nLab Deligne groupoid

Definition

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

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

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

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

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

$$exp(\mathfrak{g}^0)$$

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

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

Explicitly,

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

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

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

Examples

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

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 $infty$-category perspective?