equivariant de Rham cohomology





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The equivariant version of de Rham cohomology.



Various different dg-algebras are used to model equivariant de Rham cohomology, known as

The Weil model

reviews include (Atiyah-Bott 84, Kalkman 93, section 2.1)

The Cartan model

reviews include (Mathai-Quillen 86, Kalkman 93, section 2.2)

Let XX be a smooth manifold, GG a Lie group, and ρ:X×GX\rho : X \times G \to X a smooth action of GG on XX.


(Ω (G,𝔥 *[1]) GΩ(G,𝔥 *[1]) (\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G \hookrightarrow \Omega(G, \mathfrak{h}^\ast[1])

for the GG-invariant differential forms on GG with coefficients in the linear dual of the Lie algebra 𝔥\mathfrak{h} of HH, shifted up in degree. So for {F a}{t a}\{F^a\} \subset \{t^a\} a dual basis of 𝔥\mathfrak{h} inside a dual basis for 𝔤\mathfrak{g}, a general element of this space in degree 2p+q2 p + q is of the form

ω=F a 1F a pω a 1,,a q, \omega = F^{a_1} \wedge \cdots F^{a_p} \wedge \omega_{a_1,\cdots ,a_q} \,,

where ω \omega_{\cdots} are differential q-forms, such that for each t a𝔤t_a \in \mathfrak{g} the Lie derivative of these forms satisfies

t aω a 1,a 2,a p=C aa 1 bω b,a 2,,a p+C aa 2 bω a 1,b,,a p+, \mathcal{L}_{t_a} \omega_{a_1, a_2 \cdots , a_p} = C_{a a_1}{}^b \omega_{b , a_2, \cdots , a_p} + C_{a a_2}^{}^b \omega_{a_1 , b, \cdots , a_p} + \cdots \,,

where {C ab b}\{C_{a b}{}^b\} are the structure constants of 𝔤\mathfrak{g}, hence such that [t a,t b]=C ab ct c[t_a, t_b] = C_{a b}{}^c t_c.

Equip this graded vector space Ω (G,𝔥 *[1]) G\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G with a differential d CE(𝔤//𝔥)d_{CE(\mathfrak{g}//\mathfrak{h})} by

d CE(𝔤//𝔥):ωd dRωF aι t aω d_{CE(\mathfrak{g}//\mathfrak{h})} \colon \omega \mapsto d_{dR}\omega - F^a \iota_{t_a} \omega

(e.g. Kalkman 93 (1.15)).

The resulting dg-algebra (Ω (G,𝔥 *[1]) G,d CE(𝔤//𝔥))(\Omega^\bullet(G,\mathfrak{h}^\ast[1])^G, d_{CE}(\mathfrak{g}//\mathfrak{h})) is called the Cartan model of HH-equivariant de Rham cohomology of GG.

Observe that in the special case that X=GX = G equipped with its canonical right action of HGH \coloneqq G on itself, this construction reduces to that of the Weil algebra W(𝔤)W(\mathfrak{g}), whose cohomology is trivial (is concentrated in degree 0, where it is \mathbb{R}).

The Kalkman model

(Kalkman 93, section 3)



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

  • Jaap Kalkman, BRST model applied to symplectic geometry, Ph.D. Thesis, Utrecht, 1993 (arXiv:hep-th/9308132 (original ArXiv pdf broken)), published versions at (projectEuclid)

Discussion in the broader context of equivariant differential cohomology is in

  • Andreas Kübel, Andreas Thom, Equivariant Differential Cohomology, Transactions of the American Mathematical Society (2018) (arXiv:1510.06392)

Discussion in the context of the gauged WZW model includes

Last revised on October 24, 2018 at 12:30:31. See the history of this page for a list of all contributions to it.