The n-Category Café

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.