nLab topological recursion

Idea

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).

Related concepts

• spectral curve

• matrix model

References

Original articles

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

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

• Bertrand Eynard, Counting Surfaces, CRM Aisenstadt Chair lectures, Progress in Mathematical Physics 70, Birkhäuser, 2016

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:

• Nicolas Orantin, Topological Recursion, eight-part lecture series, 2017. youtube (1/8)

The Kontsevich-Soibelman approach (and much more) is reviewed in

• Jørgen Andersen, Gaëtan Borot, Leonid Chekhov, Nicolas Orantin, The ABCD of topological recursion (arXiv:1703.03307)

Discussion in the context of the Hitchin system includes

• David Baraglia, Zhenxi Huang, Special Kähler geometry of the Hitchin system and topological recursion (arXiv:1707.04975)

reviewed in

• David Baraglia, Topological recursion and Bergman tau functions for the Hitchin system, talk at Tau Functions of Integrable Systems and Their Applications, Banff 2018 (recording)

See also

• Gaëtan Borot, Quantum Airy structures and topological recursion talk at Geometry and Physics of Quantum Curves, Banff 2018 (recording)

Relation to topological string and mirror symmetry

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)

Geometric refinement

• Jørgen Andersen, Gaëtan Borot, Nicolas Orantin, Geometric recursion, (arXiv:1711.04729)