### Chern-Simons Lie-3-Algebra Inside Derivations of String Lie-2-Algebra

#### Posted by Urs Schreiber

The $n$-Category Café started with a discussion of the Lie 3-group underlying 11-dimensional supergravity #. In a followup #, I discussed a semistrict Lie 3-algebra $\mathrm{cs}(g)$ with the property that 3-connections taking values in it $d\mathrm{tra} : \mathrm{Lie}(P_1(X)) \to \mathrm{cs}(g)$ are Chern-Simons 3-forms with values in $g$, giving the local gauge structure of heterotic string backgrounds.

At that time I guessed that $\mathrm{cs}(g)$ is in fact equivalent to the Lie-3-algebra of inner derivations of the $\mathrm{string}_g = (\hat \Omega_k g \to P g)$ Lie-2-algebra, using the fact that there is only a 1-parameter family of possible Lie 3-algebra structures on the underlying 3-vector space.

It would be quite nice if this were indeed true.

While I still have no full proof that $\mathrm{cs}(g)$ is equivalent (tri-equivalent, if you like) to $\mathrm{inn}(\mathrm{string}_g)$, I have now checked at least one half of this statement:

there is a morphism $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g)$ and one going the other way $\mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ such that the composition $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ is the identity on $\mathrm{cs}(g)$:

So at least $\mathrm{cs}(g)$ sits inside $\mathrm{inn}(\mathrm{string}_g)$: $\mathrm{cs}(g) \subset \mathrm{inn}(\mathrm{string}_g) \,.$

The details can be found here:

$\;\;$ Chern-Simons and $\mathrm{string}_G$ Lie-3-algebras

This (rather unpleasant) computation is a generalization of that in the last section of From Loop Groups to 2-Groups, which shows the equivalence $\mathrm{string}_g \simeq g_k$.

I take this as further indication # that the structure 3-group of $G$-Chern-Simons theory is (a subgroup of) $\mathrm{INN}(\mathrm{String}_G)$.

The above notes make use of some previous computations.

The inner automorphism 3-group $\mathrm{INN}(H\to G)$ of any strict Lie 2-group $(H \to G)$ is computed in the first part of these notes on non fake flat surface transport (discussed here). The FDA description of the corresponding Lie 3-algebra is derived in the second part, in terms of the 2- and 3-form curvature of a 2-connection with values in $\mathrm{INN}(H \to G)$.

(This is a general mechanism: the equations defining the graded differential of the FDA that describes some Lie $n$-algebra are precisely the (Bianchi- and other) identities satisfied by a connection with values in that Lie $n$-algebra.)

The details of the FDA description of $\mathrm{cs}(g)$ and $\mathrm{inn}(h\to g)$, together with my notational conventions, can be found in the fda laboratory, example 10 and 12, respectively.

**Introduction**

For any Lie algebra $g$, there is a semistrict Lie-3-algebra $\mathrm{cs}(g)$ such that 3-connections $d\mathrm{tra} : \mathrm{Lie}(P_1\of{X}) \to \mathrm{cs}(g)$ (i.e. algebroid morphisms from the pair algebroid of $X$ to the 3-algebroid $\mathrm{cs}(g)$) are given by a $g$-valued 1-form $A$, its curvature 2-form and its Chern-Simons 3-form $\mathrm{CS}(A) = \langle A \wedge dA\rangle + \frac{1}{3} \langle A \wedge [A \wedge A]\rangle$ on $X$.

Another Lie 3-algebra canonically associated to $g$ is obtained as follows: The semistrict Baez-Crans Lie 2-algebra $g_k$ is equivalent to the strict Lie 2-algebra $\mathrm{string}_g := (\hat \Omega g \to P g) \,.$ For any strict Lie 2-algebra $(r \stackrel{\delta}{\to} s)$, the Lie-3-algebra $\mathrm{inn}(r \stackrel{\delta}{\to} s)$ of its inner derivations is characterized by the fact that 3-connections $d\mathrm{tra} : \mathrm{Lie}(P_1\of{X}) \to \mathrm{inn}(r\to s)$ are given by an $s$-valued 1-form $A$, an $r$-valued 2-form $B$ such that with $\beta := F_A + \delta(B)$ and $H = d_A B$ we have $d_A \beta = \delta(H)$ and $d_A H + \beta\wedge B = 0 \,.$ Had we in addition required that $\beta = 0$, then this would characterize $(r \to s)$ itself. For $r = \hat \Omega_{k} g$ and $s = P g$ we write $\mathrm{string}_g := (\hat \Omega_k g \to P g) \,.$ It is known that $\mathrm{string}_g \simeq g_k \,.$ Here we are after a generalization of this equivalence when passing from $\mathrm{string}_g$ to $\mathrm{inn}(\mathrm{string}_g)$. We fall short of actually proving an equivalence. Instead we construct a morphism $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g)$ and a morphism $\mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ such that $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ is the identity on $\mathrm{cs}(g)$. This is done for $k = -1$.

We will work throughout in terms of the Koszul dual description of semistrict Lie-$n$-algebras. Every Lie-$n$-algebra is encoded in a free differential graded algebra, and morphisms of Lie-$n$-algebras are given by maps between FDAs that are at the same time chain maps and algebra homomorphisms.

## draft commentary on fda lab

For Lie algebras and their generalizations, the graded symmetric algebra version of cochains with field coefficients is actually the Hom dual of the graded symmetric coalgebra version. The advantage of the algebra version, in addition to familiarity, is that properties of a derivation need only be checked on generators.

The graded symmetric coalgebra $S V$ is naturally a

subcoalgebra of the tensor space $T V$ i.e. is spanned by graded symmetric tensors $x_1 \vee x_2 \vee \cdots \vee x_p$ for $x_i in V$ where I use $\vee$ ratrher than $\wedge$ to emphasize the coalgebra aspect. e.g. $x \vee y = x \otimes y \pm y \otimes x$ notice: no factor of $1/2$ needed.Instead of checking a coderivation on cogenerators we check on elements with image in $V$.

So let $g$ be a Lie algebra. Define a codifferential $d: S g \to S g$ with $g$ regarded as of degree 1 by

and extend as a coderivation that means

where the sum is over $i \lt j$ and $x_i$ and $x_j$ are omitted in the $\vee$ if they are bracketed.

to check $d^2 = 0$, we need only check it on $x_1 \vee x_2 \vee x_3$

where it is readily seen to correspond to the Jacobi identity.

Now if $V$ is a dg Lie algebra, i.e. $V = V_0 \oplus V_1 \oplus \cdots$ with a differential derivation $t: V_i \to V_{i-1}$ then regrade $V$ before taking $S$ i.e. $S s V$, where $s$ denotes the shift or if you prefer $S V[\pm 1]$ - I can never remember this notation.

Define $d$ by

and

Now $d^2=0$ follows from jacobi and $t$ being a derivation and $t^2 = 0$.

Modulo an issue about one of the axioms for a Lie crossed module the latter is a dg Lie algebra.

Turning to example 7 of Urs’ fda lab, the additonal term is

The additional verification for $d^2 = 0$ follows from the invariance of $\langle \cdot,\cdot\rangle$

Now Example 8 has a surprise there is a term $d(a_1 \vee a_2 \vee a_3) = b$ in other words, we have an $L_\infty$ algebra with $l_n = 0$ for $n \gt 3$!