nLab
Deligne groupoid

Definition

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

$$\mathrm{MC}(𝔤)=\{x\in {𝔤}^1\mathrm{such}\mathrm{that}\mathrm{dx}+\frac{1}{2}[x,x]=0\}$$MC(\mathfrak{g})=\{x\in \mathfrak{g}^1 such that dx+\frac{1}{2}[x,x]=0\}

One thinks of element in this set as flat $𝔤$-connections: indeed

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

The subspace ${𝔤}^0$ of $𝔤$ 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 $𝔤$-connections (by conjugation), and this action preserves the subset of flat connections. Hence we have a gauge action of $\exp ({𝔤}^0)$ on $\mathrm{MC}(𝔤)$:

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

Explicitely,

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

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

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

Examples

For $𝔤$ a Lie algebra this is the delooping groupoid $B\exp (𝔤)=*//\exp (𝔤)$.

Related concepts

• Lie integration, deformation theory

References

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

• Alexander I. Efimov, Valery A. Lunts, Dmitri O. Orlov, Deformation theory of objects in homotopy and derived categories

  • I: general theory arXiv:math/0702838;
  • II: pro-representability of the deformation functor arXiv:math/0702839;
  • III: abelian categories arXiv:math/0702840

Parts of the above text is taken from

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