nLab connection on a bundle gerbe

• connection on a bundle

• connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe

• connection on a 3-bundle / connection on a bundle 2-gerbe

• connection on an ∞-bundle

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Idea

A connection on a bundle gerbe is a slight variant of a Cech-realization of a degree 3 Deligne cohomology cocycle.

old content, needs to be polished

Like a connection on a locally trivialized bundle is encoded in a Lie algebra-valued connection $1$-form on $Y$, the connection on a bundle gerbe gives rise to a Lie-algebra valued $2$-form on $Y$ (this shift in degree is directly related to the step from second to third integral cohomology). This $2$-form is sometimes addressed as the curving $2$-form of a bundle gerbe.

But there is more data necessary to describe a connection on a bundle gerbe. The details of the definition – which is evident for line bundle gerbes but more involved for principal bundle gerbes – can be naturally derived from a functorial concept of parallel surface transport, just like connection $1$-forms on bundles can be derived from parallel line transport.

Definitions

For line bundle gerbes

A connection (also known as “connection and curving”) on a line bundle gerbe

$$B \stackrel{p}{\to} Y^{[2]} \stackrel{\to}{\to} Y \stackrel{\pi}{\to} X$$

is

• a 2-form on $Y$

  $$B \in \Omega^2(Y)$$

• a connection $\nabla$ on the line bundle $B \to Y^{[2]}$

• such that

  $$\pi_1^*B \; -\; p_2^*B \;=\; F_\nabla$$

• together with an extension of the bundle gerbe product $\mu$ to an isomorphism

  $$\mu_\nabla \;:\; p_{12}^* (B,\nabla) \;\; \otimes p_{23}^* (B,\nabla) \;\to\; p_{13}^* (B,\nabla)$$

  of line bundles with connection.

Notice that this condition ensures that $d B$ is a $3$-form on $Y$ which agrees on double intersections

$$p_1^* d B \;\; = \;\; p_2^* d B \,.$$

This means that $d B$ actually descends to a 3-form on $X$.

The curvature associated with the connection on a line bundle gerbe is the unique 3-form on $X$

$$H \in \Omega^3(X)$$

such that

$$\pi^* H = d B \,.$$

The deRham class $[H]$ of this 3-form is the image in real cohomology of the class in integral coholomology classifying the bundle gerbe.

For principal bundle gerbes

A connection on a $G$-principal bundle gerbe is

• a $\mathrm{Lie}(G)$-valued 2-form on $Y$

  $$B \in \Omega^2(Y,\mathrm{Lie}(G))$$

• together with a $\mathrm{Lie}(\mathrm{Aut}(G))$-valued 1-form on $Y$

  $$A \in \Omega^1(Y,\mathrm{Lie}(\mathrm{Aut}(G)))$$

• and a certain twisted notion of connection on the $G$-bundle $B$

• satisfying a couple of conditions that reduce to those described above in the case $G = U(1)$.

For the case that $F_{A} + \mathrm{ad} B = 0$, these conditions are nothing but a tetrahedron law on a 2-functor from 2-paths in $Y$ to the category $\Sigma(G\mathrm{BiTor})$. This is discussed in math.DG/0511710.

For the more general case a choice for these conditions that harmonizes with the conditions found for (proper) gerbes with connection by Breen & Messing in math.AG/0106083 has been given by Aschieri, Cantini & Jurčo in
hep-th/0312154.

Surface transport

From a line bundle gerbe with connection one obtains a notion of parallel transport along surfaces in a way that generalizes the procedure for locally trivialized fiber bundles with connection.

Recall that in the case of fiber bundles, the holonomy associated to a based loop $\gamma$ is obtained by

• choosing a triangulation of the loop (i.e., a decomposition into intervals) such that each vertex sits in a double intersection $U_{ij}$ and such that each edge sits in a patch $U_i$

• choosing for each edge a lift into $Y = \sqcup_i U_i$

• choosing for each vertex a lift into $Y^{[2]} = \sqcup_{ij} U_i\cap U_j$

• assigning to each edge lifted to $U_i$ the transport computed from the connection 1-form $a_i$

• assigning to each vertex lifted to $U_i \cap U_j$ the value of the transition function $g_{ij}$ at that point

• multiplying these data in the order given by $\gamma$ .

For bundle gerbes this generalizes to a procedure that assigns a triangulation to a closed surface, that lifts faces, edges, and vertices to single, double and triple intersections, respectively, and which assigns the exponentiated integrals of the 2-form over faces, of the connection 1-form over edges, and assigns the isomorphism $\mu_{ijk}$ to vertices.

For the abelian case (line bundle gerbes) this procedure has been first described in

• K. Gawedzki & N. Reis, WZW branes and Gerbes (arXiv)

based on

• O. Alvarez, Topological quantization and cohomology Commun. Math. Phys. 100 (1985), 279-309.

Further discussion can be found in

• A. Carey, S. Johnson & M. Murray, Holonomy on D-branes, (arXiv)

Gawedzki and Reis showed this way that the Wess-Zumino term in the WZW-model is nothing but the surface holonomy of a (line bundle) gerbe.

In terms of string physics this means that the string (the $2$-particle) couples to the Kalb-Ramond field – which hence has to be interpreted as the connection (“and curving”) of a gerbe – in a way that categorifies the coupling of the electromagnetically charged ($1$-)particle to a line bundle.

The necessity to interpret the Kalb–Ramond field as a connection on a gerbe was originally discussed in

• D. Freed and E. Witten Anomalies in string theory with D-branes, Asian J. Math. 3 (1999), 819-851 (arXiv)

Underlying the Gawedzki–Reis formula is a general mechanism of transition of transport $2$-functors, described in

• Urs Schreiber, Konrad Waldorf, Connections on non-abelian gerbes and their holonomy, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (TAC, arXiv:0808.1923, web)

and similarly in

• Joao Faria Martins, Roger Picken, A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces (arXiv)

This applies to more general situations than ordinary line bundle gerbes with connection.

The generalization (“Jandl gerbes”) to unoriented surfaces (hence to type I strings)

• K. Waldorf, C. Schweigert & U. S., Unoriented WZW Models and Holonomy of Bundle Gerbes (arXiv)

Related concepts

• B-field

• Deligne cohomology

Further references

(…)

In relation to ordinary differential cohomology:

• Byungdo Park, Differential cohomology and gerbes: An introduction to higher differential geometry, lecture notes (2023) [pdf, pdf]

Discussion of transgression and higher holonomy in Deligne cohomology (for bundle gerbes with connection) over orbifolds:

• Ernesto Lupercio, Bernardo Uribe, Holonomy for Gerbes over Orbifolds, J. Geom.Phys. 56 (2006) 1534-1560 [arXiv:math/0307114, doi:10.1016/j.geomphys.2005.08.006]