nLab bundle gerbe

Idea

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

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

• a surjective submersion

  $$\array{ Y \\ \downarrow^{\mathrlap{\pi}} \\ X }$$

• together with a $U(1)$-principal bundle

  $$\array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y }$$

  over the fiber product of $Y$ with itself, i.e.

  $$\array{ L \\ \downarrow^{\mathrlap{p}} \\ Y \times_X Y &\stackrel{\overset{\pi_1}{\rightarrow}}{\underset{\pi_2}{\rightarrow}}& Y \\ && \downarrow^{\mathrlap{\pi}} \\ && X } \,,$$

• an isomorphism

  $$\mu : \pi_{12}^*L \otimes \pi_{23}^*L \to \pi_{13}^* L$$

  of $U(1)$-bundles on $Y \times_X Y \times_X Y$

• such that this satisfies the evident associativity condition on $Y\times_X Y \times_X Y \times_X Y$.

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

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

in the Cech nerve of $Y \to X$.

In a nonabelian bundle gerbe the bundle $L$ 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

• As a groupoid extension

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

• As the total space of a 2-bundle.

As a groupoid extension

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

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

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

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

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

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

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

$$P_{(Y,L,\mu)} = \left( L \stackrel{\overset{\pi_1 \circ p}{\to}}{\underset{\pi_2 \circ p}{\to}} Y \right)$$

with composition given by the composite

$$L \times_{s,t} L \stackrel{\simeq}{\to} \pi_{12}^* L \times \pi_{23}^3* L \stackrel{}{\to} \pi_{12}^* L \otimes \pi_{23}^3* L \stackrel{\mu}{\to} \pi_{13}^* L \to L \,.$$

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 $\mathbf{B}U(1)$-principal 2-bundles.

Recall from the discussion at principal infinity-bundle that the total $G$ 2-bundle space $P \to X$ classified by a cocycle $X \to \mathbf{B} G$ 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 $P_{(Y,L,\mu)}$ defined by a bundle gerbe is in ?LieGrpd? the (∞,1)-pullback

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

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

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

Proof

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

$$\mu : Y \times_X Y \times_X Y \to U(1)$$

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

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

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 $(\infty,1)$-pullbacks that defines principal $\infty$-bundles in terms of ordinary pullbacks of the universal $\mathbf{B}U(1)$-principal 2-bundle $\mathbf{E}\mathbf{B}U(1) \to \mathbf{B}^2 U(1)$.

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

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

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

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

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

$$\array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet }$$

and 2-morphisms are paper-cup diagrams

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

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

From this we read of that

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

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

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

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

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

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

$$\array{ &&& \bullet \\ & \swarrow &\swArrow_{c_1} & \downarrow &\swArrow_{c_2}& \searrow \\ \bullet &\to&& \bullet &&\to& \bullet \\ & \searrow &&\swArrow_{\mu_{i j k}(x)}&&& \swarrow } \;\;\;\; = \;\;\;\; \array{ && \bullet \\ & \nearrow &\swArrow_{c_1 c_2 \mu_{i j k}(x)}& \searrow \\ \bullet &&\to&& \bullet }$$

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

Examples

Equivariant bundle gerbes over the point

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

$$A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to} \mathbf{B}^2 A$$

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

Tautological bundle gerbe

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

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

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

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

• pick any point $x_0 \in X$;

• let $Y = P_* X$ be the based smooth path space of $X$;

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

  $$c_2 c_1^{-1} = \int_{D^3} \phi^* H \in \mathbb{R}/\mathbb{Z} \,.$$

• the composition operation $\pi_{12}^* L \otimes \pi_{23}^* L \to \pi_{13}^* L$ 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 $H^3(X,\mathbb{Z})$ has $[H]$ as its image in de Rham cohomology.

Related entries

• centrally extended groupoid

• line 2-bundle, holomorphic line 2-bundle

• transgression of bundle gerbes

• connection on a bundle gerbe

• discrete torsion

• fusion bundle

and

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle

especially

• circle n-bundle with connection, ordinary differential cohomology.

For applications in string theory see also

• B-field, WZW model.

References

The notion of bundle gerbe as such was introduced in

• Michael Murray, Bundle gerbes, J. Lond. Math. Soc. 54 (1996) pp. 403-416 (arXiv:dg-ga/9407015)

Early texts also include

• David Chatterjee, On Gerbs, 1998 (pdf)

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

• Michael Murray, Danny Stevenson, Bundle gerbes: stable isomorphism and local theory, J.Lond.Math.Soc. 62 (2000) 925-937 arXiv:math/9908135

A general picture of bundle $n$-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:alg-geom/9601025, doi:10.1007/s002220050118)

Reviews are in

• Nigel Hitchin, What is… a gerbe?, Notices of the AMS 50 no. 2 (2003) pp 218-219 pdf

• Michael Murray, An Introduction to Bundle Gerbes, In: The Many Facets of Geometry, A Tribute to Nigel Hitchin, Edited by Oscar Garcia-Prada, Jean Pierre Bourguignon, Simon Salamon, OUP, 2010. doi:10.1093/acprof:oso/9780199534920.001.0001, arXiv:0712.1651

• Severin Bunk, Gerbes in geometry, field Theory, and quantisation, Complex Manifolds 8 1 (2021) 150-182 [arXiv:2102.10406]

With emphasis on surface holonomy:

• Jürgen Fuchs, Thomas Nikolaus, Christoph Schweigert, Konrad Waldorf, Bundle Gerbes and Surface Holonomy [arXiv:0901.2085, spire:1511182]

Bundle gerbes over Lie groups (cf. the basic bundle gerbe):

• Jean-Luc Brylinski, Gerbes on complex reductive Lie groups (arXiv:math/0002158)

• Eckhard Meinrenken, The basic gerbe over a compact simple Lie group, Enseign. Math. 49 3-4 (2003) 307–333 (arXiv:math/0209194, doi:10.5169/seals-66691)

• Christoph Schweigert, Konrad Waldorf, Gerbes and Lie Groups, In: KH. Neeb, A. Pianzola (eds) Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics, vol 288. Birkhäuser Boston 2011 (arXiv:0710.5467, doi:10.1007/978-0-8176-4741-4_10)

• Derek Krepski, Basic equivariant gerbes on non-simply connected compact simple Lie groups, Journal of Geometry and Physics 133 (2018) 30-41 (arXiv:1712.08294, doi:10.1016/j.geomphys.2018.06.016)

On equivariant bundle gerbes?:

with a restrictive notion of equivariance (“strong” equivariance):

• Brylinski 2000, from p. 28

• Varghese Mathai, Danny Stevenson, Chern character in twisted K-theory: equivariant and holomorphic cases, Commun. Math. Phys. 236 161-186, 2003 (arXiv:hep-th/0201010, doi:10.1007/s00220-003-0807-7)

• Meinrenken 2002, Sec. 2.4

• Kiyonori Gomi, Reduction of strongly equivariant bundle gerbes with connection and curving (arXiv:math/0406144)

with a general notion of higher equivariance:

• Kiyonori Gomi, Yuji Terashima, Sec 5 of: Discrete Torsion Phases as Topological Actions, Commun. Math. Phys. (2009) 287: 889 (doi:10.1007/s00220-009-0736-1)

  (this article speaks in terms of Deligne cohomology)

• Thomas Nikolaus, Christoph Schweigert, Equivariance In Higher Geometry, Adv. Math., 226(4): 3367-3408, 2011 (arXiv:1004.4558, doi:10.1016/j.aim.2010.10.016)

• Michael Murray, David Michael Roberts, Danny Stevenson, Raymond Vozzo, Equivariant bundle gerbes, Advances in Theoretical and Mathematical Physics 21 (2017) no. 4 pp 921-975 (arXiv:1506.07931, doi:10.4310/ATMP.2017.v21.n4.a3)

• Krepski 2017