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String Lie 2-algebra

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\infty-Lie theory

∞-Lie theory

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∞-Lie algebroids

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Contents

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

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

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

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

When μ\mu is normalized such that it represents the image in deRham cohomology of the generator of the integral cohomology H 3(X,Spin(n))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 𝔰𝔱𝔯𝔦𝔫𝔤(n)\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 𝔤:=𝔰𝔬(n)\mathfrak{g} := \mathfrak{so}(n) in the following. The Chevalley–Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) of 𝔤\mathfrak{g} has a degree 3 element

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

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

d 𝔤μ=0. d_{\mathfrak{g}} \mu = 0 \,.

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

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

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

where

  • bb is a single new generator in degree 2;

  • the differental d 𝔰𝔱𝔯𝔦𝔫𝔤d_{\mathfrak{string}} coincides with d 𝔤d_{\mathfrak{g}} on 𝔤 *\mathfrak{g}^*:

    d 𝔰𝔱𝔯𝔦𝔫𝔤| 𝔤 *=d 𝔤 d_{\mathfrak{string}} |_{\mathfrak{g}^*} = d_{\mathfrak{g}}
  • on the new generator it is defined by

    d 𝔰𝔱𝔯𝔦𝔫𝔤:bμ. 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 μ\mu is a degree 3-cocycle of 𝔤\mathfrak{g}.

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

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

of degree 2k2-k. In addition to the Lie bracket of 𝔤\mathfrak{g}, there is only a further nontrivial bracket: it is the 3-bracket

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

given by

[x,y,z] 3=μ(x,y,z)β, [x,y,z]_3=\mu(x,y,z)\cdot \beta,

where β:b\beta:\langle b\rangle\to\mathbb{R} is the dual of bb.

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

Ω^𝔤P𝔤 \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 μ:𝔰𝔬(n)b 2𝔲(1)\mu : \mathfrak{so}(n) \to \mathbf{b}^2 \mathfrak{u}(1), hence is the canonical b𝔲(1)\mathbf{b} \mathfrak{u}(1)-principal ∞-bundle over 𝔰𝔬(n)\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(𝔰𝔬(n))CE(b 2𝔲(1)):μ 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

0 CE(𝔰𝔬(n)) μ CE(b 2𝔲(1)). \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(b 2𝔲(1))0CE(\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 b 2𝔲(1)\mathbf{b}^2 \mathfrak{u}(1)-universal principal ∞-bundle eb𝔲(1)\mathbf{e b} \mathfrak{u}(1), which is the dg-algebra

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

where bb is a generator of degree 2, cc one of degree 3 and the differential is given by

db=c d b = c

and

dc=0. d c = 0 \,.

The morphism CE(b 2𝔲(1))CE(eb𝔲(1))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(b 2𝔲(1))CE(\mathbf{b}^2 \mathfrak{u}(1)) and CE(eb𝔲(1))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(𝔤)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

CE(𝔰𝔱𝔯𝔦𝔫𝔤) CE(eb𝔲(1)) CE(𝔰𝔬(n)) CE(b 2𝔲(1)) \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 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string} as the (∞,1)-pullback

𝔰𝔱𝔯𝔦𝔫𝔤(n) * 𝔰𝔬(n) μ b 2𝔲(1). \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)𝔰𝔱𝔯𝔦𝔫𝔤𝔰𝔬(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

CE(b𝔲(1)) * CE(𝔰𝔱𝔯𝔦𝔫𝔤) CE(𝔤). \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(𝔰𝔱𝔯𝔦𝔫𝔤)CE(𝔤)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 𝔤\mathfrak{g} to a 1-object L-infinity algebroid b𝔤b \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=2n = 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\mathfrak{g} an ∞-Lie algebra and μCE(𝔤)\mu \in CE(\mathfrak{g}) an \infty-Lie algebra cocycle (a closed element in the Chevalley–Eilenberg algebra of 𝔤\mathfrak{g}) of degree kk, there is a corresponding shifted central extension

0b k2𝔲(1)𝔤 μ𝔤0. 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.

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

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

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

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

The super-L L_\infty-version of the string L 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

Revised on May 22, 2013 22:42:59 by Urs Schreiber (89.204.135.236)