∞-Lie theory

Contents

Idea

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

The corresponding Lie group is the Poincaré group.

Definition

The CE-algebra

The Chevalley-Eilenberg algebra $CE(\mathfrak{iso}(d-1,1))$ is generated from $\mathbb{R}^{d,1}$ and $\wedge^2 \mathbb{R}^{d,1}$. For $\{t_a\}$ the standard basis of $\mathbb{R}^{d-1,1}$ we write $\{\omega^{a b}\}$ and $\{e^a\}$ for these generators. With $(\eta_{a b})$ the components of the Minkowski metric we write

$\omega^{a}{}_b := \omega^{a c}\eta_{c b} \,.$

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

$d_{CE} : \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}$
$d_{CE} : e^a \mapsto \omega^{a}{}_b \wedge t^b$

Properties

Cohomology

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

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

$\omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a \in CE(\mathfrak{iso}(d-1,1)) \,.$

The volume cocycle is the volume form

$vol = \epsilon_{a_1 \cdots a_{d}} e^{a_1} \wedge \cdots \wedge e^{a_d} \in CE(\mathfrak{iso}(d-1,1)) \,.$

Invariant polynomials and Chern-Simons elements

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

We have the Bianchi identity

$d_W : r^{a b} \mapsto \omega^{a c} \wedge R_c{}^d - R^{a c} \wedge \omega_c{}^b$

and

$d_W : \theta^a \mapsto \omega^a{}_b \theta^b - R^{a}{}_b e^b \,.$

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

Another invariant polynomial is $r^{a b} \wedge r_{a b}$. This is the Killing form of $\mathfrak{so}(d-1,1)$. 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

$P_n := r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_n}{}_{a_1} \,.$

There is also an infinite series of mixed invariant polynomials

$C_{2n + 2} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge \theta^{a_n} \,.$

Chern-Simons elements for these are

$B_{2n + 1} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge e^{a_n} \,.$

Lie algebra valued forms

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

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

is

• a vielbein $E$ on $X$;

• a “spin connection$\Omega$ on $X$.

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

• the torsion $T = d E + [\Omega \wedge E]$;

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

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

The volume form is the image of the volume cocycle

$\Omega^\bullet(X) \stackrel{(E,\Omega)}{\leftarrow} W(\mathfrak{iso}(d-1,1)) \stackrel{vol}{\leftarrow} W(b^{d-1} \mathbb{R}) : vol(E) \,.$

We have

$vol(E) = \epsilon_{a_1 \cdots a_d} E^{a_1} \wedge \cdots \wedge E^{a_d} \,.$

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)