and
∞-Lie theory (higher geometry)
A super $L_\infty$-algebra is an L-∞ algebra in the context of superalgebra: the higher category theoretical/homotopy theoretical version of a super Lie algebra. For more background see at geometry of physics -- superalgebra.
In the supergravity literature the formal dual of super $L_\infty$-algebras of finite type came to be known as “FDA”s (see remark 2 below), a decade before plain L-∞ algebras were discussed in the mathematical literature. The key example in this context are extended super Minkowski spacetimes, which are super L-∞ algebras obtained by iterated higher central extension from the super Minkowski spacetime super Lie algebra. The super-L-∞ algebra cohomology of these (called “tau-cohomology” in the physics literature) turns out to classify super p-branes and serves as a tool for the construction of supergravity theories in the D'Auria-Fré formulation of supergravity. For more background on this see at geometry of physics -- fundamental super p-branes.
Abstractly, the definition is immediate:
A super $L_\infty$-algebra is an L-∞ algebra internal to the symmetric monoidal category of super vector spaces.
Explicitly this equivalently comes down to the following definition in components:
(super graded signature of a permutation)
Let $V$ be a $\mathbb{Z}$-graded super vector space, hence a $\mathbb{Z} \times (\mathbb{Z}/2)$-bigraded vector space.
For $n \in \mathbb{N}$ let
be an n-tuple of elements of $V$ of homogeneous degree $(n_i, s_i) \in \mathbb{Z} \times \mathbb{Z}/2$, i.e. such that $v_i \in V_{(n_i,s_i)}$.
For $\sigma$ a permutation of $n$ elements, write $(-1)^{\vert \sigma \vert}$ for the signature of the permutation, which is by definition equal to $(-1)^k$ if $\sigma$ is the composite of $k \in \mathbb{N}$ permutations that each exchange precisely one pair of neighboring elements.
We say that the super $\mathbf{v}$-graded signature of $\sigma$
is the product of the signature of the permutation $(-1)^{\vert \sigma \vert}$ with a factor of
for each interchange of neighbours $(\cdots v_i,v_j, \cdots )$ to $(\cdots v_j,v_i, \cdots )$ involved in the decomposition of the permuation as a sequence of swapping neighbour pairs (see at signs in supergeometry for discussion of this combination of super-grading and homological grading).
Now def. 1 is equivalent to the following def. 3. This is just the definiton for L-infinity algebras, with the pertinent sign $\chi$ now given by def. 2.
An super $L_\infty$-algebra is
a $\mathbb{Z} \times (\mathbb{Z}/2)$-graded vector space $\mathfrak{g}$;
for each $n \in \mathbb{N}$ a multilinear map, called the $n$-ary bracket, of the form
and of degree $n-2$
such that the following conditions hold:
(super graded skew symmetry) each $l_n$ is graded antisymmetric, in that for every permutation $\sigma$ of $n$ elements and for every n-tuple $(v_1, \cdots, v_n)$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ then
where $\chi(\sigma,v_1,\cdots, v_n)$ is the super $(v_1,\cdots,v_n)$-graded signature of the permuation $\sigma$, according to def. 2;
(strong homotopy Jacobi identity) for all $n \in \mathbb{N}$, and for all (n+1)-tuples $(v_1, \cdots, v_{n+1})$ of homogeneously graded elements $v_i \in \mathfrak{g}_{\vert v_i \vert}$ the followig equation holds
where the inner sum runs over all $(i,j)$-unshuffles $\sigma$ and where $\chi$ is the super graded signature sign from def. 2.
A strict homomorphism of super $L_\infty$-algebras
is a linear map that preserves the bidegree and all the brackets, in an evident sens.
A strong homotopy homomorphism (“sh map”) of super $L_\infty$-algebras is something weaker than that, best defined in formal duals, below in def. 4.
In order to define the correct homomorphisms between super $L_\infty$-algebras (“sh-maps”) as well as their super-L-∞ algebra cohomology, consider the following dualization of the above definition:
A super $L_\infty$ algebra $\mathfrak{g}$ is of finite type if the underlying $\mathbb{Z} \times (\mathbb{Z}/2)$-graded vector space is degreewise of finite dimension.
If $\mathfrak{g}$ is of finite type, then its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ is the dg-algebra whose underlying graded algebra is the super-Grassmann algebra
of the graded degreewise dual vector space $\mathfrak{g}^\ast$, equipped with the differential which on generators is the sum of the dual linear maps of the $n$-ary brackets:
and extended to all of $\wedge^\bullet \mathfrak{g}^\ast$ as a super-graded derivation of degree $(1,even)$.
Notice that here the signs in supergeometry are such that for $\alpha_i \in \mathfrak{g}^\ast_{(n_i,s_i)}$ elements of homogenous bidegree, then
and
(see at signs in supergeometry for more on this).
A strong homotopy homomorphism (“sh-map”) between super $L_\infty$-algbras of finite type
is defined to be a homomorphism of dg-algebras between their Chevalley-Eilenberg algebras going the other way:
(here $f^\ast$ is the primitive concept, and $f$ is defined as the formal dual of $f$). Hence the category of super $L_\infty$-algebras of finite type is the full subcategory
of the opposite category of dg-algebras on those that are CE-algebras as above.
Finally, the cochain cohomology of the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ of a super $L_\infty$ algebra of finite type is its L-∞ algebra cohomology with coefficients in $\mathbb{R}$:
Special cases of the general concept of super L-∞ algebras def. 3 go by special names:
Let $\mathfrak{g}$ be a super L-∞ algebra.
If $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree, it is called an L-∞ algebra.
If the only possibly non-vanishing brackets of $\mathfrak{g}$ are the unary one $[-]$ (which induces the structure of a chain complex on $\mathfrak{g}$) and the binary one, then $\mathfrak{g}$ is equivalently a (super-)dg-Lie algebra.
If $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degrees 0 to $n-1$ then it is called a super Lie n-algebra.
In particular if $\mathfrak{g}$ is concentrated in degree 0, then it is equivalently a super Lie algebra.
Combining this, if $\mathfrak{g}$ is concentrated in even $\mathbb{Z}/2$-degree and in $\mathbb{Z}$-degree 0 through $n-1$, then it is called a Lie n-algebra.
In particular if $\mathfrak{g}$ is concentrated in $\mathbb{Z}$-degree 0 and in even $\mathbb{Z}/2$-degree, then it is equivalently a plain Lie algebra.
(history of the concept of (super-)$L_\infty$ algebras)
The identification of the concept of (super-)$L_\infty$-algebras has a non-linear history:
L-∞ algebras in the incarnation of higher brackets satisfying a higher Jacobi identity, def. 3 and remark 1, were introduced in Lada-Stasheff 92, based on the example of such a structure on the BRST complex of the bosonic string that was found in the construction of closed string field theory in Zwiebach 92. Some of this history is recalled in Stasheff 16.
The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras (def. 4) is due to Lada-Markl 94. See Sati-Schreiber-Stasheff 08, around def. 13.
But in this dual incarnation, L-∞ algebras and more generally super L-∞ algebras (of finite type) had secretly been introduced within the supergravity literature already in D’Auria-Fré-Regge 80 and explicitly in van Nieuwenhuizen 82. The concept was picked up in the D'Auria-Fré formulation of supergravity (D’Auria-Fré 82) and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the supergravity literature (but beware that these dg-algebras are in general free only as graded-supercommutative superalgebras, not as differential algebras) The relation between super $L_\infty$-algebras and the “FDA”s of the supergravity literature is made explicit in (FSS 13).
higher Lie theory | supergravity |
---|---|
$\,$ super Lie n-algebra $\mathfrak{g}$ $\,$ | $\,$ “FDA” $CE(\mathfrak{g})$ $\,$ |
The construction in van Nieuwenhuizen 82 in turn was motivated by the Sullivan algebras in rational homotopy theory (Sullivan 77). Indeed, their dual incarnations in rational homotopy theory are dg-Lie algebras (Quillen 69), hence a special case of $L_\infty$-algebras (remark 1)
This close relation between rational homotopy theory and higher Lie theory might have remained less of a secret had it not been for the focus of Sullivan minimal models on the non-simply connected case, which excludes the ordinary Lie algebras from the picture. But the Quillen model of rational homotopy theory effectively says that for $X$ a rational topological space then its loop space ∞-group $\Omega X$ is reflected, infinitesimally, by an L-∞ algebra. This perspective began to receive more attention when the Sullivan construction in rational homotopy theory was concretely identified as higher Lie integration in Henriques 08. A modern review that makes this L-∞ algebra-theoretic nature of rational homotopy theory manifest is in Buijs-Félix-Murillo 12, section 2.
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.
See at geometry of physics -- fundamenal super p-branes? for more on this.
$\,$
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. (…)
In their formal dual incarnations as super-graded commutative dg-algebras (super Chevalley-Eilenberg algebras), super $L_\infty$-algebras of finite type had secretly been introduced in
Riccardo D'Auria, Pietro Fré Tullio Regge, Graded Lie algebra, cohomology and supergravity, Riv. Nuov. Cim. 3, fasc. 12 (1980) (spire)
Peter van Nieuwenhuizen, Free Graded Differential Superalgebras, in Istanbul 1982, Proceedings, Group Theoretical Methods In Physics, 228-247 and CERN Geneva - TH. 3499 (spire)
and hence a whole decade before the explicit appearance of plain (non-super) L-∞ algebras in Lada-Stasheff 92.
The concept was picked up in the D'Auria-Fré formulation of supergravity
and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the supergravity literature (where in rational homotopy theory one says “semifree dga”, since these dg-algebras are crucially not required to be free as differential algebras).
The relation between super $L_\infty$-algebras and the “FDA”s of the supergravity literature is made explicit in