A local analytic splitting of the holonomy map on flat connections
type: Item
instance of: scholarly article
title: A local analytic splitting of the holonomy map on flat connections
zbMATH Open document ID: 0797.53027
DOI: 10.1007/BF01459778
author: V. Pereyra
published in: Mathematische Annalen
publication date: 1994-07-11T00:00:00Z
full work available at URL: https://eudml.org/doc/165204
review text: 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.
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58D27
zbMATH DE Number: 590008
zbMATH Keywords: fundamental group
zbMATH Keywords: Chern-Simons invariants
zbMATH Keywords: Atiyah-Patodi-Singer \(\rho_ \alpha\) invariants
zbMATH Keywords: spectral flow
zbMATH Keywords: holonomy map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590008