Obtaining the One-Holed Torus from Pants: Duality in an SL(3,C)-Character Variety

TL;DR

This paper explores the duality of symplectic structures in SL(3,C)-character varieties associated with surfaces, specifically relating the three-holed sphere and one-holed torus, revealing a Poisson map induced by topological gluing.

Contribution

It demonstrates the symplectic duality between two Poisson structures on character varieties and describes how topological gluing induces a Poisson map.

Findings

Symplectic leaves are dual at a generic point.

Topological gluing induces a rank-preserving Poisson map.

The work relates structures of different surface types in SL(3,C)-character varieties.

Abstract

The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. The coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F imposed by the surface. In previous work, we have worked out the bracket on all generators when the surface is a three-holed sphere and when the surface is a one-holed torus. In this paper, we show how the symplectic leaves corresponding to these two different Poisson structures on X relate to each other. In particular, they are symplectically dual at a generic point. Moreover, the topological gluing map which turns the three-holed sphere into the one-holed torus induces a rank preserving Poisson map on C[X].