Parabolic refined invariants and Macdonald polynomials

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Parabolic refined invariants and Macdonald polynomials

Wu-yen Chuang, Duiliu-Emanuel Diaconescu, Ron Donagi, Tony Pantev

Abstract

A string theoretic derivation is given for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with G.

Pan. Haiman’s geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.

1 Introduction

The main goal of this paper is a string theoretic derivation of the conjecture of Hausel, Letellier and Rodriguez-Villegas [29] on the topology of character varieties of punctured Riemann surfaces. Analogous results have been obtained in [12, 11] in the absence of marked points, identifying the main conjecture of Hausel and Rodriguez-Villegas [30] with a refined Gopakumar-Vafa expansion. The same framework yields a recursion relation for Poincar´e and Hodge polynomials of Higgs bundle moduli spaces using the wallcrossing formula of Kontsevich and Soibleman [42]. A motivic version of this recursion relation is derived by Mozgovoy in [49], and proved to be in agreement with the Hausel-Rodriguez-Villegas formula.

The string theoretic construction also provides quantitative supporting evidence [11] for the P = W conjecture formulated by de Cataldo, Hausel, and Migliorini in [14], and proven in loc. cit. for rank two Higgs bundles. The present paper carries out a similar program for character varieties with marked points, the starting point being the main conjecture formulated in [29], which is briefly reviewed below.

1.1 The Hausel-Letellier-Rodriguez-Villegas formula

Let C be a smooth complex projective curve of genus g ≥ 0, and D = p1 +· · · + pk a divisor of distinct reduced marked points on C. Let γ₁, . . . , γ_k denote the generators of the fundamental group π₁(C \ D) corresponding to the marked points. For any nonempty partition µ = (µ¹, . . . , µ^l) of r≥ 1, let Cµ be a semisimple conjugacy class in GL(r,C) such that the eigenvalues of any matrix in Cµ have multiplicities {µ¹, . . . , µ^l}.

Let µ = (µ1, . . . , µk) be a collection of partitions of an integer r≥ 1. Then the character variety C(C, D; µ) is the moduli space of conjugacy classes of representations

f : π₁(C\ D) → GL(r, C)

such that f (γ_i) ∈ Cµi for all 1 ≤ i ≤ k. The character variety C(C, D; µ) actually depends on the choice eigenvalues but we will suppress this dependence from the notation since the topological invariants we compute below are independent of this choice.

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According to [29, Thm. 2.1.5], for suﬃciently generic conjugacy classes C_µ_i, C(C, D; µ) is either empty or a smooth quasi-projective variety of dimension d_µ = r²(2g − 2 + k) −

∑_k

i=1

∑_l_i

j=1(µ^j_i)² + 2, where l_i is the length of the partition µ_i, 1 ≤ i ≤ k, as above. The compactly supported cohomology H_cpt^∗ (C(C, D; µ)) carries a weight filtration W• and the mixed Poincar´e polynomial is defined by

Pc(C(C, D; µ); u, t) = ∑

i,k≥0

dim (

Gr^W_i H_cpt^k (C(C, D; µ)))

u^i/2(−t)^k. (1.1)

A priori the right hand side of (1.1) takes values in Z[u^1/2, t], but it was conjectured in [29]

that it is in fact a polynomial in (u, t).

In order to formulate the main conjecture of [29], for any partition λ let H^g_λ(z, w) = ∏

2∈λ

(z^2a(²⁾⁺¹− w^2l(²⁾⁺¹)^2g

(z^2a(²⁾⁺²− w^2l(²⁾)(z^2a(²⁾− w^2l(²⁾⁺²). (1.2) where a(2), l(2) denote the arm, respectively leg length of 2 ∈ λ. Moreover, for each marked point p_i, let x_i = (x_i,1, x_i,2, . . .) be an infinite collection of formal variables, 1 ≤ i ≤ k, and eH_λ(z², w²; x_i) be the modified MacDonald ploynomial [24, 27] labelled by λ. Then [29, Conjecture 1.2.1.(iii)] states that

Z_HLRV(z, w, x_i) = exp ( _∞

∑

k=1

∑

µ

1 k

w^−kd^µP_c(C(C, D; µ); z^−2k,−(zw)^k) (1− z^2k)(w^2k− 1)

∏k i=1

m_µ_i(x^k_i) )

(1.3) where

Z_HLRV(z, w, x_i) =∑

λ

H_λ^g(z, w)

∏k i=1

He_λ(z², w²; x_i)

and m_µ_i(x_i) are the monomial symmetric functions. For ease of exposition equation (1.3) will be referred to as the HLRV formula.

Note also that the character variety C(C, D; µ) is diﬀeomorphic to a moduli space of strongly parabolic Higgs bundles on C. By analogy with the P = W conjecture formulated in [14], one expects the weight filtration on the compactly supported cohomology on the character variety to be identified with a perverse Leray filtration for the Hitchin map on the moduli of parabolic Higgs bundles. This conjectural identification plays an important role in this paper.

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1.2 The main conjecture

In this paper we propose a program for verifying (1.3) by following a sequence of string- theoretic and geometric dualities providing identifications of various counting functions. Our main string theoretic construction relies on a conjectural identification of the generating function Z_HLRV(z, w, x) with the stable pair theory of a Calabi-Yau orbifold eY . This orbifold is constructed in Section 4 using the results of [51, 26], which identify parabolic Higgs bundles on C with Higgs bundles on a root stack. The root stack is an orbifold curve eC equipped with a natural projection to C, which makes C its coarse moduli space. Its construction depends on the discrete invariants of the parabolic structure and is reviewed in detail in Section 3.1.

In particular, note that the closed points of eC have generically trivial stabilizers, the orbifold points being in one-to-one correspondence with the marked points on C.

Given a line bundle M on C, the three dimensional Calabi-Yau orbifold eY_M is defined to be the total space of the rank two bundle eY_M := tot(

ν^∗M⁻¹⊕(

K_C_e⊗Ceν^∗M))

on eC.

In what follows we will call such three dimensional Calabi-Yau orbifolds local orbifold curves. Initially we focus on eY := eY_O := tot(

O_C_e⊕ K_C_e) .

By analogy with Pandharipande and Thomas [55], the stable pair theory of a local orbifold curve eY is defined in Section 4 as a counting theory for pairs (F, s) with F a pure dimension one sheaf on eY , and s : OYe → F a generically surjective section. The discrete invariants of F are the Euler character n = χ( eF ) and a collection of integral vectors m = (m_i)₁_≤i≤k, m_i ∈ (Z≥0)^sⁱ, s_i ≥ 1, encoding the K-theory class of F . In Section 4.1 we extend the analysis of [16]: given a line bundle M on C, we reformulate the stable pair theory of the local orbifold curve eYM in terms of parabolic ADHM sheaves on the curve C.

This yields an explicit construction of the perfect obstruction theory of the moduli space, and makes the relation with parabolic Higgs bundles more transparent.

Assuming the foundational aspects of motivic Donaldson-Thomas theory from [42], one obtains a series of refined Pandharipande-Thomas (PT) invariants for the orbifold eY_M:

Z^ref_e

YM

(q, x, y) =∑

n∈Z

∑

m

P T ( eY_M, n, m; y)qⁿ

∏k i=1

x^m_i ⁱ (1.4)

for some formal variables x =(

x₁, . . . , x_k)

, x_i = (xi,0, . . . , xi,si−1), 1≤ i ≤ k. Then the main conjecture in this paper is the following identity:

Conjecture. After a change of variables, the counting function for the refined PT invariants on the orbifold Calabi-Yau eY = tot(OCe ⊕ KCe) is identified with the combinatorial HLRV

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partition function:

Z^ref_e

Y (z⁻¹w, x, z⁻¹w⁻¹) = Z_HLRV(z, w, x). (1.5)

In Section 6 we explain how the relation (1.5) and the parabolic P = W conjecture imply that the same change of variables converts the HLRV formula (1.3) into a refined Gopakumar- Vafa expansion. Moreover, an application of the wallcrossing formula of Kontsevich and Soibelman yields a recursion relation for Poincar´e polynomials analogous to [12]. As shown in Section 7, the main arguments of [49] apply to the present case as well, proving that the solution of this recursion formula is in agreement with the predictions of the HLRV formula.

A rigorous proof of identity (1.5) is one of the most important open problems emerging from this paper. Supporting evidence for this conjecture is provided in Sections 5 and 8, which are briefly summarized below.

1.3 Macdonald polynomials via geometric engineering

Geometric engineering is used in Section 5 to relate the stable pair generating function (1.4) to a D-brane quiver quantum mechanical partition function. Analogous results in the physics literature were obtained in [37, 43, 19, 53, 33, 34, 20, 32, 41, 44, 35] while a general mathematical theory of geometric engineering is currently being developed by Nekrasov and Okounkov in [52]. The treatment in Section 5 follows the usual approach in the physics literature via IIA/M-theory duality and D-brane dynamics. A detailed comparison with the formalism of [52] is left as an open problem, as briefly explained below.

For simplicity it is assumed that there is only one marked point on C. The local orbifold curve is taken of the form form eY_M = tot(

ν^∗M⁻¹ ⊕ KCe ⊗Ce ν^∗M)

where M is a degree p ≥ 0 line bundle on C. As shown in Section 5.2, a two step chain of dualities relates the resulting stable pair theory to a series of equivariant K-theoretic invariants of nested Hilbert schemes of points in C². The construction of the K-theoretic partition function is explained in Section 5.3. The final formula recorded in equation (5.6) is a generating function of the form

Z_K(q₁, q₂; ˜y, ˜x) =∑

γ

χ^T_y_˜(V(γ))mµ(γ)(˜x) (1.6)

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where χ^T_y_˜(V(γ)) is the equivariant Hirzebruch genus of a vector bundle V(γ) on the nested Hilbert scheme N (γ). Here the sum is over all finite collections γ = (γι)₀_≤ι≤ℓ of positive integers labelling discrete invariants of flags of ideal sheaves on C², as explained in Section 5.2, above equation (5.5). For any γ, µ(γ) denotes the unordered partition of |γ| =∑_ℓ

ι=0γ_ι determined by γ, and mµ(˜x) are the monomial symmetric functions in an infinite set of variables (˜x0, ˜x1, . . .).

Note that the formalism of [52] relates the above stable pair theory with the equivariant K-theoretic stable pair theory of the product eC×C². Then one expects the partition function (1.6) to follow from this theory by virtual localization computations. In particular the bundle of fermion zero modes derived in Appendix B is expected to be naturally determined by the induced perfect obstruction theory on the fixed loci. This computation will be left as an open problem.

The main result of section 5 is identity (5.18) expressing the generating function (1.6) in terms of modified Macdonald polynomials,

Z_K(q₁, q₂; ˜x, ˜y) =∑

µ

Ω^g,p_µ (q₁, q₂, ˜y) ˜H_µ(q₂, q₁, ˜x). (1.7)

The Ω^g,p_µ (q₁, q₂, ˜y) are rational functions of the equivariant parameters (q₁, q₂) and ˜y deter- mined explicitely by a fixed point theorem, according to equation (5.17).

Formula (1.7) is proven in Section 5.5 using Haiman’s geometric construction of Macdon- ald polynomials in terms of isospectral Hilbert scheme [27, 28]. The proof also requires some geometric comparison results between nested and isospectral Hilbert schemes established in Section 5.4.

As supporting evidence for equation (1.5), it is shown in Section 6, equation (6.4), that a simple change of variables relates the right hand side of equation (1.7) to the HLRV generating function Z_HLRV(z, w; x),

Z_HLRV(z, w; x) = Z_K(

w², z²; (zw)⁻¹, (−1)^g⁻¹(zw)^gx)

. (1.8)

Further supporting evidence is provided in Section 8, which is briefly summarized below.

1.4 Parabolic conifold invariants and the equivariant index

A direct computational test of conjecture (1.5) is carried out in Section 8 using the formalism developed by Nekrasov and Okounkov in [52]. The computations are carried out for the special case where C is the projective line with one marked point p, and the local threefold

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is eY_O_C₍₁₎ i.e. the total space of the rank two bundle ν^∗OC(−1) ⊕ KCe ⊗Ce ν^∗OC(1) on eC. A conjectural relation between the equivariant index defined in [52] and orbifold refined stable pair invariants is formulated in Section 8.1, equation (1.5). This identification is checked by explicit virtual localization computations for low degree terms up to three box partitions in Section 8.2.

An important outcome of the string theoretic derivation is a new geometric construction of spectral data for parabolic Higgs bundles which lays the ground for a generalization of the HLRV formula. This is carried out in Section 3, a brief outline being provided below.

1.5 Outline of the program

For the convenience of the reader we now list all the ingredients in the physical derivation of the HLRV conjecture (1.3) in their logical sequence:

Step 1. Identify the combinatorial left hand side of the HLRV formula with the counting function for refined stable pair invariants on the three dimensional Calabi-Yau orbifold Y . This identification is provided by the conjectural formula (1.5). The constructione of the orbifold stable pair theory for this step is presented in Section 4.

Step 2. Identify the counting function for the refined stable pair invariants on eY with the generating function for the perverse Poincar´e polynomials of the moduli of parabolic Higgs bundles. This identification is a combination of two components:

(i) A geometric isomorphism of the moduli of Bridgeland stable pure dimension one sheaves on eY and the product of the moduli space of parabolic Higgs bundles on C with the aﬃne line. This identification is based on the spectral cover construction explained in Section 3.1.

(ii) A conjectural refined Gopakumar-Vafa expansion of the stable pair theory of eY generalizing the unrefined conjecture formulated in [55]. Granting identity (1.5), the specialization of the HLRV formula to Poincar´e polynomials follows recur- sively from the Kontsevich-Soibelman wall-crossing formula [42] for the variation of Bridgeland stabilities on the stable pair moduli by analogy with [12, 49]. The details are presented in Sections 6, 7.

Step 3. Identify the generating function for the perverse Poincar´e polynomials of the moduli of parabolic Higgs bundles with the generating function for the weight-refined Poincar´e

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polynomial of the character variety. This is a parabolic version of the P = W conjecture of Hausel, de Cataldo, and Migliorini. A brief discussion is provided in Section 6.

Note that the refined Gopakumar-Vafa expansion needed here was conjectured for toric Calabi-Yau threefolds in [35] and also [10] building on previous work of [25, 38]. This conjecture was extended to higher genus local curves in [11]. Here it is further extended to local orbifold curves.

In the mathematical literature, a weak form of the unrefined Gopakumar-Vafa conjecture stable pair theory was proven in [5], [59], while the full unrefined conjecture was proven in [60]. These results prove the existence of a suitable integral expansion, but do not provide a cohomological intepretation of the resulting integral invariants. The latter is also needed in the string theory derivation of the HLRV formula.

The geometric framework developed in this program admits a generalization to parabolic Higgs bundles with nontrivial eigenvalues at the marked points. This yields in particular a new orbifold spectral cover presentation for such objects, generalizing the construction in Section 3.1. This is carried out in Sections 3.2, 3.3 and 3.4, which are summarized below.

1.6 Orbifold spectral data for nontrivial eigenvalues

The orbifold spectral cover construction applies to a particular flavor of parabolic Higgs bundles introduced in Section 2.3. These are Higgs bundles with simple poles at the marked points, whose residues are ξ-parabolic maps. This condition requires each graded component of the Higgs field residue at a marked point with respect to the flag to be a specified mul- tiple of the identity. Parabolic Higgs bundles satisfying this condition are called diagonally parabolic, or, more specifically, ξ-parabolic, and form a closed substack of the moduli stack of semistable parabolic Higgs bundles.

The unordered eigenvalues of parabolic Higgs bundles are parameterized by the quotient Q of the Hitchin base defined in equation (2.12). For each point q ∈ Q, the construction in Section 3.2 produces a holomorphic symplectic orbifold surface eS_δ. The moduli space of semistable ξ-parabolic Higgs bundles is conjecturally identified with a moduli space of semistable torsion sheaves on eSδ with fixed K-theory class. The precise statement of this conjecture is given in Section 3.3, a brief outline of the proof being provided in Section 3.4.

In this geometric framework string theory arguments predict a formula of the form (1.3), where the left hand side is given by the refined stable pair theory Z_{P T}^ref( eS_δ × C) up to a change of variables. The right hand side will be a similar generating function for perverse

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Poincar´e polynomials for moduli spaces of stable ξ-parabolic Higgs bundles. As pointed out by Emmanuel Letellier and Tamas Hausel, the latter is expected to be identical with the right hand side of equation (1.3), even away from the nilpotent locus. This leads to a rather surprising conjecture stating that the refined stable pair theory Z_{P T}^ref( eS_δ× C) is independent on δ. In particular, it should be identical with the stable pair theory of eY in equation (1.4).

1.7 Open problems

We conclude the introduction with a list of open problems emerging from this work. Several such questions have already been encountered above, including:

(a) the proof of (1.5),

(b) the derivation of an analogous formula for the stable pair theory of the orbifolds Seδ× C, confirming deformation invariance, and

(c) an explicit comparison with the equivariant K-theoretic stable pair theory of eC× C² in the context of geometric engineering.

Additional possible future directions include:

(d) Section 6 presents quantitative evidence for a parabolic version of the P = W conjec- ture formulated in [14]. It would be very interesting if the parabolic P = W can be proven by direct comparison methods in certain classes of examples.

(e) Another problem is to prove the crepant resoltuion conjecture for stable pair invariants formulated in [56, Conj. 4], for the orbifolds eS_δ× C. Similar results in Donaldson-Thomas theoriers have been proven in [7, 9].

(f ) Elaborating on the same topic, a further question is whether ξ-parabolic Higgs bundles admit a spectral cover presentation in terms of torsion sheaves on the resolutions of the coarse moduli spaces. Again, the Fourier-Mukai transform should provide important input in finding the answer.

(g) Finally, a natural question is whether one can construct a TQFT formalism for (unrefined) curve counting invariants of local orbifold curves, by analogy with the results of Bryan and Pandharipande [8], and Okounkov and Pandharipande [54].

1.8 Notation and conventions

C - a smooth complex projective curve of genus g≥ 0.

D = p₁+· · · + pk - a divisor of distinct reduced marked points on C.

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µ = (µ₁, . . . , µ_k) - a collection of partitions of an integer r ≥ 1.

C(C, D; µ) - the character variety, i.e. the moduli space of conjugacy classes of representa- tions of π₁(C \ D) with values in fixed conjugacy classes at the punctures.

P_c(C(C, D; µ); u, t) - the mixed Poincar´e polynomial for the weight filtration on the com- pactly supported cohomology of the character variety.

H^g_λ(z, w) - the HLRV (z, w)-deformation of the 2g− 2 power of the hook polynomial given in equation (1.2).

Heµ(q2, q1, ˜x) - the modified MacDonald polynomial [24, 27].

ZHLRV(z, w, x) - the combinatorial HLRV partition function appearing in the left hand side of the HLRV formula.

C - an orbifold curve equipped with a morphism ν : ee C → C, which is an isomorphism outside D.

Y - a three dimensional Calabi-Yau orbifold given as ee Y = tot(

OCe⊕ KCe

). Ye_M - a three dimensional Calabi-Yau orbifold given as eY_M = tot(

ν^∗M ⊕(

K_C_e⊗ ν^∗M⁻¹)) for some line bundle M on C.

Z^ref_e

Y_M(q, y, x) - the counting function of refined stable pair invariants on eY_M.

Z_K(q₁, q₂; ˜y, ˜x) - the counting function of equivariant K-theoretic invariants of nested Hilbert schemes of points in C².

H^ssξ (C, D; m, e, α) - the moduli stack of semistable ξ-parabolic Higgs bundles on C.

Se_δ - a symplectic orbifold surface associated with a zero dimensional subscheme inside tot(K_C(D)).

Acknowledgements. We are very grateful to Ugo Bruzzo, Jim Bryan, Tamas Hausel, Mar- cos Jardim, Sheldon Katz, Ludmil Katzarkov, Emmanuel Letellier, Davesh Maulik, Sergey Mozgovoy, Alexei Oblomkov, Andrei Okounkov, Rahul Pandharipande, Fernando Rodriguez- Villegas, Vivek Shende, Andras Szenes, Richard Thomas and Zhiwei Yun for very helpful discussions. WYC and DED would like to thank Guang Pan for collaboration at an incipient

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stage of this project. WYC was supported by NSC grant 101-2628-M-002-003-MY4 and a fellowship from the Kenda Foundation. DED was partially supported by NSF grant PHY- 0854757-2009. RD acknowledges partial support by NSF grants DMS 1304962 and RTG 0636606. DED and TP also acknowledge partial support from NSF grants DMS 1107452, 1107263, 1107367 ”RNMS: GEometric structures And Representation varieties” (the GEAR Network) during the initial stages of this work. TP was partially supported by NSF RTG grant DMS-0636606 and NSF grant DMS-1302242.

2 Parabolic Higgs bundles and spectral covers

The goal of this section is to provide some background on parabolic Higgs bundles, summa- rizing the main results used throughout the paper. Let C be a smooth projective curve over C and D =∑k

i=1p_i a reduced eﬀective divisor on C. A meromorphic Higgs bundle is a pair (E, Φ) with a E a locally free sheaf and Φ : E → E ⊗ KC(D) a morphism of sheaves on C. Parabolic Higgs bundles are a refinement of meromorphic ones defined by specifying a parabolic structure at each marked point, as discussed in detail below.

2.1 Parabolic structures

In order to fix notation, let V be a finite dimensional vector space and m = (m_a)₀_≤a≤s−1 ∈ Z^s_≥0 an ordered collection of non-negative integers such that

s−1

∑

a=0

m_a = dim V. (2.1)

A flag of type m in V is a filtration

0 = V^s⊆ V^s⁻¹· · · ⊆ V¹ ⊆ V⁰ = V by vector subspaces such that

dim (V^a/V^a+1) = ma, 0≤ a ≤ s − 1. (2.2) Note that degenerate flags are allowed i.e. the inclusions do not have to be strict, but the length of the filtration is fixed.

Suppose V, W are finite dimensional vector spaces equipped with filtrations V^•, W^• of the same length s. A linear map f : V → W will be called parabolic if f(Vâ)⊆ Wâ for all 0≤ a ≤ s. The map f will be called strongly parabolic if f(Vâ)⊆ Wâ+1 for all 0≤ a ≤ s−1.

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A more refined compatibility condition can be defined when W = V ⊗ L, with L a one dimensional vector space, and W^• is the natural filtration determined by V^•. Given a collection of linear maps ξ = (

ξ_a : C → L)

0≤a≤s−1, a linear map f : V → W will be called ξ-parabolic if f is parabolic and the induced maps f_a: Vâ/Vâ+1 → Vâ/Vâ+1⊗ L are of the form

fa = 1_V^a_/V^a+1⊗ ξa.

Following the notation introduced in [23], let PHom(V, W ), SPHom(V, W ) denote the linear space of parabolic, respectively strongly parabolic linear maps. Let also

APHom(V, W ) = Hom(V, W )/PHom(V, W )

be the vector space parameterizing equivalence classes of morphisms not preserving the filtrations.

For any exact sequence

0→ V^′ → V → V^′′ → 0,

a flag V^• in V of length s induces canonical flags V^′•, V^′′• of V^′, V^′′ of the same length. In fact vector spaces equipped with flags of fixed length s form an abelian category.

Given a reduced eﬀective divisor D = ∑k

i=1p_i on the curve C, for each i ∈ {1, . . . , k}

let m_i = (m_i,a), 0 ≤ a ≤ si, s_i ≥ 1, be an ordered collection of integers of length si ≥ 1 satisfying conditions (2.1). A quasi-parabolic structure on a vector bundle E on C is a collection (E_i^•)₁_≤i≤k of flags of type m_i in the fiber E_p_i for each i ∈ {1, . . . , k}. For ease of exposition, such a quasi-parabolic vector bundle will be denoted by E^•, and its numerical type by m = (m_i)₁_≤i≤k.

For any exact sequence of vector bundles

0→ F → E → G → 0,

a quasi-parabolic structure on E at D of type m^E induces quasi-parabolic structures of types m^F, m^G on F, G such that m^F + m^G = m^E. Moreover, as explained in [23, Sect 2.2], for any two parabolic bundles E^•, F^• there is a sheaf of parabolic morphisms P Hom_C(E^•, F^•) which fits in an exact sequence

0→ P HomC(E^•, F^•)→ HomC(E, F )→ ⊕^ki=1APHom(E_p^•_i, F_p^•_i)⊗ Opi → 0 (2.3) 0→ HomC(E, F (−D)) → P HomC(E^•, F^•)→ ⊕^ki=1PHom(E_p^•_i, F_p^•_i)⊗ Opi → 0 (2.4)

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Similarly, there is a sheaf of local strongly parabolic morphisms SP Hom_C(E^•, F^•) which fits in analogous exact sequences. Note also that there is a natural duality relation [23, Prop.

2.3.i]

P Hom_C(E^•, F^•)^∨ ≃ SP HomC(F^•, E^•)⊗COC(D). (2.5) A parabolic bundle on C is a quasi-parabolic bundle E^• equipped in addition with col- lections of weights α_i = (α_i,a)₀_≤a≤s_i₋₁ ∈ R^sⁱ for each i∈ {1, . . . , l} such that

0≤ αi,0 <· · · < αi,si−1 < 1. (2.6) The data (α_i)₁_≤i≤k will be denoted by α and parabolic bundles will be denoted by (E^•, α).

There is a natural stability condition for parabolic bundles formulated in terms of parabolic slopes. The parabolic degree of (E^•, α) is defined as

deg(E^•, α) = deg(E) +

∑k i=1

si−1

∑

a=0

m_i,aα_i,a, (2.7)

and the parabolic slope is given by

µ(E^•, α) = χ(E^•, α)

rk(E) . (2.8)

with

χ(E^•, α) = deg(E^•, α)− rk(E)(g − 1) = χ(E) +

∑k i=1

si−1

∑

a=0

mi,aαi,a. (2.9) Any nontrivial proper saturated subsheaf 0⊂ E^′ ⊂ E inherits an induced parabolic structure (E^′•, α^′) on E^′. The parabolic bundle (E^•, α) is (semi)stable if any such subsheaf satisfies the parabolic slope condition

µ(E^′•, α^′) (≤) µ(E^•, α). (2.10) As shown in [47], this stability condition yields projective moduli spaces of S-equivalence classes of semistable objects. Moreover, for suﬃciently generic weights these moduli spaces are smooth.

2.2 Higgs fields, spectral covers, and foliations

A quasi-parabolic Higgs bundle on C is a quasi-parabolic vector bundle E^• equipped with a Higgs field Φ : E → E ⊗CK_C(D) such that the residue Res_p_i(Φ) : E_p_i → Epi is a parabolic

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map for each marked point. A parabolic Higgs bundle is defined by specifying in addition a collection of weights α as in the previous section.

There is a natural notion of stability for parabolic Higgs bundles, defined by imposing the parabolic slope inequality (2.10) for all proper saturated subsheaves preserved by the Higgs field. The results of [62], imply that semistable parabolic Higgs bundles with fixed numerical invariants m, deg(E) = e form an algebraic stack of finite type H^sspar(C, D; m, e, α). The stable ones form an open substack H^spar(C, D; m, e, α). Moreover there is a coarse moduli space H_par^ss (C, D; m, e, α) parameterizing S-equivalence classes of semistable objects which contains an open subspace H_par^s (C, D; m, e, α) parameterizing isomorphism classes of stable objects. According to [63], H_par^ss (C, D; m, e, α) is a normal quasi-projective variety while the stable open subspace is smooth. Note also that any semistable object must be stable for suﬃciently generic weights and primitive numerical invariants.

Similar considerations apply to strongly parabolic Higgs bundles, in which case the moduli stacks/spaces will be labelled with a subscript s-par instead of par. In addition, one can construct similarly moduli spaces of parabolic and strongly parabolic Higgs bundles where the Higgs field takes values in an arbitrary coeﬃcient line bundle M , that is Φ : E → E⊗CM (D).

In this case, the line bundle M will be specified in the notation of the moduli space e.g.

H_par^ss (C, D, M ; m, e, α).

Taking polynomial invariants of the Higgs field yields the Hitchin map

h : H_par^ss (C, D; m, e, α)→ B(C, D; r), B(C, D; r) =⊕^rl=1H⁰(C, (K_C(D))^l). (2.11) This is a surjective proper morphism, its generic fibers being disjoint unions of abelian varieties. As observed in [46, 45], the unordered eigenvalues of the Higgs field at the marked points are parameterized by the quotient B/B₀ where B₀ ⊂ B is the linear subspace

B0(C, D; r) =⊕^rl=1H⁰(C, (KC(D))^l⊗ OC(−D)) ⊂ B(C, D; r).

The moduli space is foliated by the fibers of the resulting projection,

p : H_par^ss(C, D; m, e, α)→ B(C, D; r)/B0(C, D; R). (2.12) Parabolic Higgs bundles admit a spectral cover presentation as parabolic pure dimension one sheaves on the total space P of K_C(D). Let π : P → C denote the canonical projection, Pi = π⁻¹(pi) the fiber at the marked point pi, and DP =∑k

i=1Pi. A quasi-parabolic structure F^• on a pure dimension one sheaf F on P is defined by a sequence of surjective morphisms F ⊗P P_i ↠ Fi^sⁱ⁻¹ ↠ · · · ↠ Fi¹ (2.13)

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where F_i^aare sheaves on P_ifor all 1 ≤ i ≤ k. Moreover F is required to have compact support, which implies that the sheaves F ⊗P P_i are zero dimensional. In this case ch₁(F ) = dσ with σ the class of the zero section. A parabolic structure is defined by specifying in addition parabolic weights α = (α^a_i) as above.

Any saturated sub sheaf F^′ ⊂ F inherits a natural induced parabolic structure. Then one defines a stability condition using the parabolic slope

µ(F^•, α) = 1 d

(

χ(F ) +

∑k i=1

s−1

∑

a=0

αi,a(χ(F_i^a+1)− χ(Fi^a)) )

.

This yields an algebraic moduli stack of semistable objects which is isomorphic to the moduli stack of semistable parabolic Higgs bundles on C with numerical invariants

m^a_i = d− χ(F_i^a), e = χ(F ) + d(g− 1).

This isomorphism assigns to any sheaf F the bundle E = π_∗F , the flags being determined by

E_i^a = Ker(E_p_i ↠ π_∗F_i^a).

The Higgs field Φ : E → E ⊗CK_C(D) is the pushforward Φ = π_∗y of the multiplication map F → F ⊗P π^∗KC(D) by the tautological section y ∈ H⁰(P, π^∗KC(D)).

Again, a similar spectral construction applies to parabolic Higgs bundles with coeﬃcients in a line bundle M , as defined above (2.11). In that case, P will be the total space of the line bundle M (D).

2.3 Diagonally parabolic Higgs bundles

For further reference it will be convenient to note here that the moduli stack of semistable Higgs bundles contains a closed substack where the Higgs field has ξ_i-parabolic residues at each marked point p_i, where ξ_i is a collection ξ_i = (ξ_i⁰, . . . , ξ^s_iⁱ⁻¹) ∈ KC(D)^⊕s_p_iⁱ. Using the notion introduced in Section 2.1, this means that Φ|pi : E_p_i → Epi⊗ KC(D)_p_i is parabolic, and the induced maps

E_p^a

i/E_p^a+1

i → Ep^ai/E_p^a+1

i ⊗ KC(D)_p_i

are of the form 1⊗ ξ^a_i. Such objects will be called ξ-parabolic, where ξ = (ξ^a_i), 1 ≤ i ≤ k, 0 ≤ a ≤ si. The closed substack of such objects will be denoted by H^ss_ξ_−par(C, D; m, e, α).

For ξ_i = (0, . . . , 0), 1 ≤ i ≤ k, one recovers the moduli stack of strongly parabolic Higgs bundles.

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It will be shown in Section 3 that moduli spaces of ξ-parabolic Higgs bundles occur naturally in string theory.

3 Spectral data via holomorphic symplectic orbifolds

The goal of this section is to formulate a variant of the spectral cover construction for parabolic Higgs bundles. In this variant the spectral data are torsion sheaves on holomorphic symplectic orbifold surfaces. This construction is diﬀerent from the standard spectral construction from Section 2.2 in which the spectral data are parabolic dimension one sheaves on the total space P of the line bundle K_C(D). The main motivation for this alternative approach resides in string theory, where parabolic structures must arise naturally from D-brane moduli problems rather than being specified as additional data.

For a brief outline, suppose C is a smooth projective curve equipped with a reduced divisor D = ∑k

i=1p_i of marked points. We want to describe orbifold spectral data for parabolic Higgs bundles on (C, D). Consider the moduli stack H^ss_ξ_−par(C, D; m, e, α) of diag- onally parabolic Higgs bundles introduced in Section 2.3. In this section we will show that any Higgs bundle (E, Φ) in H^ss_ξ_−par(C, D; m, e, α) can be represented by a spectral datum eG which is a Bridgeland semistable pure dimension one coherent sheaf on a certain orbifold symplectic surface eS_δ.

The surface eS_δ depends on C, the divisor D, and a zero dimensional subscheme δ inside tot(K_C(D)). To describe it let P denote the total space of K_C(D), and let P_i be the fiber over p_i, 1≤ i ≤ k. Let si ∈ Z≥1, 1≤ i ≤ k be fixed positive integers and δ = (δi)₁_≤i≤k be a fixed collection of degree si divisors

δ_i =

ℓi

∑

j=1

s_i,j℘_i,j, ℓ_i ≥ 1, si,j ≥ 1, 1 ≤ j ≤ ℓi,

on Pi for each 1 ≤ i ≤ k. By convention, set si,0 = 0 for each 1 ≤ i ≤ k. In Section 3.2 we check that the weighted blowup of P along δ produces a holomorphic symplectic orbifold surface eS_δ.

The particular δ needed for the spectral description of the Higgs bundles inH_ξ^ss_−par(C, D; m, e, α) is constructed out of the eigenvalues ξ of the residues of the Higgs fields, and the flag types m. Specifically we take s_i to be the number of steps in the parabolic filtration at the point p_i and ℓ_i to be the number of distinct entries in the vector ξ_i =(

ξ_i⁰, . . . , ξ_i^sⁱ⁻¹)

∈ KC(D)^⊕s_p ⁱ

i . We label the distinct entries of ξ_i by ℘_i,1, . . . , ℘_i,ℓ_i, and we write s_i,j for the multiplicity

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with which ℘_i,j is repeated as a coordinate inside ξ_i. In other words we choose a function ȷ :{0, . . . , si− 1} → {1, . . . , ℓi} so that

ξ_i^a= ℘i,ȷ(a), 0≤ a ≤ si− 1. (3.1)

These choices define a zero dimensional subscheme δ ⊂ P and an orbifold symplectic surface Se_δ.

Then the main result of this section is the existence of an isomorphism

H^ss_ξ_−par(C, D; m, e, α) ∼=M^ss_β ( Se_δ, d

)

(3.2)

of the moduli stackH_ξ^ss_−par(C, D; m, e, α) between semistable ξ-parabolic bundles on C with the moduli stackM^ss_β (

Se_δ, d )

of Bridgeland β-semistable pure dimension one sheaves on eS_δ with K-theory class d∈ Kc⁰( eS_δ).

To set up this isomorphism we first construct an identification of discrete invariants (r, m, e)←→ d

and an identification

α ←→ β

of the parabolic weights on the Higgs side with the Bridgeland stability parameters β on the spectral data side. These identifications are based on an explicit computation of the compactly supported K-theory of eS_δ. A precise statement is formulated in Section 3.3.

The simplest instance of this construction is ξ = 0, in which case ξ-parabolic bundles are the same as strongly parabolic bundles. In this case the construction of eS0 follows from standard root stack constructions in the literature, as explained below.

3.1 Root stacks and orbifold spectral covers

Using the construction of [2, 3], parabolic Higgs bundles have been identified with ordinary Higgs bundles on an orbicurve in [51, 26]. This section reviews the basics of this construction following the algebraic approach of [26].

Given the curve C with marked points p_i, 1 ≤ i ≤ k one first constructs an orbicurve C as follows. Let U = Ce \ {p1, . . . , p_k}. For any point pi, let Dpi, denote the formal disc centered at p_i and D^◦pi = Dpi ×C U the punctured formal disc. Let φ_i : eDpi → Dpi be the

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s_i : 1 cover given by z_i 7→ z^siⁱ. There is a natural µ_s_i-action on eDpi sending z_i 7→ ωiz_i, where ω_i = exp(2π√

−1/si). The quotient stacks [eDpi/µ_s_i] are then glued to U using the morphisms φ_i to identify the open substacks [eD^◦pi/µ_s_i] with the punctured disks D^◦pi. In characteristic zero this yields a smooth Deligne-Mumford stack eC equipped with a map ν : eC → C which identifies C with its coarse moduli space.

Following [26, Sect 2.4], a Higgs bundle on eC is a vector bundle ˜E equipped with a Higgs field ˜Φ : ˜E → ˜E ⊗Ce K_C_e. This data determines a parabolic Higgs bundle on C as follows.

For each point p_i there is a line bundle ˜L_i on eC such that ˜L^s_iⁱ = ν^∗OC(p_i). Locally, ν^∗OC(p_i) corresponds to the rank one free C[[zi]]-module generated by z_i^−sⁱ while ˜L_i corresponds to the C[[zi]]-module generated by z_i⁻¹. Now let

E = ν_∗E,˜ and

F_i^a= ν_∗(E˜⊗CeL˜^−a_i )

(3.3) for each 1≤ i ≤ k, 0 ≤ a ≤ si− 1. By the base change theorem, all direct images are locally free and the sheaves F_i^a, 0≤ a ≤ si− 1, form a filtration

E(−pi)⊆ Fi^sⁱ⁻¹ ⊆ · · · ⊆ Fi⁰ = E (3.4) for each 1≤ i ≤ k.

For concreteness, note that any locally free sheaf ˜E is locally isomorphic to a sum of line bundles of the form ⊕^rj=1L˜ⁿ_i^i,j, corresponding to the C[[zi]]-module ⊕_r

i=1z_i⁻ⁿ^i,jC[[zi]]. The morphism ν : eC → C is locally of the form ti = z_i^sⁱ, where t_i is a local coordinate on C centered at p_i. The direct image E = ν_∗E corresponds locally to the˜ C[[ti]]-module obtained by taking the µ_s_i-fixed part of ⊕_r

i=1z⁻ⁿ_i ^i,jC[[zi]]. The subsheaves F_i^a are obtained similarly by taking the µ_s_i-fixed part of ⊕_r

i=1z_i⁻ⁿ^i,j^+aC[[zi]], 0 ≤ a ≤ si− 1.

The filtration (3.4) determines a flag

E_i^a= Ker(E_p_i⊗ Opi ↠ E/Fi^a) (3.5) in the fiber E_p_i, hence one obtains a quasi-parabolic bundle E^• on C. Note also that the snake lemma yields an isomorphism

E_i^a⊗ Opi ≃ Fi^a/E(−pi) (3.6) for all 0≤ a ≤ si, 1≤ i ≤ k.

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According to [26, Prop. 2.16] assigning the quasi-parabolic bundle E^• to ˜E yields an equivalence of groupoids. Moreover, the degree of ˜E as an orbibundle equals the parabolic degree of E^• with weights α_a = _s^a

i.

Next consider a Higgs field ˜Φ : ˜E → ˜E⊗CeK_C_e on the stack and note the isomorphisms K_C_e ≃ ⊗^ki=1L˜^(s_i ⁱ⁻¹⁾⊗Ceν^∗K_C ≃ ⊗^ki=1L˜⁻¹_i ⊗Ceν^∗K_C(D).

Then Φ = ν_∗Φ : E˜ → E ⊗C Ω¹_C(D) is a Higgs field on E and equations (3.3) imply that Φ(F_i^a)⊆ Fi^a+1⊗C Ω¹_C(D)

for all 0 ≤ a ≤ si− 1, 1 ≤ i ≤ k. Using isomorphisms (3.6), this implies that the residue Respi(Φ) is strongly parabolic with respect to the flag (3.5) for each i∈ {1, . . . , k}.

According to [26, Prop. 20], the above construction yields a one-to-one correspondence between Higgs fields ˜Φ on the orbifold eC and Higgs fields Φ on C with strongly parabolic residues with respect to the flag E_i^• at each p_i, 1 ≤ i ≤ k. Furthermore, the degree of the orbifold bundle ˜E equals the parabolic degree deg(E, α) for special values of the weights

α_i,a= a

s_i, 0≤ a ≤ si− 1, 1≤ i ≤ k. (3.7) Based on this observation, it is straightforward to check that this correspondence maps (semi)stable orbibundles to (semi)stable parabolic Higgs bundles with weights (3.7). Since it works for flat families as well, it yields isomorphisms of moduli stacks.

For completeness, note that the fiber ˜E_i,0 of ˜E at the closed point 0∈ [eDpi/µ_s_i] carries a natural action of the stabilizer group µ_s_i. Hence it decomposes into irreducible representations,

E˜_i,0≃

si−1

⊕

a=0

R^⊕n_i,a^i,a, (3.8)

where R_i,a denotes the one dimensional representation of µ_s_i with character ω_i^a. Then [26, Lemma 2.19] proves that the discrete invariants mi,aof the flag (3.5) are given by mi,a = ni,a, for 0 ≤ a ≤ si − 1. In string theoretic language this means that the flag type encodes the fractional charges with respect to the twisted sector Ramond-Ramond fields.

Using the standard spectral cover construction, an orbifold Higgs bundle ( ˜E, ˜Φ) corresponds to a pure dimension one sheaf ˜F on the total space eS of K_C_e, finite with respect to the natural projection π : eS → eC. The bundle ˜E is obtained by push-forward, ˜E = π_∗F ,˜ and the Higgs field ˜Φ is the push-forward of the multiplication map ˜F → ˜F ⊗ π^∗K_C_e by

Parabolic refined invariants and Macdonald polynomials