The Chabauty-Kim algorithm is a method in Diophantine geometry for effectively computing the rational points of varieties, typically curves, over number fields. The algorithm is contingent on a dimensional inequality condition between moduli spaces of equivariant principal bundles described as non-abelian Galois cohomology functors. We'll show that these arithmetic moduli spaces exhibit several properties observed in the study of character varieties of surfaces and 3-folds with boundaries including an analogue of Goldman's symplectic structure and the theory of Lagrangian intersections associated with Heegaard splittings of 3-folds. This comports with the arithmetic-topological analogy of Mazur, Morishita, etc. This can all be done in a derived-geometric formalism which results in some features and invariants new to the arithmetic setting such as the existence of potentials whose derived critical loci constrain the rational points of varieties. This is joint work with Minhyong Kim.