New Frobenius structures on Hurwitz spaces are found. A Hurwitz space is considered as a real manifold; therefore the number of coordinates is twice as large as the number of coordinates on Hurwitzs Frobenius manifolds of Dubrovin. Simple branch points of a ramified covering and their complex conjugates play the role of canonical coordinates on the constructed Frobenius manifolds. Corresponding solutions to WDVV equations and G-functions are obtained.
Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius manifold structures on Hurwitz spaces (real doubles, deformations) is described on the level of the corresponding Riemann-Hilbert problems.
In this note we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux-Egoroff system. As an application, we explain how Shramchenko's deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental-van de Leur twisted loop group action on the space of semi-simple Frobenius manifolds.
In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.
Deformations of Dubrovin's Hurwitz Frobenius manifolds are constructed. The deformations depend on $g(g+1)/2$ complex parameters where $g$ is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of solutions to the Painlevé-VI equation with coefficients $(1/8,-1/8,1/8,3/8).$ Analogous deformations of the real doubles of the Hurwitz Frobenius manifolds are also found; these deformations depend on $g(g+1)/2$ real parameters.
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 questi…
The semisimple Frobenius manifolds related to the Hurwitz spaces $H_{g,N}(k_1, ..., k_l)$ are considered. We show that the corresponding isomonodromic tau-function $τ_I$ coincides with $(-1/2)$-power of the Bergmann tau-function which was introduced in a recent work by the authors \cite{KokKor}. This enables us to calculate explicitly the $G$-function of Frobenius manifolds related to the Hurwitz spaces $H_{0, N}(k_1, ..., k_l)$ and $H_{1, N}(k_1, ..., k_l)$. As simple consequences we get formulas for the $G$-functions of the Frobenius manifolds ${\mathbb C}^N/\tilde{W}^k(A_{N-1})$ and ${\math…
In this work we find the isomonodromic (Jimbo-Miwa) tau-function corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the G-function (solution of Getzler's equation) of the Hurwitz Frobenius manifolds. Second, in terms of this tau-function we compute the genus one correction to the free energy of hermitian two-matrix model. Third, we find the Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with quasi-permutation monodromy matrices. Finally, we get a new expression (analog of ge…
We construct Frobenius structures of "dual type" on the moduli space of ramified coverings of $\mathbb{P}^1$ with given ramification type over two points, generalizing a construction of Dubrovin. A complete hierarchy of hydrodynamic type is obtained from the corresponding deformed flat connection. This provides a suitable framework for the Whitham theory of an enlarged class of integrable hierarchies; we treat as examples the q-deformed Gelfand-Dickey hierarchy and the sine-Gordon equation, and compute the corresponding solutions of the WDVV equations.
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers corresponding algebra is an algebra of coverings of lesser dimension. For regular Hurwitz numbers the Frobenius algebra is an algebra of families of subgroups of finite groups.
Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.
This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.
The isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one are constructed explicitly. Such spaces may be equipped with the structure of a Frobenius manifold and this introduces a flat coordinate system on the manifold. The isomonodromic tau-function, and in particular the associated $G$-function, are rewritten in these coordinates and an interpretation in terms of the caustics (where the multiplication is not semisimple) is given.
A Frobenius manifold has tri-hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We focus on the case of dimension four and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painlevé VI$μ$ equation. This yields an explicit procedure associating to any semisimple Frobenius manifold of dimension three a tri-hamiltonian Frobenius manifold of dimension four. We carry out explicit examples for the case of Frobenius structures on Hurwitz spaces.
Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first …
Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces. In t…
We present expressions in terms of a double infinite series for the Stieltjes constants $γ_k(a)$. These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta function. We show that the case $γ_k(1)=γ$ corresponds to a series representation for the Riemann zeta function given much earlier by Brun. As a byproduct, we obtain a parameterized double series representation of the Hurwitz zeta function.
We study the quantum Witten-Kontsevich series introduced by Buryak, Dubrovin, Guéré and Rossi in \cite{buryak2016integrable} as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter $ε$ and a quantum parameter $\hbar$. When $\hbar=0$, this series restricts to the Witten-Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves. We establish a link between the $ε=0$ part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number non-equivalent h…
A new family of weighted double Hurwitz numbers and a new ELSV-type formula with $Ω$-classes
Center for Geometry and Physics, Institute for Basic Science (IBS)（基本科学研究所几何与物理中心）；University of Haifa（海法大学）；HSE University（高等经济学院）；Skoltech（斯科尔科沃科学技术研究所）；NRC “Kurchatov Institute” – ITEP（库尔恰托夫国家研究中心-物理技术研究所）；I.M. Krichever Scientific School, Skoltech（斯科尔科沃 I.M.克里切弗科学学校）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov-Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of $c = 1$ string theory except that the Orlov-Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermiom bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A c…
We construct local bihamiltonian structures from classical $W$-algebras associated to non-regular nilpotent elements of regular semisimple type in Lie algebras of type $A_2$ and $A_3$. They form exact Poisson pencil, admit a dispersionless limit and their leading terms define logarithmic or trivial Dubrovin-Frobenius manifolds. We calculate the corresponding central invariants which are expected to be constants. In particular, we get Dubrovin- Frobenius manifolds associated to the focused Schrödinger equation and Hurwitz space $M_{0;1,0}$ and the corresponding bihamiltonian structures of topol…
The G-function associated to the semi-simple Frobenius manifold C^n/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-function corresponding to the Frobenius manifold C^n/W^(k)(A_{n-1}) where W^(k)(A_{n-1}) is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space M_{0;k-1,n-k-1}), together with the general form of the G-function in terms of data on caustics. …
This work concerns both the semiclassical and zero temperature asymptotics of quantum weighted double Hurwitz numbers. The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distri- bution of a quantum Bose gas with vanishing fugacity. We compute the leading semi- classical term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit $\hbar \ra 0$ is shown to reproduce the simple single and double Hurwitz numbers studied by Pandharipande and …
It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend open-closed two-dimensional topological field theories to nonorientable surfaces. We call them Klein topological field theories(KTFT). We prove that KTFTs bijectively correspond to algebras with certain additional structures, called structure algebras. Semisimple structure algebra…
Double Hurwitz numbers enumerating weighted $n$-sheeted branched coverings of the Riemann sphere or, equivalently, weighted paths in the Cayley graph of $S_n$ generated by transpositions are determined by an associated weight generating function. A uniquely determined $1$-parameter family of 2D Toda $τ$-functions of hypergeometric type is shown to consist of generating functions for such weighted Hurwitz numbers. Four classical cases are detailed, in which the weighting is uniform: Okounkov's double Hurwitz numbers, for which the ramification is simple at all but two specified branch points; t…
This work concerns the semiclassical asymptotics of quantum weighted double Hurwitz numbers. We compute the leading term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit $\hbar \to 0$ is shown to reproduce the simple Hurwitz numbers studied by Pandharipande and Okounkov. The KP-Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown to converge in the classical limit to the generating function of Pandharipande and Okounkov and, with suitable scaling, so d…
In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves from the point of view of KP integrability and derive new examples of quantum curves for the families of double Hurwitz numbers.
The partition function for quantum weighted double Hurwitz numbers can be interpreted in terms of the energy distribution of a quantum Bose gas with vanishing fugacity. We compute the leading term of the partition function and the quantum weighted Hurwitz numbers in the zero temperature limit $T \rightarrow 0$, as well as the next order corrections. The leading term is shown to reproduce the case of uniformly weighted Hurwitz numbers of Belyi curves. In particular, the KP or Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown in the limit to give the one f…
We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in the transverse direction. Similar to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution to the hierarchy is different. We characterize this solution by the string equations and construct the corresponding dispersionless tau-function. This tau-function is shown to coincide with the genus-zero part of the generating function for d…
We construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes cocycle. The former is called a double construction of Frobenius algebra and the latter is called a double construction of Connes cocycle which is interpreted in terms of dendriform algebras. Both of them are equivalent to a kind of bialgebras, namely, antisymmetric infinitesimal bialgebras an…
We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O'Connell and N. J. Simm, arXiv:1805.08760] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the gener…
The KP and 2D Toda tau-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The g…
Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramif…
Landau-Ginzburg models of generalised Dubrovin-Zhang form and pole collision: Dynkin-type A
School of Mathematics and Statistics, University of Glasgow（格拉斯哥大学数学与统计学院）；Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca（米兰比可卡大学数学与应用系）；INFN sezione di Milano-Bicocca（意大利国家核物理研究所米兰比可卡分部）
We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin--Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadom…
Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $τ$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sph…
We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so fa…
This is an introduction to double algebras which is the structure modelled by the properties of the convolution product in Hopf algebras, weak Hopf algebras and in Hopf algebroids. We show that Hopf algebroids with a Frobenius integral can be characterized as the double algebras in which the two multiplications satisfy distributivity. We discuss questions of duality, antipode, Maschke theorem and examples.
Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the $2$D Toda $τ$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $τ$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rati…
This is an overview of recent results on the use of 2D Toda $τ$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$. Specializing the fermionic formula to the case of diagonal group elements gives $τ$-functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of {\…
This manuscript recounts some of the author's contributions to algebraic and enumerative combinatorics. We have focused on two types of generalizations of bipartite maps, which are bipartite graphs embedded on surfaces. Maps are known to appear in many areas of theoretical physics and discrete mathematics, but one key interest for fundamental computer science is how multi-facetted they are in the sense that multiple encodings exist which are not interchangeable, like Tutte's/loop equations, the topological recursion, the KP hierarchy and numerous bijections which made the field so rich. One ge…
A multiparametric family of 2D Toda $τ$-functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of $S_n$. The coefficients $F^{c_1, ..., c_l}_{d_1, ..., d_m}(μ, ν)$ in their series expansion over products $P_μP'_ν$ of power sum symmetric functions in the two sets of Toda flow parameters and powers of the $l+m$ auxiliary parameters are shown to enumerate $|μ|=|ν|=n$ fold branched covers of the Riemann sphere with specified ramification …
