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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 situations., Comment: v2: 29 pages, a section on KP integrability is added

For every differential graded Lie algebra $\mathfrak{g}$ one can define two different group actions on the Maurer-Cartan elements: the ubiquitous gauge action and the action of $\mathrm{Lie}_\infty$-isotopies of $\mathfrak{g}$, which we call the ambient action. In this note, we explain how the assertion of gauge triviality of a homologically trivial ambient action relates to the calculus of dendriform, Zinbiel, and Rota-Baxter algebras, and to Eulerian idempotents. In particular, we exhibit new relationships between these algebraic structures and the operad of rational functions defined by Loday., Comment: 23 pages, comments are welcome

We prove the so-called master relation in the tautological ring of the moduli space of curves that implies polynomial properties of the Dubrovin-Zhang hierarchies associated to different versions of cohomological field theories as well as their equivalences to the corresponding double ramification hierarchies.

We prove that for any initial data on a genus zero spectral curve the corresponding correlation differentials of topological recursion are KP integrable. As an application we prove KP integrability of partition functions associated via ELSV-type formulas to the $r$-th roots of the twisted powers of the log canonical bundles., Comment: 13 pages

We prove a new system of relations in the tautological ring of the moduli space of curves involving stable rooted trees with level structure decorated by the top Chern class of the Hodge bundle and $\Omega$-classes and double ramification structures. In particular, this resolves a recent conjecture on these relations as well as connects with one of the two sides of the recently established DR/DZ equivalence between the integrable hierarchies constructions of Buryak and of Dubrovin--Zhang.

We establish the Miura equivalence of two integrable systems associated to a semi-simple cohomological field theory: the double ramification hierarchy of Buryak and the Dubrovin-Zhang hierarchy. This equivalence was conjectured by Buryak and further refined by Buryak, Dubrovin, Gu\'er\'e, and Rossi.

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, group property, compatibility with topological recursion and KP integrability. As an application of these properties, we get a new and uniform proof of topological recursion for large families of weighted double Hurwitz numbers; this encompasses and significantly extends all previously known results on this matter., Comment: 28 pages

We prove a structural theorem relating the higher genera free energy functions of the Dubrovin-Zhang hierarchies to those of the trivial theory, that is, the Witten-Kontsevich free energy functions. As an important application, for any given genus $g\geq 1$, we construct a set of universal identities valid for the free energy functions of any Dubrovin-Zhang hierarchy., Comment: 24 pages. Comments are welcome!

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., Comment: 30 pages; minor corrections and modifications; Section 3.11 on the new Family III of generalized double Hurwitz numbers added

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 that relates some particular instances of topological recursion to the Mironov-Morozov-Semenoff matrix integrals., Comment: 27 pages; minor corrections

When does topological recursion applied to a family of spectral curves commute with taking limits? This problem is subtle, especially when the ramification structure of the spectral curve changes at the limit point. We provide sufficient (straightforward-to-use) conditions for checking when the commutation with limits holds, thereby closing a gap in the literature where this compatibility has been used several times without justification. This takes the form of a stronger result of analyticity of the topological recursion along suitable families. To tackle this question, we formalise the notion of global topological recursion and provide sufficient conditions for its equivalence with local topological recursion. The global version facilitates the study of analyticity and limits. For nondegenerate algebraic curves, we reformulate these conditions purely in terms of the structure of its underlying singularities. Finally, we apply this to study deformations of $ (r,s) $-spectral curves, spectral curves for weighted Hurwitz numbers, and provide several other examples and non-examples (where the commutation with limits fails)., Comment: 83 pages, 12 figures

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., Comment: 16 pages

This monograph provides an overview on the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a conceptual, exhaustive and gentle treatment of the twisting procedure, which functorially creates new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer-Cartan element. The twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras is described by means of the action of the biggest deformation gauge group ever considered. We give a criterion on quadratic operads for the existence of a meaningful twisting procedure of their associated categories of algebras. And, we introduce the twisting procedure for operads \`a la Willwacher using a new and simpler presentation, which provides us with a wide source of motivating examples related to graph homology, both recovering known graph complexes (due to Kontsevich) and introducing some new ones. This book starts with elementary surveys on gauge theory and deformation theory using differential graded Lie algebras in order to ease the way to the theory. It finishes with concise surveys on the fundamental theorem of deformation theory, higher Lie theory, rational homotopy theory, simplicial theory of homotopy algebras, and the Floer cohomology of Lagrangian submanifolds, to illustrate deep examples of applications., Comment: 183 pages, supersedes the very preliminary version arXiv:1810.02941, final author version before copyediting; will be published by Cambridge University Press & Assessment as 'Maurer-Cartan Methods in Deformation Theory: the twisting procedure' by Vladimir Dotsenko, Sergey Shadrin and Bruno Vallette

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 of this general $x-y$ swap result, we prove an explicit closed formula for the topological recursion differentials for the case of any spectral curve with unramified $y$ and arbitrary rational $x$., Comment: 51 pages; various corrections and clarifications

We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Gu\'er\'e and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case $n=1$ and arbitrary $g$ using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hern\'andez Iglesias. We also prove all the above mentioned relations in the case $g=0$ and arbitrary $n$., Comment: v3: final journal version, 44 pages

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 of formal variables. In these variables the weighted double Hurwitz numbers are presented as coefficients of expansions of some differentials that we prove to satisfy topological recursion. Our results partly resolve a conjecture that we made in [arXiv:2106.08368] and are based on a system of new explicit functional relations for the more general $(m,n)$-correlation functions, which correspond to the case when there are distinguished cycles controlled by formal variables in both special singular fibers. These $(m,n)$-correlation functions are the main theme of this paper and the latter explicit functional relations are of independent interest for combinatorics of weighted double Hurwitz numbers. We also put our results in the context of what we call the "symplectic duality", which is a generalization of the $x-y$ duality, a phenomenon known in the theory of topological recursion., Comment: 46 pages; minor modifications, section 4.5 removed, references added

We establish the functional relations between generating series of higher-order free cumulants and moments in higher-order free probability, solving an open problem posed fifteen years ago by Collins, Mingo, \'Sniady and Speicher. We propose an extension of free probability theory, which governs the all-order topological expansion in unitarily invariant matrix ensembles, with a corresponding notion of free cumulants and give as well their relation to moments via functional relations. Our approach is based on the study of a master transformation involving double monotone Hurwitz numbers via semi-infinite wedge techniques, building on the recent advances of the last-named author with Bychkov, Dunin-Barkowski and Kazarian. We illustrate our formulas by computing the first few decaying terms of the correlation functions of an ensemble of spiked GUE matrices, going beyond the law of large numbers and the central limit theorem., Comment: 59 pages

In the classical Batalin--Vilkovisky formalism, the BV operator $\Delta$ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by $\Delta$ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin--Vilkovisky algebras where the operator $\Delta$ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever $\Delta$ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s., Comment: v3: further minor changes (several formulas corrected, the explanation of acyclicity of a Koszul-type complex made more precise)

We propose a conjectural formula for $DR_g(a,-a) \lambda_g$ and check all its expected properties. Our formula refines the one point case of a similar conjecture made by the first named author in collaboration with Gu\'er\'e and Rossi, and we prove that the two conjectures are in fact equivalent, though in a quite non-trivial way., Comment: 13 pages

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 related by this duality, and gives direct tools for the analysis of singularities. As an application, we give a proof of a recent conjecture of Borot and Garcia-Failde on topological recursion for fully simple maps., Comment: 25 pages; minor corrections

The Dubrovin-Zhang hierarchy is a Hamiltonian infinite-dimensional integrable system associated to a semi-simple cohomological field theory or, alternatively, to a semi-simple Dubrovin-Frobenius manifold. Under an extra assumption of homogeneity, Dubrovin and Zhang conjectured that there exists a second Poisson bracket that endows their hierarchy with a bi-Hamiltonian structure. More precisely, they gave a construction for the second bracket, but the polynomiality of its coefficients in the dispersion parameter expansion is yet to be proved. In this paper we use the bi-Hamiltonian recursion and a set of relations in the tautological rings of the moduli spaces of curves derived by Liu and Pandharipande in order to analyze the second Poisson bracket of Dubrovin and Zhang. We give a new proof of a theorem of Dubrovin and Zhang that the coefficients of the dispersion parameter expansion of the second bracket are rational functions with prescribed singularities. We also prove that all terms in the expansion of the second bracket in the dispersion parameter that cannot be realized by polynomials because they have negative degree do vanish, thus partly confirming the conjecture of Dubrovin and Zhang., Comment: 23 pages

In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin completed cycles (a generalized version of the Giacchetto--Kramer--Lewa\'nski conjecture)., Comment: 31 pages, minor corrections

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 $\hbar^2$-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 also distinguish two large families of the Orlov-Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families we prove in addition the so-called projection property and thus the full statement of the Chekhov-Eynard-Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of $\hbar^2$-deformations of the Orlov-Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck-Riemann-Roch formula this, in turn, gives new and uniform proofs of almost all ELSV-type formulas discussed in the literature., Comment: 49 pages; several clarifications and corrections

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 case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).

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 Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as 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 small set of formal functions defined on the spectral curve., Comment: 35 pages; minor changes

We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support the conjecture., Comment: v2: minor improvements of the exposition, several typos corrected, 28 pages, accepted for publication in the Letters of Mathematical Physics

We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduce two new natural extensions of the notion of a CohFT: homotopical (necessary to structure chain-level Gromov--Witten invariants) and quantum (with examples found in the works of Buryak--Rossi on integrable systems). We introduce a new version of Kontsevich's graph complex, enriched with tautological classes on the moduli spaces of stable curves. We use it to study a new universal deformation group which acts naturally on the moduli spaces of quantum homotopy CohFTs, by methods due to Merkulov--Willwacher. This group is shown to contain both the prounipotent Grothendieck--Teichm\"uller group and the Givental group., Comment: submitted version, 57 pages, abstract and introduction rewritten

We prove the conjecture of Do and Karev that the monotone orbifold Hurwitz numbers satisfy the Chekhov-Eynard-Orantin topological recursion., Comment: 11 pages. V2: Updated grant acknowledgments of A.P. and mail address of R.K. V3: Improved exposition

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 the recent result of Alexandrov-Chapuy-Eynard-Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental's theory for cohomological field theories., Comment: 33 pages

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 $qr$-ELSV formula, proposed recently in [KLPS17]., Comment: 22 pages. Version 4: improved exposition of the proof

We identify the formulas of Buryak and Okounkov for the n-point functions of the intersection numbers of psi-classes on the moduli spaces of curves. This allows us to combine the earlier known results and this one into a principally new proof of the famous Witten conjecture / Kontsevich theorem, where the link between the intersection theory of the moduli spaces and integrable systems is established via the geometry of double ramification cycles., Comment: 11 pages, some changes in the introduction

We employ the $1/2$-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on $\mathcal{M}_g$, $g\geq 2$. We then prove several cases of the combinatorial identity, providing a new proof of Faber's formula for those cases.

This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer--Cartan element. On the way, we settle the integration theory of complete pre-Lie algebras in order to describe this twisting procedure in terms of gauge group action. We give a criterion on quadratic operads for the existence of a meaningful twisting procedure of their associated categories of (homotopy) algebras. We also give a new presentation of the twisting procedure for operads \`a la Willwacher and we perform new homology computations of graph complexes., Comment: New format (short monography), 93 pages, minor corrections, submitted version

We derive explicit recursions for the ground state generating functions of the one-dimensional Nicolai model and $\mathbb{Z}_2$ Nicolai model. Both are examples of lattice models with $\mathcal{N}=2$ supersymmetry. The relations that we obtain for the $\mathbb{Z}_2$ model were numerically predicted by Sannomiya, Katsura, and Nakayama., Comment: 6 pages

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. Our proof in particular uses a combinatorial technique developed by Han. 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 the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews., Comment: 7 pages. v2: Added a second proof of lemma 2.3, using Jucys-Murphy elements, and expanded motivation

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 the Brini-Eynard-Mari\~no spectral curve for the colored HOMFLY-PT polynomials of torus knots. This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials. We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data., Comment: 40 pages; section 10 addressing the quantum curve was added, as well as some remarks regarding Meixner polynomials thanks to T.Koornwinder

We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants., Comment: 19 pages

We prove the orbifold version of Zvonkine's $r$-ELSV formula in two special cases: the case of $r=2$ (complete $3$-cycles) for any genus $g\geq 0$ and the case of any $r\geq 1$ for genus $g=0$., Comment: 20 pages

We perform a key step towards the proof of Zvonkine's conjectural $r$-ELSV formula that relates Hurwitz numbers with completed $(r+1)$-cycles to the geometry of the moduli spaces of the $r$-spin structures on curves: we prove the quasi-polynomiality property prescribed by Zvonkine's conjecture. Moreover, we propose an orbifold generalization of Zvonkine's conjecture and prove the quasi-polynomiality property in this case as well. In addition to that, we study the $(0,1)$- and $(0,2)$-functions in this generalized case and we show that these unstable cases are correctly reproduced by the spectral curve initial data., Comment: 20 pages

We use relations in the tautological ring of the moduli spaces $\overline{\mathcal{M}}_{g,n}$ derived by Pandharipande, Pixton, and Zvonkine from the Givental formula for the $r$-spin Witten class in order to obtain some restrictions on the dimensions of the tautological rings of the open moduli spaces $\mathcal{M}_{g,n}$. In particular, we give a new proof for the result of Looijenga (for $n=1$) and Buryak et al. (for $n\geq 2$) that $\dim R^{g-1}(\mathcal{M}_{g,n}) \leq n$. We also give a new proof of the result of Looijenga (for $n=1$) and Ionel (for arbitrary $n\geq 1$) that $R^{i}(\mathcal{M}_{g,n}) =0$ for $i\geq g$ and give some estimates for the dimension of $R^{i}(\mathcal{M}_{g,n})$ for $i\leq g-2$., Comment: 24 pages

We use refined spectral sequence arguments to calculate known and previously unknown bi-Hamiltonian cohomology groups, which govern the deformation theory of semi-simple bi-Hamiltonian pencils of hydrodynamic type with one independent and \( N\) dependent variables. In particular, we rederive the result of Dubrovin-Liu-Zhang that these deformations are parametrized by the so-called central invariants, which are \( N\) smooth functions of one variable., Comment: Accepted for publication in Journal de l'\'Ecole polytechnique - Math\'ematiques

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are the necessary conditions for the Chekhov-Eynard-Orantin topological recursion., Comment: 31 pages. Version 2: simplified the computation in section 4.2, also clarifying the deduction of the diagonal part of the A-operators

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 be obtained as periods of multidifferentials globally defined on a compact Riemann surface by topological recursion. Finally, we use this construction to reply to the following question in a large class of cases: given a compact Riemann surface, what does the topological recursion compute?, Comment: 25 pages, reorganisation of some parts of paper, updated references

We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with $D$ independent variables. We find that the second and third cohomology groups are generically non-vanishing in $D>1$. Hence, in contrast with the $D=1$ case, the deformation theory in the multivariable case is non-trivial., Comment: 23 pages. Typos corrected, add explicit form of homotopy contraction operator