The n-Category Café

All you need to know is that, following the general reasoning outlined in String- and Chern-Simons $n$-Transport, the String Lie 2-group corresponding to some semisimple group $G$ is something which admits a graded differential algebra of something like left-invariant 1-forms, that looks as follows:

as a graded commutative algebra it is

$$\wedge^\bullet ( s g^* \oplus s s \mathbb{R}^* ) \,,$$

which is supposed to mean that it is the exterior algebra generated freely from ordinary left-invariant differential forms on $G$ together with one single generator in degree 2 (which you might think of either as an auxiliary 2-form, or, better, as a 1-form on a space of morphisms). This extra generator I’ll denote by

$$b \,.$$

A typical homogeneous element in this algebra hence look like

$$\omega \wedge b \wedge b \,,$$

where $\omega$ is any left-invariant $p$-form on $G$.

There is a +1-graded differential $d$ on this algebra, which squares to zero, $d^2 = 0$. It is defined to act on elements $\omega$ entirely built from left-invariant 1-forms on $G$ in the standard way. On the single generator $b$, however, it is defined to act as

$$d b := \mu = H \,.$$

Here the right hand side denotes the canonical 3-cocycle

$$\mu = \langle \cdot, [\cdot,\cdot] \rangle$$

on the semisiple Lie algebra. In order to emphasize that I want to think of this as extended to a left-invariant 3-form on $G$ I’ll equivalently write this as $H$.

Some readers will find this extremely suggestive. If you are not among those, just ignore this and accept the fact that I call the differential of $b$ by the name $H$, and that I assume that $H$ is a closed left invariant 3-form on $G$.

The differential graded commutative algebra which we have obtained this way is the Koszul-dual to the String Lie 2-algebra. More on the general background underlying these considerations can be found in

String- and Chern-Simons $n$-Transport

The general top-degree left invariant form on the String 2-group hence looks like

$$\omega = \omega_n + \omega_{n-2} \wedge b + \frac{1}{2}\omega_{n-4} \wedge b \wedge b + \frac{1}{6}\omega_{n-6} \wedge b \wedge b \wedge b + \cdots \,,$$

where $\omega_{n}$ is supposed to denote a left invariant $n$-form on $G$, for $n$ the dimension of $G$, and similarly for all the other $\omega_k$ that appear.

Hence by expanding a general top degree form on the String 2-group like this, it becomes a mere array of ordinary forms on the ordinary group, one for each even or one for each odd degree. We may similarly look at top-minus-one degree forms this way. But then nothing further is gained by looking at top-minus-two degree and so on.

Hence we find that these forms on the Lie 2-group may be thought of as coming from inhomogeneous but ordinary left invariant forms sitting in a $\mathbb{Z}_2$-grading.

Next we can – easily – work out the differential on the Lie 2-group acting on our forms as above. By using the only two rules we have, which say that

- $d$ acts on the $\omega_k$ as usual

- $d$ acts on the generator $b$ by sending it to the 3-form $H$

we immediately find

$$d \omega = (d \omega_{n-2} + H \wedge \omega_{n-4}) + \frac{1}{2}(\omega_{n-4} + H \wedge \omega_{n-6}) \wedge b + \cdots \,,$$

But this means nothing but that the differential on left-invariant forms on the Lie 2-group acts like the twisted differential

$$d + H \wedge \cdot$$

on $\mathbb{Z}_2$-graded inhomogeneous ordinary left invariant forms on the ordinary group.

Accordingly, the two cohomologies coincide. And the properties of the operator $d + H \wedge \cdot$ have attracted quite some attention.