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May 16, 2005

PSM and Algebroids, Part V

Posted by urs

[Update: The following has become section 13 of hep-th/0509163.]

Last time I discussed how the functorial definition of a p-bundle with p-connection can locally be differentiated to yield morphisms between p-algebroids. Now I think I have figured out the differential version of the transition law describing the transformation of these algebroid morphisms from one patch to the other. The result is a formalism that allows you to derive the infinitesimal cocylce relations of a nonabelian p-gerbe with curving and connection, etc. using just a couple of elementary steps.

As discussed here a p-bundle with p-connection is completely encoded in a p-holonomy p-functor from a Čech-extended p-path p-groupoid to the structure (gauge) p-group(oid).

When differentiating this statement we obtain a morphism between p-algebroids. These are conveniently handled in their dual incarnation as differential graded algebras (dg-algebras).

The source dg-algebra is essentially just that of the exterior bundle, namely

(1)(d, Γ(T *U)),

where UM is a good cover of the base space. The target dg-algebra (d 𝔤, V *) comes from a complex

(2)𝔤 *d 𝔤𝔥 *( 2 𝔤 *)d 𝔤

where g * and h * are the spaces dual to the Lie algebras g, h, … that describe the target p-group and where d 𝔤 is the dual to D= il̂ i, where l̂ i are the coderivations that define the corresponding L algebra.

Now, the original holonomy functor becomes a connection morphism between these two dg-algebras, i.e. a chain map between the corresponding complexes

(3)con: V * Γ(T *U).

If we write

(4)Q=(d,d 𝔤)

for the operator that acts on

(5) V * Γ(T *U)

as d or d 𝔤, respectively, then the property of being a chain map is equivalent to being Q-closed.

(6)[Q,con]=0 .

An (infinitesimal) gauge transformation between two such connection morphisms is just a chain homotopy


which means that the two connections differ by a Q-exact term



(9)l: V * 1 Γ(T *U),

and the bracket denotes the graded commutator. Let me call this a 1-gauge transformation. This is the differential version of a natural transformation between two p-holonomy p-functors.

But there are higher-order gauge transformations, corresponding to higher order morphisms in the p-groupoid. Given any two (n1 )-gauge transformations l n1 and l˜ n1 we can have an n-gauge transformation going between them

(10)l n1 l nl˜ n1


(11)l˜ n1 l n=[Q,l n],


(12)l n: V * nΓ(T *U),

In other words, gauge equivalence classes of n-morphisms for these p-algebroids represented as dg-algebras are nothing but Q-cohomology classes at grade n.

So now let a global connection be given by local connection morphisms con i on every patch and let them (infinitesimally) be related by 1-gauge transformations

(13)con i𝔤 ijcon j.

Then the differential version of the big diagram in section 3.6.1 of these notes looks as follows:

This says that there is a 2-gauge transformation

(14)𝔣 ijk:𝔤 ij𝔤 jk𝔤 ik

implying that

(15)𝔤 ij+𝔤 jk=𝔤 ik+[Q,𝔣 ijk].

When one works out what this simple equation says in terms of components one ideed finds that it expresses the infinitesimal version of the familiar

(16)t(f ijk)g ik=g ijg jk

as well as the otherwise rather formidable cocycle relations for the ‘2-connection transition function’ in a 2-bundle (1-gerbe) with 2-connection. This is spelled out in section 3.4 of the notes that I mentioned above.

Posted at May 16, 2005 4:05 PM UTC

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