nLab period

General

The word ‘period’ has many meanings in mathematics, most of them coming from physics: the period of an oscillation, periods in celestial mechanics, even the period of a periodic function comes from the intuition that the periodicity is in the time dimension. Functions on tori are periodic in two directions, say the Weierstrass elliptic functions on elliptic curves, so it is not a surprise that more involved kinds of periods came from the study of elliptic curves and then more general Riemann surfaces.

In differential geometry

The period of a closed differential form $\omega \in \Omega^n_{cl}(X)$ over an $n$-cycle $S$ is the integral $\int_S \omega$.

In number theory and algebraic geometry

There is another deep notion of periods in number theory and a more specific version related to specific situations in algebraic geometry. We distinguish irrational and rational numbers; complex numbers divide into algebraic and transcendental. Periods are more general than algebraic numbers: they are those (complex) numbers which can be obtained as integrals of algebraic functions (all of whose coefficients are also algebraic numbers) over semialgebraic sets. The periods form a subring of complex numbers bigger than the field of algebraic numbers. There are several operations which lead to new periods. In fact, if we view them abstractly, as integrals of some abstract function over an abstract semialgebraic set, then we can take unions of such sets, do partial integration and so on. There is a conjecture that there are no relations among periods except those of a short list of such obvious relations!

Periods appear in a number of situations in classical algebraic geometry. Specific matrices of periods are defined and important in the theory of algebraic functions, Hodge theory for algebraic cycles, the study of actions of motivic Galois groups, etc. They come as generalizations of “periods of Riemann surfaces” from 19th century.

References

Overview:

• Claire Voisin, From Analysis situs to the theory of periods, lecture at Inst. Henri Poincaré (2013) [yt]

In differential geometry

• Georges de Rham, p. 135 of: Differentiable Manifolds – Forms, Currents, Harmonic Forms, Grundlehren 266, Springer (1984) [doi:10.1007/978-3-642-61752-2]

In number theory

A general introduction to and discussion of algebraic periods is in

• Maxim Kontsevich, Don Zagier, Periods, pdf

which in section 3 discusses the appearance of periods as special values of L-functions.

A popularization is in

• Stefan Müller-Stach, What is a period ?, to appear in the Notices of the AMS, arxiv/1407.2388

See also

• A. B. Goncharov, Periods and mixed motives, math.AG/0202154

• mathoverflow: ring of periods not a field

• J. Carlson, Phillip Griffith, What is a period domain?, AMS Notices 55 11 (2008) 1418-1419 [pdf, full issue:pdf]

In perturbative quantum field theory

Discussion of motives in physics via periods as appearing in the perturbative quantum field theory, hence in correlators/scattering amplitudes, and their relation to the cosmic Galois group originates in

• Maxim Kontsevich, Operads and motives in deformation quantization, Lett.Math.Phys. 48 (1999) 35–72, math.QA/9904055

More details on this (and a good review of periods in the first place) is in

• Annette Huber, Stefan Müller-Stach, On the relation between Nori motives and Kontsevich periods, 1105.0865

and briefly in section 8.5 of

• Alain Connes, Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives

Discussion in the rigorous Lorentzian context of causal perturbation theory/perturbative AQFT is in

• Kasia Rejzner, Renormalization and periods in perturbative Algebraic Quantum Field Theory (arXiv:1603.02748)

For more see at motives in physics.