Steven B. Bradlow, Oscar Garcia-Prada, Peter B. Gothen
Surface Group Representations and U(p,q)--Higgs bundles
Preprint series:
ESI preprints
- MSC:
- 14D20 Algebraic moduli problems, moduli of vector bundles, {For analytic moduli problems, See 32G13}
- 14H60 Vector bundles on curves, See also {14F05}
- 32G13 Analytic moduli problems, {For algebraic moduli problems, See 14D20, 14D22, 14H10, 14J10}, See also {14H15,
Abstract: Using the $L^2$ norm of the Higgs field as a Morse function, we study
the moduli spaces of $\U(p,q)$-Higgs bundles over a Riemann surface.
We require that the genus of the surface be at least two, but place
no constraints on $(p,q)$. A key step is the identification of the
function's local minima as moduli spaces of holomorphic triples. In
a companion paper \cite{bradlow-garcia-gothen:2002:triples} we prove
that these moduli spaces of triples are non-empty and irreducible.
Because of the relation between flat bundles and fundamental group
representations, we can interpret our conclusions as results about the
number of connected components in the moduli space of semisimple
$\PU(p,q)$-representations. The topological invariants of the flat
bundles are used to label subspaces. These invariants are bounded by
a Milnor--Wood type inequality. For each allowed value of the
invariants satisfying a certain coprimality condition, we prove that
the corresponding subspace is non-empty and connected. If the
coprimality condition does not hold, our results apply to the closure
of the moduli space of irreducible representations.
Keywords: representations of fundamental groups, unitary groups, moduli spaces, Higgs bundles, Morse theory