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Primary invariants of Hurwitz Frobenius manifolds
P. 297–331.
Hurwitz spaces parameterizing covers of the Riemann sphere can
be equipped with a Frobenius structure. In this review, we recall the con-
struction of such Hurwitz Frobenius manifolds as well as the correspondence
between semisimple Frobenius manifolds and the topological recursion formal-
ism. We then apply this correspondence to Hurwitz Frobenius manifolds by
explaining that the corresponding primary invariants can be obtained as pe-
riods of multidifferentials globally defined on a compact Riemann surface by
topological recursion. Finally, we use this construction to reply to the follow-
ing question in a large class of cases: given a compact Riemann surface, what
does the topological recursion compute?
In book
Vol. 100: Topological Recursion and its Influence in Analysis, Geometry, and Topology. , Providence: American Mathematical Society, 2018.
Alexandrov A., Bychkov B., Dunin-Barkowski P. et al., Selecta Mathematica, New Series 2026 Vol. 32 Article 25
We revise the notion of the blobbed topological recursion by extending it to the setting of generalized topological recursion as well as allowing blobs which do not necessarily admit topological expansion. We show that the so-called non-perturbative differentials form a special case of this revisited version of blobbed topological recursion. Furthermore, we prove the KP ...
Added: April 23, 2026
Alexandrov A., Bychkov B., Dunin-Barkowski P. et al., Selecta Mathematica, New Series 2025 Vol. 31 Article 42
We discuss a universal relation that we call the x-y swap relation, which plays a prominent role in the theory of topological recursion, Hurwitz theory, and free probability theory. We describe in a very precise and detailed way the interaction of the x-y swap relation and KP integrability. As an application, we prove a recent conjecture ...
Added: May 27, 2025
Alexandrov A., Bychkov B., Dunin-Barkowski P. et al., Communications in Mathematical Physics 2025 Vol. 406 Article 94
We use the theory of x-y duality to propose a new definition/construction for the correlation differentials of topological recursion; we call it generalized topological recursion. This new definition coincides with the original topological recursion of Chekhov–Eynard–Orantin in the regular case and allows, in particular, to get meaningful answers in a variety of irregular and degenerate ...
Added: May 27, 2025
Alexandrov A., Bychkov B., Dunin-Barkowski P. et al., Journal of the European Mathematical Society 2025 P. 1–62
We prove a recent conjecture of Borot et al. that a particular universal closed algebraic formula recovers the correlation differentials of topological recursion after the swap of x and y in the input data. We also show that this universal formula can be drastically simplified (as it was already done by Hock). As an application ...
Added: May 27, 2025
Alexandrov A., Bychkov B., Dunin-Barkowski P. et al., Communications in Number Theory and Physics 2024 Vol. 18 No. 4 P. 795–841
We review the notion of symplectic duality earlier introduced in the context of topological recursion. We show that the transformation of symplectic duality can be expressed as a composition of x-y dualities in a broader context of log topological recursion. As a corollary, we establish nice properties of symplectic duality: various convenient explicit formulas, invertibility, ...
Added: March 11, 2025
Alexandrov A., Bychkov Boris, Dunin-Barkowski Petr et al., International Mathematics Research Notices 2024 Vol. 2024 No. 21 P. 13461–13487
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 ...
Added: March 11, 2025
Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., Transactions of the American Mathematical Society 2025 Vol. 378 No. 2 P. 1001–1054
We consider weighted double Hurwitz numbers, with the weight given by arbitrary rational function times an exponent of the completed cycles. Both special singularities are arbitrary, with the lengths of cycles controlled by formal parameters (up to some maximal length on both sides), and on one side there are also distinguished cycles controlled by degrees ...
Added: March 11, 2025
Bychkov B., Dunin-Barkowski P., Maxim Kazarian et al., Journal of London Mathematical Society 2024 Vol. 109 No. 6 Article e12946
We study the n-point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ℏ2-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. ...
Added: October 29, 2024
Alexandrov A., B. Bychkov, P. Dunin-Barkowski et al., Journal of Geometry and Physics 2024 Vol. 206 Article 105329
For a given spectral curve, we construct a family of symplectic dual spectral curves for which we prove an explicit formula expressing the n-point functions produced by the topological recursion on these curves via the n-point functions on the original curve. As a corollary, we prove topological recursion for the generalized fully simple maps generating functions. ...
Added: October 25, 2024
Dunin-Barkowski P., Kramer R., Popolitov A. et al., Annales Scientifiques de l'Ecole Normale Superieure 2023 Vol. 56 No. 4 P. 1199–1229
We prove the 2006 Zvonkine conjecture that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes via the so-called r-ELSV formula, as well as its orbifold generalization, the so-called qr-ELSV formula. ...
Added: October 5, 2023
Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., Communications in Mathematical Physics 2023 Vol. 402 P. 665–694
We study a duality for the n-point functions in VEV formalism that we call the ordinary vs fully simple duality. It provides an ultimate generalisation and a proper context for the duality between maps and fully simple maps observed by Borot and Garcia-Failde. Our approach allows to transfer the algebraicity properties between the systems of n-point functions ...
Added: June 29, 2023
Dunin-Barkowski P., Kazaryan M., Popolitov A. et al., Advances in Theoretical and Mathematical Physics 2022 Vol. 26 No. 4 P. 793–833
We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this ...
Added: March 20, 2023