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Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory.
For G = GL_2, PGL_2 and SL_2 we prove that the perverse filtration associated to the Hitchin map on the cohomology of the moduli space of twisted G-Higgs bundles on a Riemann surface C agrees with the weight filtration on the cohomology of the twiste
We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. F
We provide a construction of the moduli spaces of framed Hitchin pairs and their master spaces. These objects have come to interest as algebra
We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott Kahler form that realizes the Chern class of the deter
Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $mathcal M$ be the moduli space of Hitchin pairs $(E ,phiin H^0(End(E)otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d