The ABCD of topological recursion

Research output: Working paper


  • Jorgen Ellegaard Andersen
  • Gaëtan Borot, Max Planck Institute for Mathematics, GermanyLeonid O. Chekhov, Steklov Mathematical Institute, Moscow, Nicolas Orantin, École Polytechnique Fédérale de Lausanne, France
Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in T*V for some vector space V. KS topological recursion is a procedure which takes as initial data a quantum Airy structure - a family of at most quadratic differential operators on V satisfying some axioms - and gives as outcome a formal series of functions in V (the partition function) simultaneously annihilated by these operators. Finding and classifying quantum Airy structures modulo gauge group action, is by itself an interesting problem which we study here. We provide some elementary, Lie-algebraic tools to address this problem, and give some elements of classification for dim V = 2. We also describe four more interesting classes of quantum Airy structures, coming from respectively Frobenius algebras (here we retrieve the 2d TQFT partition function as a special case), non-commutative Frobenius algebras, loop spaces of Frobenius algebras and a Z2-invariant version of the latter. This Z2-invariant version in the case of a semi-simple Frobenius algebra corresponds to the topological recursion of math-ph/0702045.
Original languageEnglish
StatePublished - 9 Mar 2017

Bibliographical note

75 pages, 6 figures ; v2: sl_2 statement corrected, results added on quantum Airy structure for semi-simple Lie algebras. v3: missprints correction. v4: re-sectioning, a bit more on semi-simple Lie algebras

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