nLab
String Lie 2-algebra

Idea

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

$$0\to b𝔲(1)\to \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)\to \mathrm{𝔰𝔬}(n)\to 0$$0 \to \mathbf{b} \mathfrak{u}(1) \to \mathfrak{string}(n) \to \mathfrak{so}(n) \to 0

of the Lie algebra $\mathrm{𝔰𝔬}(n)$ by the Lie 2-algebra $b𝔲(1)$ which is classified by the canonical (up to normalization) Lie algebra 3-cocycle $\mu$ on $\mathrm{𝔰𝔬}(n)$, which may itself be understood as a morphism

$$\mu :\mathrm{𝔰𝔬}(n)\to b^2𝔲(1).$$\mu : \mathfrak{so}(n) \to b^2 \mathfrak{u}(1) \,.

When $\mu$ is normalized such that it represents the image in deRham cohomology of the generator of the integral cohomology $H^3(X,\mathrm{Spin}(n))$ of the Spin group, then the Lie integration of the String Lie 2-algebra is the String Lie 2-group.

Definition

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.

In components

As with any L-∞ algebra, we may define the String Lie 2-algebra $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ 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.

Skeletal model

Write $𝔤:=\mathrm{𝔰𝔬}(n)$ in the following. The Chevalley–Eilenberg algebra $\mathrm{CE}(𝔤)$ of $𝔤$ has a degree 3 element

$$\mu =⟨-,[-,-]⟩,$$\mu = \langle -, [-,-]\rangle \,,

well defined up to normalization ($⟨-,-⟩$ is the canonical bilinear symmetric invariant polynomial on $𝔤$ and $[-,-]$ the Lie bracket), which is closed in $\mathrm{CE}(𝔤)$

$$d_{𝔤}\mu =0.$$d_{\mathfrak{g}} \mu = 0 \,.

Hence this is the canonical (up to normalization) 3-cocycle in the Lie algebra cohomology of $𝔤$.

The Chevalley–Eilenberg algebra of $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ is

$$\mathrm{CE}(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n))=({\wedge }^{•}({𝔤}^{*}\oplus ⟨b⟩),d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}),$$CE(\mathfrak{string}(n)) = (\wedge^\bullet (\mathfrak{g}^* \oplus \langle b\rangle), d_{\mathfrak{string}}) \,,

where

• $b$ is a single new generator in degree 2;

• the differental $d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}$ coincides with $d_{𝔤}$ on ${𝔤}^{*}$:

  $$d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}{\mid }_{{𝔤}^{*}}=d_{𝔤}$$d_{\mathfrak{string}} |_{\mathfrak{g}^*} = d_{\mathfrak{g}}

• on the new generator it is defined by

  $$d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}:b\mapsto \mu .$$d_{\mathfrak{string}} : b \mapsto \mu \,.

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 $𝔤$.

One can equivalently describe the $L_{\mathrm{\infty }}$-algebra structure of $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ in terms of lots of brackets

$$[-,-,\dots ,-{]}_k:{\wedge }^k\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)\to \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n),$$[-,-,\dots,-]_k:\wedge^k \mathfrak{string}(n)\to \mathfrak{string}(n),

of degree $2-k$. In addition to the Lie bracket of $𝔤$, there is only a further nontrivial bracket: it is the 3-bracket

$$[-,-,-{]}_3:{\wedge }^3𝔤\to ⟨b{⟩}^{*}$$[-,-,-]_3:\wedge^3 \mathfrak{g}\to \langle b\rangle^*

given by

$$[x,y,z{]}_3=\mu (x,y,z)\cdot \beta ,$$[x,y,z]_3=\mu(x,y,z)\cdot \beta,

where $\beta :⟨b⟩\to ℝ$ is the dual of $b$.

Strict Lie 2-algebra model

Proposition The string Lie 2-algebra given above is equivalent to the infinite-dimensional Lie 2-algebra coming from the differential crossed module

$$\hat{\Omega }𝔤\to P𝔤$$\hat \Omega \mathfrak{g} \to P \mathfrak{g}

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.

As a homotopy fiber

Up to equivalence, the string Lie 2-algebra is the homotopy fiber of the cocycle $\mu :\mathrm{𝔰𝔬}(n)\to b^2𝔲(1)$, hence is the canonical $b𝔲(1)$-principal ∞-bundle over $\mathrm{𝔰𝔬}(n)$.

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

$$\mathrm{CE}(\mathrm{𝔰𝔬}(n))\leftarrow \mathrm{CE}(b^2𝔲(1)):\mu$$CE(\mathfrak{so}(n)) \leftarrow CE(\mathbf{b}^2 \mathfrak{u}(1)) : \mu

and the homotopy fiber in question is dually modeled by the homotopy pushout

$$\begin{array}{ccc}& & 0\\ & & \uparrow \\ \mathrm{CE}(\mathrm{𝔰𝔬}(n))& \stackrel{\mu }{\leftarrow }& \mathrm{CE}(b^2𝔲(1))\end{array}.$$\array{ && 0 \\ && \uparrow \\ CE(\mathfrak{so}(n)) &\stackrel{\mu}{\leftarrow}& CE(\mathbf{b}^2 \mathfrak{u}(1)) } \,.

By the general rules for computing homotopy pushouts, this may be computed by an ordinary pushout if we choose a resolution of $\mathrm{CE}(b^2𝔲(1))\to 0$ 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 $b^2𝔲(1)$-universal principal ∞-bundle $eb𝔲(1)$, which is the dg-algebra

$$\mathrm{CE}(eb𝔲(1))=({\wedge }^{•}(⟨b⟩\oplus ⟨c⟩),d)$$CE(\mathbf{e b} \mathfrak{u}(1)) = (\wedge^\bullet( \langle b\rangle \oplus \langle c\rangle ), d)

where $b$ is a generator of degree 2, $c$ one of degree 3 and the differential is given by

$$db=c$$d b = c

and

$$dc=0.$$d c = 0 \,.

The morphism $\mathrm{CE}(b^2𝔲(1))\to \mathrm{CE}(eb𝔲(1))$ is the one that identifies the two degree-3 generators.

Now $\mathrm{CE}(b^2𝔲(1))$ and $\mathrm{CE}(eb𝔲(1))$ are Sullivan algebras, hence are cofibrant objects in the model structure on dg-algebras. The dg-algebra $\mathrm{CE}(𝔤)$ 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

$$\begin{array}{ccc}\mathrm{CE}(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤})& \leftarrow & \mathrm{CE}(eb𝔲(1))\\ \uparrow & & \uparrow \\ \mathrm{CE}(\mathrm{𝔰𝔬}(n))& \leftarrow & \mathrm{CE}(b^2𝔲(1))\end{array}$$\array{ CE(\mathfrak{string}) &\leftarrow& CE(\mathbf{e b} \mathfrak{u}(1)) \\ \uparrow && \uparrow \\ CE(\mathfrak{so}(n)) &\leftarrow& CE(\mathbf{b}^2 \mathfrak{u}(1)) }

is a pushout diagram. Dually, this exhibits $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}$ as the (∞,1)-pullback

$$\begin{array}{ccc}\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)& \to & *\\ \downarrow & & \downarrow \\ \mathrm{𝔰𝔬}(n)& \stackrel{\mu }{\to }& b^2𝔲(1)\end{array}.$$\array{ \mathfrak{string}(n) &\to& * \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& \mathbf{b}^2 \mathfrak{u}(1) } \,.

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

$$b𝔲(1)\to \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to \mathrm{𝔰𝔬}(n)$$\mathbf{b} \mathfrak{u}(1) \to \mathfrak{string} \to \mathfrak{so}(n)

is a fiber sequence. By analogous reasoning as before, we see that this is modeled by the ordinary pushout

$$\begin{array}{ccc}\mathrm{CE}(b𝔲(1))& \leftarrow & *\\ \uparrow & & \uparrow \\ \mathrm{CE}(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤})& \leftarrow & \mathrm{CE}(𝔤)\end{array}.$$\array{ CE(\mathbf{b} \mathfrak{u}(1)) &\leftarrow& * \\ \uparrow && \uparrow \\ CE(\mathfrak{string}) &\leftarrow& CE(\mathfrak{g}) } \,.

This is indeed a homotopy pushout even without resolving the point, because $\mathrm{CE}(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤})\leftarrow \mathrm{CE}(𝔤)$ is already a cofibration, being the pushout of a cofibration by the above.

As the prequantum line 2-bundle of a Courant algebroid

The delooping of a semisimple Lie algebra $𝔤$ to a 1-object L-infinity algebroid $b𝔤$ 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.

As a Heisenberg Lie 2-algebra

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.

Generalizations

More generally, for $𝔤$ an ∞-Lie algebra and $\mu \in \mathrm{CE}(𝔤)$ an $\mathrm{\infty }$-Lie algebra cocycle (a closed element in the Chevalley–Eilenberg algebra of $𝔤$) of degree $k$, there is a corresponding shifted central extension

$$0\to b^{k-2}𝔲(1)\to {𝔤}_{\mu }\to 𝔤\to 0.$$0 \to \mathbf{b}^{k-2} \mathfrak{u}(1) \to \mathfrak{g}_\mu \to \mathfrak{g} \to 0 \,.

For instance the supergravity Lie 3-algebra is such an extension of the super Poincare Lie algebra by a super Lie algebra 4-cocycle.

Related concepts

• special orthogonal Lie algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra

• type II supergravity Lie 2-algebra

• supergravity Lie 3-algebra, supergravity Lie 6-algebra

References

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

• John Baez, Alissa Crans, Higher-dimensional Algebra V: Lie 2-algebras, Theory and Applications of Categories 12 (2004), 492-528. (web) (arXiv:math.QA/0307263)

In

• Andre Henriques, Integrating $L_{\mathrm{\infty }}$-algebras (arXiv:0603563)

the string Lie 2-algebra is integrated to the string 2-group using the general abstract method described at Lie integration.

In

• John Baez, Alissa Crans, Urs Schreiber and Danny Stevenson, From loop groups to 2-groups, Homotopy, Homology and Applications 9 (2007), 101-135. (arXiv:math.QA/0504123)

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

• Sati, Schreiber, Stasheff, $L_{\mathrm{\infty }}$-algebra connections

where also the relation to the supergravity Lie 3-algebra and other structures is discussed.

The super-$L_{\mathrm{\infty }}$-version of the string $L_{\mathrm{\infty }}$-algebra was considered in

• John Baez, John Huerta, Division Algebras and Supersymmetry II (arXiv:1003.3436).

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

• differential cohomology in a cohesive topos

and example 3.5.4 in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles (arXiv:1304.6292)