### PSM and Algebroids, Part IV

#### Posted by Urs Schreiber

Last time I discussed how Lie $p$-algebroids (and hence Lie $p$-algebras) and dg-algebras on graded vector spaces of maximal grade $p$ are two aspects of the same thing. This goes a long way towards merging the study of $p$-bundles with $p$-connections with the study of algebroid morphisms as they arise in the Poisson $\sigma $-model, Dirac $\sigma $-models and other field theories.

Here are more details.

In these notes I describe how (weak) principal $p$-bundles with $p$-connection over categorically trivial base $p$-spaces $M$ (and hence also (weakened) ($p-1$)-bundle-gerbes with connection, curving, etc.) are completely encoded in a single $p$-functor

from the *lifted path $p$-groupoid* ${P}_{p}^{C}(U)$
(where $U\to M$ is a good cover) to the weak structure $p$-group(oid) ${G}_{p}$.
This will be called the **global holonomy $p$-functor**.

It is discussed how all the cocylce relations describing a $p$-bundle with $p$-connection transparently follow from the functoriality of ${\mathrm{hol}}_{p}$ and how the $p$-gauge transformations of the $p$-bundle with $p$-connection come from natural transformations between different such global $p$-holonomy functors.

This should be called the **integral picture** and is discussed in section 2.
By going to a ‘differential version’ of the ${\mathrm{hol}}_{p}$-functor
one should arrive at something that should be called the
**differential picture** of a $p$-bundle with $p$-connection, where
the above $p$Â-groupoids are replaced by $p$-algebroids and where the
functor between them becomes an algebroid morphism

from some sort of path $p$-algebroid to the $p$-algebroid ${\U0001d524}_{p}$ coming from the structure $p$-group(oid) ${G}_{p}$.

I do not know yet how to do this
differentiation *globally* (if possible at all), but locally
(i.e. on a given patch ${U}_{i}$ of the good cover $U$)
this should essentially be what Thomas Strobl and collaborators
have been studying,
motivated by studies of the Poisson $\sigma $-model and other
topological field theories.

In section 3 I essentially review aspects this approach, trying to put it in context with the integral picture. I discuss how the consistency conditions known from the integral picture (like the vanishing of the fake curvature) arise in the differential picture and how the notion of gauge transformations (locally) in both pictures coincide.

At the currently, certainly incomplete, level of understanding, one can easily see certain things in one picture but not, or not so easily, in the other. For instance in the intergal picture the global issues related to transitions from one patch to another are very transparent, while they remain to be fully understood in the differential picture. On the other hand, due to the intricacies of weak $p$-group(oid)s it is hard to translate the general diagrams that we discuss into formulas for local data when the structure group is weak and/or really a groupoid. But the analogous generalization, namely from strict $p$-algebras to (not weak but) semistrict $p$-algebroids, is straightforwardly done in the differential picture.