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Partial abelianization of GL_n-local systems and non-commutative A-coordinates

  • Kineider, Clarence
Publication Date
Dec 20, 2023
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In this thesis, we aim to study moduli spaces of G-local systems over a ciliated surface S for various Lie groups G. We generalize a construction of Gaiotto-Moore-Neitzke called abelianization, allowing this procedure to be conducted ''partially''. The result of this generalized procedure allow us to describe the topology of open dense subspaces inside the moduli space of GL_n-local systems, and together with Eugen Rogozinnikov we extended further the abelianization procedure to symplectic local systems, allowing us to describe the topology of open dense subspaces inside the moduli space of symplectic local systems. In particular, we describe the topology of the set of maximal representations of a punctured surface group into a symplectic group Sp(A,s) over a symmetric algebra (A,s). Given the strong relations between the original construction by Gaiotto-Moore-Neitzke and Fock-Goncharov cluster coordinates, another benefit of the generalized abelianization construction is that it help us define and study a non-commutative generalization of Fock-Goncharov A-coordinates. When G = GL_{2n}(R), we show that these non-commutative A-coordinates are a representation of the non-commutative algebra A_S introduced by Berenstein-Retakh. These non-commutative A-coordinates also restrict to non-commutative coordinates on the space of symplectic representations with lagrangian framing.

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