Abstract
The moduli spaces of flat {text{SL}}_2- and {text{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 Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”.
Highlights
Let C be a compact Riemann surface of genus g with base point c ∈ C, and G be either SLr or PGLr
In [9], Hausel and Thaddeus showed that the de Rham moduli spaces MDd R(C, SLr) and MDd R(C, PGLr) are mirror partners in the sense of Strominger–Yau–Zaslow mirror
According to the general mirror symmetric framework, it is reasonable to expect a symmetry between their Hodge numbers
Summary
The intersection cohomology of a complex variety X with compact support, middle perversity and rational coefficients is denoted by IHc∗(X). Recall that IHc∗(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 by Γ. The intersection cohomology of X decomposes under the action of Γ into isotypic components:. Topological mirror symmetry for rank two character varieties. ∑ ) = j wj , where acts on the normal bundle of X in X with eigenvalues e2 iwj with wj ∈ (0, 1)
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