nLab
super L-infinity algebra

Idea

A super $L_{\mathrm{\infty }}$-algebra is an L-∞ algebra in the context of superalgebra: the higher category theoretical/homotopy theoretical analog of a super Lie algebra.

Definition

Definition

A super $L_{\mathrm{\infty }}$-algebra is an L-∞ algebra internal to super vector spaces.

The category of super $L_{\mathrm{\infty }}$-algebras is

$$S{L}_{\mathrm{\infty }}\mathrm{Alg}:=({\mathrm{ScdgAlg}}_{\mathrm{sf}}^{+}{)}^{\mathrm{op}}$$S L_\infty Alg := (ScdgAlg^+_{sf})^{op}

the opposite category of semi-free dg-algebras in super vector spaces: commutative monoids in the category of cochain complexes of super vector spaces whose underlying commutative graded algebra is free on generators in positive degree.

For $𝔤$ a super $L_{\mathrm{\infty }}$-algebra we write $\mathrm{CE}(𝔤)$ for the corresponding dg-algebra: its Chevalley-Eilenberg algebra.

Properties

• Every super $L_{\mathrm{\infty }}$-Lie algebra has a Lie integration to a super ∞-groupoid and a smooth super ∞-groupoid. See there for more.

Examples

In the context of supergravity/string theory the

• super Poincare Lie algebra

and its super-$L_{\mathrm{\infty }}$-extensions to the

• supergravity Lie 3-algebra (m2brane)

• supergravity Lie 6-algebra

• type II supergravity Lie 2-algebra

play a central role. Their exceptional infinity-Lie algebra cohomology governs the consistent Green-Schwarz action functionals for super-$p$-branes. (See the discusson of the brane scan) there.

Moreover, the BRST complex of the superstring might form a super $L_{\mathrm{\infty }}$-algebra whose brackets give the n-point function of the string, in analogy to what happens for the bosonic string in Zwiebach’s string field theory. (…)