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

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

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

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

When $\mu$ is normalized such that it represents the image in deRham cohomology of the generator of the integral cohomology $H^3(X,Spin(n))$ 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 $\mathfrak{string}(n)$ 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.

Skeletal model

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

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

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 $CE(\mathfrak{g})$

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

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

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

where

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

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

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

• on the new generator it is defined by

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

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 $\mathfrak{g}$.

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

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

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

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

given by

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

where $\beta:\langle b\rangle\to\mathbb{R}$ 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

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

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

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

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

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

By the general rules for computing homotopy pushouts, this may be computed by an ordinary pushout if we choose a resolution of $CE(\mathbf{b}^2 \mathfrak{u}(1)) \to 0$ 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

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

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

$$d b = c$$

and

$$d c = 0 \,.$$

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

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

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

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

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

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

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

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

This is indeed a homotopy pushout even without resolving the point, because $CE(\mathfrak{string}) \leftarrow CE(\mathfrak{g})$ 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 $\mathfrak{g}$ 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.

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 (Baez-Rogers 10). See at 2-plectic geometry for more.

Generalizations

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

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

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

• fivebrane Lie 6-algebra

• type II supergravity Lie 2-algebra

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

• string Lie algebroid

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 $L_\infty$-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, hha:1201127333]

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 as the Heisenberg Lie 2-algebra on the group $G$ is discussed in

• John Baez, Chris Rogers, Categorified Symplectic Geometry and the String Lie 2-Algebra, Homology Homotopy Appl. Volume 12, Number 1 (2010), 221-236. (arXiv:0901.4721,euclid).

The string Lie 2-algebra is also considered in a certain context in

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

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

• 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)

More on this in

• Lennart Schmidt, Twisted Weil Algebras for the String Lie 2-Algebra, in Christian Saemann, Urs Schreiber, Martin Wolf (eds.) Higher Structures in M-Theory Durham Symposium 2018, Fortschritte der Physik 2019 (arXiv:1903.02873)