### ?

## 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

Dunin-Barkowski P., Kazaryan M., Orantin N. et al., Advances in Mathematics 2015 Vol. 279 P. 67-103

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of ...

Added: September 24, 2015

Kazaryan M., Zograf P., Letters in Mathematical Physics 2015 Vol. 105 No. 8 P. 1057-1084

We compute the number of coverings of CP1∖{0,1,∞} with a given monodromy type over ∞ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev–Petviashvili) hierarchy and satisfies a topological recursion ...

Added: January 19, 2016

Dunin-Barkowski P., Orantin N., Popolitov A. et al., International Mathematics Research Notices 2018 Vol. 2018 No. 18 P. 5638-5662

We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of ...

Added: December 22, 2016

Dunin-Barkowski P., , in: Proceedings of Symposia in Pure Mathematics. Vol. 100: Topological Recursion and its Influence in Analysis, Geometry, and Topology.: Providence: American Mathematical Society, 2018.. P. 231-295.

We describe a way of producing local spectral curves for arbitrary
semisimple cohomological field theories (and Gromov-Witten theories in particular) and global spectral curves for semisimple cohomological field theories
satisfying certain conditions. By this we mean that applying the topological
recursion procedure on the spectral curve reproduces the total potential of the
corresponding cohomological field theory. ...

Added: February 20, 2019

Dunin-Barkowski P., Lewanski D., Popolitov A. et al., Journal of London Mathematical Society 2015 Vol. 92 No. 3 P. 547-565

In this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is ...

Added: November 16, 2015

Dunin-Barkowski P., Norbury P., Orantin N. et al., Journal of the Institute of Mathematics of Jussieu 2019 Vol. 18 No. 3 P. 449-497

We apply the spectral curve topological recursion to Dubrovin's universal Landau-Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold. ...

Added: December 22, 2016

Providence: American Mathematical Society, 2018

This volume contains the proceedings of the 2016 AMS von Neumann Symposium on Topological Recursion and its Influence in Analysis, Geometry, and Topology, which was held from July 4–8, 2016, at the Hilton Charlotte University Place, Charlotte, North Carolina.
The papers contained in the volume present a snapshot of rapid and rich developments in the emerging ...

Added: February 20, 2019

Dunin-Barkowski P., Mulase M., Norbury P. et al., Journal fuer die reine und angewandte Mathematik 2017 Vol. 2017 No. 726 P. 267-289

We construct the quantum curve for the Gromov–Witten theory of the complex projective line. ...

Added: March 3, 2015

Dunin-Barkowski P., Kramer R., Popolitov A. et al., Journal of Geometry and Physics 2019 Vol. 137 P. 1-6

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of ...

Added: February 20, 2019

Bychkov B., Dunin-Barkowski P., Shadrin S., Combinatorics of Bousquet-M\'elou--Schaeffer numbers in the light of topological recursion / Cornell University. Series arXiv "math". 2019.

In this paper we prove, in a purely combinatorial way, a structural quasi-polynomiality property for the Bousquet-M\'elou--Schaeffer numbers. Conjecturally, this property should follow from the Chekhov-Eynard-Orantin topological recursion for these numbers (or, to be more precise, the Bouchard-Eynard version of the topological recursion for higher order critical points), which we derive in this paper from ...

Added: October 8, 2019

Dunin-Barkowski P., Popolitov A., Shadrin S. et al., Working papers by Cornell University. Series math "arxiv.org" 2017 Vol. 1712 No. 08614 P. 1-38

We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on ...

Added: January 2, 2018

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., Explicit closed algebraic formulas for Orlov-Scherbin n-point functions / Cornell University. Series "Working papers by Cornell University". 2020.

We derive a new explicit formula in terms of sums over graphs for the n-point correlation functions of general formal weighted double Hurwitz numbers coming from the Orlov-Scherbin partition functions. Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain ...

Added: October 6, 2020

Bychkov B., Dunin-Barkowski P., Shadrin S., European Journal of Combinatorics 2020 Vol. 90 P. 103184

In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this paper from ...

Added: September 22, 2020