Timeline for Principal Bundle with given Curvature
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2011 at 13:24 | vote | accept | Tobias Diez | ||
| Sep 7, 2011 at 15:11 | comment | added | Deane Yang | David, sorry about that. I withdraw all of my comments. I didn't notice that you were allowing for different bundles. I don't know what a "symmetric 2-group" is, but the space of complex line bundles with $U(1)$-connections does indeed have an abelian group structure by taking the tensor product of two bundles and the naturally induced connection defined using the product rule. The group inverse is given by the dual bundle with the naturally induced connection, and the identity element is the trivial bundle with the trivial connection. So you can add two elements this way. | |
| Sep 7, 2011 at 5:22 | comment | added | David Roberts♦ | Hi Deane - that is only true if you consider connections on the same bundle. If you permit different bundles, the difference of two bundles-with-connection with the same curvature is a third bundle, equipped with a flat connection. This is what the exact sequence I give says. The statement about the difference of two connections on the same bundle that give rise to the same curvature is more connected with the other exact sequence I mention. | |
| Sep 7, 2011 at 4:40 | comment | added | Deane Yang | I think you were closer to the mark when you said that the difference between two connections with the same curvature is a 1-form. The difference between two connections with the same curvature is not a flat connection. The space of connections is an affine space (no distinguished origin), and the difference between any two $U(1)$-connections is a real-valued $1$-form. Two connections have the same curvature if and only if their difference is a closed $1$-form. Two $U(1)$- connections are gauge equivalent if and only if their difference is an exact $1$-form. | |
| Sep 7, 2011 at 4:30 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Sep 7, 2011 at 3:16 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
added 108 characters in body
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| Sep 7, 2011 at 3:06 | comment | added | David Roberts♦ | Good point. I'll edit. | |
| Sep 7, 2011 at 2:09 | comment | added | Deane Yang | How can you add two connections? | |
| Sep 7, 2011 at 0:46 | history | answered | David Roberts♦ | CC BY-SA 3.0 |