How many "elementary" characterizations of twisted SU(2) representation varieties are known? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:14:21Z http://mathoverflow.net/feeds/question/122811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122811/how-many-elementary-characterizations-of-twisted-su2-representation-varieties How many "elementary" characterizations of twisted SU(2) representation varieties are known? Nate Bottman 2013-02-24T16:38:38Z 2013-03-08T21:58:15Z <p>If $\Sigma_g$ is a genus-$g$ surface, $g \geq 2$, then let $\mathcal{M}(\Sigma_g)$ be its twisted SU(2) representation variety, i.e. $$\mathcal{M}(\Sigma_g) := \{ (A_1, B_1, \ldots, A_g, B_g) \in SU(2)^{2g} \:|\: [A_1,B_1]\cdots[A_g,B_g] = -I \}/SO(3),$$ where $SO(3) = SU(2)/\{\pm 1\}$ acts on these tuples by conjugating each component. Then $\mathcal{M}(\Sigma_g)$ is a smooth, compact $(6g-6)$-dimensional symplectic manifold, and it's symplectomorphic to the moduli space of stable bundles over a genus-$g$ Riemann surface with rank $2$ and determinant equal to some fixed line bundle of odd degree (this moduli space is a Kaehler manifold).</p> <p>In his 1968 paper "Stable bundles of rank 2 and odd degree over a curve of genus 2", Peter Newstead showed that $\mathcal{M}(\Sigma_2)$ is isomorphic to the intersection of two generic quadric hypersurfaces in $\mathbb{P}^5$. In <a href="http://arxiv.org/abs/1006.1099" rel="nofollow">a recent paper</a>, Ivan Smith used this characterization to get a lot of information about the Fukaya category of $\mathcal{M}(\Sigma_2)$ in terms of the Fukaya category of $\Sigma_2$. (My impression is that people care a lot about the Lagrangian intersection theory of $\mathcal{M}(\Sigma_g)$ because if you understand that, you understand the instanton Floer homology of 3-manifolds.) My question is:</p> <blockquote> <p>are there any similarly elementary characterizations of $\mathcal{M}(\Sigma_g)$ for $g \geq 3$?</p> </blockquote> <p>I know that bits and pieces about $\mathcal{M}(\Sigma_g)$ are known, e.g. its symplectic volume and characteristic classes (maybe even its cohomology?); I would really like to know about more elementary things that are known to be isomorphic to it as symplectic/Kaehler manifolds.</p> http://mathoverflow.net/questions/122811/how-many-elementary-characterizations-of-twisted-su2-representation-varieties/124022#124022 Answer by Sam Lewallen for How many "elementary" characterizations of twisted SU(2) representation varieties are known? Sam Lewallen 2013-03-08T21:28:48Z 2013-03-08T21:58:15Z <p>I think Nate might know this by now, but in case anyone else is curious, there is a generalization of the intersection-of-quadrics-in-$\mathbb{P}^5$ picture to general genus, proved first in this paper, I believe</p> <blockquote> <p>Classification of Vector Bundles of Rank 2 on Hyperelliptic Curves, U.V. Desale and S. Ramanan, Inventiones</p> </blockquote> <p>The story is that $\mathcal{M}_g$ is homeomorphic (in fact, symplectomorphic), to the space of $g-2$ dimensional linear subspaces in the intersection of two quadrics in $\mathbb{P}^{2g+1}$.</p>