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

$${\wedge }^{•}(s{g}^{*}\oplus ss{ℝ}^{*}),$$

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

$$db:=\mu =H.$$

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.