  We introduce a new variant of Hochschild's two-sided bar construction for the
setting of curved differential graded algebras. One can geometrically think of
the classical bar complex as elements from the algebra positioned along
different points in the closed interval $[0,1]$. In this paper, we start with a
curved differential graded algebra and define a new ``zigzag algebra'' that,
informally, consists of algebra elements arranged on a zigzag of intervals
going back and forth between $0$ and $1$. We focus on two curved differential
graded algebras: the de Rham algebra of differential forms with values in the
endomorphism bundle associated to a vector bundle with connection, and its
induced zigzag algebra. We define a curved version of Chen's iterated integral
that incorporates parallel transport and maps this zigzag algebra of
bundle-valued forms to bundle-valued forms on the path space. This iterated
integral is proven to be a homotopy equivalence of curved differential graded
algebras, and for real-valued forms it factors through the usual Chen iterated
integral.

Algebraic Topology