∞-Lie theory

# Contents

## Definition

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

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

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

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

$\mathrm{exp}\left({𝔤}^{0}\right)$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 $\mathrm{exp}\left({𝔤}^{0}\right)$ on $\mathrm{MC}\left(𝔤\right)$:

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

Explicitely,

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

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

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

## Examples

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

## 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

Parts of the above text is taken from

Revised on March 8, 2011 18:26:28 by Urs Schreiber (188.201.208.83)