I've been studying some gauge theories approach to problems in mechanics in order to get a better understanding of the ideas from gauge theories and to see some applications of fibre bundle theory. The problem is that the articles I've been studying speak quite loosely of things and rarely present some mathematial rigor.
In the moment I'm facing the following situation: let $M$ be a smooth manifold and $c : [0,1]\to M$ a singular $1$-cube in $M$. If $\omega \in \Omega^1(M)$ is a one-form defined in $M$ then the integral of this one-form is perfectly well defined as
$$\int_c \omega = \int_{[0,1]} c^{\ast}\omega.$$
Now on the article I'm reading appears one strange object known as the reverse path ordered exponential, which I've already asked here. The point is that this path ordered exponential, is defined in the article as
$$\bar{P}\exp \left[\int_{t_1}^{t_2}A(t)dt\right]=1+\int_{t_1<t<t_2}A(t)dt+\int_{t_1<t<t'<t_2}A(t)A(t')dtdt' + \cdots$$
and I simply can't understand what those integrals on the right hand side means.
I mean, $A$ is supposed to be the gauge potential. So as far as I know, the gauge potential is the connection one-form of a principal bundle pulled back by a local trivialization. That is, if $G$ is the group of the bundle and $\mathfrak{g}$ is it's Lie algebra, then $A\in \Omega^1(M,\mathfrak{g})$ is a $\mathfrak{g}$-valued one-form.
In that case what the integrals
$$\int_{t_1<t<t_2} A(t)dt$$
$$\int_{t_1<t<t'<t_2}A(t)A(t')dtdt'$$
means rigorously? How they are defined? I've never encountered this kind of integral before, so I'm really not understanding the idea here.
The further I've got was the following: if we are integrating over a singular $1$-cube $c$, then
$$\int_c A = \int_{[t_1,t_2]}c^\ast A,$$
but $c^\ast A$ is a $1$-form on $[t_1,t_2]$ so it is really $c^{\ast} A = \tilde{A} dt$ for some $\tilde{A}$. So in that case
$$\int_c A = \int_{[t_1,t_2]} \tilde{A}dt,$$
so if $A$ on the integral I presented is really $\tilde{A}$ then the first integral makes sense. But the second one, with $A(t)A(t')$ I still cannot understand.
Thanks very much in advance.