nLab Deligne groupoid



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For 𝔤= i𝔤 i\mathfrak{g}=\bigoplus_{i}\mathfrak{g}^i a differential graded Lie algebra, let MC(𝔤)MC(\mathfrak{g}) be the set of Maurer-Cartan elements, i.e.,

MC(𝔤)={x𝔤 1|dx+12[x,x]=0} MC(\mathfrak{g})=\{x\in \mathfrak{g}^1 \;|\; dx+\frac{1}{2}[x,x]=0\}

One thinks of elements in this set as flat 𝔤\mathfrak{g}-connections: indeed

xMC(𝔤)(d+[x,]) 2=0. x\in MC(\mathfrak{g}) \Leftrightarrow (d+[x,-])^2=0.

The subspace 𝔤 0\mathfrak{g}^0 of 𝔤\mathfrak{g} is a Lie algebra; the group

exp(𝔤 0) exp(\mathfrak{g}^0)

acts on as a group of gauge transformations on the set of 𝔤\mathfrak{g}-connections (by conjugation), and this action preserves the subset of flat connections. Hence we have a gauge action of exp(𝔤 0)exp(\mathfrak{g}^0) on MC(𝔤)MC(\mathfrak{g}):

e a(d+[x,])e a=d+[e a*x,]. e^{a}(d+[x,-])e^{-a}=d+[e^a*x,-].


e a*x=x+ n=0 ([a,]) n(n+1)!([a,x]da) e^a*x=x+\sum_{n=0}^\infty \frac{([a,-])^n}{(n+1)!}([a,x]-da)

The Deligne groupoid Del(𝔤)Del(\mathfrak{g}) of the dgla 𝔤\mathfrak{g} is the action groupoid

Del(𝔤)=MC(𝔤)//exp(𝔤 0) Del(\mathfrak{g})=MC(\mathfrak{g})// exp(\mathfrak{g}^0)


For 𝔤\mathfrak{g} a Lie algebra this is the delooping groupoid Bexp(𝔤)=*//exp(𝔤)\mathbf{B} exp(\mathfrak{g})=*//exp(\mathfrak{g}).


Some of the ideas of Deligne on deformation theory were transmitted via

  • W. M. Goldman, J. J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43–96, MR90b:32041, numdam

but the later study of the Deligne 2-groupoid is from a letter of Deligne to Breen from 1994 (see Ezra Getzler‘s webpage; the letter page is not to be linked). See related

  • E. Getzler, A Darboux theorem for Hamiltonian operators in the formal calculus of variations, Duke Math. J. 111, n. 3 (2002), 535-560, MR2003e:32026, doi

Other references

  • V. Hinich, Descent of Deligne groupoids, Int. Math. Research Notices, 1997, n. 5, 223-239, alg-geom/9606010; DG coalgebras as formal stacks, J. Pure Appl. Algebra 162 (2001), 209-250, pdf

  • Amnon Yekutieli, MC elements in complete DG Lie algebras, arXiv/1103.1035

A careful analysis extends the assignment of the Deligne groupoid to a Maurer-Cartan pseudofunctor, see part 2 of

Parts of the above text is taken from

  • Domenico Fiorenza, The periods map of a complex projective manifold. An infty infty -category perspective?

Last revised on July 1, 2020 at 15:17:47. See the history of this page for a list of all contributions to it.