# 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 $𝔤$ a Lie 2-algebra the 2-groupoid of $𝔤$-valued forms is the 2-groupoid whose objects are differential forms with values in $𝔤$, 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 $\left({G}_{2}\stackrel{\delta }{\to }{G}_{1}\right)$ with action $\alpha :{G}_{1}\to \mathrm{Aut}\left({G}_{2}\right)$. Write $BG$ for the corresponding delooping 2-groupoid, the one coming from the crossed complex

$\left[BG\right]=\left({G}_{2}\stackrel{\delta }{\to }{G}_{1}\stackrel{\to }{\to }*\right)\phantom{\rule{thinmathspace}{0ex}}.$[\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1 \stackrel{\to}{\to} *) \,.

Write $\left[{𝔤}_{2}\stackrel{{\delta }_{*}}{\to }{𝔤}_{1}\right]$ for the corresponding differential crossed module with action ${\alpha }_{*}:{𝔤}_{1}\to \mathrm{der}\left({𝔤}_{2}\right)$

###### Definition

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

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

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

• An object is a pair

$A\in {\Omega }^{1}\left(U,{𝔤}_{1}\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}B\in {\Omega }^{2}\left(U,{𝔤}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$A \in \Omega^1(U,\mathfrak{g}_1)\,, \;\;\; B \in \Omega^2(U,\mathfrak{g}_2) \,.
• A 1-morphism $\left(g,a\right):\left(A,B\right)\to \left(A\prime ,B\prime \right)$ is a pair

$g\in {C}^{\infty }\left(U,{G}_{1}\right)\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}a\in {\Omega }^{1}\left(U,{𝔤}_{2}\right)$g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)

such that

$A\prime ={g}^{-1}Ag+{g}^{-1}dg+{g}^{-1}{\delta }_{*}ag$A' = g^{-1} A g + g^{-1} d g + g^{-1} \delta_* a g

and

$B\prime ={\alpha }_{{g}^{-1}}\left(B+da+\left[a\wedge a\right]+{\alpha }_{*}\left(A\wedge a\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$B' = \alpha_{g^{-1}}( B + d a + [a \wedge a] + \alpha_*(A \wedge a) ) \,.

The composite of two 1-morphisms

$\left(A,B\right)\stackrel{\left({g}_{1},{a}_{1}\right)}{\to }\left(A\prime ,B\prime \right)\stackrel{\left({g}_{2},{a}_{2}\right)}{\to }\left(A″,B″\right)$(A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to} (A'', B'')

is given by the pair

$\left({g}_{1}{g}_{2},{a}_{1}+\left({\alpha }_{{g}_{2}}{\right)}_{*}{a}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(g_1 g_2, a_1 + (\alpha_{g_2})_* a_2) \,.
• a 2-morphism $f:\left(\lambda ,a\right)\to \left(\lambda \prime ,a\prime \right)$ is a function

$f\in {C}^{\infty }\left(U,{G}_{2}\right)$f \in C^\infty(U,G_2)

such that

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

and

$a\prime ={f}^{-1}df+{f}^{-1}af+{f}^{-1}\left({r}_{f}^{-1}\circ {\alpha }_{f}{\right)}_{*}\left(a\right)f$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 $𝔤$ a general Lie 2-algebra.

Let ${𝔤}_{0}$ and ${𝔤}_{1}$ be the two vector spaces involved and let

$\left\{{t}^{a}\right\}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left\{{b}^{i}\right\}$\{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

$\mathrm{CE}\left(𝔤\right)\in {\mathrm{cdgAlg}}_{ℝ}$CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}

with these generators.

We thus have

${d}_{\mathrm{CE}\left(𝔤\right)}{t}^{a}=-\frac{1}{2}{C}^{a}{}_{bc}{t}^{b}\wedge {t}^{c}-{r}^{a}{}_{i}{b}^{i}$d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i
${d}_{\mathrm{CE}\left(𝔤\right)}{b}^{i}=-{\alpha }_{aj}^{i}{t}^{a}\wedge {b}^{j}-{r}_{abc}{t}^{a}\wedge {t}^{b}\wedge {t}^{c}\phantom{\rule{thinmathspace}{0ex}},$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 $\left\{{C}^{a}{}_{bc}\right\}$ (the bracket on ${𝔤}_{0}$) and $\left\{{r}_{a}^{i}\right\}$ (the differential ${𝔤}_{1}\to mathgfrak{g}_{0}$) and $\left\{{alph}^{i}{}_{aj}\right\}$ (the action of ${𝔤}_{0}$ on ${𝔤}_{1}$) and $\left\{{r}_{abc}\right\}$ (the “Jacobiator” for the bracket on ${𝔤}_{0}$).

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

$\left({d}_{\mathrm{CE}\left(𝔤\right)}{\right)}^{2}=0\phantom{\rule{thinmathspace}{0ex}}.$(d_{CE(\mathfrak{g})})^2 = 0 \,.

Over a test space $U$ a $𝔤$-valued form datum is a morphism

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

from the Weil algebra $W\left(𝔤\right)$.

This is given by a 1-form

$A\in {\Omega }^{1}\left(U,{𝔤}_{0}\right)$A \in \Omega^1(U, \mathfrak{g}_0)

and a 2-form

$B\in {\Omega }^{2}\left(U,{𝔤}_{1}\right)\phantom{\rule{thinmathspace}{0ex}}.$B \in \Omega^2(U, \mathfrak{g}_1) \,.

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

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

${H}^{i}={d}_{\mathrm{dR}}{B}^{i}+{\alpha }^{i}{}_{aj}{A}^{a}\wedge {B}^{j}+{t}_{abc}{A}^{a}\wedge {A}^{b}\wedge {A}^{c}\phantom{\rule{thinmathspace}{0ex}}.$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 $\left(A,B\right)$ for which the 2-form curvature

${\delta }_{*}B-dA+\left[A\wedge A\right]=0$\delta_* B - d A + [A \wedge A] = 0

and the 3-form curvature

$dB+\left[A\wedge B\right]=0$d B + [A \wedge B] = 0

vanishes is a resolution of the underlying discrete Lie 2-groupoid $♭BG$ of the Lie 2-groupoid $BG$.

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

###### Proposition

Let ${\Pi }_{2}:\mathrm{CartSp}\to 2\mathrm{LieGrpd}$ be the smooth 2-fundamental groupoid functor and let ${P}_{2}:\mathrm{CartSp}\to 2\mathrm{LieGrpd}$ be the path 2-groupoid functor, taking values in the 2-catgeory $2\mathrm{Grpd}\left(\mathrm{Difeol}\right)$ 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

${\mathrm{Hom}}_{2\mathrm{Grpd}\left(\mathrm{Diffeol}\right)}\left({\Pi }_{2}\left(-\right),BG\right):{\mathrm{CartSp}}^{\mathrm{op}}\to 2\mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}};$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

${\mathrm{Hom}}_{2\mathrm{Grpd}\left(\mathrm{Diffeol}\right)}\left({P}_{2}\left(-\right),BG\right):{\mathrm{CartSp}}^{\mathrm{op}}\to 2\mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}};$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 $\left\{{U}_{i}\to X\right\}$ a good open cover the 2-groupoid, let $X\stackrel{\simeq }{←}C\left(\left\{{U}_{i}\right\}\right)$ be the corresponding Cech nerve smooth 2-groupoid. Then

${\mathrm{Hom}}_{2\mathrm{Grpd}\left(\mathrm{Diffeol}\right)}\left(C\left(\left\{{U}_{i}\right\}\right),\overline{B}G\right)$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

$\mathrm{Hom}\left(C\left(\left\{{U}_{i}\right\}\right),\mathrm{Hom}\left({P}_{2}\left(-\right),BG\right)\right)\simeq \mathrm{Hom}\left({P}_{2}\left(C\left(\left\{{U}_{i}\right\}\right)\right),BG\right)\phantom{\rule{thinmathspace}{0ex}},$Hom( C(\{U_i\}), Hom(P_2(-), \mathbf{B}G)) \simeq Hom(P_2(C(\{\U_i\})), \mathbf{B}G) \,,

where ${P}_{2}\left(C\left(\left\{{U}_{i}\right\}\right)\in 2\mathrm{LieGrpd}$ 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.