A local analytic splitting of the holonomy map on flat connections
Let \(X\) be a compact manifold and let \(\text{Hom}(\pi_ 1 X,G)\) denote the real-algebraic variety of representations of the fundamental group of \(X\) into a compact lie group \(G\). There are a number of constructions of functions on \(\text{Hom}(\pi_ 1X,G)\) which begin with the observation that the moduli space of gauge-equivalence classes of flat connections on \(X\) is in 1-1 correspondence with the \(G\)-conjugacy classes in \(\text{Hom}(\pi_ 1X,G)\). Examples include Chern-Simons invariants of representations, Atiyah-Patodi-Singer \(\rho_ \alpha\) invariants, and the spectral flow of the family of Dirac operators \(\pm(d_{a_ t} * - * d_{a_ t})\) from the given connection to a fixed flat connection. These invariants are defined using a representative flat connection, and then showing that the resulting invariant depends only on the gauge equivalence class. In certain situations, it is desirable to choose representative flat connections canonically. This is done using a slice for the gauge group action on the space of connections. However, this leaves open the question of whether the algebraic or analytic structure of the representation variety can be transferred to the space of connections. In particular, the perturbation theory of analytic paths of operators is much simpler than merely continuous or smooth families of operators, and so for example one would like to know that analytic paths of representations determine analytic paths of connections. In this paper we construct a (local) analytic function from \(\text{Hom}(\pi_ 1 X,G)\) to the space of connections that splits the holonomy map.
Atiyah-Patodi-Singer \(\rho_ \alpha\) invariants
scientific article; zbMATH DE number 590008
scientific article