Experimental evidence

where

T = χ(E, E)− Q1χ(E, E⊗C L1)− Q2χ(E, E⊗C L2)

+ Q₁Q₂χ(E, E⊗C L₁⊗CL₂)− χ(OC, E)− Q1Q₂χ(E, L₁⊗C L₂)

(8.21)

and

P =(1− Q1)APEnd(E_p^•) + Q₂(Q₁− 1)SPEnd(Ep^•)⊗ OC(p)_p. (8.22) Using the canonical exact sequence

0→ OC z1

−→OC(p) → OC(p)_p → 0,

one finds that OC(p)_p = Z⁻¹ in the representation ring of eG. Moreover, there is a natural G-equivariant isomorphisme

SPEnd(E_p^•)≃ APEnd(Ep^•)^∨. Therefore equation (8.22) yields

P = (1− Q1)APEnd(E_p^•) + Q₂(Q₁ − 1)Z⁻¹APEnd(E_p^•)^∨. (8.23)

straightforward computation yields the following expressions W₍₁₁₎(q, y) = 2y¹²+ 2y¹⁴

y¹³ q + 3y¹¹+ 3y¹⁵+ 4y¹³ y¹³ q² + 4y¹⁰+ 6y¹²+ 4y¹⁶+ 6y¹⁴

y¹³ q³+· · · W₍₂₁₎(q, y) = y¹³+ y¹⁵

y¹⁴ q + 2y¹⁶+ 2y¹²+ 4y¹⁴ y¹⁴ q² + 8y¹³+ 4y¹¹+ 4y¹⁷+ 8y¹⁵

y¹⁴ q³+· · · W₍₁₁₁₎(q, y) = 3y¹⁴+ 3y¹⁶

y¹⁵ q + 6y¹³+ 6y¹⁷+ 9y¹⁵ y¹⁵ q² + 10y¹²+ 18y¹⁴+ 10y¹⁸+ 18y¹⁶

y¹⁵ q³+· · ·

(8.24)

A sample computation will be displayed below for µ = (1, 1, 1) and e = 1. Employing the results of Section 8.2 the fixed loci are in this case classified as follows. Taking into account inequalities (8.8)–(8.10), there are six fixed points:

1) E =OC ⊕ Q⁻¹₂ ZOC+ Q⁻²₂ Z²OC(∞)

Φ2(0, 0) = z⁻¹, Φ2(0, 1) = z⁻¹, ψ(0, 0) = 1, E_p² = E(0, 2), E_p¹ = E(0, 2)⊕ E(0, 1).

2.a) E =OC ⊕ Q⁻¹2 ZOC + Q⁻²₂ Z²OC(∞)

Φ₂(0, 0) = z⁻¹, Φ₂(0, 1) = 1, ψ(0, 0) = 1, E_p² = E(0, 2), E_p¹ = E(0, 2)⊕ E(0, 1).

2.b) E =OC ⊕ Q⁻¹₂ ZOC + Q⁻²₂ Z²OC(∞)

Φ2(0, 0) = z⁻¹, Φ2(0, 1) = 1, ψ(0, 0) = 1, E_p² = E(0, 1), E_p¹ = E(0, 2)⊕ E(0, 1).

2.c) E =OC ⊕ Q⁻¹2 ZOC + Q⁻²₂ Z²OC(∞)

Φ₂(0, 0) = z⁻¹, Φ₂(0, 1) = 1, ψ(0, 0) = 1, E_p² = E(0, 1), E_p¹ = E(0, 0)⊕ E(0, 1).

3.a) E =OC ⊕ Q⁻¹OC(∞) + Q⁻¹₂ Z²OC

Φ1(0, 0) = 1, Φ2(0, 0) = z⁻¹, ψ(0, 0) = 1, E_p² = E(1, 0), E_p¹ = E(1, 0)⊕ E(0, 1).

3.b) E =OC ⊕ Q⁻¹OC(∞) + Q⁻¹2 Z²OC

Φ₁(0, 0) = 1, Φ₂(0, 0) = z⁻¹, ψ(0, 0) = 1, E_p² = E(0, 1), E_p¹ = E(1, 0)⊕ E(0, 1).

The underlying vector bundle E is encoded in a decorated Young diagram of the form 1

0 0 for cases (1)− (2.c), respectively

0 0 1

for cases (3.a)− (3.b). The expression (8.21) takes the form

T₁ = 2 + Z⁻¹Q₂+ ZQ⁻¹₂ − Z²Q₁Q²₂− ZQ1Q⁻¹₂ + Z⁻²Q₁Q₂− Z + Z⁻³Q²₂

− 2Z⁻¹Q1Q2− Z⁻²Q1Q²₂− Q1

for case (1),

T₂ = 1 + Z⁻¹Q₂+ Q⁻¹₂ − ZQ1Q⁻²₂ − Q1Q⁻¹₂ − Z⁻¹Q₁Q₂+ Z⁻²Q³₂ + Z⁻¹Q²₂− Q1− Z⁻¹Q₁Q²₂

for cases (2.a)-(2.c), respectively

T₃ = 1 + Z⁻¹Q⁻¹₁ Q₂+ Q²₁Q⁻¹₂ − Z⁻¹Q⁻¹₁ Q²₂− Z⁻¹Q₁Q₂− Q²₁ for cases (3.a) and (3.b).

The expression (8.22) specializes to, respectively,

P₁ = − 2 + ZQ1Q⁻¹₂ + 2Q₁− ZQ⁻¹₂ + 2Z⁻¹Q₁Q₂+ Z⁻²Q₁Q²₂− 2Z⁻¹Q₂− Z²Q²₂ P2.a= − 1 + Q1Q⁻¹₂ + Z⁻¹Q1+ Q1− Z⁻¹− Q⁻¹2 + Z⁻¹Q1Q2

+ Q₁Q₂+ Z⁻¹Q₁Q²₂− Z⁻¹Q₂− Q2− Z⁻¹Q²₂

P_2.b = − 1 + Q1+ Z⁻¹Q₁Q₂+ 2Q₁Q⁻¹₂ + 2Z⁻¹Q₁Q²₂− Z⁻¹Q₂− 2Q⁻¹2 − 2Z⁻¹Q²₂ P_2.c = − 1 − Z⁻¹Q₂− Q⁻¹2 − ZQ⁻²2 + Z⁻¹Q₁Q₂+ Q₁Q⁻¹₂ + ZQ₁Q⁻²₂ − Z²Q³₂

− Z⁻¹Q²₂+ Z⁻²Q₁Q³₂+ Z⁻¹Q₁Q²₂+ Q₁

P_3.a= − 1 − ZQ1Q⁻¹₂ + Q²₁+ ZQ²₁Q⁻¹₂ + Z⁻¹Q₁Q₂− Z⁻¹Q⁻¹₁ Q₂

− Z⁻²Q⁻¹₁ Q²₂ + Z⁻²Q²₂

P_3.b = − 1 − Q1− 2Z⁻¹Q⁻¹₁ Q₂ + Z⁻¹Q₂+ 2Q²₁+ Z⁻¹Q₁Q₂

Let F_m denote the right hand side of equation (8.7). Then, using the above computations, one obtains

F₁ = R³Z⁻²Q⁻¹₁ Q₂(1− Q⁻¹1 )(1− Z²Q⁻¹₁ Q⁻¹₂ )(1− Z³Q2⁻³) (1− Z⁻¹)(1− ZQ⁻¹2 )(1− Z⁻²Q⁻¹₁ Q²₂) F_2.a= R³Q⁻¹₁ Q₂(1− Q⁻¹1 Q⁻¹₂ )(1− ZQ⁻¹1 )(1− Z²Q⁻³₂ )

(1− Q⁻¹2 )(1− Z)(1 − Z⁻¹Q⁻¹₁ Q²₂) F_2.b = R³Q⁻¹₁ Q₂(1− ZQ⁻¹1 Q⁻²₂ )(1− Q⁻¹1 Q₂)(1− Z²Q⁻³₂ )

(1− Q2)(1− ZQ⁻²₂ )(1− Z⁻¹Q⁻¹₁ Q²₂) F_2.c = RZ⁻¹Q²₂1− Z²Q⁻¹₁ Q⁻³₂

1− Z⁻¹Q²₂

F_3.a = R³Z²Q₁Q⁻³₂ (1− Q⁻²1 Q₂)(1− Z²Q2⁻²)(1− Z⁻¹Q⁻²₁ Q₂) (1− Z⁻¹Q⁻¹₁ Q₂)(1− Z²Q₁Q⁻²₂ )(1− ZQ1Q2⁻²) F_3.b = R³Z²Q₁Q⁻³₂ (1− Q⁻²₁ Q₂)(1− Q⁻²₁ )(1− ZQ⁻¹₂ )

(1− Q⁻¹1 )(1− ZQ1Q⁻²₂ )(1− ZQ1Q⁻¹₂ ) where R = Z⁻¹Q₁Q₂. Adding all local contributions yields

I_(1,1,1),1=−3(

R^1/2+ R^−1/2) ,

confirming conjecture (8.5) in this case. Similar computations confirm the conjecture for (µ, e) = ((1, 1), 1), ((1, 1), 2), ((1, 1), 3), ((2, 1), 1), ((2, 1), 2), ((2, 1), 3),

((1, 1, 1), 2), ((1, 1, 1), 3).

A Degree zero ADHM sheaves

This section proves a result used in the main text stating that the underlying vector bundle of any rank r, degree 0 asymptotically flat ADHM sheaf E must be trivial, E ≃ OC^⊕r.

If r = 1, the claim is obvious since deg(E) = 0 and there is a nonzero morphism ψ : OC → E.

Suppose r ≥ 2 and E is slope semistable. For any (n1, n₂) ∈ (Z≥0)², let E(n₁, n₂) = Φⁿ₁¹Φⁿ₂²ψ(OC)⊂ E. Note that E(n1, n₂) is either the zero sheaf or isomorphic to OC since it is a locally free quotient ofOC. Let

E^′ = ∑

(n1,n2)∈Z²_≥0

E(n₁, n₂)⊆ E.

The asymptotic stability condition implies that E/E^′ is a zero dimensional sheaf on C. By construction there there exists a finite set ∆⊂ (Z_≥0)² and a surjective morphism

V_∆= ⊕

(n1,n2)∈∆

E(n₁, n₂)↠ E^′.

Since E is semistable of degree 0, it follows that the resulting morphism V_∆ → E must be surjective as well, hence E ≃ E^′.

Now let

0 = J E₀ ⊂ JE1 ⊂ · · · ⊂ JEn = E

be a Jordan-H¨older filtration of E. Obviously, there is a commutative triangle of surjective morphisms

V_∆ ^//

J%%J JJ JJ JJ

JJ E



E/J E_n−1.

This implies that there is at least one direct summand E(m₁, m₂) ⊂ V∆ which fits into a commutative triangle

E(m₁, m₂) ^//

N ''N NN NN NN NN

N E



E/J E_n−1

with all maps nontrivial. Moreover, the horizontal map must be in fact injective. Since E/J E_n−1 is stable of degree 0, and E(m₁, m₂) ≃ OC, it follows that E/J E_n−1 ≃ OC, and the map E(m₁, m₂)→ E/JEn−1 is an isomorphism. This implies that there is a splitting

E ≃ E/JEn−1⊕ JEn−1≃ OC⊕ JEn−1

By construction, J E_n−1 is degree 0 slope semistable and there is a surjective morphism

⊕

(n1,n2)∈∆\{(m¹,m2)}

E(n1, n2)↠ JEn−1.

Repeating the above argument shows that J E_n−1/J E_n−2≃ OC and there is a splitting J E_n−1 ≃ OC⊕ JEn−2

Proceeding recursively, one finds that E ≃ O^⊕rC in a finite number of steps.

To finish the proof, suppose E is not slope semistable. Then it It will be shown below that this leads to a contradiction. By assumption, E has a a Harder-Narasimhan filtration

0 = HE₀ ⊂ HE1 ⊂ · · · ⊂ HEl = E with l ≥ 2.

The first observation is that Φj(Ek)⊆ Ek for all 1≤ j ≤ 2 and 1 ≤ k ≤ l. Suppose this fails for some 1 ≤ k ≤ l − 1 and some 1 ≤ j ≤ 2, and let k be minimal with this property i.e. Φ_j(E_k′) ⊆ Ek^′ for all k^′ < k. Then let k^′′ > k be minimal such that Φ_j(E_k) ⊆ Ek^′′ for all j ∈ {1, 2} and Φj(E_k) ⊈ Ek^′′−1 for at least one value of j ∈ {1, 2}. Then Φj yields a nontrivial morphism Φ_j : E_k/E_k−1 → Ek^′′/E_k′′−1 contradicting the defining property of the Harder-Narasimhan filtration.

Since Φ₁, Φ₂ preserve the Harder-Narasimhan filtration, the asymptotic stability condi-tion for ADHM sheaves implies that ψ(OX) ⊈ El−1. Hence ψ yields a nontrivial morphism OX → E/Eh−1. Since E/E_l−1is semistable, this implies µ(E/E_l−1)≥ 0, again contradicting the properties of the Harder-Narasimhan filtration which imply that µ(E/E_l−1) < µ(E) = 0.

In conclusion, the underlying bunde of an asymptotically stable degree 0 ADHM sheaf must be indeed isomorphic to O^⊕rC .

B Fermion zero modes

The goal of this section is to determine the bundle of fermion zero modes on the moduli space of supersymmetric D2-D6 configurations found in Section 5. As proven in Section 5.2, supersymmetry constraints require the Chan-Paton bundle on r such D2-branes to be isomorphic to the trivial rank r bundle, and all field configurations to be constant. This shows that the low energy effective action of such a configuration is reduced to supersymmetric quantum mechanics. The detailed action of a similar system has been written in [6, Sect 2.2] as the dimensional reduction of a two dimensional (0, 2) gauged linear sigma model.

Analogous considerations will yield the action in the present case by dimensional reduction of a two dimensional (0, 4) gauged linear sigma model. Omitting the details, note the resulting

quantum mechanical system will have a moduli space of flat directions isomorphic to N (γ), as expected. Using standard (0, 2) sigma model technology [61], the bundle of fermion zero modes is isomorphic to the middle cohomology of a monad complex, as shown below.

In absence of the orbifold point p, the D2-D6 moduli space is isomorphic to the Hilbert scheme of pointsH^r. For any stable ADHM data (A1, A2, I), the space of fermion zero modes is isomorphic to the middle cohomology group of the complex F(A1,A2,I)

0→ H¹(End_C(E))−→^d1

H¹(End_C(E))^⊕2

⊕ H¹(E)

⊕ H¹(E^∨)

d2

−→H¹(End_C(E))→ 0 (B.1)

where

d₁(α) =(

[α, A₁], [α, A₂], αI) d2(β1, β2, γ, δ) = [β1, A2] + [A1, β2] + Iδ.

(B.2) Since E ≃ O^⊕rC , one can easily prove using Serre duality that F(A1,A2,I,J ) is left and right exact while its middle cohomology is isomorphic to(

T_(A^∗_{1,A2,I,J )}H^r)_⊕g

, whereT_(A^∗_{1,A2,I,J )}H^r is the fiber of the cotangent bundle to the Hilbert schemeH^r at the point (A1, A2). Using the same argument in flat families of stable ADHM data, it follows that the bundle of fermion zero modes is isomorphic to the direct sum(

T^∗H^r)_⊕g .

Now suppose there is an orbifold point, in which case the supersymmetric configurations are in one-to-one correspondence with stable parabolic ADHM data (A₁, A₂, I, J ; V^•) as shown in Section 5.2. The space of fermion zero modes will then given by the middle cohomology of a complexF(A1,A2,I,J ;V^•) of the form (B.1), where E is replaced with an orbi-bundle ˜E on eC. Using the correspondence described in Section 4, this complex can be written in terms of parabolic data as follows.

Recall that there is a root line bundle ˜L on ˜C such that ˜L^s≃ ν^∗OC(p), where ν : eC → C is the natural projection. Moreover, the canonical class of eC is given by

K_Ce ≃ ν^∗K_C ⊗CeL˜^(s−1) ≃ ν^∗K_C(p)⊗CeL˜⁻¹. Then Serre duality on the stack eC yields the following isomorphisms

H¹(End_Ce( ˜E))≃ H⁰(End_Ce( ˜E)⊗Ceν^∗KC(p)⊗CeL˜⁻¹)^∨ H¹( ˜E^∨)≃ H⁰( ˜E⊗Ceν^∗K_C(p)⊗CeL˜⁻¹)^∨.

Now recall that the pushforward E = ν_∗E is a vector bundle on C equipped with a filtration˜ by subsheaves F_a = ν_∗( ˜E⊗CeL˜⁻¹), a≥ 1. This filtration determines a flag Ep^• in the fiber E_p, hence a parabolic structure on E at p. Moreover the higher direct images R^kν_∗E are trivial˜ and there is a one-to-one correspondence between morphisms ˜Φ : ˜E → ˜E and parabolic morphisms Φ : E^• → E^•. Therefore one obtains isomorphisms of the form

H¹(End_Ce( ˜E))≃ H⁰(SP End_C(E)⊗Cν^∗K_C(p))^∨ H¹( ˜E^∨)≃ H⁰(F₁⊗Cν^∗K_C(p))^∨

H¹( ˜E)≃ H¹(E)≃ H⁰(E^∨⊗C K_C) Then dual complex is isomorphic to

0→ H⁰(SP End_C(E^•)⊗C K_C(p))−→^d′1

H⁰(SP End_C(E^•)⊗C K_C(p))^⊕2

⊕

H⁰(E^∨⊗C K_C(p))

⊕

H⁰(F₁⊗CK_C(p))

d^′₂

−→H⁰(SP EndC(E^•)⊗C KC(p)) → 0

(B.3)

where SP End_C(E^•) denotes the sheaf of strongly parabolic endomorphisms of E^•. The expressions of the differentials are formally identical with the ones given in (B.2).

Next note that by construction there is an exact sequence

0→ EndC(E)⊗CK_C → SP EndC(E^•)⊗ KC(p)→ SP End(Ep^•)⊗C Op(p) → 0 (B.4) of sheaves on C. Moreover, the inclusions

0⊂ E(−p) ⊂ F1 ⊂ E yield inclusions of vector spaces

0⊂ H⁰(E⊗C K_C)⊆ H⁰(F₁⊗C K_C(p))⊆ H⁰(E⊗CK_C).

However since E ≃ O^⊕r_C , there is an isomorphism

H⁰(E⊗C KC)≃ H⁰(E⊗C KC(p)).

Therefore there is an isomorphism

H⁰(E⊗C K_C)≃ H⁰(F₁⊗C K_C(p)). (B.5)

Using the exact sequence (B.4) and isomorphism (B.5) a straightforward computation shows that there is an exact sequence of complexes

0→ F(A^∨1,A2,I) → F(A^∨1,A2,I;V^•) → D(A1,A2;V^•)→ 0 (B.6) where D(A1,A2;V^•) is the three term complex

0→ SPEnd(V^•)−→SPEnd(V^{δ1 •})^{⊕2 δ}−→SPEnd(V^{2 •})→ 0.

The differentials δ₁, δ₂ are given by

δ₁(f ) =(

[f, A₁], [f, A₂]) δ₂(g₁, g₂) = [g₁, A₂] + [A₁, g₂].

Now note that under the current assumptions δ₁ is injective and δ₂ is surjective, hence the complex D(A1,A2;V^•) has trivial cohomology.

To prove this claim, recall that (A₁, A₂) is by assumption a cyclic commuting pair pre-serving the flag V^•. In particular (A₁, A₂)is regular i.e. the subspace f ∈ End(C^r) such that [f, A₁] = [f, A₂] is isomorphic to a Cartan subalgebra of End(C^r). On the other hand if f ∈ SPEnd(V^•), it follows that f is nilpotent, hence it must be trivial. This shows that Ker(δ₁) = 0. Surjectivity of δ₂ follows by an analogous argument for the dual morphism

δ^∨₂ : SPEnd(V^•)^∨ → SPEnd(V^•)^∨⊕ SPEnd(V^•)^∨

The dual vector space SPEnd(V^•)^∨ is isomorphic to a space of strongly parabolic maps on the dual vector space V^∨ equipped with the dual flag

V_s^∨_−a = Ker(

V^∨ ↠ (V^a)^∨)

, 0≤ a ≤ s.

That is SPEnd(V^•)^∨ ≃ SPEnd(V_•^∨). Moreover, δ₂^∨(ξ) =(

[ξ, A^∨₁], [A^∨₂, ξ])

=(

[A₁, ξ^∨]^∨, [ξ^∨, A₂]^∨) . Then the same argument shows that Ker(δ₂^∨) = 0, hence δ2 is surjective.

In conclusion, the exact sequence (B.6) implies that the complexes F_(A^∨_1,A2,I;V•) and F_(A^∨_1,A2,I) are quasi-isomorphic.

C Some basic facts on nested Hilbert schemes

The goal of this section is to prove that the nested Hilbert scheme N (γ) used in Section 5.2 is reduced and connected. The proof relies on an alternative presentation of N (γ) given in [4] as a moduli space of stable framed quiver representations. Namely, consider the moduli space of stable framed quiver representations of the form:

C^rℓ

Aℓ,1



Aℓ,2

YY ^fℓ−1,ℓ //C^{rℓ−1 fℓ−2,ℓ−1 //}

Aℓ−1,1



Aℓ−2,2

YY · · · C^r0

A0,1



A0,2

YY C^Ioo (C.1)

with quadratic relations

[A_0,1, A_0,2] = 0, A_ı,1f_ı,ı+1− fı,ı+1A_ı,1= 0, A_ı,2f_ı,ı+1− fı,ı+1A_ı,2 = 0. (C.2) The discrete invariants r_ı, 0≤ ı ≤ ℓ, are given by

rı =

∑ℓ ȷ=ı

γȷ.

For generic King stability parameters (θı, θ_∞)∈ R^ℓ+2 satisfying

θ_∞ =−

∑ℓ ı=0

n_ıθ_ı, θ_ı > 0, 0≤ ı ≤ ℓ.

a representation of the above quiver is semistable if and only if the ADHM data (A_0,1, A_0,2, I) is stable and the linear maps f_ı,ı+1 are injective for all 0 ≤ ı ≤ ℓ − 1. Here ∞ denotes the framing node corresponding to the tail of the arrow I in the above diagram. In particular for (θı, θ_∞) sufficiently generic there are no strictly semistable objects and the stabilizer group of any stable framed representation is trivial.

Let A(γ) denote the linear space of all linear maps of the form (C.1), not subject to any stability condition or relations. Then the subset of stable quiver representations is an open subspace U (γ) ⊂ A(γ). Let V (γ) ⊂ A(γ) be the closed subscheme determined by the quadratic equations (C.2), and V_U(γ) its restriction to U (γ). Since all stabilizers are trivial, V_U(γ) is a principal G(γ) bundle over N (γ), where G(γ) = ×^ℓ_ı=0GL(n_ı,C). Therefore in order to conclude thatN (γ) is reduced it suffices to prove that V (γ) is reduced. Now recall that any ideal I ∈ C[x1, . . . , x_N] generated by irreducible polynomials is a radical ideal. This

statement can be easily proven by induction on N . Then it suffices to prove that all quadrics in equation (C.2) are irreducible. A straightforward computation shows that any quadric in (C.2) is of the form

∑s i=1

x_iy_i with s≥ 2, which is indeed irreducible.

In order to prove N (γ) is connected, recall that the morphism ρred : N (γ) → eH^r_red constructed in diagram (5.8) was shown there to have connected fibers for γ = (1, . . . , 1).

This implies that N (γ) is connected since ρred is also surjective and eH^rred is connected. The above quiver moduli space yields a natural morphism N (1, . . . , 1) → N (γ) for any ordered partition γ of r. Using the Jordan normal for the linear maps A_ı,1, A_ı,2, 0 ≤ ı ≤ ℓ, it is straightforward to show that this morphism is surjective. Therefore N (γ) must also be connected, as required in the proof of equation (5.14).

D A compactness result

This section proves that the moduli spaces of asymptotically stable parabolic ADHM sheaves in the example considered in Section 8 are proper. In that case C ≃ P¹ and there is a single orbifold point p, which is one of the fixed points of the canonical torus action on C. The second fixed point is denoted by∞. The orbifold eY is the total space of the rank two bundle K_Ce ⊗Ceν^∗OC(∞) ⊕ ν^∗OC(−∞). Therefore one has a moduli space of asymptotically stable parabolic ADHM sheaves on C with coefficient line bundles OC(−∞), KC ⊗C O(−∞) ⊗C

OC(p). The underlying vector bundle E of any such ADHM sheaf E splits as a direct sum

E ≃ ⊕^l_j=1OC(e_j∞)^⊕rj with

0≤ d1 < d₂ <· · · < dl.

Positivity follows from asymptotic ADHM stability, which requires E to be generically gen-erated by the image of the section ψ : OC → E as a quiver sheaf. The Higgs fields Φ1, Φ₂ have components

Φ₁(j, j^′) :OC(e_j∞)^⊕rj → OC((e_j′ − 1)∞)^⊕rj′

Φ2(j, j^′) :OC(ej∞)^⊕rj → OC((ej^′ − 1)∞)^⊕rj′ ⊗C OC(p)

For degree reasons Φ₁(j, j^′) = 0 for all j^′ ≤ j, and Φ2(j, j^′) = 0 for all j^′ < j. Moreover, note that the diagonal components Φ₂(j, j) must be constant maps. Since the residue Res_pΦ₂ must be nilpotent, it follows that the components Φ₂(j, j) must vanish as well. This implies that all polynomial invariants of the quiver sheafE are identically zero since ϕ : E → OC is identically zero. Since the generalized Hitchin map determined by the polynomial invariants is proper, it follows that the moduli space is proper.

References

[1] L. ´Alvarez-C´onsul and O. Garc´ıa-Prada. Hitchin-Kobayashi correspondence, quivers, and vortices. Comm. Math. Phys., 238(1-2):1–33, 2003.

[2] I. Biswas. Parabolic bundles as orbifold bundles. Duke Math. J., 88(2):305–325, 1997.

[3] N. Borne. Fibr´es paraboliques et champ des racines. Int. Math. Res. Not. IMRN, (16):Art. ID rnm049, 38, 2007.

[4] P. Borgas dos Santos and M. Jardim. ADHM description of flag Hilbert Schemes. in preparation.

[5] T. Bridgeland. Hall algebras and curve-counting invariants. J. Amer. Math. Soc.

24(4):969-998, 2011.

[6] U. Bruzzo, W.-y. Chuang, D.-E. Diaconescu, M. Jardim, G. Pan, et al. D-branes, surface operators, and ADHM quiver representations. Adv.Theor.Math.Phys., 15:849–911, 2011.

arXiv:1012.1826.

[7] J. Bryan and T. Graber. The crepant resolution conjecture. Algebraic geometrySeattle 2005. Part 1, 2342, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Provi-dence, RI, 2009. arXiv:0610129.

[8] J. Bryan and R. Pandharipande. The local Gromov-Witten theory of curves. J. Amer.

Math. Soc., 21(1):101–136 (electronic), 2008. With an appendix by Bryan, C. Faber, A.

Okounkov and Pandharipande.

[9] B. Young. Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds. with an appendix by J. Bryan Duke Math. J. 152 (2010), no. 1, 115153. arXiv:0802.3948.

[10] J. Choi, S. Katz, and A. Klemm. The refined BPS index from stable pair invariants.

2012. arXiv:1210.4403.

[11] W. Chuang, D. Diaconescu, and G. Pan. BPS states and the P=W conjecture.

arXiv:1202.2039, to appear in Moduli Spaces, Cambridge Univ. Press.

[12] W.-y. Chuang, D.-E. Diaconescu, and G. Pan. Wallcrossing and Cohomology of The Moduli Space of Hitchin Pairs. Commun.Num.Theor.Phys., 5:1–56, 2011.

[13] I. Ciocan-Fontanine and M. Kapranov. Virtual fundamental classes via dg-manifolds.

Geom. Topol., 13(3):1779–1804, 2009.

[14] M. A. A. de Cataldo, T. Hausel, and L. Migliorini. Topology of Hitchin systems and Hodge theory of character varieties: the case A₁. Ann. of Math. (2), 175(3):1329–1407, 2012.

[15] M. A. A. de Cataldo and L. Migliorini. The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Amer. Math. Soc. (N.S.), 46(4):535–633, 2009.

[16] D. E. Diaconescu. Moduli of ADHM sheaves and local Donaldson-Thomas theory. J.

Geom. Phys., (62):763–799.

[17] D.-E. Diaconescu. Chamber structure and wallcrossing in the ADHM theory of curves, I. J. Geom. Phys., 62(2):523–547, 2012.

[18] R. Dijkgraaf, C. Vafa, and E. Verlinde. M-theory and a topological string duality. 2006.

hep-th/0602087.

[19] T. Eguchi and H. Kanno. Five-dimensional gauge theories and local mirror symmetry.

Nucl. Phys., B586:331–345, 2000.

[20] T. Eguchi and H. Kanno. Topological strings and Nekrasov’s formulas. JHEP, 12:006, 2003.

[21] B. Fantechi and L. G¨ottsche. Riemann-Roch theorems and elliptic genus for virtually smooth schemes. Geom. Topol., 14(1):83–115, 2010.

[22] W. Fulton, J. Harris. Representation theory. A first course. Graduate Texts in Mathe-matics, 129. Springer-Verlag, Berlin, New York, 1991. xvi+551 pp.

[23] O. Garc´ıa-Prada, P. B. Gothen, and V. Mu˜noz. Betti numbers of the moduli space of rank 3 parabolic Higgs bundles. Mem. Amer. Math. Soc., 187(879):viii+80, 2007.

[24] A.M. Garsia, M. Haiman. A graded representation model for Macdonald’s polynomials.

Proc. Nat. Acad. Sci. U.S.A., 90(8):3607-3610, 1993.

[25] R. Gopakumar, C. Vafa. M theory and topological strings II. arXiv:9812127.

[26] M. Groechenig. Hilbert schemes as moduli of Higgs bundles and local systems.

arXiv:1206.5116.

[27] M. Haiman. Macdonald polynomials and geometry. In New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97), volume 38 of Math. Sci. Res. Inst. Publ., pages 207–254. Cambridge Univ. Press, Cambridge, 1999.

[28] M. Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J.

Amer. Math. Soc., 14(4):941–1006 (electronic), 2001.

[29] T. Hausel, E. Letellier, and F. Rodriguez-Villegas. Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J., 160(2):323–400, 2011.

[30] T. Hausel and F. Rodriguez-Villegas. Mixed Hodge polynomials of character varieties.

Invent. Math., 174(3):555–624, 2008. With an appendix by Nicholas M. Katz.

[31] N. Hitchin. Stable bundles and integrable systems. Duke Math. J., 54(1):91–114, 1987.

[32] T. J. Hollowood, A. Iqbal, and C. Vafa. Matrix Models, Geometric Engineering and Elliptic Genera. JHEP, 03:069, 2008.

[33] A. Iqbal and A.-K. Kashani-Poor. Instanton counting and Chern-Simons theory. Adv.

Theor. Math. Phys., 7:457–497, 2004.

[34] A. Iqbal and A.-K. Kashani-Poor. SU(N) geometries and topological string amplitudes.

Adv. Theor. Math. Phys., 10:1–32, 2006.

[35] A. Iqbal, C. Kozcaz, and C. Vafa. The refined topological vertex. JHEP, 10:069, 2009.

[36] D. Joyce and Y. Song. A theory of generalized Donaldson-Thomas invariants. Mem.

Amer. Math. Soc. 217 (2012), no. 1020, iv+199 pp. arxiv.org:0810.5645.

[37] S.H. Katz, A. Klemm and C. Vafa. Geometric engineering of quantum field theories.

Nucl. Phys. B497:173-195, 1997. hep-th/9609239.

[38] S.H. Katz, A. Klemm and C. Vafa. M theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3:1445-1537, 1999.

[39] Y. Kawamata. Francia’s flip and derived categories. In Algebraic geometry, pages 197–

215. de Gruyter, Berlin, 2002.

[40] G. Kerr. Weighted blowups and mirror symmetry for toric surfaces. Adv. Math., 219(1):199–250, 2008.

[41] Y. Konishi. Topological strings, instantons and asymptotic forms of Gopakumar-Vafa invariants. hep-th/0312090.

[42] M. Kontsevich and Y. Soibelman. Stability structures, Donaldson-Thomas invariants and cluster transformations. arXiv.org:0811.2435.

[43] A. E. Lawrence and N. Nekrasov. Instanton sums and five-dimensional gauge theories.

Nucl. Phys., B513:239–265, 1998.

[44] J. Li, K. Liu, and J. Zhou. Topological string partition functions as equivariant indices.

Asian J. Math., 10(1):81–114, 2006.

[45] M. Logares and J. Martens. Moduli of parabolic Higgs bundles and Atiyah algebroids.

J. Reine Angew. Math., 649:89–116, 2010.

[46] E. Markman. Spectral curves and integrable systems. Compositio Math., 93(3):255–290, 1994.

[47] M. Maruyama and K. Yokogawa. Moduli of parabolic stable sheaves. Math. Ann., 293(1):77–99, 1992.

[48] D. Maulik. Motivic Residues and Donaldson-Thomas theory. to appear.

[49] S. Mozgovoy. Solutions of the motivic ADHM recursion formula. Int. Math. Res. Not.

IMRN, (18):4218–4244, 2012.

[50] H. Nakajima. Lectures on Hilbert schemes of points on surfaces, volume 18 of University Lecture Series. American Mathematical Society, Providence, RI, 1999.

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