nLab
iterated integral

Contents

Idea

An iterated integral is a expression involving nested integrals, such as

x=a b( y=g(x) h(x)f(x,y)dy)dx, \int_{x=a}^b \bigg(\int_{y=g(x)}^{h(x)} f(x,y) \,\mathrm{d}y\bigg) \,\mathrm{d}x ,

where the grouping parentheses are usually left out. Iterated integrals are the subject of Fubini theorems.

Iterated integrals include nested integration of differential forms of the kind as it appears in the formulation of parallel transport for nonabelian-value connection form (known as the Dyson formula in physics).

Applied to higher degree forms iterated integrals serve to express generalized transgression of differential forms to loop spaces and other mapping spaces relaized as diffeological spaces.

References

The notion of iterated integration of differential forms originates in informal observation like the Dyson formula for parallel transport. It was formalized in the context of differential geometry on diffeological spaces (Chen spaces) (especially loop spaces) in

Essentially these formulas are used in

for expressing higher holonomy of certain flat infinity-connections given by infinity-representations of tangent Lie algebroids.

A review of this and more discussion in the context of a higher Riemann-Hilbert correspondence is in

A relation to algebraic cycles is discussed in

Last revised on August 7, 2017 at 09:27:48. See the history of this page for a list of all contributions to it.