Timeline for Cartan 3-form on a Lie group G
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| May 12, 2014 at 16:30 | answer | added | user50728 | timeline score: 2 | |
| Apr 26, 2011 at 12:48 | comment | added | Theo Buehler | This question was also asked on math.SE and in the meantime a further pointer to the literature has been added in an answer: math.stackexchange.com/questions/35137/… | |
| Apr 26, 2011 at 8:13 | comment | added | Konrad Waldorf | The general story (for non-simply connected Lie groups) is: the (exponentiated) WZ-term is the holonomy of a gerbe-connection around the worldsheet. The 3-form is the curvature of the gerbe-connection. You may look into papers of Gawedzki, e.g. "WZW branes and gerbes". | |
| Apr 26, 2011 at 6:01 | comment | added | Somnath Basu | Look at the one of the volumes of Connections, Curvature, and Cohomology by Grueb et al; perhaps vol. 2 titled "Lie Groups". | |
| Apr 26, 2011 at 5:15 | comment | added | Kevin Wray | Thanks for the info, do you have a reference by any chance? | |
| Apr 26, 2011 at 5:03 | comment | added | José Figueroa-O'Farrill | (And the calculation of $H^3(\mathfrak{g})$, of course!) | |
| Apr 26, 2011 at 5:02 | comment | added | José Figueroa-O'Farrill | If $G$ is compact and simple, then $H^3(G;\mathbb{R}) \cong \mathbb{R}$, with generator the 3-form you have written down. This follows from the isomorphism between $H^3(G;\mathbb{R})$ and $H^3(\mathfrak{g})$, with $\mathfrak{g}$ the Lie algebra of $G$. | |
| Apr 26, 2011 at 4:29 | comment | added | Kevin Wray | Also, I have tried to go through Witten's paper, but he really doesn't discuss the math behind the $3$-form, $\langle \phi_g[\phi_g\wedge\phi_g]\rangle$, but will go back and check again. | |
| Apr 26, 2011 at 4:28 | history | edited | Kevin Wray | CC BY-SA 3.0 |
changed the words "action/Lagrangian" to "term"
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| Apr 26, 2011 at 4:26 | comment | added | Kevin Wray | Yes, of course, sorry about that. I will edit it now. | |
| Apr 26, 2011 at 4:13 | comment | added | José Figueroa-O'Farrill | What you have written is only the Wess-Zumino term. The WZW lagrangian also has a standard sigma model term. Why not read Witten's original paper Non-abelian bosonization in two dimensions (CMP 1984)? | |
| Apr 26, 2011 at 3:58 | history | asked | Kevin Wray | CC BY-SA 3.0 |