# nLab 2-groupoid of Lie 2-algebra valued forms

∞-Lie theory

## Examples

### $\infty$-Lie algebras

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

For $\mathfrak{g}$ a Lie 2-algebra the 2-groupoid of $\mathfrak{g}$-valued forms is the 2-groupoid whose objects are differential forms with values in $\mathfrak{g}$, whose morphisms are gauge transformations between these, and whose 2-morphisms are higher order gauge transformations of those.

This naturally refines to a non-concrete Lie 2-groupoid is the 2-truncated ∞-Lie groupoid whose $U$-parameterized smooth families of objects are smooth differential forms with values in a Lie 2-algebra, and whose morphisms are gauge transformations of these.

This is the higher category generalization of the groupoid of Lie-algebra valued forms.

A cocycle with coefficients in this 2-groupoid is a connection on a 2-bundle.

## Definition

### For strict Lie 2-algebras

Consider a Lie strict 2-group $G$ corresponding to a Lie crossed module $(G_2 \stackrel{\delta}{\to} G_1)$ with action $\alpha : G_1 \to Aut(G_2)$. Write $\mathbf{B}G$ for the corresponding delooping 2-groupoid, the one coming from the crossed complex

$[\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1 \stackrel{\to}{\to} *) \,.$

Write $[\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1]$ for the corresponding differential crossed module with action $\alpha_* : \mathfrak{g}_1 \to der(\mathfrak{g}_2)$

###### Definition

The 2-groupoid of Lie 2-algebra valued forms is defined to be the 2-stack

$\bar \mathbf{B}G : CartSp{}^{op} \to 2Grpd$

which assigns to $U \in CartSp$ the following 2-groupoid:

• An object is a pair

$A \in \Omega^1(U,\mathfrak{g}_1)\,, \;\;\; B \in \Omega^2(U,\mathfrak{g}_2) \,.$
• A 1-morphism $(g,a) : (A,B) \to (A',B')$ is a pair

$g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)$

such that

$A' = g^{-1} A g + g^{-1} d g + g^{-1} \delta_* a g$

and

$B' = \alpha_{g^{-1}}( B + d a + [a \wedge a] + \alpha_*(A \wedge a) ) \,.$

The composite of two 1-morphisms

$(A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to} (A'', B'')$

is given by the pair

$(g_1 g_2, a_1 + (\alpha_{g_2})_* a_2) \,.$
• a 2-morphism $f : (\lambda,a) \to (\lambda', a')$ is a function

$f \in C^\infty(U,G_2)$

such that

$g' = \delta(f)^{-1} \cdot g$

and

$a' = f^{-1} d f + f^{-1} a f + f^{-1}(r_f^{-1} \circ \alpha_f)_*(a)f$

and composition is defined as follows

(…)

### For general Lie 2-algebras

We consider now $\mathfrak{g}$ a general Lie 2-algebra.

Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let

$\{t^a\} \,, \;\;\; \{b^i\}$

be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general Chevalley-Eilenberg algebra

$CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}$

with these generators.

We thus have

$d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i$
$d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j - r_{a b c} t^a \wedge t^b \wedge t^c \,,$

for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathgfrak{g}_0$) and $\{\alph^i{}_{a j}\}$ (the action of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the “Jacobiator” for the bracket on $\mathfrak{g}_0$).

These constants are subject to constraints (the weak Jacobi identity and its higher coherence laws) which are precisely equivalent to the condition

$(d_{CE(\mathfrak{g})})^2 = 0 \,.$

Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism

$\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,B)$

from the Weil algebra $W(\mathfrak{g})$.

This is given by a 1-form

$A \in \Omega^1(U, \mathfrak{g}_0)$

and a 2-form

$B \in \Omega^2(U, \mathfrak{g}_1) \,.$

The curvature of this is $(\beta, H)$, where the 2-form component (“fake curvature”) is

$\beta^a = d_{dR} A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c + r^a{}i B^i$

and whose 3-form component is

$H^i = d_{dR} B^i + \alpha^i{}_{a j} A^a \wedge B^j + t_{a b c} A^a \wedge A^b \wedge A^c \,.$

## Properties

###### Proposition

(flat Lie 2-algebra valued forms)

The full sub-2-groupoid on flat Lie 2-algebra valued forms, i.e. those pairs $(A,B)$ for which the 2-form curvature

$\delta_* B - d A + [A \wedge A] = 0$

and the 3-form curvature

$d B + [A \wedge B] = 0$

vanishes is a resolution of the underlying discrete Lie 2-groupoid $\mathbf{\flat} \mathbf{B}G$ of the Lie 2-groupoid $\mathbf{B}G$.

This is discussed at ∞-Lie groupoid in the section strict Lie 2-groups – differential coefficients.

###### Proposition

Let $\mathbf{\Pi}_2 : CartSp \to 2LieGrpd$ be the smooth 2-fundamental groupoid functor and let $P_2 : CartSp \to 2LieGrpd$ be the path 2-groupoid functor, taking values in the 2-catgeory $2Grpd(Difeol)$ of 2-groupoids internalization to diffeological spaces. Then

• the 2-groupoid of Lie 2-algebra valued forms for which both 2- and 3-form curvature vanish is canonically equivalent to

$Hom_{2Grpd(Diffeol)}(\Pi_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd \,;$
• the 2-groupoid of Lie 2-algebra valued forms for which the 2-form curvature vanishes is canonically equivalent to

$Hom_{2Grpd(Diffeol)}(P_2(-), \mathbf{B}G) : CartSp^{op} \to 2Grpd \,;$

The equivalence is given by 2-dimensional parallel transport. A proof is in SchrWalII.

The following proposition asserts that the Lie 2-groupoid of Lie 2-algebra valued forms is the coefficient object for for differential nonabelian cohomology in degree 2, namely for connections on principal 2-bundles and in particular on gerbes.

###### Proposition

(2-bundles with connection)

For $X$ a paracompact smooth manifold and $\{U_i \to X\}$ a good open cover the 2-groupoid, let $X \stackrel{\simeq}{\leftarrow} C(\{U_i\})$ be the corresponding Cech nerve smooth 2-groupoid. Then

$Hom_{2Grpd(Diffeol)}( C(\{U_i\}), \bar \mathbf{B}G)$

is equivalent to the 2-groupoid of $G$-principal 2-bundles with 2-connection.

This is discussed and proven in SchrWalII for the case where the 2-form curvature is restricted to vanish. In this case the above can be written as

$Hom( C(\{U_i\}), Hom(P_2(-), \mathbf{B}G)) \simeq Hom(P_2(C(\{\U_i\})), \mathbf{B}G) \,,$

where $P_2(C(\{\U_i\}) \in 2LieGrpd$ is a resolution of the path 2-groupoid of $X$.

## References

The 2-groupoid of Lie 2-algebra valued forms described in definition 2.11 of

• Schreiber, Waldorf, Smooth functors versus differential forms (web).

There are many possible conventions. The one reproduced above is supposed to describe the bidual opposite 2-category of the 2-groupoid as defined in that article, with the direction of 1- and 2-morphisms reversed.