nLab
connection on a bundle gerbe

Context

\infty-Chern-Weil theory

Differential cohomology


Contents

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 11-form on YY, the connection on a bundle gerbe gives rise to a Lie-algebra valued 22-form on YY (this shift in degree is directly related to the step from second to third integral cohomology). This 22-form is sometimes addressed as the curving 22-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 11-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

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

is

  • a 2-form on YY

    BΩ 2(Y) B \in \Omega^2(Y)
  • a connection \nabla on the line bundle BY [2]B \to Y^{[2]}

  • such that

    π 1 *Bp 2 *B=F \pi_1^*B \; -\; p_2^*B \;=\; F_\nabla
  • together with an extension of the bundle gerbe product μ\mu to an isomorphism

    μ :p 12 *(B,)p 23 *(B,)p 13 *(B,) \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 dBd B is a 33-form on YY which agrees on double intersections

p 1 *dB=p 2 *dB. p_1^* d B \;\; = \;\; p_2^* d B \,.

This means that dBd B actually descends to a 3-form on XX.

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

HΩ 3(X) H \in \Omega^3(X)

such that

π *H=dB. \pi^* H = d B \,.

The deRham class [H][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 GG-principal bundle gerbe is

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

    BΩ 2(Y,Lie(G)) B \in \Omega^2(Y,\mathrm{Lie}(G))
  • together with a Lie(Aut(G))\mathrm{Lie}(\mathrm{Aut}(G))-valued 1-form on YY

    AΩ 1(Y,Lie(Aut(G))) A \in \Omega^1(Y,\mathrm{Lie}(\mathrm{Aut}(G)))
  • and a certain twisted notion of connection on the GG-bundle BB

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

For the case that F A+adB=0F_{A} + \mathrm{ad} B = 0, these conditions are nothing but a tetrahedron law on a 2-functor from 2-paths in YY to the category Σ(GBiTor)\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 γ\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 ijU_{ij} and such that each edge sits in a patch U iU_i

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

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

  • assigning to each edge lifted to U iU_i the transport computed from the connection 1-form a ia_i

  • assigning to each vertex lifted to U iU jU_i \cap U_j the value of the transition function g ijg_{ij} at that point

  • multiplying these data in the order given by γ\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 μ ijk\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 22-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 (11-)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 22-functors, described in

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 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 (arXiv)

References

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Revised on April 25, 2014 02:27:28 by Urs Schreiber (89.204.137.64)