## December 23, 2005

### Equivariant Structures on Gerbes

#### Posted by Urs Schreiber

In the last entry I announced our new paper on gerbe holonomy for unoriented surfaces, referring to this as an instance of a (generalized ) notion of equivariant structure on a gerbe. There will be more to say about this, but lest the impression arises that my statement suggests that equivariant structures on gerbes haven’t been considered before, I hasten to emphasize that quite the opposite is true. Here is some selected literature.

It was notably Eric Sharpe who emphasized in a series of papers the fact that the phenomenon known as discrete torsion, which was encountered in the form of certain phase ambiguities in the study of string amplitudes on orbifolds

C. Vafa
Modular Invariance and Discrete Torsion on Orbifolds
Nucl. Phys. B273 (1986) 592-606

C. Vafa & E. Witten
On orbifolds with discrete torsion
J. Geom. Phys. 15 (1995), 189-214

has an interpretation in terms of choices of something like an equivariant structure on the gerbe of the Kalb-Ramond field:

Eric Sharpe
Discrete Torsion
hep-th/0008154

Eric Sharpe
Recent Developments in Discrete Torsion
hep-th/0008191

based on

Eric Sharpe
Discrete Torsion and Gerbes I & II
hep-th/9909108, hep-th/9909120 .

This work was phrased in terms of bundle gerbes and Deligne cocycles. It motivated a long series of papers by Ernesto Lupercio and Bernado Uribe, who gave a description of gerbes on orbifolds using more mathematical language, see for instance

E. Lupercio & B. Uribe
Deligne Cohomology for Orbifolds, Discrete Torsion and B-Fields
hep-th/0201184

E. Lupercio & B. Uribe
An Introduction to Gerbes on Orbifolds
math.DG/0402318

E. Lupercio & B. Uribe
Holonomy for Gerbes over Orbifolds
math.AT/0307114

Lupercio and Uribe mostly use groupoid-theoretic tools to address gerbes on orbifolds. For a good covering by open sets ${U}_{i}$ of an ordinary base manifold, we can construct the Čech-groupoid of the covering, whose objects are points $\left(x,i\right)$ in the covering (if $x\in {U}_{i}$), and which has precisely one morphism

(1)$\left(x,i\right)\to \left(x,j\right)$

if $x\in {U}_{i}\cap {U}_{j}$. A (local trivialization of a) bundle can be understood as a functor from this groupoid to the structure group. Similarly for a gerbe. Either you enlarge the Č-groupoid in a natural way to what I call a Čech-2-groupoid and look at 2-functors from that to a structure 2-group, or, pretty much equivalently, you look at pseudo-functors from the Čech(-1-)groupoid to a 2-group. (Aspects of this I have mentioned here and here).

Either way, there is a natural possibility to generalize this to orbifolds. Let there be a group $K$ acting on base space, choose, for simplicity, a covering which respects this group action and add arrows

(2)$\left(x,i\right)\to \left(\mathrm{kx},i\right)$

to your Čech-groupoid whenever $\mathrm{kx}$ is the image of $x$ under the aciton of some $k\in K$. Then hit this with a 2-/pseudo-functor and obtain a (locally trivialized) gerbe on an orbifold. Roughly.

There is more literature on gerbes on orbifolds, but right now I’ll leave it at that. If anyone feels there is an important paper which should also be mentioned, please drop a comment.

The crucial point of our recent paper, as I said, is that it generalizes equivariance of (bundle) gerbes (and their surface holonomy) from orbifolds to orientifolds. With a comprehensive understanding of ‘equivariance of 2-bundles’, I claim that the orientifold case formalism is unified with the orbifold case formalism.

Posted at December 23, 2005 11:18 AM UTC

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