Which bundles does the character vareity parameterize? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T04:24:12Z http://mathoverflow.net/feeds/question/76573 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76573/which-bundles-does-the-character-vareity-parameterize Which bundles does the character vareity parameterize? John Mangual 2011-09-27T22:49:13Z 2011-09-28T06:26:22Z <p>For any Riemann surface with punctures $C$, and lie group $G$, the character variety is the space of maps $\mathrm{Hom}(\pi_1(C), G)$. </p> <p>I know that if $G= S_n$ (not a lie group), then $\mathrm{Hom}(\pi_1(C), S_n)//S_n$ parametrizes <a href="http://www.math.ucdavis.edu/~osserman/rfg/290W/branched-covers.pdf" rel="nofollow">branched covers</a> of $M$. Here $S_n$ acts by conjugation (permuting the various copies of $C$.)</p> <p>If $G = \mathrm{GL}(n,\mathbb{C})$, is $\mathrm{Hom}(\pi_1(C), G)$ parameterizing vector bundles over $C$? What is the equivalence relation here?</p> http://mathoverflow.net/questions/76573/which-bundles-does-the-character-vareity-parameterize/76607#76607 Answer by Sam Gunningham for Which bundles does the character vareity parameterize? Sam Gunningham 2011-09-28T06:26:22Z 2011-09-28T06:26:22Z <p>In general, the set $\mathrm{Hom}(\pi_1(C),G)/G$ (where $G$ acts by conjugation) naturally parameterizes $G$-local systems. </p> <p>For example, if $G=GL_n(\mathbb C)$, these are just ordinary local systems of vector spaces: a map from $\pi_1(C)$ to $GL_n(\mathbb C)$ describes the monodromy around loops in $C$, and conjugate maps correspond to isomorphic local systems. If $G=U(n)$, the character variety naturally parametrizes unitary local systems (i.e. the fibres have an inner product such that the monodromy is unitary). </p> <p>As Mike comments, there are various theorems generally known as non-abelian Hodge theory which relate these topological objects to holomorphic objects. For example, on a closed curve, the Narashimhan-Seshadhri theorem gives a bijection between isomorphism classes of unitary local systems and degree 0 holomorphic vector bundles (with some stability condition).</p> <p>Similarly, there is a bijection between (iso classes of) all local systems and degree 0 Higgs bundles (with stability conditions).</p> <p>These can be thought of as giving diffeomorphisms between the character variety and the moduli spaces which naturally parameterize these objects (note that for $G=GL_n(\mathbb C)$, the character variety has it's own complex structure, which does not pull back to the natural complex structure on the moduli of Higgs bundles under these diffeomorphisms).</p> <p>There are similar results when the curve $C$ is not compact (i.e. has punctures) you have to be a bit more careful, and you may want to include extra data like a filtration of the bundle at the punctures, and constrain the monodromy around the punctures to lie in a particular conjugacy class.</p> <p>I think this is discussed in the appendix to Wells' <em>Differential Analysis on Complex Manifolds</em> (written by Oscar Garcia-Prada), as well as in the papers of Narashimhan-Seshadhri, Hitchin, Donaldson, Corlette, Simpson etc... </p>