# nLab differential string structure -- proofs

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

# Contents

## 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 $(b \mathbb{R} \to \mathfrak{string})$

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

### 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

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

• $d b = - \mu + c + h$;

• $d c = g$,

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

• $d h = \sigma \mu - g$;

• $d g = 0$.

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. 3 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.

Revised on September 8, 2011 13:52:11 by Urs Schreiber (131.211.239.172)