Log topological recursion through the prism of $x-y$ swap
Alexander Alexandrov, Boris Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, Sergey Shadrin
TL;DR
This work extends topological recursion to settings with logarithmic singularities by introducing Logarithmic Topological Recursion (LogTR), which preserves the core x–y swap symmetry in a generalized framework. LogTR sits inside blobbed topological recursion and uses a logarithmic projection principle to control singularities, enabling a unified treatment of diverse enumerative problems. The authors prove a universal LogTR x–y swap theorem, provide a recomputation procedure linking LogTR to the original TR, and illustrate the framework through Lambert Hodge integrals, Hurwitz numbers, BMS numbers, and Theta/kappa insertions, among others. They also discuss KP integrability and nontrivial duals (Family III), highlighting the broad applicability of LogTR to modern enumerative geometry and potential new ELSV-type formulas. Overall, the paper significantly broadens the reach of TR methods to logarithmic geometries and solidifies the theoretical underpinnings of the x–y duality in this extended setting.
Abstract
We introduce a new concept of logarithmic topological recursion that provides a patch to topological recursion in the presence of logarithmic singularities and prove that this new definition satisfies the universal $x-y$ swap relation. This result provides a vast generalization and a proof of a very recent conjecture of Hock. It also uniformly explains (and conceptually rectifies) an approach to the formulas for the $n$-point functions proposed by Hock.