Does anyone have a reference to learn more about the Cartan $3$-form on a group manifold $G$? I have read that the WZW term is nothing more than the integral of the pullback of the Cartan $3$-form via $g:W\rightarrow G$

$WZW = -\frac{1}{6}\int_W \langle \phi_g\wedge[\phi_g\wedge\phi_g]\rangle$,

where $\phi_g=g^\ast(\phi)$ is the pullback of the Maurer-Cartan form, and would like to learn more about the math behind WZW terms. For eg., why is it the generator of $H^3(G,\mathbb{R})$ when $G$ is a connected, simply connected, compact Lie group?

  • $\begingroup$ 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)? $\endgroup$ Apr 26, 2011 at 4:13
  • $\begingroup$ Yes, of course, sorry about that. I will edit it now. $\endgroup$
    – Kevin Wray
    Apr 26, 2011 at 4:26
  • $\begingroup$ 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. $\endgroup$
    – Kevin Wray
    Apr 26, 2011 at 4:29
  • $\begingroup$ 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 $G$. $\endgroup$ Apr 26, 2011 at 5:02
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    $\begingroup$ Thanks for the info, do you have a reference by any chance? $\endgroup$
    – Kevin Wray
    Apr 26, 2011 at 5:15

1 Answer 1


You should check out Theodore Frankel's Geometry of Physics. There is a section on the gauge group . The cartan three form is the three form from the Chern Simons term at a flat connection. A flat connection is when the potential A is equal to the $g^{-1}dg$. In the Chern Simons 3 form you have A^A^A. At pure gauge this becomes $Tr(g^{-1}dg\wedge g^{-1}dg\wedge g^{-1}dg)$. This term is proportional to the cartan 3 form. I hope this helped.


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