nLab
super L-infinity algebra

Context

Supergeometry

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

The category of super L L_\infty-algebras is

SL Alg:=(ScdgAlg sf +) 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\mathfrak{g} a super L L_\infty-algebra we write CE(𝔤)CE(\mathfrak{g}) for the corresponding dg-algebra: its Chevalley-Eilenberg algebra.

Properties

Examples

In the context of supergravity/string theory the

and its super-L L_\infty-extensions to the

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

Moreover, the BRST complex of the superstring might form a super L L_\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. (…)

References

Revised on August 22, 2013 03:47:13 by Urs Schreiber (151.201.35.138)