nLab differential string structure -- proofs

This page contains technical details to be used at the main page differential string structure . See there for context.

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Factorization of the $L_\infty$-cocycle

Proposition

The $L_\infty$-algebra cocycle

$$\mu : \mathfrak{so} \to b^2 \mathbb{R}$$

factors as

$$\mathfrak{so} \stackrel{}{\to} (b \mathbb{R} \to \mathfrak{string}) \stackrel{}{\to} b^2 \mathbb{R}$$

given dually on CE-algebras by

$$CE(\mathfrak{so}) \stackrel{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} CE(b \mathbb{R} \to \mathfrak{string}) \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} CE(b^2 \mathbb{R}) \,.$$

The left morphism is a quasi-isomorphism.

Proof

To see that we have a quasi-isomorphism, notice that the dg-algebra $CE(b \mathbb{R} \to \mathfrak{string})$ is isomorphic to the one with generators $\{t^a, b, c'\}$ and differentials

$$\begin{aligned} d|_{\mathfrak{g}^*} & = [-,-]^* \\ d b & = c' \\ d c' & = 0 \end{aligned} \,,$$

where the isomorphism is given by the identity on the $t^a$s and on $b$ and by

$$c \mapsto c' + \mu \,.$$

The primed dg-algebra is the tensor product $CE(\mathfrak{g}) \otimes CE( inn(b \mathbb{R}))$, where the second factor is manifestly cohomologically trivial.

Factorization of the differential $L_\infty$-cocycle

We now give a concrete construction showing

Proposition

The factorization

$$CE(\mathfrak{so}) \stackrel{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} CE(b \mathbb{R} \to \mathfrak{string}) \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} CE(b^2 \mathbb{R})$$

from above lifts to a factorization of differential $L_\infty$-algebraic cocycles

$$\array{ CE(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} & CE(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} & CE(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{} \\ W(\mathfrak{so}) & \underoverset{\simeq}{ }{\leftarrow} & W(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ }{\leftarrow} & W(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{} \\ inv(\mathfrak{so}) & \underoverset{\simeq}{ }{\leftarrow} & \tilde inv(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ }{\leftarrow} & inv(b^2 \mathbb{R}) } \,.$$

Proof

This is at its heart trivial, but potentially a bit tedious. We proceed in two steps:

1. consider a “modified Weil algebra” of the twisted string Lie 3-algebra $(b \mathbb{R} \to \mathfrak{string})$

   in The modified Weil algebra;

2. construct the desired factorization by factoring itself through two fairly evident morphisms into and out of the modified Weil algebra,

   in The differential lift.

The modified Weil algebra

Our factorization below makes use of an isomorphic copy of the Weil algebra $W(b\mathbb{R} \to \mathfrak{g}_\mu)$.

Proposition

The Weil algebra $W(b\mathbb{R} \to \mathfrak{g}_\mu)$ of $(b^2 \mathbb{R} \to \mathfrak{g})$ is given on the extra shifted generators $\{r^a := \sigma t^a, h := \sigma b, g := \sigma c\}$ (where $\sigma$ is the shift operator extended as a graded derivation, see Weil algebra) by

• $d t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$;

• $d b = - \mu + c + h$;

• $d c = g$,

with Bianchi identities

• $d r^a = - C^a{}_{b c} t^b \wedge r^c$

• $d h = \sigma \mu - g$;

• $d g = 0$.

Let $\tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)$ be the dg-algebra with the same underlying graded algebra as $W(b\mathbb{R} \to \mathfrak{g}_\mu)$ but with the differential modified as follows

• $d t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$;

• $d r^a = - C^a{}_{b c} t^b \wedge r^a$;

• $d b = - cs + c + \tilde h$;

• $d \tilde h = \langle -,-\rangle - g$;

• $d c = g$ .

• $d g = 0$,

where “$\tilde h$” is the new name for the generator that used to be called “$h$”

There is an isomorphism

$$W(b\mathbb{R} \to \mathfrak{g}_\mu) \to \tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)$$

in dgAlg that is the identity on all generators except on $h$, where it is given by

$$h \mapsto \tilde h + (\mu - cs) \,.$$

Note

Where the formula for the differential of $W(b\mathbb{R}\to \mathfrak{g}_\mu)$ has the 3-cocycle $\mu$ that for $\tilde W(b\mathbb{R}\to \mathfrak{g}_\mu)$ has the Chern-Simons element $cs$. The shift by $cs-\mu$ is precisely what shifts the curvature characteristic $d_{W(\mathfrak{g})}\mu$ into the shifted copy of $\mathfrak{g}^*$ in the Weil algebra, thus exhibiting the modified $h$ as an invariant polynomial.

Corollary

The invariant polynomials on $(b \mathbb{R} \to \mathfrak{g}_\mu)$ are generated from those of $\mathfrak{g}_\mu$ together with $\tilde h$ and $g$:

$$\tilde inv(b \mathbb{R} \to \mathfrak{string}) = (inv(\mathfrak{so})\otimes \langle \tilde h, g\rangle)/(d \tilde h = \langle -,-\rangle - g) \,.$$

The differential lift

We now use the isomorphism

$$W(b \mathbb{R} \to \mathfrak{string}) \stackrel{\simeq}{\to} \tilde W(b \mathbb{R} \to \mathfrak{string})$$

from prop. and obtain the desired factorization, as the composite

$$\array{ CE(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} & CE(b \mathbb{R} \to \mathfrak{string}) & = & CE(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} & CE(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{i^*_{\mathfrak{so}}} && \uparrow^\mathrlap{i^*_{(b\mathbb{R} \to \mathfrak{string})}} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{i^*_{b^2 \mathbb{R}}} \\ W(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto cs \\ r^a \mapsto r^a \\ \tilde h \mapsto 0 \\ g \mapsto \langle-,-\rangle } \right) }{\leftarrow} & W(b \mathbb{R} \to \mathfrak{string}) & \underoverset{\simeq}{ \left( \array{ \tilde h \mapsto h + (cs - \mu) } \right) } {\leftarrow} & \tilde W(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c \\ g \mapsto g } \right) }{\leftarrow} & W(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{p^*_{\mathfrak{so}}} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{p^*_{(b \mathbb{R} \to \mathfrak{string})}} && \uparrow^\mathrlap{p^*_{b^2 \mathbb{R}}} \\ inv(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ \tilde h \mapsto 0 \\ g \mapsto \langle -,-\rangle \\ \langle \cdots \rangle \mapsto \langle \cdots \rangle } \right) }{\leftarrow} & \tilde inv(b \mathbb{R} \to \mathfrak{string}) & = & \tilde inv(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ g \mapsto g } \right) }{\leftarrow} & inv(b^2 \mathbb{R}) } \,.$$

Here

• the unlabelled vertical morphisms are defined as the unique ones that make the respective square commute;

• the notation $\langle \cdots \rangle$ stands for all the invariant polynomials of $\mathfrak{so}$ and $\langle-,-\rangle$ specifically for the Killing form.