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B. Ruba, Analyticity of the free energy in quantum Airy structures , J. Phys. A: Math. Theor. 53
Original
1906
Earlier work this paper cites.
V. Bouchard, P. Ciosmak, L. Hadasz, K. Osuga, B. Ruba, and P. Sulkowski, Super quantum Airy structures , Commun. Math. Phys. 380
Original
1907
Earlier work this paper cites.
L. Hadasz and B. Ruba, Airy structures for semisimple Lie algebras , Commun. Math. Phys. 385
Original
1911
Earlier work this paper cites.
G. Hochschild and J-P. Serre, Cohomology of Lie algebras , Ann. Math. Second Series 57
1953
Earlier work this paper cites.
A. Nijenhuis and R.W. Richardson Jr., Cohomology and deformations in graded Lie algebras , Bull. Amer. Math. Soc. 72
1966
Earlier work this paper cites.
J. Patera, P. Winternitz, and H. Zassenhaus, Maximal abelian subalgebras of real and complex symplectic Lie algebras , J. Math. Phys. 24
1973
Earlier work this paper cites.
J.F. Adams, Lectures on Lie groups , Univ. of Chicago Pr., 1982
1982
Earlier work this paper cites.
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Commun. Math. Phys. 147
1992
Earlier work this paper cites.
M. Kontsevich and Yu. Manin, Gromov–Witten classes, quantum cohomology and enumerative geometry , Commun. Math. Phys. 164
1994
Earlier work this paper cites.
L. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras , J. Knot. Th. Ram. 5
1996
Earlier work this paper cites.
B. Dubrovin, Geometry of 2 d 2d topological field theories , Lecture Notes in Mathematics, vol. 1620, pp. 120–348, Springer, Berlin, 1996, hep-th/9407018
1996
Earlier work this paper cites.
A.B. Givental, Gromov–Witten invariants and quantization of quadratic hamiltonians , Mosc. Math. J. 1
2001
Earlier work this paper cites.
C.I. Lazaroiu, On the structure of open-closed topological field theory in two dimensions , Nucl. Phys. B 603
2001
Earlier work this paper cites.
G.W. Moore, Some comments on branes, G-flux and K-theory , Int. J. Mod. Phys. A 16
2001
Earlier work this paper cites.
B. Eynard, All genus correlation functions for the hermitian 1 1 -matrix model , JHEP (2004), no. 0411:031, hep-th/0407261
2004
Earlier work this paper cites.
A. Alexeevski and S. Natanzon, Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves , Sel. Math. New Series 12
2006
Earlier work this paper cites.
L.O. Chekhov, B. Eynard, and N. Orantin, Free energy topological expansion for the 2 2 -matrix model , JHEP (2006), no. 0612:053, math-ph/0603003
2006
Earlier work this paper cites.
V. Bouchard and M. Mariño, Hurwitz numbers, matrix models and enumerative geometry , From Hodge Theory to Integrability and tQFT: tt*-geometry (R. Donagi and K. Wendland, eds.), Proc. Symp. Pure Math., AMS, 2007, math.AG/0709.1458
Original
2007
Earlier work this paper cites.

Similar

V. Bouchard and K. Osuga, 𝒩 = 1 \mathcal{N}=1 super topological recursion , Lett. Math. Phys. 111
Original
2007
Cited alongside, same era.
T. Coates and A.B. Givental, Quantum Riemann–Roch, Lefschetz and Serre , Ann. Math. 165
2007
Cited alongside, same era.
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion , Commun. Numb. Th. Phys. 1
2007
Cited alongside, same era.
M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces , Invent. Math. 167
2007
Cited alongside, same era.
V. Bouchard, A. Klemm, M. Mariño, and S. Pasquetti, Remodeling the B-model , Commun. Math. Phys. 287
Original
2009
Cited alongside, same era.

Then

J.E. Andersen, L.O. Chekhov, P. Norbury, and R.C. Penner, Topological recursion for Gaussian means and cohomological field theories , Theor. Math. Phys. 185
Original
2015
Later among the works it cites.
G. Borot, B. Eynard, and N. Orantin, Abstract loop equations, topological recursion, and applications , Commun. Numb. Th. Phys. 9
Original
2015
Later among the works it cites.
Y. Soibelman, Airy structures and topological recursion , (July 2016), Talk at AMS Symposium "Topological recursion and applications"
2016
Later among the works it cites.
J.E. Andersen, G. Borot, and N. Orantin, Geometric recursion , (2017), math.GT/1711.04729
Original
2017
Closest in time.
P. Dunin-Barkowski, P. Norbury, N. Orantin, A. Popolitov, and S. Shadrin, Primary invariants of Hurwitz–Frobenius manifolds , Topological recursion and its influence in analysis, geometry and topology (C.C.M. Liu and M. Mulase, eds.), vol. 100, 2018, math.AG/1605.07644 , pp. 297–331

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2017

The ABCD of topological recursion

Andersen, Jorgen Ellegaard, Borot, Gaëtan, Chekhov, Leonid O. et al.

Understand

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 ${\rm dim}\,V = 2$.

Built on

B. Ruba, Analyticity of the free energy in quantum Airy structures , J. Phys. A: Math. Theor. 53
Original
1906
Earlier work this paper cites.
V. Bouchard, P. Ciosmak, L. Hadasz, K. Osuga, B. Ruba, and P. Sulkowski, Super quantum Airy structures , Commun. Math. Phys. 380
Original
1907
Earlier work this paper cites.
L. Hadasz and B. Ruba, Airy structures for semisimple Lie algebras , Commun. Math. Phys. 385
Original
1911
Earlier work this paper cites.
G. Hochschild and J-P. Serre, Cohomology of Lie algebras , Ann. Math. Second Series 57
1953
Earlier work this paper cites.
A. Nijenhuis and R.W. Richardson Jr., Cohomology and deformations in graded Lie algebras , Bull. Amer. Math. Soc. 72
1966
Earlier work this paper cites.
J. Patera, P. Winternitz, and H. Zassenhaus, Maximal abelian subalgebras of real and complex symplectic Lie algebras , J. Math. Phys. 24
1973
Earlier work this paper cites.
J.F. Adams, Lectures on Lie groups , Univ. of Chicago Pr., 1982
1982
Earlier work this paper cites.
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Commun. Math. Phys. 147
1992
Earlier work this paper cites.
M. Kontsevich and Yu. Manin, Gromov–Witten classes, quantum cohomology and enumerative geometry , Commun. Math. Phys. 164
1994
Earlier work this paper cites.
L. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras , J. Knot. Th. Ram. 5
1996
Earlier work this paper cites.
B. Dubrovin, Geometry of 2 d 2d topological field theories , Lecture Notes in Mathematics, vol. 1620, pp. 120–348, Springer, Berlin, 1996, hep-th/9407018
1996
Earlier work this paper cites.
A.B. Givental, Gromov–Witten invariants and quantization of quadratic hamiltonians , Mosc. Math. J. 1
2001
Earlier work this paper cites.
C.I. Lazaroiu, On the structure of open-closed topological field theory in two dimensions , Nucl. Phys. B 603
2001
Earlier work this paper cites.
G.W. Moore, Some comments on branes, G-flux and K-theory , Int. J. Mod. Phys. A 16
2001
Earlier work this paper cites.
B. Eynard, All genus correlation functions for the hermitian 1 1 -matrix model , JHEP (2004), no. 0411:031, hep-th/0407261
2004
Earlier work this paper cites.
A. Alexeevski and S. Natanzon, Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves , Sel. Math. New Series 12
2006
Earlier work this paper cites.
L.O. Chekhov, B. Eynard, and N. Orantin, Free energy topological expansion for the 2 2 -matrix model , JHEP (2006), no. 0612:053, math-ph/0603003
2006
Earlier work this paper cites.
V. Bouchard and M. Mariño, Hurwitz numbers, matrix models and enumerative geometry , From Hodge Theory to Integrability and tQFT: tt*-geometry (R. Donagi and K. Wendland, eds.), Proc. Symp. Pure Math., AMS, 2007, math.AG/0709.1458
Original
2007
Earlier work this paper cites.

Similar

V. Bouchard and K. Osuga, 𝒩 = 1 \mathcal{N}=1 super topological recursion , Lett. Math. Phys. 111
Original
2007
Cited alongside, same era.
T. Coates and A.B. Givental, Quantum Riemann–Roch, Lefschetz and Serre , Ann. Math. 165
2007
Cited alongside, same era.
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion , Commun. Numb. Th. Phys. 1
2007
Cited alongside, same era.
M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces , Invent. Math. 167
2007
Cited alongside, same era.
V. Bouchard, A. Klemm, M. Mariño, and S. Pasquetti, Remodeling the B-model , Commun. Math. Phys. 287
Original
2009
Cited alongside, same era.