group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
∞-Lie theory (higher geometry)
The equivariant version of de Rham cohomology.
Various different dg-algebras are used to model equivariant de Rham cohomology, known as
reviews include (Atiyah-Bott 84, Kalkman 93, section 2.1)
reviews include (Mathai-Quillen 86, Kalkman 93, section 2.2)
Let $X$ be a smooth manifold, $G$ a Lie group, and $\rho : X \times G \to X$ a smooth action of $G$ on $X$.
Write
for the $G$-invariant differential forms on $G$ with coefficients in the linear dual of the Lie algebra $\mathfrak{h}$ of $H$, shifted up in degree. So for $\{F^a\} \subset \{t^a\}$ a dual basis of $\mathfrak{h}$ inside a dual basis for $\mathfrak{g}$, a general element of this space in degree $2 p + q$ is of the form
where $\omega_{\cdots}$ are differential q-forms, such that for each $t_a \in \mathfrak{g}$ the Lie derivative of these forms satisfies
where $\{C_{a b}{}^b\}$ are the structure constants of $\mathfrak{g}$, hence such that $[t_a, t_b] = C_{a b}{}^c t_c$.
Equip this graded vector space $\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G$ with a differential $d_{CE(\mathfrak{g}//\mathfrak{h})}$ by
(e.g. Kalkman 93 (1.15)).
The resulting dg-algebra $(\Omega^\bullet(G,\mathfrak{h}^\ast[1])^G, d_{CE}(\mathfrak{g}//\mathfrak{h}))$ is called the Cartan model of $H$-equivariant de Rham cohomology of $G$.
Observe that in the special case that $X = G$ equipped with its canonical right action of $H \coloneqq G$ on itself, this construction reduces to that of the Weil algebra $W(\mathfrak{g})$, whose cohomology is trivial (is concentrated in degree 0, where it is $\mathbb{R}$).
(…)
The Weil model is discussed for instance in
A good account of the Cartan model is in
A review of the Weil model and the Cartan model and the introduction of the “BRST model” (Kalkman model) is in
Discussion in the broader context of equivariant differential cohomology is in
Discussion in the context of the gauged WZW model includes
Edward Witten, appendix of On holomorphic factorization of WZW and coset models, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. (EUCLID)
José Figueroa-O'Farrill, S Stanciu, Gauged Wess-Zumino terms and Equivariant Cohomology, Phys.Lett. B341 (1994) 153-159 (arXiv:hep-th/9407196)
José de Azcárraga, J. C. Perez Bueno, On the general structure of gauged Wess-Zumino-Witten terms (arXiv:hep-th/9802192)
Last revised on October 24, 2018 at 12:30:31. See the history of this page for a list of all contributions to it.