The String Coffee Table

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

where $U\to M$ is a good cover of the base space. The target dg-algebra $(d^{𝔤},{\bigwedge }^{•}{V}^{*})$ comes from a complex

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 $D={\sum }_i{\hat{l}}_i$, where ${\hat{l}}_i$ are the coderivations that define the corresponding $L_{\mathrm{\infty }}$ algebra.

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

If we write

for the operator that acts on

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

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

where

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 $(n-1)$-gauge transformations $l_{n-1}$ and ${\tilde{l}}_{n-1}$ we can have an $n$-gauge transformation going between them

iff

where

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 ${\mathrm{con}}_i$ on every patch and let them (infinitesimally) be related by 1-gauge transformations

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

implying that

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

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.