∞-Lie theory (higher geometry)
The string Lie 2-algebra is the infinitesimal approximation to the Lie 2-group that is called the string 2-group.
It is a shifted ∞-Lie algebra central extension
of the Lie algebra $\mathfrak{so}(n)$ by the Lie 2-algebra $\mathbf{b} \mathfrak{u}(1)$ which is classified by the canonical (up to normalization) Lie algebra 3-cocycle $\mu$ on $\mathfrak{so}(n)$, which may itself be understood as a morphism
When $\mu$ is normalized such that it represents the image in deRham cohomology of the generator of the integral cohomology $H^3(X,Spin(n))$ of the Spin group, then the Lie integration of the String Lie 2-algebra is the String Lie 2-group.
We spell out first an explicit algebraic realization of the string Lie 2-algebra and then give its abstract definition as a homotopy fiber or principal ∞-bundle.
As with any L-∞ algebra, we may define the String Lie 2-algebra $\mathfrak{string}(n)$ equivalently in terms of its Chevalley–Eilenberg algebra.
There are various equivalent models we discuss a small one with a trinary bracket, and an infinite dimensional model which is however strict in that it comes from a differential crossed module.
Write $\mathfrak{g} := \mathfrak{so}(n)$ in the following. The Chevalley–Eilenberg algebra $CE(\mathfrak{g})$ of $\mathfrak{g}$ has a degree 3 element
well defined up to normalization ($\langle - ,- \rangle$ is the canonical bilinear symmetric invariant polynomial on $\mathfrak{g}$ and $[-,-]$ the Lie bracket), which is closed in $CE(\mathfrak{g})$
Hence this is the canonical (up to normalization) 3-cocycle in the Lie algebra cohomology of $\mathfrak{g}$.
The Chevalley–Eilenberg algebra of $\mathfrak{string}(n)$ is
where
$b$ is a single new generator in degree 2;
the differental $d_{\mathfrak{string}}$ coincides with $d_{\mathfrak{g}}$ on $\mathfrak{g}^*$:
on the new generator it is defined by
That the differential defined this way is indeed of degree +1 and squares to 0 is precisely the fact that $\mu$ is a degree 3-cocycle of $\mathfrak{g}$.
One can equivalently describe the $L_\infty$-algebra structure of $\mathfrak{string}(n)$ in terms of lots of brackets
of degree $2-k$. In addition to the Lie bracket of $\mathfrak{g}$, there is only a further nontrivial bracket: it is the 3-bracket
given by
where $\beta:\langle b\rangle\to\mathbb{R}$ is the dual of $b$.
Proposition The string Lie 2-algebra given above is equivalent to the infinite-dimensional Lie 2-algebra coming from the differential crossed module
of the universal central extension of the loop Lie algebra? mapping into the path Lie algebra, which acts on the former in the evident way.
This is proven in BCSS.
Up to equivalence, the string Lie 2-algebra is the homotopy fiber of the cocycle $\mu : \mathfrak{so}(n) \to \mathbf{b}^2 \mathfrak{u}(1)$, hence is the canonical $\mathbf{b} \mathfrak{u}(1)$-principal ∞-bundle over $\mathfrak{so}(n)$.
Here we take by definition the (∞,1)-category of ∞-Lie algebroids to be that presented by the opposite (after passing to Chevalley-Eilenberg algebras) of the model structure on dg-algebras. For a detailed discussion of the recognition of this homotopy fiber see section 3.1 and specifically example 3.5.4 of (Fiorenza-Rogers-Schreiber 13).
In terms of dg-algebras, the cocycle is dually a morphism
and the homotopy fiber in question is dually modeled by the homotopy pushout
By the general rules for computing homotopy pushouts, this may be computed by an ordinary pushout if we choose a resolution of $CE(\mathbf{b}^2 \mathfrak{u}(1)) \to 0$ by a cofibration and ensure that all three objects in the pushout diagram are cofibrations.
For the resolution we take the standard one by the CE-algebra of the $\mathbf{b}^2 \mathfrak{u}(1)$-universal principal ∞-bundle $\mathbf{e b} \mathfrak{u}(1)$, which is the dg-algebra
where $b$ is a generator of degree 2, $c$ one of degree 3 and the differential is given by
and
The morphism $CE(\mathbf{b}^2 \mathfrak{u}(1)) \to CE(e b \mathfrak{u}(1))$ is the one that identifies the two degree-3 generators.
Now $CE(\mathbf{b}^2 \mathfrak{u}(1))$ and $CE(\mathbf{e b} \mathfrak{u}(1))$ are Sullivan algebras, hence are cofibrant objects in the model structure on dg-algebras. The dg-algebra $CE(\mathfrak{g})$ is not quite a Sullivan algebra, but almost: it is a semifree dga and only fails to have the filtering property on the differential. This is sufficient for computing the desired homotopy fiber, as discussed at ∞-Lie algebra cohomology – Extensions.
One observes now that
is a pushout diagram. Dually, this exhibits $\mathfrak{string}$ as the (∞,1)-pullback
And this may be taken to be the abstract definition of the string Lie 2-algebra.
By the general logic of fiber sequences this implies that also
is a fiber sequence. By analogous reasoning as before, we see that this is modeled by the ordinary pushout
This is indeed a homotopy pushout even without resolving the point, because $CE(\mathfrak{string}) \leftarrow CE(\mathfrak{g})$ is already a cofibration, being the pushout of a cofibration by the above.
The delooping of a semisimple Lie algebra $\mathfrak{g}$ to a 1-object L-infinity algebroid $b \mathfrak{g}$ carries the Killing form as a quadratic bilinear invariant polynomial and is as such a symplectic Lie n-algebroid over the point for $n = 2$, hence a Courant Lie 2-algebroid over the point.
As described at symplectic infinity-groupoid one can consider the higher analog of geometric quantization of these objects. This is again the homotopy fiber as above.
The String Lie 2-algebra identifies also with the Heisenberg Lie 2-algebra of the string sigma-model for the specialization to the WZW model. See at 2-plectic geometry for more.
More generally, for $\mathfrak{g}$ an ∞-Lie algebra and $\mu \in CE(\mathfrak{g})$ an $\infty$-Lie algebra cocycle (a closed element in the Chevalley–Eilenberg algebra of $\mathfrak{g}$) of degree $k$, there is a corresponding shifted central extension
For instance the supergravity Lie 3-algebra is such an extension of the super Poincare Lie algebra by a super Lie algebra 4-cocycle.
string Lie 2-algebra
In one incarnation or other the String Lie 2-algebra has been considered in literature of dg-algebras, but its Lie theoretic interpretation as a Lie 2-algebra has been made fully explicit only in
In
the string Lie 2-algebra is integrated to the string 2-group using the general abstract method described at Lie integration.
In
the equivalent strict model given by a differential crossed module is found, which is then integrated termwise as ordinary Lie algebras to a crossed module of Frechet-Lie groups, hence to a Lie strict 2-group model of the String Lie 2-group.
The string Lie 2-algebra is also considered in a certain context in
where also the relation to the supergravity Lie 3-algebra and other structures is discussed.
The super-$L_\infty$-version of the string $L_\infty$-algebra was considered in
See also division algebra and supersymmetry.
Discussion of the string Lie 2-algebra as the homotopy fiber of the underlying 3-cocycle is around prop. 3.3.96 in
and example 3.5.4 in