(Yα,i−j+1)2+aαβ}

Y

(i,j)∈Y_β



1 − ye^{−(Yα,jt −i+1)1+(Yβ,i−j)2+aαβ}^g



1 − e^{−(Yα,jt −i+1)1+(Yβ,i−j)2+aαβ} where for any Young tableau Y , Yi, i ∈ Z^≥1 denotes the length of the i-th column and Y^t denotes the transpose of Y . If i is greater than the number of columns of Y , Y_i = 0. Moreover a_αβ= a_α− aβ for any α, β = 1, 2.

Let Z_(Y^(g,p)

1,Y2)(q₁, q₂, Q_f, y) be the expression obtained by setting q₁= e^−1, q2= e^−2 and Qf = e^a12, and e^a1= −1 in Z_(Y^(g,p)

1,Y₂)(1, 2, a1, a2; y).