nLab
2-groupoid of Lie 2-algebra valued forms

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

\infty-Chern-Weil theory

Contents

Idea

For 𝔤\mathfrak{g} a Lie 2-algebra the 2-groupoid of 𝔤\mathfrak{g}-valued forms is the 2-groupoid whose objects are differential forms with values in 𝔤\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 UU-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 GG corresponding to a Lie crossed module (G 2δG 1)(G_2 \stackrel{\delta}{\to} G_1) with action α:G 1Aut(G 2)\alpha : G_1 \to Aut(G_2). Write BG\mathbf{B}G for the corresponding delooping 2-groupoid, the one coming from the crossed complex

[BG]=(G 2δG 1*). [\mathbf{B}G] = (G_2 \stackrel{\delta}{\to} G_1 \stackrel{\to}{\to} *) \,.

Write [𝔤 2δ *𝔤 1][\mathfrak{g}_2 \stackrel{\delta_*}{\to} \mathfrak{g}_1] for the corresponding differential crossed module with action α *:𝔤 1der(𝔤 2)\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

B¯G:CartSp op2Grpd \bar \mathbf{B}G : CartSp{}^{op} \to 2Grpd

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

  • An object is a pair

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

    gC (U,G 1),aΩ 1(U,𝔤 2) g \in C^\infty(U,G_1)\,,\;\;\; a \in \Omega^1(U,\mathfrak{g}_2)

    such that

    A=g 1Ag+g 1dg+g 1δ *ag A' = g^{-1} A g + g^{-1} d g + g^{-1} \delta_* a g

    and

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

    The composite of two 1-morphisms

    (A,B)(g 1,a 1)(A,B)(g 2,a 2)(A,B) (A,B) \stackrel{(g_1,a_1)}{\to} (A',B') \stackrel{(g_2,a_2)}{\to} (A'', B'')

    is given by the pair

    (g 1g 2,a 1+(α g 2) *a 2). (g_1 g_2, a_1 + (\alpha_{g_2})_* a_2) \,.
  • a 2-morphism f:(λ,a)(λ,a)f : (\lambda,a) \to (\lambda', a') is a function

    fC (U,G 2) f \in C^\infty(U,G_2)

    such that

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

    and

    a=f 1df+f 1af+f 1(r f 1α f) *(a)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\mathfrak{g} a general Lie 2-algebra.

Let 𝔤 0\mathfrak{g}_0 and 𝔤 1\mathfrak{g}_1 be the two vector spaces involved and let

{t a},{b i} \{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(𝔤)cdgAlg CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}

with these generators.

We thus have

d CE(𝔤)t a=12C a bct bt cr a ib 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 CE(𝔤)b i=α aj it ab jr abct at bt c, 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 bc}\{C^a{}_{b c}\} (the bracket on 𝔤 0\mathfrak{g}_0) and {r a i}\{r^i_a\} (the differential 𝔤 1mathgfrakg 0\mathfrak{g}_1 \to \mathgfrak{g}_0) and {alph i aj}\{\alph^i{}_{a j}\} (the action of 𝔤 0\mathfrak{g}_0 on 𝔤 1\mathfrak{g}_1) and {r abc}\{r_{a b c}\} (the “Jacobiator” for the bracket on 𝔤 0\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(𝔤)) 2=0. (d_{CE(\mathfrak{g})})^2 = 0 \,.

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

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

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

This is given by a 1-form

AΩ 1(U,𝔤 0) A \in \Omega^1(U, \mathfrak{g}_0)

and a 2-form

BΩ 2(U,𝔤 1). B \in \Omega^2(U, \mathfrak{g}_1) \,.

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

β a=d dRA a+12C a bcA bA c+r aiB 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 dRB i+α i ajA aB j+t abcA aA bA c. 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)(A,B) for which the 2-form curvature

δ *BdA+[AA]=0 \delta_* B - d A + [A \wedge A] = 0

and the 3-form curvature

dB+[AB]=0 d B + [A \wedge B] = 0

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

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

Proposition

Let Π 2:CartSp2LieGrpd\mathbf{\Pi}_2 : CartSp \to 2LieGrpd be the smooth 2-fundamental groupoid functor and let P 2:CartSp2LieGrpdP_2 : CartSp \to 2LieGrpd be the path 2-groupoid functor, taking values in the 2-catgeory 2Grpd(Difeol)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)(Π 2(),BG):CartSp op2Grpd; 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(),BG):CartSp op2Grpd; 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 XX a paracompact smooth manifold and {U iX}\{U_i \to X\} a good open cover the 2-groupoid, let XC({U i})X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) be the corresponding Cech nerve smooth 2-groupoid. Then

Hom 2Grpd(Diffeol)(C({U i}),B¯G) Hom_{2Grpd(Diffeol)}( C(\{U_i\}), \bar \mathbf{B}G)

is equivalent to the 2-groupoid of GG-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(),BG))Hom(P 2(C({U i})),BG), 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})2LieGrpdP_2(C(\{\U_i\}) \in 2LieGrpd is a resolution of the path 2-groupoid of XX.

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.

See also

differential cohomology in an (∞,1)-topos – survey - connections on 2-bundles.

Revised on October 26, 2012 15:29:07 by Urs Schreiber (80.187.201.44)