On partial abelianization of framed local systems

Abstract Gaiotto, Moore and Neitzke introduced spectral networks to understand the framed G -local systems over punctured surfaces for G a split Lie group via a procedure called abelianization. We generalize this construction to groups G of the form $${{\,\textrm{GL}\,}}_2(A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>GL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where A is a unital associative ring, and to some of its subgroups. This relies on a precise analysis of the two-fold ramified coverings associated with spectral networks and triangulations and on a matrix reinterpretation of their path lifting rules; along the way we provide another proof of the Laurent phenomenon brought to light by Berenstein and Retakh. The partial abelianization enables us to gives parametrizations of the moduli spaces of decorated G -local systems and of framed G -local systems over punctured surfaces. For $$(A, \sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> a Hermitian involutive $${\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> -algebra the group $$G={{\,\textrm{Sp}\,}}_2(A, \sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>Sp</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is a classical Hermitian Lie group of tube type, and we are able to identify and parametrize the moduli space of maximal framed G -local systems.