nLab sheaf of L-∞ algebras



\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids



Related topics


\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



A sheaf of L 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 XX may be thought of as an XX-parameterized formal moduli problem (eg. Hinich 03).

Local L L_\infty-algebras


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

  1. A graded vector bundle LXL\rightarrow X,

  2. A square 00 differential operator d:Γ(X,L)Γ(X,L)\mathrm{d}:\Gamma(X,L)\rightarrow\Gamma(X,L) of cohomological degree 11,

  3. A collection {l n} n2\{ l_n \}_{n \geq 2} of poly-differential operators

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

    which are alternating, of cohomological degree 2n2-n and such that they endow Γ(X,L)\Gamma(X,L) with an L L_\infty structure.

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


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

CE((U))= n0Hom(((U)[1]) ^n,) S n \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, Hom\mathrm{Hom} denotes the dg-vector space of continuous linear maps and the subscript S nS_n denotes the coinvariants respect to the action of the symmetric group S nS_n.


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

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

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

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


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

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

where 𝒥ℯ𝓉(L)\mathscr{Jet}(L) is the local L L_\infty-algebra, in the category of D XD_X-modules, of sections of the jet bundle of LL and mathscrDens Xmathscr{Dens}_X is the right D XD_X-module of densities on XX.

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


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

Quasi-coherent sheaves of dg-Lie algebras appear in

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

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

Last revised on January 23, 2023 at 10:14:01. See the history of this page for a list of all contributions to it.