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Does anyone have a reference to learn more about the Cartan $3$-form on a group manifold $G$? I have read that the WZW Lagrangian 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 actions. For eg., why is it the generator of $H^3(G,\mathbb{R})$ when $G$ is a connected, simply connected, compact Lie group?

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    This question has been crossposted to [MO](http://mathoverflow.net/questions/62998/cartan-3-form-on-a-lie-group-g). Further pointers to the literature and clarifying remarks can be found in the comments there.2011-04-26

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A quite self contained treatment of the de Rham cohomology of Lie groups is given in the first chapter of Algebraic models in geometry by Yves Felix, John Oprea, Daniel Tarne.(Here is an open access copy of the first chapter)

In paticular, the Cartan 3-form as a generator of $H^3(G)$ is treated in theorem 1.47 (pages 23-24).

The basic idea for the calculation of the cohomology groups due to Cartan, Weil, Chevalley is:

  1. To establish an isomorphism between the de Rham cohomology of a compact connected Lie group and the space of left and right invariant forms by averaging over the group manifold.

  2. To establish an isomoprphism of the space of left and right invariant forms to the Lie algebra cohomology by restriction to the unit element. (This part is developed in exercise 1.7 page 53).

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    The link to the first chapter is broken.2014-05-12