nLab iterated integral

Idea

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

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

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 realized as diffeological spaces.

Related concepts

• Dyson formula

• Chen connection

• higher holonomy

• ordinary homology – description in terms of higher linear algebra

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

• 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

• Kiyoshi Igusa, Iterated integrals of superconnections (arXiv:0912.0249)

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

• Jonathan Block, Aaron Smith, A Riemann–Hilbert correspondence for infinity local systems (arXiv:0908.2843)

A relation to algebraic cycles is discussed in

• Richard Hain, Iterated integrals and algebraic cycles: examples and prospects (arXiv:math/0109204)

Chen iterated integrals in the context of stochastic analysis

• Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 2 (1998) 215–310 [pdf]
• Terry J. Lyons, Michael Caruana, Thierry Lévy, Differential equations driven by rough paths, Springer LNM 1908 (2007)

Kapranov’s work on “noncommutative Fourier transform” uses some ideas from Chen’s work:

• Mikhail Kapranov, Noncommutative geometry and path integrals, In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics 270, doi arXiv:math.QA/0612411

• Mikhail Kapranov, Free Lie algebroids and the space of paths, Sel. math., New ser. 13, 277 (2007) doi arXiv:math:AG/0702584

• Mikhail Kapranov, Membranes and higher groupoids, arXiv:1502.06166

Iterated sums and iterated integrals over semirings, where the case of tropical semiring is a central, with applications (including in machine learning):

• Joscha Diehl, Kurusch Ebrahimi-Fard, Nikolas Tapia, Tropical time series, iterated-sums signatures, and quasisymmetric functions, SIAM Journal on Applied Algebra and Geometry 6:4 (2022) arxiv:2009.08443 doi