+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Bundles +-- {: .hide} [[!include bundles - contents]] =-- #### Cohomology +--{: .hide} [[!include cohomology - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## 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 <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#BnU1">circle 2-group</a>. 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 [[principal U(1)-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 {#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](#AsGroupoidExtension) and then describe how this is the total space of a principal 2-bundle in * [As the total space of a 2-bundle](#AsTotalSpace). ### As a groupoid extension {#AsGroupoidExtension} 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 {#AsTotalSpace} 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]].) +-- {: .num_prop} ###### 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} ###### 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 omega-groupoid|strict 2-groupoip]]s. 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 <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#IntegrationOfCocycles">Lie integration of oo-Lie algebroid cocycles</a> 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](http://arxiv.org/abs/dg-ga/9407015)) Early texts also include * [[David Chatterjee]], _On Gerbs_, 1998 ([pdf](https://people.maths.ox.ac.uk/hitchin/files/StudentsTheses/chatterjee.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](https://arxiv.org/abs/math/9908135) A general picture of bundle $n$-gerbes (with connection) as [[circle n-bundle with connection|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](http://arxiv.org/abs/alg-geom/9601025), [doi:10.1007/s002220050118](https://doi.org/10.1007/s002220050118)) Reviews are in * [[Nigel Hitchin]], _What is... a gerbe?_, Notices of the AMS **50** no. 2 (2003) pp 218-219 [pdf](http://www.ams.org/notices/200302/what-is.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](http://dx.doi.org/10.1093/acprof:oso/9780199534920.001.0001), arXiv:[0712.1651](https://arxiv.org/abs/0712.1651) * [[Severin Bunk]], _Gerbes in geometry, field Theory, and quantisation_, Complex Manifolds **8** 1 (2021) 150-182 &lbrack;[arXiv:2102.10406](https://arxiv.org/abs/2102.10406)&rbrack; With emphasis on [[surface holonomy]]: * [[Jürgen Fuchs]], [[Thomas Nikolaus]], [[Christoph Schweigert]], [[Konrad Waldorf]], _Bundle Gerbes and Surface Holonomy_ &lbrack;[arXiv:0901.2085](https://arxiv.org/abs/0901.2085), [spire:1511182](https://inspirehep.net/literature/1511182)&rbrack; Bundle gerbes over [[Lie groups]] (cf. the [[basic bundle gerbe]]): * {#Brylinski00} [[Jean-Luc Brylinski]], *Gerbes on complex reductive Lie groups* ([arXiv:math/0002158](https://arxiv.org/abs/math/0002158)) * {#Meinrenken02} [[Eckhard Meinrenken]], *The basic gerbe over a compact simple Lie group*, Enseign. Math. **49** 3-4 (2003) 307--333 ([arXiv:math/0209194](https://arxiv.org/abs/math/0209194), [doi:10.5169/seals-66691](https://dx.doi.org/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](https://arxiv.org/abs/0710.5467), [doi:10.1007/978-0-8176-4741-4_10](https://doi.org/10.1007/978-0-8176-4741-4_10)) * {#Krepski17} [[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](https://arxiv.org/abs/1712.08294), [doi:10.1016/j.geomphys.2018.06.016](https://doi.org/10.1016/j.geomphys.2018.06.016)) {#ReferencesEquivariant} On [[equivariant infinity-bundle|equivariant bundle gerbes]]: with a restrictive notion of equivariance ("strong" equivariance): * [Brylinski 2000, from p. 28](#Brylinski00) * [[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](https://arxiv.org/abs/hep-th/0201010), [doi:10.1007/s00220-003-0807-7](https://doi.org/10.1007/s00220-003-0807-7)) * [Meinrenken 2002, Sec. 2.4](#Meinrenken02) * [[Kiyonori Gomi]], *Reduction of strongly equivariant bundle gerbes with connection and curving* ([arXiv:math/0406144](https://arxiv.org/abs/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](https://doi.org/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](https://arxiv.org/abs/1004.4558), [doi:10.1016/j.aim.2010.10.016](https://doi.org/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](https://arxiv.org/abs/1506.07931), [doi:10.4310/ATMP.2017.v21.n4.a3](https://dx.doi.org/10.4310/ATMP.2017.v21.n4.a3)) * [Krepski 2017](#Krepski17) * [[Krzysztof Gawedzki]]. *Square root of gerbe holonomy and invariants of time-reversal-symmetric topological insulators* (2017). ([arXiv:1702.06179](https://arxiv.org/abs/1702.06179)). [[!redirects bundle gerbes]]