MIRROR SYMMETRY FOR PARABOLIC HIGGS BUNDLES VIA p -ADIC INTEGRATION

. Applying the technique of p -adic integration, we prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli spaces of (strongly) parabolic Higgs bundles for the structure groups SL n and PGL n , building on previous work of Groechenig-Wyss-Ziegler on the non-parabolic case. We also prove the E -polynomial of the smooth moduli space of parabolic GL n -Higgs bundles is independent of the degree of the underlying vector bundles.

1. Introduction 1.1.Topological mirror symmetry for Higgs bundles.The notion of a Higgs bundle was introduced by Hitchin in his seminal paper [Hit87] with the motivation of studying certain differential equations from gauge theory.These geometric objects have an extremely rich structure and draw intense research interest from different areas of mathematics.One of the most intriguing feature of the moduli space M of Higgs bundles over a smooth projective curve X is that it has the structure of a completely integrable system.This structure is induced by the so-called Hitchin map from M to an affine space A: coordinate functions on A induce Poissoncommuting functions on M, and a general fiber of this map is an abelian variety, i.e. a compact torus which is also a complex algebraic variety.
Let G be a complex reductive group and G its Langlands dual group.The moduli space M G of G-Higgs bundles and the moduli space M G of G-Higgs bundles are mapped to the same affine space A via the Hitchin map: These two integrable systems are dual to each other, in the sense that for a general a ∈ A, the fibers h −1 G (a) and h −1 G (a) are dual abelian varieties 1 .This statement is first proved by Hausel and Thaddeus [HT03] for G = SL n and G = PGL n , later generalized to all G by Donagi and Pantev [DP12].Motivated by this duality between Hitchin integrable systems and the Strominger-Yau-Zaslow picture of mirror symmetry, Hausel and Thaddeus proposed in [HT03] the so-called topological mirror Shiyu Shen has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No. 101034413.
1 Strictly speaking, these Hitchin fibers are gerbes over abelian varieties, and this duality should be interpreted as a duality between commutative group stacks.
symmetry conjecture, in which they predict a correspondence between the (appropriately defined) Hodge numbers of those moduli spaces M d SLn and M e PGLn (here d and e stands for the degree of the underlying vector bundles, and we assume they are coprime to n).This conjecture (in the non-parabolic setting) has been proved by Groechenig, Wyss and Ziegler [GWZ20a] through the application of p-adic integration to the Hitchin integrable systems.Developing on the beautiful connections between p-adic integration, point-counting and Betti numbers established by Weil, they reduce the job of matching the Hodge numbers of M d SLn and M e PGLn to matching the integral of certain measures on the p-adic version of the moduli spaces.The structure of the Hitchin fibrations allows this comparison to be further reduced to a number theoretic computation concerning dual abelian varieties over local fields.See Section 3 for the structure of this argument.In [MS21], the authors provided a separate proof of this conjecture using sheaf-theoretic methods.In [GO19], the authors established the topological mirror symmetry for parabolic Higgs bundles of rank n = 2, 3 with full flags.1.2.Mirror symmetry for parabolic Higgs bundles.The goal of this paper is to prove the topological mirror symmetry conjecture for the moduli spaces of parabolic Higgs bundles.Let X be a smooth projective curve over the field C of complex numbers.Let D = q 1 + q 2 + • • • + q m be an effective reduced divisor on X, and let P D = (P 1 , P 2 , . . ., P m ) be an ordered m-tuple of parabolic subgroups of GL n (C).We fix a line bundle M on X that either satisfies deg M > 2g − 2 or equals the canonical bundle K.We consider the moduli space M d,α GLn of semistable M -twisted parabolic GL n -Higgs bundles of degree d on X, the moduli space M d,α SLn of semistable M -twisted parabolic "SL n "-Higgs bundles of degree d and the moduli stack M d,α PGLn of semistable M -twisted parabolic PGL n -Higgs bundles of "degree" d.The precise definition of those moduli spaces are given in Section 4.Here α is a set of real numbers called parabolic weights which plays a role in defining the notion of stability on those objects.We say α is generic for degree d if a semistable parabolic vector bundle of degree d is automatically stable.
Let α be a set of parabolic weights generic for degree d.With this assumption, both M d,α GLn and M d,α SLn are smooth quasi-projective varieties, and M d,α PGLn is a smooth Deligne-Mumford stack.All three moduli spaces admit the Hitchin map to an affine space A P .The generic fiber of the parabolic Hitchin map is described in [She18]: there exists an open subset A 0 P ⊂ A P of the Hitchin base such that the Hitchin fiber for the GL n -moduli space M α GLn is isomorphic to the relative Picard scheme Pic(Σ/A), where Σ/A is a flat family of smooth projective curves constructed from the family of spectral curves by consecutive blow-ups, see Theorem 4.6.For the SL n -moduli space M d,α SLn and the PGL n -moduli stack M d,α PGLn , the generic Hitchin fibers over A 0 P are torsors for dual abelian schemes.Let α (resp.α ′ ) be a set of parabolic weights generic for degree d (resp.e), we prove the following 2 SLn ; u, v) = E st (M e,α ′ PGLn , α d ; u, v) of E-polynomials.The right-hand side of 2 Our proof relies on the assumption that a certain open subset of the parabolic Hitchin base defined in [She18] is non-empty.See Remark 4.7.This is always satisfied when g X ≥ 2. When the genus is 1 or 0, we need to put restrictions on the number of marked points and the type of parabolic subgroups involved, see [She18] Section 1.2.
the equality stands for the twisted stringy E-polynomial with respect to a gerbe α d on M e,α ′ PGLn .(b) The E-polynomial E(M d,α GLn ; u, v) is independent of the degree d and the parabolic weights α.
When X is a smooth projective curve over a finite field k, assuming the set of parabolic weights α is generic for d, we also have the following result concerning point-counts Theorem 1.2 (cf.Theorem 4.4 (b)).Assume char(k) > n.Then the point-count GLn (k) is independent of d and α.Our proof of Theorem 1.1 and Theorem 1.2 is based on the same strategy as in [GWZ20a], but some new ingredients come into play.The arguments in [GWZ20a] rely heavily on the existence of the so-called regular locus M reg ⊂ M3 of the moduli space, which is characterized by the Higgs field being regular everywhere on the curve X.This open subset M reg forms a torsor for a certain commutative group scheme over the entire Hitchin base, and the complement satisfies codim(M\M reg ) ≥ 2. This locus M reg serves multiple purposes in [GWZ20a]: (a) it is used to construct the gauge form ω on M that they do p-adic integration with.This gauge form ω automatically satisfies the property that the p-adic volume of a general Hitchin fiber is independent of the degree of the underlying vector bundle (as long as the Hitchin fiber is non-empty).(b) when working over a local field F with ring of integers O F , the regular locus is used to show that for both the GL nand the PGL n -moduli space, the Hitchin fiber admits an O F -rational point over any O F -point of the Hitchin base, and this O F -rational point can be chosen such that the underlying vector bundle is of any degree.This statement plays an important role in the proof of both the topological mirror symmetry and the independence on degree for E-polynomials and point-counts in the non-parabolic setting.
Unfortunately, we don't have this regular locus in the parabolic setting unless all parabolic subgroups involved are Borel, and even if all parabolic subgroups are Borel, we don't have codim(M\M reg ) ≥ 2. In Section 5, we construct a gauge form ω on M4 using a different strategy: we relate the moduli space M with the cotangent bundle T * N of the moduli space N of vector bundles with certain types of parabolic and level structures.We obtain the desired gauge form by manipulating the symplectic form on T * N .
The relation between stringy E-polynomials, stringy point-counts and p-adic integration reviewed in Section 2 reduces the comparison of stringy E-polynomials and point-counts in Theorem 1.1 and Theorem 1.2 to the comparison of the p-adic integral of ω on the involved moduli spaces.By throwing away a subset of zero measure, this is further reduced to comparing the p-adic volume of the Hitchin fibers over F -points of A 0 P ⊂ A P .For Theorem 1.1 (b) and Theorem 1.2, the main result of [BB07] plays a crucial role to guarantee that the gauge form ω behaves well when we change the degree d of the underlying vector bundles, see Subsection 5.2.For Theorem 1.1 (a), we establish a stacky version of the torsor-gerbe duality in [HT03] (Proposition 3.2 and Proposition 3.6), see Theorem 4.27.This Theorem 4.27 together with computations in [GWZ20a] guarantees the equality of p-adic integrals on the SL n -and PGL n -side.1.3.Structure of the article.In Section 2, we review the relation between stringy E-polynomials, stringy point-counts and p-adic integration over certain smooth abelian Deligne-Mumford stacks.In Section 3, we describe the framework for the application of p-adic integration to Hitchin systems.In Section 4, we first define the GL n -, SL n and PGL n -moduli spaces involved in our main theorems.Then we describe the generic Hitchin fiber in all three cases.Then we establish the stacky version of the torsor-gerbe duality in [HT03], see Theorem 4.27.In Section 5, we first construct a gauge form on the moduli space of parabolic Higgs bundles.Then we show this gauge form behaves well when we change the degree d of the underlying vector bundles.Both Theorem 1.1 and Theorem 1.2 are proved in Section 6.
1.4.Acknowledgements.I would like to thank Michael Groechenig for many helpful discussions on this subject, and for his contribution to the proof of Proposition 5.1.I would like to thank Tamas Hausel, Anton Mellit and André Oliveira for helpful conversations on related subjects.
2. E-polynomials, point-counting and p-adic integration 2.1.E-polynomials and point-counting.The goal of this subsection is to review the relation between stringy E-polynomials and stringy point-counts described in [GWZ20a] Section 2. We start by recalling the definition of E-polynomial of complex quasi-projective varieties.Let X be a complex smooth projective variety.The E-polynomial of X is defined to be E(X; u, v) = (−1) p+q h p,q (X)u p v q , where h p,q (X) is the (p, q)-th Hodge number of X.This definition extends to all complex quasi-projective varieties by imposing the condition that for any closed subvariety Z ⊂ X, we have The precise formula of E(X; u, v) in terms of mixed Hodge numbers can be found in [HRV08] Definition 2.1.4.For a quotient stack of the form X = [Y /Γ], where Y is a complex quasi-projective variety with the action of a finite group Γ, the E-polynomial of X is defined to be the Γ-invariant part of E(Y ; u, v).Now we assume X is smooth.Motivated by stringy Hodge numbers considered in [BD96], Hausel and Thaddeus [HT03] defined the stringy E-polynomial of X to be where the first summation is over the set Γ conj of conjugacy classes in Γ, the second summation is over connected components of the quotient stack [Y γ /C(γ)], and F (γ, Z) is the so-called fermionic shift, see [GWZ20a] Definition 2.2.If we shift our base field from C to a finite field F q of order q, similar formula as in (2.1) is used to define the stringy point-count Here the point-count #Z(F q ) is defined by , where Z(F q ) iso stands for the set of isomorphic classes in Z(F q ).With an eye towards the formulation of topological mirror symmetry, we further consider a smooth quotient stack X = [Y /Γ] together with a µ r -gerbe α over X .By the discussion in [HT03] Section 4, this µ r -gerbe α leads to a µ r -torsor over the inertia stack IX = γ∈Γconj [Y γ /C(γ)], which we denote by L. This µ r -torsor L can be used to define a twisted version of the E-polynomial in (2.1) and the point-count in (2.2), which leads to the so-called twisted stringy E-polynomial E st (X , α; u, v) for X over C and twisted stringy point-count # α st (X ) for X over F q .We refer the readers to [GWZ20a] Definition 2.12 and 2.13 for the precise definition of E st (X , α; u, v) and # α st (X ).Now we record the relation between twisted stringy E-polynomial and twisted stringy point-count.
Theorem 2.1 (cf.[GWZ20a] Theorem 2.19).Let R ⊂ C be a subalgebra of finite type over Z.Let Y 1 and Y 2 be two smooth R-varieties acted on by two finite abelian groups Γ 1 and Γ 2 respectively.For i = 1, 2, let X i = [Y i /Γ i ] be the corresponding quotient stack, and let α i be a µ r -gerbe on X i .If for any ring homomorphism R −→ F q from R to a finite field F q we have equality # α1 st (X 1 × R F q ) = # α2 st (X 2 × R F q ) of twisted stringy point-counts, then we also have the following equality of twisted stringy E-polynomials: 2.2.Point-counting and p-adic integration.In this subsection we review the theory of p-adic integration over certain Deligne-Mumford stacks and its relation to stringy point-counts.The main references are [GWZ20a], [GWZ20b] and [Yas17].We fix F to be a non-Archimedean local field with ring of integers O F and residue field k F of characteristic p and order q.For any positive integer n, F n equipped with the non-Archimedean absolute value becomes a locally compact topological group, therefore admits a unique Haar measure µ that satisfies µ(O n F ) = 1.Now let X be a smooth variety over F with a volume form ω ∈ Γ(X, Ω top X/F ).We say ω is a gauge form if ω is a trivialising section of the invertible sheaf Ω top X/F .For each gauge form ω, we associate with it a measure µ ω on the set X(F ) of F -points on X.The measure µ ω is defined locally: for any open chart U ֒→ F n of X(F ), assuming where the right-hand side stands for the integral of the absolute value of f with respect to the Haar measure µ.Now let X be a smooth variety over O F .In this case, the p-adic measure µ ω does not depend on the choice of the gauge form ω ∈ Γ(X, Ω top X/OF ) when restricted to the set X(O F ) of O F -points on X. Therefore p-adic measures on local charts of X glue together to form a measure on X(O F ) which we call µ can .The following theorem of Weil builds the bridge between point-counting and p-adic integration.Theorem 2.2 (cf.[Wei82] Theorem 2.2.5).Let X be a smooth variety over O F of dimension n.Then we have dµ can .
We record the follow properties of the p-adic measure µ ω that play a significant role in simplifying the computation of p-adic integration for Hitchin systems.(b) Let h : X → Y be a smooth morphism between smooth F -varieties.Let ω X (resp.ω Y ) be a gauge form on X (resp.Y ).Let θ ∈ Γ(X, Ω top X/Y ) be the unique relative top-degree form that satisfies ω X = h * ω Y ∧ θ.For any integrable function f on X(F ), we have the following equality where θ y ∈ Γ(h −1 (y), Ω top h −1 (y)/F ) stands for the pull-back of θ to h −1 (y).
The theory of p-adic integration for smooth F -varieties is generalized to certain Deligne-Mumford stacks in [Yas17].Following [GWZ20a], we restrict ourselves to the following situation: where Y is smooth quasi-projective O F -variety with a generically free action by a finite abelian group Γ, such that the order |Γ| of Γ is coprime to p and O F contains all the |Γ|-th roots of unity.
Let U ⊂ Y be the locus where the action of Γ is free.Let M be the geometric quotient of Y by Γ and let pr : Y −→ M be the quotient map.In this setting, one can define a measure µ orb on M (O F ) ♯ := M (O F ) ∩ pr(U )(F ) which is called the orbifold measure.We refer the readers to [GWZ20a] and [Yas17] for the precise definition of this measure µ orb .For our applications to Higgs bundles, we will only need the following simple description of µ orb in a special setting.
Lemma 2.5 (cf.[GWZ20a] Remark 4.13).We assume Ω top Y admits a Γ-invariant trivializing section ω.Then ω descends to a section ω orb of Ω top pr(U) , and the orbifold measure µ orb is given by integrating |ω orb | on M (O F ) ♯ ⊂ pr(U )(F ).Now let α be a µ r -gerbe on an admissible finite abelian quotient stack M. We further assume that O F contains all the r-th roots of unity and r is coprime to the characteristic of the residue field.Following [GWZ20a], we define a function f α : We have the following pull-back diagram Since the Γ-action on U is free, the vertical arrow on the left-hand side gives a Γ-torsor over Spec(F ).By the definition of the quotient stack M = [Y /Γ], this pull-back diagram gives a lifting of x F ∈ pr(U )(F ) to an F -point xF ∈ M(F ) of M. Pulling back along xF , we get a µ r -gerbe x * F α over Spec(F ), which gives an element in the Brauer group Br(F ).The Brauer group of a non-Archimedean local field F is isomorphic to Q/Z via the Hasse invariant inv : The relation between stringy point-count and p-adic integration is described in the following theorem.
Theorem 2.6 (cf.[GWZ20a] Corollary 5.28).Let M be an admissible finite abelian quotient stack over O F with a µ r -gerbe α and the associated function f α as described above.Then the stringy point-count of the special fiber M kF := M × OF k F can be computed using the following equation where fα denotes the complex conjugate of f α .

p-adic integration on Hitchin systems
The goal of this section is to apply the theory of p-adic integration discussed in Section 2.2 to Hitchin systems.The topological mirror symmetry equality in [GWZ20a] Theorem 6.11 relies on the fact that for both the SL n -side and PGL nside, there exists a large enough open subset that forms a torsor for certain commutative group scheme over the entire Hitchin base (see Remark 3.2).With an eye towards application in the parabolic setting, we proceed our discussion without assuming the existence of this open locus.The main conclusion we record here is Theorem 3.7.

Weak Abstract Hitchin systems.
Definition 3.1.Let A be a smooth R-variety.Let M be an admissible finite abelian group stack over R together with a proper map h : M −→ A. We call (M, A) a weak abstract Hitchin system if there exists an open dense subset A 0 ⊂ A and an abelian A 0 -scheme P such that the restriction M 0 = M × A A 0 of M to A 0 is a P-torsor and codim(M\M 0 ) ≥ 1.
Remark 3.2.In [GWZ20a] Definition 6.8, an abstract Hitchin system is defined to be a weak abstract Hitchin system in Definition 3.1 with an open dense substack M ′ ⊂ M that forms a torsor for a commutative group scheme Q over A, with the assumption that Q × A A 0 = P and codim(M\M ′ ) ≥ 2.
Let (M, A) be a weak abstract Hitchin system over R such that r is invertible in R, and let α be a µ r -gerbe on M. We denote by Split ′ (M 0 /A 0 , α) the principal component of the space of relative splittings of α on M 0 , see [GWZ20a] Definition 6.4.We briefly recall the definition of Split ′ (M 0 /A 0 , α) as follows.We first associate with α its stack of relative splittings Split µr (M 0 /A 0 , α): for any map of R-schemes f : S −→ A 0 , it classifies splittings of the µ r -gerbe f * α on S × A 0 M 0 .The stack Split µr (M 0 /A 0 , α) is naturally a pseudo torsor for the moduli stack Tor µr (M 0 /A 0 ) of µ r -torsors.We denote by Split µr (M 0 /A 0 , α) the µ r -rigidification of Split µr (M 0 /A 0 , α) in the sense of [ACV03], Section 5, which becomes a pseudo torsor for the scheme Pic(M 0 /A 0 )[r] of r-torsion points in the relative Picard scheme Pic(M 0 /A 0 ).Now we define where Pic τ (M 0 /A 0 ) is the torsion component in Pic(M 0 /A 0 ).Since M 0 is a Ptorsor over A 0 , we have Pic τ (M 0 /A 0 ) ∼ = P ∨ , where P ∨ is the dual abelian scheme of P. Therefore Split ′ (M 0 /A 0 , α) is a pseudo P ∨ -torsor.
Definition 3.3.We consider two weak abstract Hitchin systems (M 1 , M 2 , A) over the same base A, such that M i becomes a P i -torsor when restricted to the same open dense subset A 0 ⊂ A. Let α i be a µ r -gerbe on M i for i = 1, 2, and we assume r is invertible in R and R contains all the r-th roots of unity.We call (M 1 , M 2 , A, α 1 , α 2 ) a dual pair of weak abstract Hitchin systems if the following conditions hold.(a) P 1 and P 2 are dual abelian schemes over A 0 , and there is an étale isogeny φ : P 1 −→ P 2 such that the degree of φ is invertible in R.
(b) there are isomorphisms of P i -torsors for i = 1, 2 Now we consider a weak abstract Hitchin system (M, A) over R such that when restricted to A 0 ⊆ A, M 0 is a torsor for an abelian A 0 -scheme P. Since P −→ A 0 is smooth, h : M 0 −→ A 0 is also smooth, therefore we have the following isomorphism of invertible sheaves Since P is an abelian A 0 -scheme, there exist translation invariant trivializing sections of Ω top P/A 0 (at least locally on A 0 ).As observed in [GWZ20a] Lemma 6.13, for any translation invariant global section ω ∈ Γ(P, Ω top P/A 0 ), one can associate with it a translation invariant global section of Ω top M 0 /A 0 as follows.We pick some étale cover U → A 0 that trivializes the P-torsor M 0 , i.e. there exists an isomorphism M 0 U ∼ = P U of P U -torsors.Through this isomorphism we get a section of Ω top U /U descends to a translation invariant section of Ω top M 0 /A 0 , and the resulting section does not depend on the choice of the étale cover U → X and the trivialization.
Lemma 3.4.The map from translation invariant relative volume forms on P/A 0 to translation invariant relative volume forms on M 0 /A 0 described above induces an isomorphism Proof.The statement follows from the fact that for any Zariski open subset V ⊆ A 0 , pulling back along functions.This is because both maps P −→ A 0 and M 0 −→ A 0 are proper with geometrically reduced and geometrically connected fibers.
For the rest of this paper, for any global section ω ∈ Γ(M 0 , Ω top M 0 /A 0 ), we will denote by ω′ the corresponding section in Γ(P, Ω top P/A 0 ), and vice versa.If we assume that the invertible sheaf Ω top A 0 /R is trivial and fix a trivializing section ω A 0 , it follows from isomorphism (3.2) and Lemma 3.4 that any trivializing section ω ∈ Γ(M 0 , Ω top M 0 /R ) can be written uniquely as where ω ∈ Γ(M 0 , Ω top M 0 /A 0 ) is a translation invariant trivializing section.3.2.p-adic integration and mirror symmetry.In this subsection, we restrict ourselves to the case when R = O F , where O F is the ring of integers of a non-Archimedean local field F .Let (M, A) be a weak abstract Hitchin system over O F such that when restricted to the open subset A 0 ⊂ A, M 0 becomes a torsor for an abelian A 0 -scheme P. Since we assume M is admissible, it admits a presentation where Y is a smooth quasi-projective O F -variety with a generically free Γ-action.We denote by pr : Y −→ M = Y /Γ the quotient map to the coarse moduli space M .We denote by U ⊂ Y the locus where the Γ-action is free.Note that since M 0 /A 0 is a P-torsor, the open substack M 0 ⊂ M is actually a scheme over O F .It follows that M 0 maps isomorphically to an open subscheme of pr(U ), which we still denote by M 0 .Definition 3.5.We define a dual pair of weak abstract Hitchin systems over O F .For i = 1, 2, when restricted to the open subset A 0 ⊂ A, M 0 i becomes a torsor for an abelian A 0 -scheme P i , and we have an étale isogeny φ : P 1 −→ P 2 .We denote by f αi : M i (O F ) ♯ −→ C the function corresponding to α i as defined in (2.3).We further assume that there is a trivializing section ω A 0 of the invertible sheaf Ω top A 0 /OF , and that there is a trivializing section ω i of Ω top pr(Ui)/OF for i = 1, 2. By the discussion at the end of Subsection 3.1, the topdegree form ω i can be written uniquely as a wedge product when restricted to M 0 i : ) is a translation invariant trivializing section.We denote by ω′ i the corresponding section in Γ(P i , Ω top Pi/A 0 ), see Lemma 3.4.Theorem 3.7.If we assume φ * ω′ 2 = ω′ 1 , then we have the following equality of p-adic integrals Proof.Combining Proposition 2.3 and Remark 3.6, we have Besides the assumptions for Theorem 3.7, if we further assume that for i = 1, 2, the top-degree form pr * ω i extends to a Γ i -invariant trivializing section of Ω top Yi/OF , then by Lemma 2.5, the equality in Theorem 3.7 becomes By Theorem 2.6, this equality implies the following equality of stringy point-counts of the corresponding special fibers 4. Moduli spaces of parabolic Higgs bundles 4.1.Moduli space of parabolic GL n -Higgs bundles.We fix a Noetherian integral scheme B. We assume B is of finite type over a universally Japanese ring.
The main examples that will be relevant to our applications are when B = Spec(C), B = Spec(R) where R ⊂ C is a finite generated Z-subalgebra, B = Spec(F q ) where F q is a finite field and B = Spec(F q [[t]]).Let X be a smooth projective curve over B, i.e.X −→ B is smooth projective of relative dimension one with geometrically connected fibers.We assume the genus of X satisfies g X ≥ 2. We assume X admits a marked point q, i.e. an element q ∈ X(B).Let K be the relative canonical bundle of X over B. We fix a line bundle M on X such that M ⊗ K −1 is either trivial or of strictly positive degree.We will consider parabolic Higgs bundles of rank n and multiplicities m = (m 1 , m 2 , . . ., m r ), where m i are positive integers that sum up to n.We recall the following definitions concerning parabolic Higgs bundles.
, where E is a vector bundle of rank n on X, φ : and E • q is a partial flag structure of type m on the fiber E q of E at q ∈ X(B): We further require that when restricted to the fiber E q , the Higgs field φ q is nilpotent with respect to the partial flag structure, i.e. φ q (E i q ) ⊂ E i−1 q .(b) An M -twisted parabolic Higgs bundle is an M -twisted quasi-parabolic Higgs bundle E together with parabolic weights α = (α 0 , α 1 , . . ., α r ), which is a tuple of real numbers that satisfies We define the parabolic degree of an M -twisted parabolic Higgs bundle to be Let F be a subbundle of E. The partial flag structure E • q on E q naturally induces a partial flag structure F • q on F q .The induced parabolic weights for (F, If F is further assumed to be φ-invariant, we get a parabolic Higgs subbundle F of E .Similarly, the partial flag structure on E q induces a partial flag structure on the quotient (E/F ) q with parabolic weights We have the following equality of parabolic degrees It is called stable if the inequality is always strict.(b) We denote by M α GLn the moduli space of semistable M -twisted parabolic Higgs bundles with parabolic weights α.We denote by M d,α GLn the degree GLn is a quasi-projective variety over B, see [Yok93], Corollary 1.6 and Corollary 4.7.
We say α is generic for degree d if there exists no parabolic vector bundle with degree d and parabolic weights α that is semistable but not stable.More precisely, we require that the equality (4.1) doesn't have any solutions, where (m ′ , α ′ ) are the parabolic weights and multiplicities induced from a proper non-zero subbundle F ⊂ E of rank n ′ and degree d ′ .
For a set of parabolic weights α generic for degree d, it is shown in [Yok93] that the moduli space M d,α GLn is a smooth quasi-projective variety over B. The following lemma gives a criterion for the existence of generic parabolic weights for a given degree d.This criterion will be used in the proof of Theorem 4.4.
Proof.Let V be a vector space of dimension n with a partial flag structure of type m.For different choices of integer d ′ and subspace V ′ ⊂ V of dimension n ′ , (4.1) imposes countably many restrictions on α = (α 1 , α 2 , . . ., α r ).Since all α i take values from real numbers, there exists no generic parabolic weights for degree d if and only if (4.1) is automatically satisfied for some choices of d ′ and V ′ .In other words, there exists a generic α for degree d if and only if the following family of equations (4.2) has no solutions, where d ′ is an integer, n ′ is a positive integer smaller than n, and ) is a partition of n ′ .Now we prove the "if" direction of the lemma.Suppose gcd(m 1 , m 2 , . . ., m r , d) = 1.If there exists a set of solutions (d ′ , n ′ , m ′ ) for (4.2), then we have n|dn ′ and n|m i n ′ for each i.Since gcd(m 1 , m 2 , . . ., m r , d) = 1, we have n|n ′ , which contradicts our assumption that n ′ is a positive integer smaller than n.
For the other direction, suppose gcd(m 1 , m 2 , . . ., m s , d gives a set of solutions for (4.2).
Now we state the first main theorem of the paper, which we prove in Section 6.
Theorem 4.4.Let X be a smooth projective curve over a field k.Let α be a set of parabolic weights generic for degree d.By taking the characteristic polynomial of the Higgs field, we get the parabolic Hitchin map h : where Γ(X, M (q) i ) 5 .The map h satisfies the following property: Theorem 4.5 (cf.[Yok93] Corollary 1.6 and Corollary 5.12).The parabolic Hitchin map h is proper.
The (scheme-theoretic) image of h is an affine subspace A P ⊂ A GL n which depends on the parabolic multiplicities m.In order to describe A P , we introduce the following notations.Let λ = (λ 1 , λ 2 , . . ., λ s ), λ 1 ≥ λ 2 , . . ., ≥ λ s be the partition of n conjugate to m.We define λ 0 = 0. Let t = (t 1 , t 2 , . . .t n ) be the ordered n-tuple of integers defined by t i = j if and only if 5 More precisely, A GLn is the scheme of sections of n i=1 Γ(X, M (q) i ), which is an affine B-space.
i.e. the first λ 1 elements are 1, the next λ 2 elements are 2,. . ., and the last λ s elements are s.It is shown in [BK18] that (4.3) For the rest of this section, we assume the base scheme B satisfies the property that for any algebraically closed field k such that B( k) is non-empty, we have char( k) = 0 or char( k) > n.The generic fiber of the parabolic Hitchin map is described in the following theorem.
where Pic(Σ/A 0 P ) is the relative Picard scheme of Σ over A 0 P .This family of curves admits a map p X : Σ −→ X such that the corresponding Σ −→ X × A 0 P is finite, and the isomorphism above is induced by pushing-forward along p X .Remark 4.7.When m = (1, 1, . . ., 1), i.e. we consider complete flags at the marked point, A 0 P is the locus where the spectral curves are smooth.In all the other cases, the spectral curves are always singular above the marked point q.For any geometric point a ∈ A 0 P ( k), the spectral curve at a locally looks like above q, and Σ a is the normalization of this spectral curve.Since A 0 P lies in the locus where the spectral curves are integral, M α GLn × AP A 0 P lies in the stable locus of M α GLn .
Remark 4.8.The degree of a parabolic Higgs bundle and the degree of its corresponding spectral sheaf on Σ a differ by the degree of the line bundle det((p X ) * O Σa ).The line bundle det((p X ) * O Σ ) on X × A is isomorphic to the pull-back of a line bundle on X, which by abuse of notation we still denote by det((p X ) * O Σ ).We denote d m = deg(det((p X ) * O Σ )) which depends on the parabolic multiplicities m.
It follows from the construction of Σ that Proposition 4.9.For any geometric point a ∈ A 0 P ( k), the pull-back map p * X : Pic(Xk) −→ Pic(Σ a ) is injective.
We would like to apply the same argument as in [BNR89] Remark 3.10.For that purpose, we need the following lemma: Lemma 4.10.For any a ∈ A 0 P ( k), there exists a stable parabolic Higgs bundle E such that h(E) = a and the underlying vector bundle E decomposes as a direct sum of line bundles of distinct degree.
Proof.We write a = (a 1 , a 2 , . . ., a n ), a i ∈ Γ(Xk, M i ((i − t i )q)).We consider the following matrix acting on E = (L n , L n−1 , . . ., L 1 ) t , where L i = M i ((i − t i )q).When restricted to the fiber E q , the Higgs field φ q is a matrix with Jordan blocks of type λ, which lies in the Richardson orbit corresponding to the parabolic multiplicities m.It follows that there exists a (unique) partial flag structure of type m on E q that is compatible with φ q .
Proof of Proposition 4.9.It follows from Theorem 4.6 that there exists a line bundle L i defined in the proof of Lemma 4.10.
Let N be a line bundle on Xk such that p * Since L i are line bundles of distinct degree, N must be the trivial line bundle on Xk.

4.2.
The SL n -side and the PGL n -side.In this subsection, we introduce the moduli space of parabolic SL n -and PGL n -Higgs bundles.Remark 4.12.(a) Let α be generic parabolic weights for degree deg(L).Since the only automorphisms that a stable parabolic SL n -Higgs bundle admits is multiplying by the n-th roots of unity, the natural map M L,α SLn −→ M L,α SLn gives the moduli stack M L,α SLn the structure of a µ n -gerbe over M L,α SLn .We denote this µ n -gerbe by α.(b) Let Γ = Pic 0 (X)[n] be the group scheme of n-torsion points in the Picard scheme Pic(X).The moduli space M L,α SLn admits a natural Γ-action.For each γ ∈ Γ, this action is defined by mapping the underlying vector bundle E to E ⊗ γ and modifying the other data accordingly.
(c) Consider the case when X is a smooth complex projective curve.Let L 1 and L 2 be two line bundles on X of the same degree d.Tensoring with an n-th root of SLn .Now we consider the SL n -version of the parabolic Hitchin map ȟ : M L,α SLn −→ A, where A ⊂ A P is the affine subspace defined by the factor in Γ(X, M (q)) being zero.Let A 0 = A ∩ A 0 P and let Σ/A 0 be the family of curves that appears in Theorem 4.6 with p X : Σ −→ X.In order to describe the generic fiber of ȟ, we consider the norm map Definition 4.13.Let P = Nm −1 Σ/X×A 0 (O X ).For a line bundle L on X, let PL = Nm −1 Σ/X×A 0 (L).This commutative group scheme P is the so-called relative Prym scheme of Σ/X × A 0 .We have the following description of the parabolic Hitchin fiber above A 0 .Proposition 4.14.(a) (M L,α SLn ) 0 := M L,α SLn × A A 0 is isomorphic to PL⊗det −1 ((pX ) * OΣ) , (b) P is an abelian scheme over A 0 , and (M L,α SLn ) 0 is a P-torsor.Proof.Part (a) follows from Theorem 4.6.For part (b), the only thing left to check is that P is geometrically connected over A 0 .Note that P is the kernel of the surjective homomorphism of abelian schemes over A 0 .The dual morphism is given by the pull-back map The desired statement that P is geometrically connected over A 0 follows from Proposition 4.9, see [Mum70] Section 15, Theorem 1.Now we describe the PGL n -side of the Hitchin fibration.Definition 4.15.We denote by M d,α PGLn the moduli stack of M -twisted quasiparabolic PGL n -Higgs bundles that admits a presentation as a semistable parabolic GL n -Higgs bundle of degree d and parabolic weights α on each geometric fiber of X/B.Remark 4.16.(a) For each line bundle L on X of degree d, we have an isomorphism GLn is the subspace characterized by the Higgs field having zero trace.
(b) The marked point q ∈ X(B) induces a lifting of the Γ-action from M L,α SLn to the moduli stack M L,α SLn , see Example 4.22.It follows that the µ n -gerbe α on M L,α SLn descends to a µ n -gerbe on M d,α PGLn .Let L d be the degree d line bundle defined by PGLn that comes from the µ n -gerbe α on M L d ,α SLn .(c) Via isomorphisms (4.4), the SL n -version of the parabolic Hitchin map descends to the PGL n -version of the parabolic Hitchin map ĥ : M d,α PGLn −→ A.
Recall that P is an abelian scheme over A 0 that lies in the following short exact sequence 0 therefore P is an abelian scheme over A 0 that is dual to P. Combining with Proposition 4.14, we have (b) P is an abelian scheme over A 0 that is dual to P, and (M d,α PGLn ) 0 is a Ptorsor.Now we assume the base scheme B = Spec(R) is an affine integral scheme such that n is invertible in R and R contains all the n 2g -th roots of unity.We also assume the finite group scheme Pic 0 (X)[n] is constant over B. Recall that for any µ n -gerbe α and integer d, we can construct a new µ n -gerbe α d by taking the induced torsor for the map [d] : Bµ n → Bµ n defined by mapping a line bundle to its d-th power.We have the following proposition which we prove in Section 4.3.
Applying p-adic integration to this dual pair of weak abstract Hitchin systems leads to the second main theorem of this paper, which we prove in Section 6.
Theorem 4.20.Let X be a smooth complex projective curve.Let L be a line bundle on X of degree d and let α 1 (resp.α 2 ) be a set of parabolic weights that is generic for degree d (resp.e).Then we have the following equality of E-polynomials: 4.3.Torsor-gerbe duality for Hitchin systems.In this subsection, we discuss the duality relation between the two µ n -gerbes α and α on the moduli space of parabolic SL n -and PGL n -Higgs bundles.Our discussion uses the framework of duality for certain "good" commutative stacks.We refer the readers to [Ari08] and [BB07]  ( (5) G = P is an abelian scheme, then G ∨ = P ∨ is the dual abelian scheme.
In fact, all the commutative group stacks we are going to consider are étale locally isomorphic to a finite product of commutative group stacks that appear in the Example 4.21 above.
Example 4.22.Let X be a smooth projective curve over B. We denote by Pic(X) (resp.Pic(X)) the Picard stack (resp.Picard scheme) of X.The natural map Pic(X) −→ Pic(X) gives Pic(X) the structure of a G m -gerbe over Pic(X).If we further assume there exists a marked point q ∈ X(B), then Pic(X) can be identified with the moduli stack of line bundles on X with a trivialization at q.This identification induces a splitting Pic(X) ∼ = Pic(X) × BG m of the G m -gerbe.Since the marked point q also induces an isomorphism Pic(X) ∼ = Pic 0 (X) × Z, we have Pic(X) ∼ = Pic 0 (X) × BG m × Z.Note that a marked point always exists étale locally.Since Pic 0 (X) is a self-dual abelian scheme and (BG m , Z) are dual to each other as commutative group stacks, we have an isomorphism Pic(X) ∨ ∼ = Pic(X).

It is well-known that this isomorphism does not depend on the choice of the marked point, hence it can be glued to a global isomorphism. We include a proof of this fact for completion of the exposition.
Lemma 4.23.The isomorphism Pic(X) ∨ ∼ = Pic(X) in Example 4.22 does not depend on the choice of the marked point q ∈ X(B).
Proof.Let q 1 and q 2 be two marked points of X.Let ρ 1 (resp.ρ 2 ) be the isomorphisms between Pic(X) and Pic 0 (X) × BG m × Z described in Example 4.22 using q 1 (resp.q 2 ).Let S be a scheme over B and let (L, F , d) be an S-point of Pic 0 (X)×BG m ×Z, where d ∈ Z, L ∈ Pic gp (X× B S)/p * S Pic gp (S) and F ∈ Pic gp (S).Here Pic gp (Y ) stands for the Picard group of Y , and p S : X × B S → S is the projection map.The transition automorphism Our goal is to show that under the identification 1 .This desired statement can be deduced from the observation that the map Z → Pic 0 (X) Now we consider a commutative group stack P over B together with a µ n -gerbe α : P → P on it.We assume this gerbe α is étale locally trivial over B. We say α admits a group structure if it can be put into a short exact sequence of commutative group stacks as follows: We associate with such a µ n -gerbe α the following scheme Split It follows from the definition above that Split c (P, α) is isomorphic to σ −1 (1), which is a P ∨ -torsor.Now we assume P is an abelian scheme over B. Let P ′ be a P-torsor and let P ′ be its µ n -rigidification.The natural map α ′ : P ′ −→ P ′ gives P ′ the structure of a µ n -gerbe over P ′ .We associate with α ′ the principal component Split ′ (P ′ , α ′ ) of relative splittings defined in (3.1), which is a P ∨ -torsor.From P ′ we can construct a commutative group stack P Z with a group morphism σ : P Z → Z such that σ −1 (0) = P, σ −1 (1) = P ′ and each σ −1 (m) is a P-torsor for any m ∈ Z.The µ n -rigidification of P Z is a commutative group scheme P Z that fits into a short exact sequence The commutative group stack P Z forms a µ n -gerbe over P Z which we denote by α Z .Note that the stack of splittings Split From a µ n -gerbe with a group structure α : P → P over an abelian scheme P, we've constructed three P ∨ -torsors: Split ′ (P ′ , α ′ ) in 3.1, Split c (P, α) in Definition 4.24 (b), and Split c (P Z , α Z ) in the previous paragraph.The following Lemma shows that those three P ∨ -torsors are isomorphic to each other.
It follows from Remark 4.25 that Split c ( PZ /A 0 , αe Z ) ∼ = σ −1 (e).Note that the commutative group stack PZ × Bµn BG m lies in the short exact sequence (4.9) where Pic dZ (Σ/A 0 ) := m∈Z Pic dm (Σ/A 0 ), and Nm is the composite of Nm with the projection Pic(X) → Pic 0 (X) in the isomorphism (4.8).Note that Pic dZ (Σ/A 0 ) lies in the following short exact sequence and the dual short exact sequence is given by as the commutative group stack Pic(Σ/A 0 ) is self dual.We also have the short exact sequence Taking the dual short exact sequence of (4.9), we get Here the morphism Pic 0 (X × A 0 /A 0 ) − → Pic dZ (Σ/A 0 ) ∨ is given by the composite where ι is inclusion from (4.8), p * X is pull-back along the projection p X : Σ → X and π is the map in (4.10).Combining with (4.11), we get and the map ( PZ × Bµn BG m ) ∨ σ − → Z is given by the degree map of Pic(Σ/A 0 ).It follows that we have the following isomorphisms of ( PZ ) ∨ -torsors [d] BG m .Now we turn to the second isomorphism.The G m -gerbe induced from αZ lies in the following short exact sequence The dual short exact sequence is given by 0 and the dual short exact sequence is given by By (4.11), we have Pic eZ (Σ/A 0 ) ∨ ∼ = Pic(Σ/A 0 ) × BGm,[e] BG m .The map η is the composite where the first morphism is the rigidification map and the third morphism maps L to L ⊗ O X (− deg(L)q).It follows that we have isomorphisms of ( PZ ) ∨ -torsors Proof of Proposition 4.19.We first note that the moduli stack M e,α2 PGLn is admissible in the sense of Definition 2.4.This follows from the isomorphism M e,α2 SLn /Γ] in (4.4) and the observation that Γ acts freely on (M Le,α2 SLn ) 0 , which is a consequence of Proposition 4.9.Properness of the Hitchin morphisms ȟ and ĥ follows from Theorem 4.5.Torsor structures over A 0 and Definition 3.3 Part (a) follows from discussions in Section 4.2.Part (b) follows from Theorem 4.27 and Lemma 4.26.For part (c), let a ∈ A(O F ) ∩ A 0 (F ) with corresponding a F ∈ A 0 (F ).We need to show that if both Hitchin fibers ȟ−1 (a F ) = ( PO(d ′ q) ) aF and ĥ−1 (a F ) = ( Pe ′ ) aF have F -rational points, then both G m -gerbes induced from αe ′ and αd ′ split over a F .For the SL n -side, note that since the Hitchin map ȟ is proper, an F -point of the Hitchin fiber ĥ−1 (a F ) extends to an O F -point.The desired splitting of the BG m -gerbe induced from αe ′ |ȟ−1 (aF ) follows from Lemma 6.5 in [GWZ20a].Now we turn to the PGL n -side.Since the BG m -gerbe induced from αe ′ |ȟ−1 (aF ) splits, an F -point in the Hitchin fiber ( PO(d ′ q) ) aF lifts to an F -point in ( PO(d ′ q) × Bµn,[e ′ ] BG m ) aF , which in turn leads to a splitting of the BG m -gerbe induced from αd ′ | ĥ−1 (aF ) by Theorem 4.27.
Remark 4.28.When m = (1, 1, ..., 1), i.e. we consider complete flags at the marked point, the family of curves Σ/A 0 admits a global section q ∈ Σ(A 0 ) that lies above the marked point q.The existence of this global section implies that PO(dq) is isomorphic to P × Bµ n By Theorem 4.27, this further implies that the BG mgerbe induced from αd ′ splits over (M e,α2 PGLn ) 0 = M e,α2 PGLn × A A 0 .It follows that the gerbe αd ′ does not affect the computation of twisted E-polynomial, and the equality of E-polynomials in Theorem 4.20 becomes PGLn ; u, v).This agrees with the observation in [GO19] that the gerbe on the PGL n -side is not needed to formulate the topological mirror symmetry for parabolic Higgs bundles with complete flags.

Algebraic Volume forms on moduli spaces of parabolic Higgs bundles
5.1.Existence of algebraic gauge forms.Let X be a smooth projective curve over a field k with a marked point q ∈ X(k).Let M be a line bundle on X.In this subsection, we denote by M L ⊆ M L,α SLn the stable in the moduli space of semistable M -twisted parabolic SL n -Higgs bundles with parabolic weights α and determinant L. For generic parabolic weights α, we have M L = M L,α SLn .Recall that Γ = Pic(X)[n] acts on M L via tensor product.
Proposition 5.1.The moduli space M L admits a Γ-invariant gauge form.
The proof of Proposition 5.1 will occupy the rest of this subsection.We denote by M the moduli stack of stable M -twisted parabolic SL n -Higgs bundles with parabolic weights α and determinant L. Since M is a µ n -gerbe over M, proving Proposition 5.1 is equivalent to showing that Ω top M admits a Γ-invariant trivializing section.
We fix a presentation M = K(D 1 − D 2 ), where D 1 and D 2 are effective Weil divisors disjoint from q.We define the following stacks which will play an role in the proof of Proposition 5.1.Definition 5.2.(a) We denote by M D1 the moduli stack of K(D 1 )-twisted quasiparabolic SL n -Higgs bundles of determinant L.
(b) We denote by N the moduli stack of quasi-parabolic vector bundles of determinant L together with a D 1 -level structure.More precisely, if we let D 1 = a i p i , the moduli stack N classifies vector bundles on X of determinant L together with a partial flag structure at q and a trivialization on the a i -th formal neighborhood of p i for each i.
Lemma 5.3.Both M D1 and N are algebraic stacks locally of finite type over k.
Proof.Let Bun n be the moduli stack of vector bundles of rank n on X.We consider the forgetful maps M D1 → Bun n and N → Bun n .We note that those forgetful maps are representable and locally of finite presentation.Since Bun n is an algebraic stack locally of finite type over k, both M D1 and N are algebraic stacks locally of finite type over k.
Remark 5.4.(a) The cotangent stack T * N classifies K(D 1 )-twisted quasi-parabolic SL n -Higgs bundles of determinant L together with a D 1 -level structure.By forgetting this D 1 -level structure, we get a natural map π D1 : T * N −→ M D1 , which gives T * N the structure of a SL n (D 1 )-torsor over M D1 .By Cohen's structure theorem, we get an isomorphism where k i is the residue field of p i .
(b) There is a natural inclusion map ι : of Higgs fields, where End 0 (E) is the sheaf of trace zero endomorphisms of E.
(c) Each stack M D1 and T * N admits a Γ-action that is defined similarly as the Γ-action on M. Both maps π D1 : T * N −→ M D1 and ι : M −→ M D1 are Γ-equivariant with respect to those Γ-actions.Now we study the deformation of T * N at E = (E, φ, E • q , ϕ, θ), where (E, φ, E • q , ϕ) is a K(D 1 )-twisted quasi-parabolic SL n -Higgs bundle and θ is a D 1 -level structure of E. We denote by Par 0 (E) the sheaf of trace zero endomorphisms of E that map E i q to E i q at q, and by SPar 0 (E) the sheaf of trace zero endomorphisms that map E i q to E i−1 q at q.The deformation of E in T * N is governed by the complex (5.1) sitting at degree −1 and degree 0: the hypercohomology H 0 (F • E ) gives the deformation of E, and the hypercohomology H −1 (F • E ) gives the lie algebra of the automorphism group of E. The killing form on sl n induces an isomorphism Par 0 (E) ∼ = (SPar 0 (E) ⊗ O X (q)) * .Under this isomorphism, the dual complex (F sitting at degree 0 and degree 1.We identify (F By Serre duality, we have an isomorphism We denote by (T * N) ′ ⊂ T * N the maximal open Deligne-Mumford substack.We note that Indeed, smoothness of (T * N) ′ can be checked by proving the lifting property for small extensions of finite-generated Artinian local k-algebras, and the obstruction for the existence of such liftings lies in given by Serre duality induces a symplectic form on (T * N) ′ .This symplectic form is invariant under the action of Γ and SL n (D 1 ).We denote by ω D1 the top exterior product of this symplectic form, which gives a Γ-invariant trivializing section of Ω top (T * N) ′ .
The goal is to construct a Γ-invariant trivializing section of Ω top M from ω D1 .Our construction relies on the following theorem of M. Rosenlicht.
Theorem 5.5 (cf.[Ros61] Theorem 3).Let G be a connected algebraic group over k with identity element e.Let f : G −→ G m be a map of k-schemes such that f (e) = 1.Then f is a character.
Corollary 5.6.The only invertible regular functions on SL n (k[x]/(x a )) are constant functions.
Proof.We consider the short exact sequence where π is induced by the quotient map k[x]/(x a ) −→ k.We note that the kernel U is a unipotent group over k.Indeed, for any u ∈ U , u − I is a matrix with entries in the ideal xk[x]/(x a ), therefore (u − I) a = 0.It follows that there are no nontrivial characters on U .It follows from 5.5 that all invertible regular functions on SL n (k[x]/(x a )) come from regular functions on SL n (k).Since SL n (k) is semi-simple, there are no non-trivial characters on SL n (k).Again it follows from Theorem 5.5 that there are no non-constant invertible regular functions on SL n (k).The differential form ι ρ(τ ) ω D1 is both horizontal and SL n (D 1 )-invariant, therefore descends to a top-degree form ω D1 on M ′ D1 , which is a Γ-invariant trivializing section.
Step 2. In this step we construct a trivializing section of Ω top M from the topdegree form ω D1 on M ′ D1 .Recall there is a natural inclusion map ι : M −→ M D1 .The image lies in the maximal Deligne-Mumford substack M ′ D1 ⊂ M D1 since M classifies stable Higgs bundles.Let E = (E, φ, E • q , ϕ) be a point of M. The deformation of E in M is governed by sitting at degree −1 and degree 0. The deformation of E in M D1 is governed by sitting at degree −1 and degree 0. We note that when restricted to M, the hypercohomology for both complexes are zero at degree −1 and 1.Indeed, gives the lie algebra of the automorphism group for E in M (resp.M D1 ).We have = 0 since E lies in the maximal Deligne-Mumford locus.The fact that H 1 (H • E ) = 0 follows from H 1 (F • E ) = 0 for the complex F • E in (5.1) and the long exact sequence for hypercohomology.The fact that H 1 (G • E ) = 0 follows from computation of the dimension of M and M D1 .It follows from the long exact sequence for hypercohomology that we have an isomorphism where End 0 (E D2 ) is the locally free sheaf on M for which the fiber at E consists of trace zero endomorphisms of E D2 .Now we consider the SL n (D 2 )-torsor π : M −→ M defined by adding a D 2level structure.More precisely, M classifies quintuples E = (E, φ, E • q , ϕ, θ), where (E, φ, E • q , ϕ) is an element of M and θ is a D 2 -level structure of E. The D 2 -level structure induces a canonical trivialization We fix a non-zero vector σ ∈ ∧ top sl n (D 2 ), which induces a trivialization of π * (∧ top End 0 (E D2 )) through isomorphism (5.3).It follows from Corollary 5.6 that this trivialization of π * (∧ top End 0 (E D2 )) descends to a trivialization By isomorphism (5.2), the trivializing section ω D1 of Ω top induces a trivializing section ω of Ω top M which is still Γ-invariant.5.2.Comparison between degrees.Recall that we denote by M L the moduli space of stable M -twisted parabolic SL n -Higgs bundles with parabolic weights α and determinant L. In Subsection 5.1, we have constructed an algebraic gauge form ω L on M L .The goal of this subsection is to discuss the relation between this ω L for different choices of the line bundle L.
The moduli space M L admits the Hitchin map ȟ : M L −→ A such that when restricted to the open subscheme A 0 ⊂ A, M 0 L := M L × A A 0 becomes a torsor for the relative Prym scheme P/A 0 .By the discussion at the end of Subsection 3.1, if we fix a trivializing section ω A ∈ Γ(A, Ω top A ), the top-degree form ω L can be written uniquely as a wedge product when restricted to M 0 L : (5.4) ) is a translation invariant trivializing section.We denote by ω′ L the corresponding section in Γ( P, Ω top P/A 0 ), see Lemma 3.4.Then we have the following proposition.
Proposition 5.7.The section ω′ L ∈ Γ( P, Ω top P/A 0 ) does not depend on the choice of the line bundle L.
It is enough to prove Proportion 5.7 when the base field k is algebraically closed, therefore we assume k = k.Recall we denote by N the moduli stack of quasiparabolic vector bundles of determinant L together with a D 1 -level structure.We will use the notation N L when we want to emphasize the choice of L. Let Proof.Let E be a point of (T * N) ′ .We denote . By the spectral sequence for hypercohomology, we have the following short exact sequence ).Let π : T * N → N be the projection map, and let ι : T * π(E) N → T * N be the inclusion of the cotangent fiber at π(E).Note that we have H ) is given by the tangent map of ι, and the map H 0 (F ) is given by the tangent map of π.Now the desired statement follows from the definition of the two symplectic forms.
Since T * N classifies K(D 1 )-twisted quasi-parabolic SL n -Higgs bundles together with a D 1 -level structure, we have the Hitchin map Let X be the total spectral curve inside T * X(D 1 + q) × A D1 , and let p X : X −→ X be the projection map.We introduce the following notations.Definition 5.9.(a) We denote by X 0 be the open curve defined by (b) We denote by A 0 D1 ⊂ A D1 be the locus where the spectral curves are smooth and connected above X 0 .Let (T * N) 0 = T * N × AD 1 A 0 D1 and X 0 = p −1 X (X 0 ).Remark 5.10.Let (E, φ, E • q ) be a K(D 1 )-twisted quasi-parabolic Higgs bundle that is mapped to a ∈ A 0 D1 under the Hitchin map.Since the spectral curve X a is smooth above X 0 , by the BNR correspondence [BNR89], the spectral sheaf corresponding to (E, φ) restricts to a line bundle on X 0 a .Now we fix a closed point p ∈ X 0 (k).Let X 0 (p) be the restriction of by modifying the spectral sheaf above X 0 .The following proposition is an analogue of Theorem 4.12 in [BB07].
Proof.The proof is essentially the same as in [BB07].We consider the moduli stack Hecke 1 p of triples (E, F , i) where E ∈ N L , F ∈ N L(p) and i : E ֒→ F is an inclusion of the underlying vector bundles such that F/E is the simple skyscraper sheaf at p ∈ X 0 (k) and the partial flag structures and D 1 -level structures on E and F coincide under i.We consider the following maps: where q maps the triple to F and p maps the triple to E. Both p and q are smooth.
Consider the following pull-back diagram: The stack Z 0 classifies the following data ((E, φ E ), (F , φ F ), i : ) 0 and (E, F , i) ∈ Hecke 1 p .Since the twisted Higgs bundles (E, φ E ) and (F, φ F ) are isomorphic away from p, they map to the same point a ∈ A 0 D1 under the Hitchin map and the corresponding spectral sheaves on Σ a differ by a simple skyscraper sheaf at some p ′ ∈ Σ a that maps to p under the projection map p X : Σ a −→ X. Recall that Σ a is the normalization of the spectral curve X a and they are isomorphic above X 0 .It follows that Z where pr 2 is the projection GLn that is a P-torsor, where P = Pic 0 (Σ/A 0 P ).We have the following Corollary 5.12.Assume that char(k) and n are coprime.For each integer d, there exists a gauge form ω d on the stable locus of the moduli space M d,α GLn , such that if we fix a gauge form on A P , the corresponding ω′ d ∈ Γ(P, Ω top P/A 0

P
) is independent of d.
Proof.Let L be a line bundle on X at degree d.The moduli spaces M d,α GLn and M L,α SLn are related by the following map: Φ : M L,α SLn × H 0 (X, M ) × Pic 0 (X) −→ M d,α

GLn
((E, φ, E • q , ϕ), s, F ) → (E ⊗ F, φ ⊗ id + id ⊗s, E • q ⊗ F q , ).The scheme on the left-hand side admits a free Γ-action defined by γ • (E, s, F ) = (E ⊗ γ −1 , s, F ⊗ γ), and Φ can be identified with the quotient map of this Γ-action.In Proposition 5.1, we've constructed a Γ-invariant gauge form ω L on M L,α SLn .We fix a gauge form ω 2 on H 0 (X, M ) and a translation invariant gauge form ω 3 on Pic 0 (X).The gauge form p * 1 ω L ∧ p * 2 ω 2 ∧ p * 3 ω 3 is Γ-invariant, therefore descends to a gauge form ω d on M d,α GLn .We still need to show that when fixing a gauge form on A P , the corresponding ).Let a ∈ A P ( k) be a geometric Since M L d ,α SLn (X) is a smooth complex variety, its stringy E-polynomial doesn't depend on the choice of the gerbe, therefore it is equivalent to show (6.2) E st (M L d ,α SLn (X), αe ′ ; u, v) = E st (M e,α ′ PGLn (X), αd ′ ; u, v).By Theorem 2.1, in order to prove the desired equality (6.2) of (stringy) E-polynomials, it is enough to show that for any ring homomorphism R → F q , the SL n and PGL n -moduli space have the same twisted point-count.By Theorem 2.6, equality of stringy point-counts can be reduced to proving the equality of p-adic integrals on the SL n -and PGL n -moduli space M L d ,α SLn (X OF ) and M e,α ′ PGLn (X OF ), which are constructed from M L d ,α SLn (X R ) and M e,α ′ PGLn (X R ) by base-change through R → F q → F q [[t]] = O F .Note that by Proposition 4.19, SLn (X R ), M e,α ′ PGLn (X R ), A R , αe ′ , αd ′ ) forms a dual pair of weak abstract Hitchin systems, therefore we can apply discussions in Subsection 3.2 to simplify the computation of p-adic integrals.Recall that we've constructed Γ-invariant gauge forms ω R,d on M L d ,α SLn (X R ) and ω R,e on M Le,α ′ SLn (X R ).At the end of Subsection 3.1, we've associated with them relative gauge forms ω′ R,d and ω′ R,e on PR /A R .By Theorem 3.7 and Remark 3.8, in order to prove the desired equality of p-adic integrals for any base-change R → F q [[t]], it is enough to show that ω′ R,d = ω′ R,e .In Proposition 5.7, we have proved the corresponding equality ω′ d = ω′ e of relative gauge forms on P ∼ = PR ⊗ R C. Since PR is smooth over R, the equality over C implies the desired equality over R.
For simplicity of notations, we write M di,αi GLn (X) = M di (X).Let X R be an Rmodel for X such that the gauge form ω di constructed in Corollary 5.12 extends to a gauge form ω R,di on X R for i = 1, 2. By Theorem 2.1 and Theorem 2.6, proving 6.3 is reduced to showing the following equality of p-adic integrals (6.4) |ω OF ,d2 | for every ring homomorphism R −→ O F = F q [[t]] where F q is a finite field.For i = 1, 2, Proposition 2.3 implies Theorem 1.1 (cf.Theorem 4.20 and Theorem 4.4 (a)).(a) We have an equality E(M d,α Proposition 2.3 ([Yas17] Lemma 4.3 and [GWZ20a] Proposition 4.1).(a) Let X be a smooth O F -variety with a gauge form ω on X F := X × OF F , and let Y ⊂ X be a closed subscheme of positive codimension.Then µ ω (Y (O F )) = 0.
(a) Let k = C. Then the E-polynomial E(M d,α GLn ; u, v) is independent of d and α.(b) Let k = F q such that char(k) > n.Then the point-count #M d,α GLn (k) is independent of d and α.
Theorem 4.6 (cf.[She18] Theorem 2.11.).There exists an open dense subset A 0 P ⊂ A P and a flat family of smooth projective curves Σ −→ A 0 P such that

Definition 4 .
11. (a) Let L be a line bundle on X.A M -twisted parabolic SL n -Higgs bundle of determinant L is a pair (E , ϕ), where E is a M -twisted parabolic Higgs bundle with trace zero Higgs field, and ϕ : det(E) ∼ = L is an isomorphism of line bundles.(b)We denote by M L,α SLn (resp.M L,α SLn ) the moduli space (resp.moduli stack) of semistable M -twisted parabolic SL n -Higgs bundles of determinant L and parabolic weights α.

Proposition 4. 19 .
Let α 1 (resp.α 2 ) be a set of parabolic weights that is generic for degree d (resp.e).Then the quintuple (M L d ,α1 SLn , M e,α2 PGLn , A, αe ′ , αd ′ ) is a dual pair of weak abstract Hitchin systems over B in the sense of Definition 3.3.Here d ′ = d − d m , e ′ = e − d m and L d is the degree d line bundle defined by for a detailed account of this theory.Let B be a Noetherian scheme such that n is invertible on B. Let G be a commutative group stack locally of finite type over B. The dual commutative group stack G ∨ = Hom(G, BG m ) classifies 1-morphisms of group stacks from G to BG m .The main examples we are going to consider are: Example 4.21.

c
(P, α) which is closely related to the principal component of relative splittings defined in (3.1).Definition 4.24.(a) We define Split c µn (P, α) to be the stack that classifies 1morphisms of group stacks between P and Bµ n that induce splittings of the short exact sequence (4.5).(b) We define Split c (P, α) to be the stack that classifies 1-morphisms of group stacks between P and BG m that induce splittings of the short exact sequence (4.6) 0 − → BG m − → P × Bµn BG m − → P − → 0. This short exact sequence comes from the G m -gerbe induced from α. Remark 4.25.(a) The stack Split c µn (P, α) is a torsor for P ∨ [n] := Hom(P, Bµ n ) and we have an isomorphism Split c (P, α) ∼ = Split c µn (P, α) × P ∨ [n] P ∨ .(b) Consider the dual short exact sequence of (4.6): 0 − → P ∨ − → (P × Bµn BG m ) ∨ σ − → Z − → 0.
c µn (P Z , α Z ) is a P ∨ Z [n]-torsor, and by the short exact sequence (4.7),P ∨ Z [n] is étale locally isomorphic to P ∨ [n] × Bµ n .It follows that Split c µn (P Z , α Z ) forms a µ n -gerbe over its µ n -rigidification denoted by Split c µn (P Z , α Z ), and Split c µn (P Z , α Z ) is a P ∨ [n]-torsor.Similarly Split c (P Z , α Z ) forms a G m -gerbe over Split c (P Z , α Z ), and Split c (P Z , α Z ) is a P ∨ -torsor.
Proof of Proposition 5.1.Step 1.We denote by M ′ D1 ⊂ M D1 the maximal Deligne-Mumford substack.The first step is to construct a trivializing section of Ω top M ′ D 1 from the trivializing section ω D1 of Ω top (T * N) ′ .Recall that T * N is a SL n (D 1 )-torsor over M D1 .Let T T * N/MD 1 be the relative tangent sheaf.The SL n (D 1 )-action on T * N induces a map ρ : sl n (D 1 ) −→ T T * N/MD 1 .We fix a non-zero vector τ ∈ ∧ top sl n (D 1 ) in the top exterior product of the Lie algebra sl n (D 1 ).It follows from Corollary 5.6 that the polyvector field ρ(τ ) ∈ ∧ top T T * N/MD 1 is SL n (D 1 )-invariant.It is also Γ-invariant since the Γ-action on T * N commutes with the SL n (D 1 )-action.Now we consider the contraction ι ρ(τ ) ω D1 of the top-degree form ω D1 with ρ(τ ).By definition, ι ρ(τ ) ω D1 is characterized by the property that < ι ρ(τ ) ω D1 , Y >=< ω D1 , ρ(τ ) ∧ Y > for any polyvector field Y of the correct degree.Recall that T * N is a SL n (D 1 )-torsor over M D1 , see Remark 5.4 (a).
p N : T * N −→ N be the projection map.The cotangent stack T * N admits a tautological 1-form θ N θ N = dp N • δ : T * N −→ T * (T * N) defined by the composition of the differential of p N dp N : T * N × N T * N −→ T * (T * N) with the diagonal map δ : T * N −→ T * N × N T * N. We have the following Lemma 5.8.Over the maximal open Deligne-Mumford locus (T * N) ′ ⊂ T * N, the 2-form dθ N is equal to the canonical symplectic form defined using Serre duality of the complex in (5.1).