Moduli spaces of local systems and higher Teichmüller theory |
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Authors: | Vladimir Fock Alexander Goncharov |
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Institution: | (1) ITEP, B. Cheremushkinskaya 25, 117259 Moscow, Russia;(2) Department of Mathematics, Brown University, Providence, RI 02912, USA |
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Abstract: | Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic;
the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When
S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they
carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant
under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have
positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their
positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical
semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces.
It is related to the motivic dilogarithm. |
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