nLab
topological recursion

Contents

Contents

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).

References

Original articles

The method was introduced in

Review and exposition

Review and exposition includes

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:

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

Discussion in the context of the Hitchin system includes

reviewed in

See also

Relation to topological string and mirror symmetry

The relation to the topological string B-model, Gromov-Witten invariants and mirror symmetry is due to

Geometric refinement

Last revised on September 10, 2018 at 12:30:41. See the history of this page for a list of all contributions to it.