Modeling bundle-valued forms on the path space with a curved iterated integral

TL;DR

This paper develops a new algebraic framework using curved differential graded algebras and a zigzag construction to extend Chen's iterated integral to bundle-valued forms on path spaces, establishing a homotopy equivalence.

Contribution

It introduces a curved version of the Hochschild bar construction and Chen's iterated integral for bundle-valued forms, expanding the algebraic tools for path space analysis.

Findings

Defined a zigzag algebra for curved differential graded algebras.

Constructed a curved Chen iterated integral incorporating parallel transport.

Proved the integral is a homotopy equivalence of curved dg algebras.

Abstract

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.