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## Topological recursion and Givental’s formalism: Spectral curves for Gromov-Witten theories

P. 231-295.

We describe a way of producing local spectral curves for arbitrary

semisimple cohomological field theories (and Gromov-Witten theories in par-

ticular) 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.

### In book

Vol. 100: Topological Recursion and its Influence in Analysis, Geometry, and Topology. , Providence : American Mathematical Society, 2018

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

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

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., 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., , 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. 297-331.

Hurwitz spaces parameterizing covers of the Riemann sphere can
be equipped with a Frobenius structure. In this review, we recall the construction of such Hurwitz Frobenius manifolds as well as the correspondence
between semisimple Frobenius manifolds and the topological recursion formalism. We then apply this correspondence to Hurwitz Frobenius manifolds by
explaining that the corresponding primary invariants can ...

Added: February 20, 2019

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., / Cornell University. Series math "arxiv.org". 2021. No. 2106.08368.

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: April 20, 2022

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

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

Dunin-Barkowski Petr, Kazarian Maxim, 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

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., / Cornell University. Series math "arxiv.org". 2020. No. 2012.14723.

We study the n-point differentials corresponding to Kadomtsev-Petviashvili 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. We ...

Added: April 20, 2022

Buryak A., Moscow Mathematical Journal 2023 Vol. 23 No. 3 P. 309-317

An algorithm to determine all the Gromov–Witten invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro type constraints that allow to determine all the other Gromov–Witten invariants in terms of the stationary ones. In the case of an ...

Added: November 20, 2023

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

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

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

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

Bryan J., Pandharipande R., Faber C. et al., Journal of the American Mathematical Society 2008 No. 21 P. 101-136

Added: September 30, 2014

Oblomkov A., Okounkov A., Pandharipande R., Communications in Mathematical Physics 2020 Vol. 374 No. 3 P. 1321-1359

We propose an explicit formula for the GW/PT descendent correspondence in the stationary case for nonsingular connected projective threefolds. The formula, written in terms of vertex operators, is found by studying the 1-leg geometry. We prove the proposal for all nonsingular projective toric threefolds. An application to the Virasoro constraints for the stationary descendent theory of ...

Added: May 15, 2020

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

Basalaev A., Oberwolfach Reports 2015 Vol. 12 No. 2 P. 1-2

We show that all elliptic orbifolds have the property of being modular. Namely, that there is a non--trivial Givental's action that leaves it unaffected. ...

Added: February 26, 2019

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

Basalaev A., Journal of Geometry and Physics 2014 Vol. 77 P. 30-42

We study the relation between the Frobenius manifolds of GW theory and the Hurwitz– Frobenius manifold.Weprove that the Frobenius manifold given by the orbifold GW theory of P^1
(2, 2, 2, 2) is isomorphic to the submanifold in the Hurwitz–Frobenius manifold
of ramified coverings of the sphere by the genus 1 curve with the ramification profile (2, ...

Added: February 26, 2019