arXiv subjects

S. M. Natanzon

Publications and source records attributed to S. M. Natanzon.

14 recordsLinked to original sources

arXiv2021

Around spin Hurwitz numbers

We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. They are related to peculiar $Q$ Schur functions, which are actually related to characters of the Sergeev group. This allows one to put the whole story into the modern context of matrix models and integrable hierarchies. Hurwitz partition functions are actually broader than conventional $τ$-functions, but reduce to them in particular circumstances. We explain, how a special $d$-soliton $τ$-functions of KdV and Veselov-Novikov hierarchies generate the spin Hurwitz numbers $H^\pm\left( Γ^r_d \right)$ and $H^\pm\left( Γ^r_d,Δ\right)$. The generating functions of the spin Hurwitz numbers are hypergeometric $τ$-functions of the BKP integrable hierarchy, and we present their fermionic realization. We also explain how one can construct $τ$-functions of this type entirely in terms of the $Q$ Schur functions. An important role in this approach is played by factorization formulas for the $Q$ Schur functions on special loci.

math-ph↗

arXiv2019

Integrals of tau functions

We consider integrals of tau functions of Zakharov-Shabat systems whose higher times are related to the eigenvalues of products of random matrices. Apart of random matrices there is the set of $n$ pairs of given matrices which play the role of parameters. In terms of these matrices we introduce the notions of words, dressed words and dual words, these notions are related to the graphs and dual graphs drawn on a Riemann surface $Σ$. The integrals of tau functions over independent ensembles of random matrices can be computed in form of series over partitions of products of the Schur polynomials with a multiplier which depends on the Euler characteristic of the Riemann surface. This form allows to compare the integrals of tau functions with correlation functions of certain quantum models. We present a tau function whose integral is equal to the correlation function of the Wilson loops of the two-dimensional Yang-Mills model on $Σ$.

nlin.SI↗

arXiv2014

Hurwitz numbers and BKP hierarchy

We consider special series in ratios of the Schur functions which are defined by integers $\textsc{f}\ge 0$ and $\textsc{e} \le 2$, and also by the set of $3k$ parameters $n_i,q_i,t_i,\,i=1,..., k$. These series may be presented in form of matrix integrals. In case $k=0$ these series generates Hurwitz numbers for the $d$-fold branched covering of connected surfaces with a given Euler characteristic $\textsc{e}$ and arbitrary profiles at $\textsc{f}$ ramification points. If $k>0$ they generate weighted sums of the Hurwitz numbers with additional ramification points which are distributed between color groups indexed by $i=1,...,k$, the weights being written in terms of parameters $n_i,q_i,t_i$. By specifying the parameters we get sums of all Hurwitz numbers with $\textsc{f}$ arbitrary fixed profiles and the additional profiles provided the following condition: both, the sum of profile lengths and the number of ramification points in each color group are given numbers. In case $\textsc{e}=\textsc{f}=1,2$ the series may be identified with BKP tau functions of Kac and van de Leur of a special type called hypergeometric tau functions. Sums of Hurwitz numbers for $d$-fold branched coverings of ${\mathbb{RP}}^2$ are related to the one-component BKP hierarchy. We also present links between sums of Hurwitz numbers and one-matrix model of the fat graphs.

nlin.SI↗

arXiv2014

Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 double Hurwitz numbers. They can also serve as a starting point for a constructive approach to the Riemann mapping problem and the inverse potential problem in 2D.

math.CO↗

arXiv2009

Disk single Hurwitz numbers

It is investigated Hurwitz numbers, that correspond to covering of disk with single non-simple boundary critical value. It is found differential equations, that describe a generating function for these numbers.

math.GT↗

arXiv2009

Cyclic Foam Topological Field Theories

This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories one-to-one correspond to graph-Cardy-Frobenius algebras, that are families $(A,B_\star,ϕ)$, where $A=\{A^s|s\in S\}$ are families of commutative associative Frobenius algebras, $B_\star = \bigoplus_{σ\inΣ} B_σ$ is an graduated by graphes, associative algebras of Frobenius type and $ϕ=\{ϕ_σ^s: A^s\to (B_σ)|s\in S,σ\in Σ\}$ is a family of special representations. There are constructed examples of Cyclic Foam Topological Field theories and its graph-Cardy-Frobenius algebras

math.GT↗

arXiv2008

Test for reality of algebraic functions

In this paper is proved that a complex algebraic function on complexification of a real algebraic curve is equivalent to real algebraic function, if and only if the divisor of preimage of critical values is stable under the involution of complex conjugation.

math.AG↗

arXiv2007

Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups

Analogue of classical Hurwitz numbers is defined in the work for regular coverings of surfaces with marked points by seamed surfaces. Class of surfaces includes surfaces of any genus and orientability, with or without boundaries; coverings may have certain singularities over the boundary and marked points. Seamed surfaces introduced earlier are not actually surfaces. A simple example of seamed surface is book-like seamed surface: several rectangles glued by edges like sheets in a book. We prove that Hurwitz numbers for a class of regular coverings with action of fixed finite group $G$ on cover space such that stabilizers of generic points are conjugated to a fixed subgroup $K\subset G$ defines a new example of Klein Topological Field Theory (KTFT). It is known that KTFTs are in one-to-one correspondence with certain class of algebras, called in the work Cardy-Frobenius algebras. We constructed a wide class of Cardy-Frobenius algebras, including particularly all Hecke algebras for finite groups. Cardy-Frobenius algebras corresponding to regular coverings of surfaces by seamed surfaces are described in terms of group $G$ and its subgroups. As a result, we give an algebraic formula for introduced Hurwitz numbers.

math.GT↗

arXiv2005

Extention cohomological fields theories and noncommutative Frobenius manifolds

We construct some extension ({\it Stable Field Theory}) of Cohomological Field Theory. The Stable Field Theory is a system of homomorphisms to some vector spaces generated by spheres and disks with punctures. It is described by a formal tensor series, satisfying to some system of "differential equations". In points of convergence the tensor series generate special noncommutative analogues of Frobenius algebras, describing 'Open-Closed' Topological Field Theories.

math-ph↗

arXiv2004

Towards an Effectivisation of the Riemann Theorem

The Riemann Theorem states, that for any nontrivial connected and simply connected domain on the Riemann sphere there exists some its conformal bijection to the exterior of the unit disk. In this paper we find an explicit form of this map for a broad class of domains with analytic boundaries.

math.CV↗

arXiv2001

Integrable systems and effectivisation of Riemann theorem about domaims of the complex plane

Consider a closed analytic curve $γ$ in the complex plane and denote by > $D_+$ and $D_-$ the interior and exterior domains with respect to the curve. The point $z=0$ is assumed to be in $D_+$. Then according to Riemann theorem there exists a function $w(z)=\frac 1r z+\sum_{j=0}^\infty p_j z^{-j}$, mapping $D_-$ to the exterior of the unit disk $\{w\in C|| w | >1\}$. It is follow from [arXiv : hep-th /0005259] that this function is described by formula $\log w=\log z-\partial_{t_0} (\frac 12\partial_{t_0}+\sum\limits_{k\geqslant 1}\frac{z^{-k}}{k} \partial_{t_k})v$, where $v=v(t_0, t_1, \bar t_1, t_2, \bar t_2,...)$ is a function from the area $t_0$ of $D_+$ and the momemts $t_k$ of $D_-$. Moreover, this function satisfies the dispersionless Hirota equation for 2D Toda lattice hierarchy. Thus for an effectivisation of Riemann theorem it is sufficiently to find a representation of $v$ in the form of Taylor series $v=\sum N(i_0 | i_1,...,i_k| \bar i_1,...,\bar i_{\bar k})t_0 t_{i_1},...,t_k \bar t_{\bar i_1},...,\bar t_{\bar i_{\bar k}}$. The numbers $N(i_0 | i_1,...,i_k | \bar i_1, ..., \bar i_{\bar k})$ for $i_α, \bar i_β\leqslant 2$ is found in [arXiv: hep-th/0005259]. In this paper we find some recurrence relations that give a possible to find all $N(i_0\bigl| i_1,...,i_k|\bar i_1,...,\bar i_{\bar k})$.

math.CV↗

arXiv2000

Witten solution of the Gelfand-Dikii hierarchy

Among solutions of n-Gelfand-Dikii's hierarchy there exists a remarkable solution W, which satisfies the string equation. We call it Witten's solution because according to the Witten conjecture the function F(x_1, x_2, x_3,...) = W(x_1,(x_2)/2, (x_3)/3,...) is the generating function for intersection nambers of Mumford-Morita-Muller cohomological classes of the moduli space of n-spin Riemann surfaces. This conjecture was proved by Kontsevich for n=2 and by Witten himself for surfaces of genus 0. In this paper we find recurrence relations between coefficients of Taylor series of W. This reduces the Witten's conjecture to conjecture that the Mamford-Morita-Muller numbers satisfy to the same relations. These relations give also an algorithm for calculation of $n$-spin Mamford-Morita-Muller numbers in assuming that the Witten conjecture is true. Moreover we prove that F(x_1,x_2,...,x_{n-1},0,0,...)= W(x_1, (x_2)/2,...,(x_{n-1})/(n-1),0,0,...).

math.AG↗

arXiv1999

Formulas for A_n and B_n-solutions of WDVV equations

The simplest non-trivial solutions of WDVV equations are A_n and B_n-potentials, which describe metrics of K.Saito on spaces of versal deformation of A_n and B_n-singularities. These are some polynomials, which were known for $n\leqslant$ 4. We find some recurrence relations, which give a possibility to find all A_n and B_n-potentials. In passing we give recurrence formulas for coefficients of dispersionless KP hierarchy.

hep-th↗

arXiv1999

Geometry and algebra of real forms of complex curves

Let Y be a complex algebraic curve and let [Y]={X_1,...,X_n} be the set of all real algebraic curves X_i with complexification X_i(C)=Y, such that the real points X_i(R) divide X_i(C). We find all such families [Y]. According to Harnak theorem a number |X_i| of connected components of X_i(R) satifies by the inequality |X_i|<= g+1, where g is the genus of Y. We prove that SUM |X_i| <= 2g-(n-9) 2^{n-3}-2 <= 2g+30 and these estimates are exact.

math.CV↗