superalgebra

and

supergeometry

∞-Lie theory

Contents

Idea

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

The category of super $L_\infty$-algebras is

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

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

Examples

In the context of supergravity/string theory the

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

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 $L_\infty$-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)