nLab sheaf of L-∞ algebras

Idea

A sheaf of $L_\infty$-algebras is a suitable sheaf/stack/∞-stack of L-∞ algebras/dg-Lie algebras.

These structures appear in the deformation theory of the given base space/site: where a single L-∞ algebra is equivalently a formal moduli problem (hence “over the point), so a sheaf of these over some base $X$ may be thought of as an $X$-parameterized formal moduli problem (eg. Hinich 03).

Local $L_\infty$-algebras

Definition

A local $L_\infty$-algebra on a smooth manifold $X$ is defined by the following data:

1. A graded vector bundle $L\rightarrow X$,

2. A square $0$ differential operator $\mathrm{d}:\Gamma(X,L)\rightarrow\Gamma(X,L)$ of cohomological degree $1$,

3. A collection $\{ l_n \}_{n \geq 2}$ of poly-differential operators

   $$l_n \,:\; \Gamma(X,L)^{\otimes n} \rightarrow \Gamma(X,L),$$

   which are alternating, of cohomological degree $2-n$ and such that they endow $\Gamma(X,L)$ with an $L_\infty$ structure.

One usually denotes by $\mathcal{L}$ the sheaf of sections of the bundle $L$.

Definition

The Chevalley-Eilenberg cochain complex $\mathrm{CE}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by

$$\mathrm{CE}\left(\mathcal{L}(U)\right) \;=\; \prod_{n\geq 0} \mathrm{Hom}\left( (\mathcal{L}(U)[1])^{\hat{\otimes} n}, \,\mathbb{R} \right)_{S_n}$$

equipped with the usual Chevalley-Eilenberg differential; where $\hat{\otimes}$ denotes the completed projective tensor product, $\mathrm{Hom}$ denotes the dg-vector space of continuous linear maps and the subscript $S_n$ denotes the coinvariants respect to the action of the symmetric group $S_n$.

Definition

The reduced Chevalley-Eilenberg cochain complex $\mathrm{CE}_{\mathrm{red}}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by the kernel

$$\mathrm{CE}_{\mathrm{red}}(\mathcal{L}) \subset \mathrm{CE}(\mathcal{L})$$

of the natural augmentation map $\mathrm{CE}\left(\mathcal{L}(U)\right) \rightarrow \mathbb{R}$.

Both $\mathrm{CE}(\mathcal{L})$ and $\mathrm{CE}_{\mathrm{red}}(\mathcal{L})$ are differentiable pro-cochain complexes.

Definition

The local Chevalley-Eilenberg cochain complex $\mathrm{CE}_{\mathrm{loc}}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by

$$\mathrm{CE}_{\mathrm{loc}}(\mathcal{L}) \;\coloneqq\; \mathscr{Dens}_X \otimes_{D_X} \mathrm{CE}_{\mathrm{red}}(\mathscr{Jet}(L))$$

where $\mathscr{Jet}(L)$ is the local $L_\infty$-algebra, in the category of $D_X$-modules, of sections of the jet bundle of $L$ and $mathscr{Dens}_X$ is the right $D_X$-module of densities on $X$.

The cochain complex $\mathrm{CE}_{\mathrm{loc}}(\mathcal{L})$ can be thought as the sheaf of Lagrangian densities on the graded vector bundle $L$.

Related concepts

• descent for L-∞ algebras

• factorization algebras

• formal moduli problems

• L-∞ algebroid

• homotopy theory of L-∞ algebras

• sheaf of spectra

References

Discussion in the context of deformation theory/parameterized formal moduli problems is in

• Vladimir Hinich, Deformations of sheaves of algebras (arXiv:math/0310116)

Quasi-coherent sheaves of dg-Lie algebras appear in

• Amnon Yekutieli, appendix A of Twisted Deformation Quantization of Algebraic Varieties (arXiv:0905.0488)

• I. Ciocan-Fontanine, Mikhail Kapranov, Derived Quot schemes (arXiv:math/9905174)

Discussion in physics, of sheaves of $L_\infty$-algebras of physical fields in prequantum field theory includes

• Anton Zeitlin, Conformal Field Theory and Algebraic Structure of Gauge Theory (arXiv:0812.1840)

Local $L_\infty$-algebras are defined and discussed by

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory