∞-Lie theory

# Contents

## Idea

The Poincaré Lie algebra $\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)$ is the semidirect product of the special orthogonal Lie algebra $\mathrm{𝔰𝔬}\left(d-1,1\right)$ and the abelian translation Lie algebra ${ℝ}^{d-1,1}$.

The corresponding Lie group is the Poincaré group.

## Definition

### The CE-algebra

The Chevalley-Eilenberg algebra $\mathrm{CE}\left(\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)\right)$ is generated from ${ℝ}^{d,1}$ and ${\wedge }^{2}{ℝ}^{d,1}$. For $\left\{{t}_{a}\right\}$ the standard basis of ${ℝ}^{d-1,1}$ we write $\left\{{\omega }^{ab}\right\}$ and $\left\{{e}^{a}\right\}$ for these generators. With $\left({\eta }_{ab}\right)$ the components of the Minkowski metric we write

${\omega }^{a}{}_{b}:={\omega }^{ac}{\eta }_{cb}\phantom{\rule{thinmathspace}{0ex}}.$\omega^{a}{}_b := \omega^{a c}\eta_{c b} \,.

In terms of this the CE-differential that defines the Lie algebra structure is

${d}_{\mathrm{CE}}:{\omega }^{ab}={\omega }^{a}{}_{c}\wedge {\omega }^{cb}$d_{CE} : \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}
${d}_{\mathrm{CE}}:{e}^{a}↦{\omega }^{a}{}_{b}\wedge {t}^{b}$d_{CE} : e^a \mapsto \omega^{a}{}_b \wedge t^b

## Properties

### Cohomology

We discuss some elements in the Lie algebra cohomology of $\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)$.

The canonical degree-3 $\mathrm{𝔰𝔬}\left(d-1,1\right)$-cocycle is

${\omega }^{a}{}_{b}\wedge {\omega }^{b}{}_{c}\wedge {\omega }^{c}{}_{a}\in \mathrm{CE}\left(\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a \in CE(\mathfrak{iso}(d-1,1)) \,.

The volume cocycle is the volume form

$\mathrm{vol}={ϵ}_{{a}_{1}\cdots {a}_{d}}{e}^{{a}_{1}}\wedge \cdots \wedge {e}^{{a}_{d}}\in \mathrm{CE}\left(\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$vol = \epsilon_{a_1 \cdots a_{d}} e^{a_1} \wedge \cdots \wedge e^{a_d} \in CE(\mathfrak{iso}(d-1,1)) \,.

### Invariant polynomials and Chern-Simons elements

With the basis elements $\left({e}^{a},{\omega }^{ab}\right)$ as above, denote the shifted generators of the Weil algebra $W\left(\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)\right)$ by ${\theta }^{a}$ and ${r}^{ab}$, respectively.

We have the Bianchi identity

${d}_{W}:{r}^{ab}↦{\omega }^{ac}\wedge {R}_{c}{}^{d}-{R}^{ac}\wedge {\omega }_{c}{}^{b}$d_W : r^{a b} \mapsto \omega^{a c} \wedge R_c{}^d - R^{a c} \wedge \omega_c{}^b

and

${d}_{W}:{\theta }^{a}↦{\omega }^{a}{}_{b}{\theta }^{b}-{R}^{a}{}_{b}{e}^{b}\phantom{\rule{thinmathspace}{0ex}}.$d_W : \theta^a \mapsto \omega^a{}_b \theta^b - R^{a}{}_b e^b \,.

The element ${\eta }_{ab}{\theta }^{a}\wedge {\theta }^{b}\in W\left(\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)\right)$ is an invariant polynomial. A Chern-Simons element for it is $\mathrm{cs}={\eta }_{ab}{e}^{a}\wedge {\theta }^{b}$. So this transgresses to the trivial cocycle.

Another invariant polynomial is ${r}^{ab}\wedge {r}_{ab}$. This is the Killing form of $\mathrm{𝔰𝔬}\left(d-1,1\right)$. Accordingly, it transgresses to a multiple of ${\omega }^{a}{}_{b}\wedge {\omega }^{b}{}_{c}\wedge {\omega }^{c}{}_{a}$.

This is the first in an infinite series of Pontryagin invariant polynomials

${P}_{n}:={r}^{{a}_{1}}{}_{{a}_{2}}\wedge {r}^{{a}_{2}}{}_{{a}_{3}}\wedge \cdots \wedge {r}^{{a}_{n}}{}_{{a}_{1}}\phantom{\rule{thinmathspace}{0ex}}.$P_n := r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_n}{}_{a_1} \,.

There is also an infinite series of mixed invariant polynomials

${C}_{2n+2}:={\theta }_{{a}_{1}}\wedge {r}^{{a}_{1}}{}_{{a}_{2}}\wedge {r}^{{a}_{2}}{}_{{a}_{3}}\wedge \cdots \wedge {r}^{{a}_{n-1}}{}_{{a}_{n}}\wedge {\theta }^{{a}_{n}}\phantom{\rule{thinmathspace}{0ex}}.$C_{2n + 2} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge \theta^{a_n} \,.

Chern-Simons elements for these are

${B}_{2n+1}:={\theta }_{{a}_{1}}\wedge {r}^{{a}_{1}}{}_{{a}_{2}}\wedge {r}^{{a}_{2}}{}_{{a}_{3}}\wedge \cdots \wedge {r}^{{a}_{n-1}}{}_{{a}_{n}}\wedge {e}^{{a}_{n}}\phantom{\rule{thinmathspace}{0ex}}.$B_{2n + 1} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge e^{a_n} \,.

### Lie algebra valued forms

A Lie algebra-valued form with values in $\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)$

${\Omega }^{•}\left(X\right)←W\left(\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)\right):\left(E,\Omega \right)$\Omega^\bullet(X) \leftarrow W(\mathfrak{iso}(d-1,1)) : (E,\Omega)

is

• a vielbein $E$ on $X$;

• a “spin connection$\Omega$ on $X$.

The curvature 2-form $\left(T,R\right)$ consists of

• the torsion $T=dE+\left[\Omega \wedge E\right]$;

• the Riemannian curvature $R=d\Omega +\left[\Omega \wedge \Omega \right]$.

If the torsion vanishes, then $\Omega$ is a Levi-Civita connection for the metric ${E}^{a}\otimes {E}^{b}{\eta }_{ab}$ defined by $E$.

The volume form is the image of the volume cocycle

${\Omega }^{•}\left(X\right)\stackrel{\left(E,\Omega \right)}{←}W\left(\mathrm{𝔦𝔰𝔬}\left(d-1,1\right)\right)\stackrel{\mathrm{vol}}{←}W\left({b}^{d-1}ℝ\right):\mathrm{vol}\left(E\right)\phantom{\rule{thinmathspace}{0ex}}.$\Omega^\bullet(X) \stackrel{(E,\Omega)}{\leftarrow} W(\mathfrak{iso}(d-1,1)) \stackrel{vol}{\leftarrow} W(b^{d-1} \mathbb{R}) : vol(E) \,.

We have

$\mathrm{vol}\left(E\right)={ϵ}_{{a}_{1}\cdots {a}_{d}}{E}^{{a}_{1}}\wedge \cdots \wedge {E}^{{a}_{d}}\phantom{\rule{thinmathspace}{0ex}}.$vol(E) = \epsilon_{a_1 \cdots a_d} E^{a_1} \wedge \cdots \wedge E^{a_d} \,.

If the torsion vanishes, this is indeed a closed form.

Revised on August 22, 2011 16:43:39 by Urs Schreiber (82.113.99.25)