Topological mirror symmetry for rank two character varieties of surface groups

The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tam\'as Hausel in Remark 3.30 of"Global topology of the Hitchin system".

Let C be a compact Riemann surface of genus g with base point c ∈ C, and G be either SL r or PGL r . We study the following moduli spaces (cf [13]): • the de Rham moduli space of principle flat G-bundles on C \ c with holonomy e 2πid/r around c; • the Dolbeault moduli space of semistable G-Higgs bundles of degree d, i.e. semistable pairs (E, φ) consisting of a principal G-bundle E of degree d and a section φ ∈ H 0 (C, ad(E) ⊗ K C ), where K C is the canonical bundle; • the Betti moduli space parametrising G-representations of the fundamental group of C \ c with monodromy e 2πid/r around c.
These moduli spaces are denoted respectively M d DR (C, G), M d Dol (C, G) and M d B (C, G). For convenience, we simply write M (C, G) when we refer indifferently to M 0 Dol (C, G), M 0 DR (C, G) or M 0 B (C, G). In [8], Hausel and Thaddeus showed that the de Rham moduli spaces M d DR (C, SL r ) and M d DR (C, PGL r ) are mirror partners in the sense of Strominger-Yau-Zaslow mirror symmetry. According to the general mirror symmetric framework, it is reasonable to expect a symmetry between their Hodge numbers.
Hausel and Thaddeus conjectured the equality of the stringy E-polynomials for (d, r) = (e, r) = 1, and they prove it for r = 2, 3. The conjecture is now a theorem due to [6] or [11].
In [7,Remark 3.30] Hausel asked what cohomology theory we should compute on M d DR (C, SL r ), with (d, r) = 1, to accomplish the agreement (1). We propose to use intersection cohomology. As first piece of evidence, we show the topological mirror symmetry conjecture in rank two, and degree zero, i.e. when we turn off the B-fields 1 B andB.

Intersection stringy E-polynomial
The intersection cohomology of a complex variety X with compact support, middle perversity and rational coefficients is denoted by IH * c (X). Recall that IH * c (X) carries a canonical mixed Hodge structure, and so we can define the intersection E-polynomial of X as Suppose that X is endowed with the action of a finite abelian group Γ, and denote the group of characters of Γ byΓ. The intersection cohomology of X decomposes under the action of Γ into isotypic components: Then, if we pose Define also the intersection stringy E-polynomial by • X γ is the fixed-point set of γ ∈ Γ.
• F (γ) is the Fermionic shift, defined as F (γ) = j w j , where γ acts on the normal bundle of X γ in X with eigenvalues e 2πiwj with w j ∈ (0, 1).

Topological mirror symmetry
Let Γ := Pic 0 (C)[r] ≃ (Z/rZ) 2g be the group of r-torsion line bundles over the compact Riemann surface C of genus g, endowed with the canonical flat connection. The group Γ acts by tensorisation on M d Dol (C, SL r ) and M d DR (C, SL r ). Via the non-abelian Hodge correspondence, the action corresponds to the algebraic action of the characters Γ ≃ Hom( We identify w : Γ →Γ through Poincaré duality (also known as Weil pairing) where γ = w(κ). In particular, we obtain For any γ ∈ Γ \ {0}, we have an associated 2-torsion line bundle L γ . Consider the étale double cover π γ : C γ → C consisting of the square root of a non-zero section of L ⊗2 γ ≃ O C in the total space of L γ , and let ι be its deck transformation. For any L ∈ M (C γ , GL 1 ), the rank-two vector bundle L ⊕ ι * L is a ι-invariant object in M (C γ , GL 2 ), which descends to an object L ι ∈ M (C, GL 2 ). Hence, the pushforward morphism The determinant map det π γ, * can be identified with the norm map Therefore, the fixed-point set M (C, SL 2 ) γ admits the following geometric characterization: where the last term is the connected component of On the Dolbeault side, M Dol (C, SL 2 ) γ is isomorphic to the quotient by Z/2Z of the cotangent bundle of an abelian variety of dimension g − 1, as M Dol (C γ , GL 1 ) is isomorphic to T * Pic 0 (C γ ); see also the proof of Theorem 3.2. On the Betti side, we have The involution defining the Z/2Z-quotient is the inverse of the group law.
Since the Γ-module IH * c (M (C, SL 2 )) is a direct sums of copies of the trivial and regular representations by [ Note that the Fermionic shift F (γ) equals half of the codimension of M Dol (C, SL 2 ) γ in M Dol (C, SL 2 ), since γ is an involution. Hence, for γ = 0 we have indeed Finally, the same argument of [8, §6]

Remark 2.3 (Failure of topological mirror symmetry for ordinary cohomology).
In general the equality (3) fails for ordinary cohomology. For instance, for κ = 1, γ = w(κ) and q = uv we have where the first equality follows from [9,Theorem 2] [12, §3]. To the best of the author's knowledge, this is not available in higher rank, and so it is unclear if the arguments above extend in higher rank. However, remarkable progress in this direction have been made in [11] and [10].

Perverse topological mirror symmetry
The intersection cohomology of M Dol (C, SL r ) and M Dol (C, SL 2 ) γ are filtered by the weight filtration W of Deligne's mixed Hodge structure, and by the perverse filtration P associated to the Hitchin fibrations  where dim = 2(r 2 − 1)(g − 1).
We conjecture the exchange of the perverse Hodge numbers.  Proof. By Relative Hard Lefschetz, it is enough to show P IE(M Dol (C, SL 2 )) κ = P IE(M Dol (C, SL 2 ) γ /Γ; u, v)(uvq) F (γ) for any κ ∈Γ and γ = w(κ). As in Theorem 2.2, the case κ = 1, alias γ = 0, is trivial. Suppose then κ = 1 and γ = 0. The perverse filtration on IH d (M Dol (C, SL 2 )) κ is concentrated in degree d − 2g + 2 by [12,Theorem 5.5]. Moreover, the Hitchin map χ γ is a quotient by the inverse of the group law of the projection where Prym is the connected component of the identity of the kernel of the norm map Nm : Pic 0 (C γ ) → Pic 0 (C), given by Nm(L) = L ⊗ ι * L. Hence, the perverse filtration on IH d (M Dol (C, SL 2 ) γ /Γ), with γ = 0, is concentrated in degree d; cf proof of [4, Theorem 6.6]. Then one easily see that Conjecture 3.1 for r = 2 is equivalent to Theorem 2.2.