Year: 2024
Type: preprint
Abstract: Goldman defined a symplectic form on the smooth locus of the $G$-character variety of a closed, oriented surface $S$ for a Lie group $G$ satisfying very general hypotheses. He then studied the Hamiltonian flows associated to $G$-invariant functions $G \to \mathbb R$ obtained by evaluation on a simple closed curve and proved that they are generalized twist flows. In this article, we investigate the Hamiltonian flows on (subsets of the) $G$-character variety induced by evaluating a $G$-invariant multi-function $G^k \to \mathbb R$ on a tuple $ \underlineα \in π_1(S)^k$. We introduce the notion of a subsurface deformation along a supporting subsurface $S_0$ for $\underlineα$ and prove that the Hamiltonian flow of an induced invariant multi-function is of this type. We also give a formula for the Poisson bracket between two functions induced by invariant multi-functions and prove that they Poisson commute if their supporting subsurfaces are disjoint. We give many examples of functions on character varieties that arise in this way and discuss applications, for example, to the flow associated to the trace function for non-simple closed curves on $S$.
Source: arXiv (Cornell University)
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: RNMS: Geometric structures and representation varieties, G4812722889, TRR 191: Symplektische Strukturen in Geometrie, Algebra und Dynamik, RNMS: Geometric Structures and Representation Varieties, Positivity in Lie groups and representation varieties, RNMS: Geometric Structures and Representation Varieties
OpenAlex ID: W4403964564