The ABCD of topological recursion

Abstract

Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in $T^{\ast }V$ 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 ${\mathbb{Z}}_2$-invariant version of the latter. This ${\mathbb{Z}}_2$-invariant version in the case of a semi-simple Frobenius algebra corresponds to the topological recursion of math-ph/0702045.

75 pages, 6 figures ; v2: sl_2 statement corrected, results added on quantum Airy structure for semi-simple Lie algebras. v3: misprints correction. v4: re-sectioning, a bit more on semi-simple Lie algebras. v5: typos correcting, proof of Lemma 6.7 fixed

References in corpus (12)

• Remodeling the B-model
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• Weighted Hurwitz numbers and topological recursion: an overview
• A short overview of the "Topological recursion"
• Weighted Hurwitz numbers and topological recursion
• Higher Airy structures and topological recursion for singular spectral curves
• Super Quantum Airy Structures
• Higher Airy structures, W algebras and topological recursion
• Lecture notes on topological recursion and geometry
• Airy structures for semisimple Lie algebras
• Airy structures and deformations of curves in surfaces

Cited by in corpus (21)

• Topological recursion for Masur-Veech volumes
• Higher Airy structures and topological recursion for singular spectral curves
• Super Quantum Airy Structures
• Higher Airy structures, W algebras and topological recursion
• The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Topological Recursion in The Ramond Sector
• Lecture notes on topological recursion and geometry
• Relations on ${\overline{\mathcal{M}}}_{g,n}$ and the negative $r$-spin Witten conjecture
• Topological recursion for hyperbolic string field theory
• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks
• Spectral curves for hypergeometric Hurwitz numbers
• Large genus behavior of topological recursion
• $\mathcal{N}=1$ Super Topological Recursion
• Topological Recursion, Airy structures in the space of cycles
• Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursions
• The factorial growth of topological recursion
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers
• Seiberg-Witten Theory and Topological Recursion
• A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}({\mathfrak{gl}}_r)$-Algebras