Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
An iterated integral is a expression involving nested integrals, such as
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.
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
Kuo Tsai Chen, Iterated integrals of differential forms and loop space homology, Ann. Math. 97 (1973), 217–246. JSTOR
Kuo Tsai Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 83–98. doi:10.1090/S0002-9947-1975-0377960-0
Kuo Tsai Chen, Iterated path integrals. Bull. Amer. Math. Soc., 83(5):831-879, 1977 doi:10.1090/S0002-9904-1977-14320-6
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 June 4, 2020 at 13:54:08. See the history of this page for a list of all contributions to it.