nLab
bundle gerbe

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A bundle gerbe is a special model for the total space Lie groupoid of a BU(1)\mathbf{B}U(1)-principal 2-bundle for BU(1)\mathbf{B}U(1) the circle 2-group.

More generally, for GG a more general Lie 2-group (often taken to be the automorphism 2-group G=AUT(H)G = AUT(H) of a Lie group HH), a nonabelian bundle gerbe for GG is a model for the total space groupoid of a GG-principal 2-bundle.

The definition of bundle gerbe is not in fact a special case (nor a generalization) of the definition of gerbe, even though there are equivalences relating both concepts.

Definition

A bundle gerbe’* over a smooth manifold XX is

  • a surjective submersion

    Y π X \array{ Y \\ \downarrow^{\mathrlap{\pi}} \\ X }
  • together with a U(1)U(1)-principal bundle

    L p Y× XY \array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y }

    over the fiber product of YY with itself, i.e.

    L p Y× XY π 2π 1 Y π X, \array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y &\stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}}& Y \\ && \downarrow^{\mathrlap{\pi}} \\ && X } \,,
  • an isomorphism

    μ:π 12 *Lπ 23 *Lπ 13 *L \mu : \pi_{12}^*L \otimes \pi_{23}^*L \to \pi_{13}^* L

    of U(1)U(1)-bundles on Y× XY× XYY \times_X Y \times_X Y

  • such that this satisfies the evident associativity condition on Y× XY× XY× XYY\times_X Y \times_X Y \times_X Y.

Here π 12,π 23,π 13\pi_{12}, \pi_{23}, \pi_{13} are the three maps

Y [3]Y [2] Y^{[3]} \stackrel{\stackrel{\rightarrow}{\rightarrow}}{\rightarrow} Y^{[2]}

in the Cech nerve of YXY \to X.

In a nonabelian bundle gerbe the bundle LL is generalized to a bibundle.

Interpretation

A bundle gerbe may be understood as a specific model for the total space Lie groupoid of a principal 2-bundle.

We first describe this Lie groupoid in

and then describe how this is the total space of a principal 2-bundle in

As a groupoid extension

Give a surjective submersion π:YX\pi : Y \to X, write

C(Y):=(Y× XYY) C(Y) := \left( Y \times_X Y \stackrel{\to}{\to} Y \right)

for the corresponding Cech groupoid. Notice that this is a resolution of the smooth manifold XX itself, in that the canonical projection is a weak equivalence (see infinity-Lie groupoid for details)

C(Y) X. \array{ C(Y) \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.

The data of a bundle gerbe (Y,L,μ)(Y,L,\mu) induces a Lie groupoid P (Y,L,μ)P_{(Y,L,\mu)} which is a BU(1)\mathbf{B}U(1)-extension of C(Y)C(Y), exhibiting a fiber sequence

BU(1)P (Y,L,μ)X. \mathbf{B}U(1) \to P_{(Y,L,\mu)} \to X \,.

This Lie groupoid is the groupoid whose space of morphisms is the total space LL of the U(1)U(1)-bundle

P (Y,L,μ)=(Lπ 2pπ 1pY) P_{(Y,L,\mu)} = \left( L \stackrel{\overset{\pi_1 \circ p}{\to}}{\underset{\pi_2 \circ p}{\to}} Y \right)

with composition given by the composite

L× s,tLπ 12 *L×π 23 3*Lπ 12 *Lπ 23 3*Lμπ 13 *LL. L \times_{s,t} L \stackrel{\simeq}{\to} \pi_{12}^* L \times \pi_{23}^3* L \stackrel{}{\to} \pi_{12}^* L \otimes \pi_{23}^3* L \stackrel{\mu}{\to} \pi_{13}^* L \to L \,.

As the total space of a principal 2-bundle

We discuss how a bundle gerbe, regarded as a groupoid, is the total space of a BU(1)\mathbf{B}U(1)-principal 2-bundles.

Recall from the discussion at principal infinity-bundle that the total GG 2-bundle space PXP \to X classified by a cocycle XBGX \to \mathbf{B} G is simply the homotopy fiber of that cocycle. This we compute now.

(For more along these lines see infinity-Chern-Weil theory introduction. For the analogous nonabelian case see also nonabelian bundle gerbe.)

Proposition

The Lie groupoid P (Y,L,μ)P_{(Y,L,\mu)} defined by a bundle gerbe is in ∞LieGrpd the (∞,1)-pullback

P (Y,L,μ) * X g B 2U(1) \array{ P_{(Y,L,\mu)} &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}^2 U(1) }

of a cocycle [g]H(X,B 2U(1))H 3(X,)[g] \in H(X,\mathbf{B}^2 U(1)) \simeq H^3(X,\mathbb{Z}).

In fact a somewhat stronger statement is true, as shown in the following proof.

Proof

We can assume without restriction that the bundle LL in the data of the bundle gerbe is actually the trivial U(1)U(1)-bundle L=Y× XY×U(1)L = Y \times_X Y \times U(1) by refining, if necessary, the surjective submersion YY by a good open cover. In that case we may identify μ\mu with a U(1)U(1)-valued function

μ:Y× XY× XYU(1) \mu : Y \times_X Y \times_X Y \to U(1)

which in turn we may identify with a smooth 2-anafunctor

C(U) μ B 2U(1) X. \array{ C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,.

From here on the computation is a special case of the general theory of groupoid cohomology and the extensions classified by it.

Then recall from universal principal infinity-bundle that we model the (,1)(\infty,1)-pullbacks that defines principal \infty-bundles in terms of ordinary pullbacks of the universal BU(1)\mathbf{B}U(1)-principal 2-bundle EBU(1)B 2U(1)\mathbf{E}\mathbf{B}U(1) \to \mathbf{B}^2 U(1).

We may model all this in the case at hand in terms of strict 2-groupoips. Then using an evident cartoon-notation we have

B 2U(1)={ cU(1) } \mathbf{B}^2 U(1) = \left\{ \array{ & \nearrow \searrow \\ \bullet &\Downarrow^{\mathrlap{c \in U(1)}}& \bullet \\ & \searrow \nearrow } \right\}

and EBU(1)\mathbf{E}\mathbf{B}U(1) is the 2-groupoid whose morphisms are diagrams

c \array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet }

in B 2U(1)\mathbf{B}^2 U(1) with composition given by horizontal pasting

c 1 c 2 \array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet }

and 2-morphisms are paper-cup diagrams

c k = ck . \array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet \\ & \searrow &\swArrow_{k}& \swarrow \\ && \bullet } \;\;\;\;\; = \;\;\;\;\; \array{ && \bullet \\ & \nearrow &\swArrow_{c k}& \searrow \\ \bullet &&\to&& \bullet } \,.

So EBU(1)\mathbf{E}\mathbf{B}U(1) is the Lie 2-groupoid with a single object, with U(1)U(1) worth of 1-morphisms and unique 2-morphism between these.

From this we read of that

P (Y,L,μ) EBU(1) C(U) μ B 2U(1) X \array{ P_{(Y,L,\mu)} &\to& \mathbf{E} \mathbf{B}U(1) \\ \downarrow && \downarrow \\ C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ X }

is indeed a pullback square (in the category of simplicial presheaves over CartSp). The morphisms of the pullback Lie groupoid are pairs of diagrams

c (x,i) (x,j) \array{ && \bullet \\ & \nearrow &\swArrow_{c}& \searrow \\ \bullet &&\to&& \bullet \\ \\ (x,i) &&\to&& (x,j) }

hence form a trivial U(1)U(1)-bundle over the morphisms of C(U)C(U), and the 2-morphims are pairs consisting of 2-morphisms

(x,j) (x,i) (x,k) \array{ && (x,j) \\ & \nearrow &\swArrow& \searrow \\ (x,i) &&\to&& (x,k) }

in C(U)C(U) and paper-cup diagrams of the form

c 1 c 2 μ ijk(x) = c 1c 2μ ijk(x) \array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet \\ & \searrow &&\swArrow_{\mu_{i j k}(x)}&&& \swarrow } \;\;\;\; = \;\;\;\; \array{ && \bullet \\ & \nearrow &\swArrow_{c_1 c_2 \mu_{i j k}(x)}& \searrow \\ \bullet &&\to&& \bullet }

in B 2U(1)\mathbf{B}^2 U(1), which exhibits indeed the composition operation in P (Y,L,μ)P_{(Y,L,\mu)}.

Examples

Equivariant bundle gerbes over the point

For AG^GA \to \hat G \to G a group extension by an abelian group GG classified by a 2-cocycle cc in group cohomology, which we may think of as a 2-functopr c:BGB 2Ac : \mathbf{B}\mathbf{G} \to \mathbf{B}^2 A, the corresponding fiber sequence

AG^GBABG^BGcB 2A A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to} \mathbf{B}^2 A

exhibits BG^\mathbf{B}\hat G as the bundle gerbe over BG\mathbf{B}G (in equivariant cohomology of the point, if you wish) with Dixmier-Douady class cc.

Tautological bunde gerbe

Let XX be a simply connected smooth manifold and HΩ 3(X) cl,intH \in \Omega^3(X)_{cl, int} a degree 3 differential form with integral periods.

We may think of this a cocycle in ∞-Lie algebroid cohomology

H:TXb 2. H : T X \to b^2 \mathbb{R} \,.

By a slight variant of Lie integration of oo-Lie algebroid cocycles we obtain from this a bundle gerbe on XX by the following construction

  • pick any point x 0Xx_0 \in X;

  • let Y=P *XY = P_* X be the based smooth path space of XX;

  • let LY× XYL \to Y \times_X Y be the U(1)U(1)-bundle which over an element (γ 1,γ 2)(\gamma_1,\gamma_2) in Y× XYY \times_X Y – which is a loop in XX assigns the U(1)U(1)-torsor whose elements are equivalence class of pairs (Σ,c)(\Sigma,c), where Σ\Sigma is a surface cobounding the loop and where cU(1)c \in U(1), and where the equivalence relation is so that for any 3-ball ϕ:D 3X\phi : D^3 \to X cobounding two such surfaces Σ 1\Sigma_1 and Σ 2\Sigma_2 we have that (Σ 1,c 1)(\Sigma_1,c_1) is equivalent to (Σ 2,c 2)(\Sigma_2, c_2) the difference of the labels differs by the integral of the 3-form

    c 2c 1 1= D 3ϕ *H/. c_2 c_1^{-1} = \int_{D^3} \phi^* H \in \mathbb{R}/\mathbb{Z} \,.
  • the composition operation π 12 *Lπ 23 *Lπ 13 *L\pi_{12}^* L \otimes \pi_{23}^* L \to \pi_{13}^* L is loop-wise the evident operation that on loops removes from a figure-8 the inner bit and whch is group multiplication of the labels.

This produces a bundle gerbe whose class in H 3(X,)H^3(X,\mathbb{Z}) has [H][H] as its image in de Rham cohomology.

and

especially

For applications in string theory see also

References

The notion of bundle gerbe as such was introduced in

Early texts also include

(notice that the title here suppresses one “e” intentionally).

A general picture of bundle nn-gerbes (with connection) as circle (n+1)-bundles with connection classified by Deligne cohomology is in

  • Pawel Gajer, Geometry of Deligne cohomology Invent. Math., 127(1):155–207 (1997) (arXiv)

Revised on May 29, 2014 23:58:06 by Urs Schreiber (31.55.15.191)