nLab integrable system

Idea

Models of theoretical physics are often given in the form of differential equations. Informally, integrability is the property of a concrete model which enables one to solve these equations in a closed form, or in terms of quadratures (expressions in terms of ordinary integrals).

Classical integrable systems

In classical mechanics, a sort of integrability follows if there are generalized “angle-action” coordinates; the evolution of a system is then of a very special form, hence such “Liouville integrability” (existence of a maximal Poisson-commuting invariants) leads to a special type of global structure of orbits. In addition to finite-dimensional integrable models, many classical PDEs of mathematical physics can be exactly solved, which can often be expressed in terms of infinite-dimensional Hamiltonian systems. There are many methods and approaches which are often simultaneously applicable, and under some assumptions essentially equivalent; we list just keywords: Lax pairs, bihamiltonian systems, spectral curve, classical r-matrix, inverse scattering method, Riemann–Hilbert method and Birkhoff decomposition (which also provide connections to the subject of special functions)…

Much of the knowledge about the algebras of differential operators on curves has been obtained from research on Calogero–Moser systems and generalizations (and their connections to the KP hierarchy). An important class of integrable systems are special cases of Hitchin system.

Quantum integrability

Often classically integrable systems after quantization may be also exactly solved, or show some other form of integrability, but there are no general rules when this is possible. Quantum integrability usually means that we can find a complete system of commuting operators whose common eigenstates will form a basis of the Hilbert space of the model. Many examples are found making manipulations with algebras of operators; as a consequence generalizations of existing models often involve combinatorial data with algebaric meaning; say for example generalized Calogero models related to the root systems. Quantum R-matrices and quantum groups were invented with a view toward quantum integrable systems systematizing analogues of a classical quantum integrability trick called the Bethe Ansatz.

Related entries

• Liouville integrability,

• integrable PDE

• soliton

• spin chain, Toda lattice, Bethe ansatz

• sine-Gordon model, KdV equation, KP equation

• quasitriangular bialgebra, Yangian, quantum group

• Lax equation, spectral curve

• Burchnall-Chaundy theory, argument shift method

• tau-function, Sato Grassmannian

• Calogero model, Dunkl operator

• topological recursion

References

General

Introduction and overview:

• O. Babelon, D. Bernard, M. Talon: Introduction to classical integrable systems, Cambridge Univ. Press 2003.

• V.E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Univ. Press 1997.

• Wikipedia, Integrable system

With focus on the role of the Yang-Baxter equation:

• Michio Jimbo: Yang-Baxter Equation in Integrable Systems, Advanced Series in Mathematical Physics 10, World Scientific (1990) [doi:10.1142/1021]

See also:

• Ron Donagi, Eyal Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, pp. 1-119 in Integrable systems and quantum groups (expanded CIME lectures, Montecatini Terme, 1993), Lecture Notes in Math. 1620 (1996) arXiv:alg-geom/9507017, MR97h:14017

• Eyal Markman, Spectral curves and integrable systems Compositio Math. 93 no. 3 (1994), p. 255-290, numdam

• M.R. Adams, J. Harnad, J. Hurtubise, Integrable Hamiltonian systems on rational coadjoint orbits of loop algebras, Hamiltonian systems, transformation groups and spectral transform methods, Proc. CRM Workshop, Montreal 1989 (1990) pp. 19–32

• Michèle Audin, Hamiltonian systems and their integrability, SMF/AMS Texts and Monographs, vol. 15 (2008)

• A.Levin, M.Olshanetsky, A.Smirnov, A.Zotov, Characteristic classes and integrable systems. General construction, arxiv/1006.0702

• Yuji Kodama, Lauren Williams, KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. 108 (22) 8984–8989 arxiv/1105.4170 doi

• Pierre Van Moerbeke, David Mumford, The spectrum of difference operators and algebraic curves, Acta Math. 143, n. 1, 93–154, MR80e:58028, doi

• B. A. Dubrovin, Completely integrable Hamiltonian systems that are associated with matrix operators, and Abelian varieties, Funkcional. Anal. i Priložen. 11 (1977), no. 4, 28–41, 96, MR0650114

• Boris A. Dubrovin, Integrable systems and Riemann surfaces lecture notes, pdf

  ~dubrovin/rsnleq_web.pdf)

Via semi-holomorphic 4d Chern-Simons theory:

• Kevin Costello, Edward Witten, Masahito Yamazaki, Gauge theory and integrability I, arxiv:1709.09993

Discussion in the context of harmonic maps:

• Shabnam Beheshti, A. Shadi Tahvildar-Zadeh, Integrability and vesture for harmonic maps into symmetric spaces (arXiv:1209.1383)

• Michael Gekhtman, Anton Izosimov, Integrable systems and cluster algebras, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2403.07287]

On using integrable systems for solving combinatorics problems:

• Paul Zinn-Justin, Integrability and combinatorics, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2404.13221]

In gravity

In gravity:

• Dmitry Korotkin, Henning Samtleben, Integrability and Einstein’s Equations, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2311.07900]

In string/M-theory

On integrability in Seiberg-Witten theory:

• Alexander Gorsky, Igor Krichever, Andrei Marshakov, Andrei Mironov, Andrey Morozov, Integrability and Seiberg-Witten Exact Solution, Phys. Lett. B 355 (1995) 466-474 [arXiv:hep-th/9505035, doi:10.1016/0370-2693(95)00723-X]

• Andrei Marshakov, Seiberg-Witten Curves and Integrable Systems [arXiv:hep-th/9903252]

On integrability in topological string theory:

• Mina Aganagic, Robbert Dijkgraaf, Albrecht Klemm, Marcos Marino, Cumrun Vafa, pp. 7 of: Topological Strings and Integrable Hierarchies, Commun. Math. Phys. 261 (2006) 451-516 [arXiv:hep-th/0312085, doi:10.1007/s00220-005-1448-9]

• Mina Aganagic, Miranda Cheng, Robbert Dijkgraaf, Daniel Krefl, Cumrun Vafa, §2 in: Quantum Geometry of Refined Topological Strings, J. High Energ. Phys. 2012 19 (2012) [arXiv:1105.0630, doi:10.1007/JHEP11(2012)019]

On possible structures in M2-brane dynamics and M2-M5-brane bound states which could be M-theory-lifts of the familiar integrability of the Green-Schwarz superstring on $AdS_5$ $\times$ $S^5$:

• Kirill Gubarev, Edvard Musaev, Integrability structures in string theory [arXiv:2301.06486]