### Bundle Gerbes: Connections and Surface Transport

#### Posted by Urs Schreiber

A continuation of my proposed addition # to the Wikipedia entry on bundle gerbes.

By the way, if anyone reading this is versed in Wikipedia editing and feels like inserting these $n$-Café entries nicely formatted into Wikipedia, I’d very much appreciate it. I can offer the source code on request, if that helps.

**Connection on a bundle gerbe - General.**

Like bundle gerbes are a categorification of transitions in fiber bundles, bundle gerbes with connection are a categorification of transitions in fiber bundles with connection.

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.

**Connection on a bundle gerbe - Definition.**

*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.

*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*

hep-th/0205233

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*

hep-th/0204199.

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.

hep-th/9907189.

Underlying the Gawedzki-Reis formula is a general mechanism of transition of transport 2-functors. This applies to more general situations than ordinary line bundle gerbes with connection.

The generalization to unoriented surfaces (hence to type I strings) was given in

K. Waldorf, C. Schweigert & U. S.

Unoriented WZW Models and Holonomy of Bundle Gerbes

hep-th/0512283.

## Re: Bundle Gerbes: Connections and Surface Transport

Yeah, I can incorporate these into the Wikipedia article. I assume that you wrote these in iTeX; the source for that (or anything else in TeX) would be helpful. Do you want me to wait for them to be hashed out further through discussions here?

I hope that you also learn how to write Wikipedia articles (easy), including using TeX (trickier, due to some bugs and Bad Things in texvc) yourself! But contributing to Wikipedia through me is better than not contributing at all.