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The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications, for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves.
The irreducibility of the space of curves of given genus
P. Deligne and D. Mumford · 1969
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Two-dimensional gravity and intersection theory on moduli space
E. Witten · 1990
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Intersection theory on the moduli space of curves and the matrix Airy function
M. Kontsevich · 1992
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On hurwitz numbers and hodge integrals
T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein · 1999
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Hodge integrals and Gromov-Witten theory
C. Faber and R. Pandharipande · 2000
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Graphs on surfaces and their applications
S. K. Lando and A. K. Zvonkin · 2004
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Invariants of algebraic curves and topological expansion
B. Eynard and N. Orantin · 2007
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B. Eynard and N. Orantin · 2007
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Explicit closed algebraic formulas for Orlov–Scherbin n n -point functions
B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin · 2008
Earlier work this paper cites.
Hurwitz numbers, matrix models and enumerative geometry
V. Bouchard and M. Marino · 2008
Cited alongside, same era.
V. Bouchard, A. Klemm, M. Marino, and S. Pasquetti · 2009
Cited alongside, same era.
The laplace transform of the cut-and-join equation and the bouchard-marino conjecture on hurwitz numbers
B. Eynard, M. Mulase, and B. Safnuk · 2009
Cited alongside, same era.
Intersection numbers of spectral curves
B. Eynard · 2011
Cited alongside, same era.
Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin · 2012
Counting Surfaces
B. Eynard · 2016
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Topological recursion for fully simple maps from ciliated maps
G. Borot, S. Charbonnier, and E. Garcia-Failde · 2021
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Analytic theory of higher order free cumulants
G. Borot, S. Charbonnier, E. Garcia-Failde, F. Leid, and S. Shadrin · 2021
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B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin · 2021
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A universal formula for the x − y x-y swap in topological recursion
A. Alexandrov, B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin · 2022
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Cited alongside, same era.
The Laplace Transform, Mirror Symmetry, and the Topological Recursion of Eynard–Orantin
M. Mulase · 2012
Cited alongside, same era.
About the x-y symmetry of the F g F_{g} algebraic invariants
B. Eynard and N. Orantin · 2013
Cited alongside, same era.
Invariants of spectral curves and intersection theory of moduli spaces of complex curves
B. Eynard · 2014
Cited alongside, same era.
A short overview of the ”Topological recursion”
B. Eynard · 2014
Cited alongside, same era.
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Symplectic duality for topological recursion
B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin · 2022
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A simple formula for the x x - y y symplectic transformation in topological recursion
A. Hock · 2022
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On the x x - y y Symmetry of Correlators in Topological Recursion via Loop Insertion Operator
A. Hock · 2022
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