Timeline for stable bundle on Calabi-Yau 3-fold
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 4, 2016 at 7:55 | comment | added | Ben McKay | It is not that fancy; you don't need to think about $G_2$ manifolds or Calabi-Yau manifolds. If $G$ is a Lie group and $E \to X$ is a principal $G$-bundle over any Hausdorff topological space $X$, not necessarily a manifold, then the associated line bundles are $L=E \times^{\rho} \mathbb{C}$ where $\rho \colon G \to \mathbb{C}^{\times}$ is a representation of $G$. But if $G$ is semisimple, then $G$ has only on 1-dimensional representation: the trivial one. So $\rho(g)=1$ and therefore $L$ is trivial and admits a flat connection, and so $c_1(L)=0$. | |
| Jul 4, 2016 at 6:36 | comment | added | user94640 | Through pull back the bundle $E$ over CY-3 fold to $\pi^{\ast}E$ over the $G_{2}$ manifold $CY^{3}\times S^{1}$. I had proved this assumed, but I can't belive my result. | |
| Jul 3, 2016 at 20:11 | comment | added | Ben McKay | Do I contradict myself? Very well, then I contradict myself, I am large, I contain multitudes. Walt Whitman | |
| Jul 3, 2016 at 19:26 | comment | added | Ben McKay | But if $G$ is semisimple, then yes because semisimple Lie groups have no 1-dimensional nontrivial reps. | |
| Jul 3, 2016 at 19:26 | history | answered | Ben McKay | CC BY-SA 3.0 |