Topological recursion (Chekhov-Eynard-Orantin 06,Eynard-Orantin 07) is a universal recursion formula that controls asymptotic expansion of many integrable systems such as matrix models or the Hitchin system (Baraglia-Huang 17).
Maxim Kontsevich and Yan Soibelman reformulated and slightly generalised it, seeing it as a quantization of certain quadratic Lagrangians in the cotangent bundle of some vector space (reviewed in Andersen-Borot-Chekhov-Orantin 17, section 2).
There is some deep relation to the topological string B-model and Gromov-Witten invariants (Bouchard-Klemm-Marino-Pasquetti 09). It yields proofs of mirror symmetry in certain cases, valid at all genera (Eynard-Orantin 12, Fang-Liu-Zong 13).
A geometric refinement of topological recursion is known as geometric recursion and developed in (Andersen-Borot-Orantin 17).
The method was introduced in
Leonid Chekhov, Bertrand Eynard, Nicolas Orantin, Free energy topological expansion for the 2-matrix model, JHEP 0612:053,2006 (arXiv:math-ph/0603003)
Bertrand Eynard, Nicolas Orantin, Invariants of algebraic curves and topological expansion (arXiv:math-ph/0702045)
Review and exposition includes
Bertrand Eynard, Nicolas Orantin, Topological recursion in random matrices and enumerative geometry, J. Phys. A: Mathematical and Theoretical , 42(29), 2009 (arXiv:0811.3531)
Bertrand Eynard, A short overview of the “Topological recursion” (arXiv:1412.3286)
Gaëtan Borot, Lecture notes on topological recursion and geometry, (arXiv:1705.09986)
A textbook developing various topics related to topological recursion (in particular, discussing the links to enumeration of maps) is
A series of lectures on topological recursion were delivered by Nicolas Orantin during the Institut Henri Poincaré‘s thematic trimester “Combinatorics and interactions”, and are available online:
The Kontsevich-Soibelman approach (and much more) is reviewed in
Discussion in the context of the Hitchin system includes
reviewed in
See also
The relation to the topological string B-model, Gromov-Witten invariants and mirror symmetry is due to
Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178, 2009 (arXiv:0709.1453)
Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string I: Deriving the matrix model (arXiv:1003.1737)
Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string II: The spectral curve and mirror geometry (arXiv:1007.2194)
Bertrand Eynard, Nicolas Orantin, Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture (arXiv:1205.1103)
Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds (arXiv:1310.4818)