Timeline for Rank 2 flat bundles on an elliptic curve, via extensions
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 22, 2011 at 19:27 | comment | added | Peter Dalakov | Of course, in my 3-rd sentence "respectively" should be replaced by "in reverse order". | |
| Aug 22, 2011 at 19:01 | comment | added | Peter Dalakov | Yes! This is the moduli space of $SL(2,\mathbb{C})$ local systems, which is homeomorphic to the moduli space of (top.trivial) Higgs bundles. For an elliptic curve and connected reductive $G$ these are described in a beautiful paper by Michael Thaddeus. The normalisations of the two spaces are respectively $T^\vee E\otimes \Lambda /W$ and $J\otimes \Lambda /W$, where $\Lambda$ is the coweight lattice, $W$ is the Weyl group, and $J$ is the unique group extension of $E$ by $\mathbb{C}$. The normalisation of the rep.variety is $H\times H/W$, $H=\mathbb{C}^\times \otimes \Lambda$. | |
| Aug 22, 2011 at 18:06 | comment | added | Michael Thaddeus | Yes. Actually every object is strictly semistable. And indeed, this is what the Higgs moduli space does. | |
| Aug 22, 2011 at 17:40 | comment | added | Sam Lewallen | Thanks! Is it possible to put all this together into a coherent moduli space of $SL(2,C)$ connections? Naively this seems fraught, because of the semisimple guys where these $L\otimes I_2$'s pop up. Is this what the Higgs moduli space does, somehow? I really should read some of these papers... | |
| Aug 21, 2011 at 20:26 | history | answered | Peter Dalakov | CC BY-SA 3.0 |