connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe
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.
A connection (also known as “connection and curving”) on a line bundle gerbe
is
a 2-form on $Y$
a connection $\nabla$ on the line bundle $B \to Y^{[2]}$
such that
together with an extension of the bundle gerbe product $\mu$ to an isomorphism
of line bundles with connection.
Notice that this condition ensures that $d B$ is a $3$-form on $Y$ which agrees on double intersections
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 $X$
such that
The deRham class $[H]$ of this 3-form is the image in real cohomology of the class in integral coholomology classifying the bundle gerbe.
A connection on a $G$-principal bundle gerbe is
a $\mathrm{Lie}(G)$-valued 2-form on $Y$
together with a $\mathrm{Lie}(\mathrm{Aut}(G))$-valued 1-form on $Y$
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.
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
based on
Further discussion can be found in
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
Underlying the Gawedzki–Reis formula is a general mechanism of transition of transport $2$-functors, described in
and similarly in
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
(…)