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Hurwitz numbers from matrix integrals over Gaussian measure

We explain how Gaussian integrals over ensemble of complex matrices with source matrices generate Hurwitz numbers of the most general type, namely, Hurwitz numbers with arbitrary orientable or non-orientable base surface and arbitrary profiles at branch points. We use the Feynman diagram approach. The connections with topological theories and also with certain classical and quantum integrable theories in particular with Witten's description of two-dimensional gauge theory are shown.

Here, we consider three tasks. The first is to present models that allow the generation of Hurwitz numbers with arbitrary given sets of profiles at isolated points, that describe the enumeration of nonisomorphic covering maps of any closed surface. The second is to connect such models with topological field theories. Finally, we look at relations between Hurwitz numbers and quantum integrable systems. Such a relation was noticed by Dubrovin in connection with the quantum dispersionless KdV equation. The second connection with integrable systems is that the correlation functions of two-dimensional quantum gauge theory (2D Yang-Mills theory), defined on a closed orientable or non-orientable surface found by E.Witten [99], can also be considered as generating functions for Hurwitz numbers.
In Section 2 we give necessary definitions and present known facts about Hurwitz numbers, topological field theories and relations with representations of symmetric group. We collect the necessary consequences of axioms of the related topological field theory in the form of Proposition 1. The notion of independent Ginibre ensembles is explained. Section 3.1 is central. There, in the case of orientable surfaces, we present another way of determining the Hurwitz numbers, namely, as an enumeration of the number of possible ways to glue surfaces from sets of polygons. Polygons are constructed from basic polygons using Young diagrams attached to them. Gluing the basic polygons leads to the formation of a base surface. The enumeration of possible ways of gluing is determined using Wick's rule, applicable to Gaussian integrals. Note that the Wick rule is widely used by physicists to interpret various matrix integrals. It was first used by t'Hooft [101] for QCD matrix models, and then it was successfully used for various problems in physics and mathematical physics, see [43], [16], [52], [14] and many others. For our purpose, we need a model of independent Ginibre ensembles with two sets of matrices: a set that we call the set of random matrices and a set, which we call the set of source matrices. The last one is needed to get specific Hurwitz numbers, see Theorem 3. Let us note, that Ginibre ensembles are popular models to study problems in quantum chaos [29] and information transfer problems [3], [11].
To obtain Hurwitz numbers in case of non-orientable surface, we modify the measure of integration, see Theorem 4 in Subsection 3.2, and then we the axioms of Hurwitz topological field theory. To modify the measure we use tau functions. The tau function is the central object in the theory of classical integrable models.
In Section 4, the Hurwitz numbers are related to specific quantum solvable models of two different types, considered respectively in subsections 4.1 and 4.3. In Subsection 4.1, we recall Dubrovin's observation that completed cycles (which were introduced into the theory of Hurwitz numbers by Okounkov in [84])) turn out to be the eigenvalues of quantum Hamiltonian operators, which are the Hamiltonians of "higher" quantum dispersionless KdV equations. We study this relation using Jucys-Murphy elements and a Toda lattice analogue of the dispersionless KdV equations. We consider the action of such a Toda lattice Hamiltonian on our matrix models. This action generates the additional dependence of Hurwitz numbers on the completed cycles introduced by Okounkov [85]. Next, we select all such integrals that can be equated to tau functions, see Propositions 5 and 6 (such tau functions describe the lattice of solitons in classical integrable systems). Subsection 4.3 shows that the partition function of the two-dimensional quantum gauge theory with fixed holonomy around marked points, presented by Witten in [102] can be considered as integral over complex matrices with sources, and this partition function also generates Hurwitz numbers. In this case, we consider the integrands that are tau-functions of a special family found in [58], [88].

Definition of Hurwitz Numbers.
Hurwitz number is a weighted number of branched coverings of a surface with a prescribed topological type of critical values. Hurwitz numbers of oriented surfaces without boundaries were introduced by Hurwitz at the end of the 19th century. Later it turned out that they are closely related to the study of the module spaces of Riemann surfaces [24], to the integrable systems [83], to modern models of mathematical physics [matrix models], and to closed topological field theories [19]. In this paper we will consider only the Hurwitz numbers over compact surfaces without boundary. The definition and important properties of Hurwitz numbers over arbitrary compact (possibly with boundary) surfaces were suggested in [10].
Clarify the definition. Consider a branched covering ϕ : P → Ω of degree d over a compact surface without boundary. In the neighborhood of each point z ∈ P , the map ϕ is topologically equivalent to the complex map u → u p in the neighborhood of u = 0 ∈ C. The number p = p(z) is called degree of the covering ϕ at the point z. The point z ∈ P is called branch point or critical point if p(z) = 1. There is only a finite number of critical points. The image ϕ(z) of any critical point is called critical value.
Let us associate with a point s ∈ Ω all points z 1 , . . . , z ∈ P such that ϕ(z i ) = s. Let p 1 , . . . , p be the degrees of the map ϕ at these points. Their sum d = p 1 + · · · + p is equal to the degree d of ϕ. Thus, to each point s ∈ S there corresponds a partition d = p 1 + · · · + p of the number d. Having ordered the degrees p 1 ≥ · · · ≥ p > 0 at each point s ∈ Ω, we can introduce the Young diagram ∆ s = [p 1 , . . . , p ] of weight d with = (∆ s ) lines of length p 1 . . . , p . This Young diagram ∆ s is called topological type of the value s. (We consider s to be a critical value if not all p i are equal to 1.) Let us note that the Euler characteristics e(P ) and e(Ω) of the surfaces P and Ω are related via the Riemann-Hurwitz relation: or, which is the same, where s 1 , . . . , s f are critical values. We say that coverings ϕ 1 : P 1 → Ω and ϕ 2 : P 2 → Ω are equivalent if there exists a homeomorphism F : P 1 → P 2 such that ϕ 1 = ϕ 2 F ; in case ϕ 1 = ϕ 2 the homeomorphism F is called automorphism of the covering. Automorphisms of a covering ϕ form the group Aut(ϕ) of a finite order |Aut(ϕ)|. Equivalent coverings have isomorphic groups of automorphisms.
Let us choose points s 1 , . . . , s f ∈ Ω and Young diagrams ∆ 1 , . . . , ∆ f of weight d. Consider the set Φ of all equivalence classes of the coverings for which s 1 , . . . , s f is the set of all critical values, and ∆ 1 , . . . , ∆ f are the topological types of these critical values. Hereinafter, unless otherwise indicated we consider that the surface Ω is connected. Hurwitz number is the number It is easy to prove that the Hurwitz number is independent of the positions of the points s 1 , . . . , s f on Ω. It depends only on the Young diagrams of ∆ 1 , . . . , ∆ f and the Euler characteristic e = e(Ω).

Hurwitz Numbers and topological field theory.
The notion of closed topological field theory was proposed by M. Atiayh in [15]. This theory deals with closed oriented surfaces without boundary. We also fix a finite-dimensional vector space A with a basis α 1 , . . . , α N . Consider now an arbitrary set of points p 1 , p 2 , . . . , p f ∈ Ω. Let us assign a vector a i ∈ A to each point p i .

Figure 1:
A closed topological field theory attributes a number < a 1 , a 2 , . . . , a f > Ω called correlator, to the dataset described above. We assume that the numbers < a 1 , a 2 , . . . , a f > Ω are invariant with respect to any autohomomorphism of the surface. Moreover, we assume that correlators < a 1 , a 2 , . . . , a f > Ω generate a system of polylinear forms on A, which satisfies the axiom of nondegeneracy and the cutting axioms: Axiom of Nondegeneracy means that the bilinear form < a 1 , a 2 > Ω is nondegenerate. Let us denote the inverse matrix to Axioms of cutting describes the evolution of correlators < a 1 , a 2 , . . . , a f > Ω as the result of the cutting the surface along a contour γ ⊂ Ω with the subsequent contraction of each boundary contour to a point. Two possible topological types of contours give two axioms of cutting. Suppose γ splits the surface Ω into 2 surfaces Ω and Ω ( Figure 2). ( If γ does not split Ω (figure3).
The first corollary of the axioms of a topological field theory is the existance of an algebra structure on A. Namely, the multiplication is defined as < a 1 a 2 , a 3 > S 2 =< a 1 , a 2 , a 3 > S 2 . Thus, the structure constants for this algebra in the basis The axiom of the cutting gives ( Figure 4 ) Therefore, s,t c s ij c t sk = s,t c s jk c t si , that is, A is an associative algebra. Vector i < α i > S 2 F αi,αj A α j is the unit of algebra A. The linear form l A (a) =< a > S 2 generates non-degenerate invariant bilinear form (a 1 , a 2 ) A = l(a 1 a 2 ) =< a 1 , a 2 > S 2 . The topological invariance makes all marked points p i equivalent and, therefore, A is a commutative algebra.
Thus, A is a commutative Frobenius algebra [26], that is, an algebra with the unit and the invariant nondegenerate scalar product generated by the linear functional l A . We will call such pairs (A, l A ) commutative Frobenius pairs. Moreover, the described construction generates functor F from the category of closed topological field theories to the category of commutative Frobenius pairs. Theorem 1. [20] The functor F is the equivalence between the category of closed topological field theories and the category of commutative Frobenius pairs.
The structure of the Frobenius pair and the cutting axiom give the following explicit formula for correlators: where K A = ij F αi,αj A α i α j and g is the genus of the surface Ω.
An extension of topological field theories to non-orientable surfaces was proposed in [10]. At the present paper, we consider surfaces without boundaries, and we will call such theories closed Klein topological field theory. A closed Klein topological theory is determined by the same scheme as the closed one but here the correlator < a 1 , . . . , a f > Ω is defined for both orientable and non-orientable surfaces Ω. In addition, each point of p i is supplied with the orientation of its small neighborhood, and the algebra A is supplied with an involution : A → A. Besides, we assume that the change of the orientation of the neighborhood of a point p i changes the element a ∈ A placed at p i to a = (a) ∈ A.
For non-orientable surfaces, in addition to the above-described cuts, there are 2 more types of cuts. The type depends on whether the contour is divided into one or two contours after the cut. In the first case, the cut is called the Möbius cut, and in the second the Klein cut ( [10]).
Gluing along the Klein cut transforms correlators by the same rule as for non-dividing contours of closed topological field theory: Gluing along the Möbius cut transforms correlators according to the rule where the linear functional D(a) =< a > RP 2 : A → K is defined by the Klein topological field theory for the real projective plane. Let us denote by U the element of algebra conjugate with respect to the (defined above) metric on A to the linear functional D(a). Then it turns out that the quad (A, l A , U, ) satisfies the following properties: 1) (A, l A ) is the commutative Frobenius pair; 2) the involution : A → A generates an automorphism of algebra; Theorem 2.
[10] The functor F is extended to the category of closed Klein topological field theories and defines an equivalence between the category of closed Klein topological field theories and the quadruplet (A, l A , U, ), with properties 1) -4). Moreover < a 1 , a 2 , ..., a f > Ω = l A (a 1 a 2 ...a f (U ) 2−e(Ω) ), where e(Ω) is the Euler characteristic of the surface Ω.
The closed Klein topological field theory can be extended to the Klein topological field theory, which includes surfaces with a boundary [10]. Its algebraic description is given by a more complex algebraic structure, called in [7,8] equipped Cardy-Frobenius algebra. Moreover, it was proved in [10] that the category of the Klein topological field theories is equivalent to the category of the equipped Cardy-Frobenius algebras. This algebraic structure naturally arises also in the representation theory of finite groups [60]. Moreover, Klein topological field theory can be continued on foams arising in string theory and algebraic geometry [76], [69] [18], [31].
Cohomological field theory is equivalent to a flat family of closed topological field theories forming the Frobenius-Dubrovin manifold [21,62].
Following the pattern of Klein topological field theory, we can construct more general topological field theories with values in functors [75]. For some special functor this theory gives the Kontsevich-Manin cohomological field theory and Gromov-Witten invariants [57]. Cohomological field theory is equivalent a flat family of closed topological field theories forming a Frobenius-Dubrovin manifold [21,62]. Dubrovin discovered that these manifolds play an important role in various branches of mathematics. Moreover, Frobenius-Dubrovin manifolds are one-to-one correspond to quasihomogeneous solutions of the WDVV hierarchy of differential equations, which appear in theory of quantum gravity.
We now return to the Hurwitz numbers. Dijkgraaf [19] noticed that the Hurwitz numbers for coverings of degree d generate certain closed topological field theory with the vector space Y d generated by Young diagrams {∆} of weight d. Dijkgraaf considered correlators and proved, that they satisfy all the axioms of closed topological field theory. If we continue this definition of correlators for the case of non-orientable surfaces, we get closed Klein topological field theory ( [10]). In this case, the operator * is trivial. Further, speaking of correlators , we will always mean the correlators < * , . . . , * > Ω of the Hurwitz topological field theory. Put < * , . . . , * >=< * , . . . , * > S 2 . The linear functional < ∆ > on Y d is equal to The definition of Hurwitz numbers can also be expanded onto a surface with a boundary in such a way that they generate Klein topological field theory [7][8][9][10]. The relationship of these new Hurwitz numbers with algebraic geometry and mathematical physics is still not well understood (some achievements in this direction are contained in [55,78], [80]). The correspondence ∆ ↔ C ∆ gives an isomorphism between the vector spaces Y d and ZC[S d ]. It transfers the structure of the algebra to Y d . We will keep it in mind in this section, speaking about multiplication on Y d .
Consider now a covering ϕ : Ω → S 2 of a given type (∆ 1 , . . . , ∆ f ) with the critical values p 1 . . . p f . The complete preimage of ϕ −1 (p) consists of d points q 1 , . . . , q d . The path along the contour γ i results in a permutation σ i ∈ S d of q 1 , . . . , q d . The conjugacy class of σ i is given by ∆ i . Moreover, the product σ 1 . . . σ f is the identical permutation. Thus, any covering of S 2 of type (∆ 1 , . . . , ∆ f ) generates an element of the set Moreover, equivalent coverings generate elements of M that conjugated by some permutation σ ∈ S d .
Construct now the inverse correspondence from the classes of M (∆ 1 , . . . , ∆ f ) to equivalent classes of coverings ϕ : Ω → S 2 of type (∆ 1 , . . . , ∆ f ) with the critical values p 1 , . . . , p f . Cuts r i ⊂ S 2 between points p and p i inside the contour γ i generate a cut sphere Correspond now the covering which corresponds to (σ 1 , . . . , σ f ) ∈ M . For this we consider d copies of the cut sphere S, number them, and glue its boundaries according to the permutations σ 1 , . . . , σ f . This gives a compact surface P . Moreover, the correspondances between the copies of S and S generate the covering ϕ : where M is the set of the conjugated classes of M by S d and Aut(σ 1 , . . . , σ f ) is the stabilizer of (σ 1 , . . . , σ f ) by these conjugations.
On the other hand, Thus, For an arbitrary closed connected surface Ω this relation turns into [10] H where U = σ∈S d σ 2 [8,10].
The proof for arbitrary Ω is almost the same as for the sphere. You only need to replace the relation σ 1 . . . σ f = 1 by the relations for the generators in π 1 (Ω, p). For an orientable Ω, the new relation is [

Hurwitz numbers and representation theory.
Formula (10) allows to describe Hurwitz numbers in terms of characters of the symmetric groups. The corresponding formula is where the summation ranges over all the characters of the irreducible representations of the group S d and where |C ∆ | is the cardinality of the set of elements of the cyclic type ∆. The first versions of the formula in the language of symmetric groups appeared in the works of Frobenius and Schur [27,28] (these were the cases later related to the unbranched coverings of S 2 and RP 2 ). The geometric consideration for the case of arbitrary surfaces appeared in [63,64]. Now we give a sketch of the proof of the formula (11).
Any partition λ of weight d generates the irreducible representation of S d of dimension dim λ. Let χ λ be the character of this representation. Then dim λ = χ λ (C [1,...,1] ). For any Young diagram ∆, we consider normalized character : The known orthogonality relations for the characters are [61] and where d = |∆| = |λ| and is the order of the automorphism group of the Young diagram ∆. (In this formula m i is the number of rows of length i in ∆.) Elements form the basis of idempotents of ZC[S d ], that is The transition between the two basises is given by and therefore Moreover and Therefore that is equivalent to (11). From (22) and (13) we get and for symmetric group A = S d and vectors ∆ 1 , ∆ we obtain This formula has a simple geometric explanation in the framework of Hurwitz topological field theory. Quadratic form F ∆,∆ A is used in the cutting axiom for gluing two parts of a surface. It is diagonal form kI, where k is inverse to the number of all admissible ways of gluing between the boundaries of the two parts. Moreover, gluing along a contour on which the covering has the same degree is admissible. Therefore k = z ∆ from (15). According to our definitions, Hurwitz topological field theory generates the Frobenius algebra with the structure constants in the basis of Young diagrams {∆}. According to (8), these structure constants coincide with structure constants of Z(C[S d ]) in basis C ∆ . The correspondence ∆ ↔ C ∆ generates the isomorphism between the Frobenius algebra of Hurwitz topological field theory and the centre of group algebra of symmetric group. (see [10] for details).
For the number D(∆) describing Möbius cut (6), we get where H 1 (∆) is the Hurwitz number counting the coverings of the real projective plane RP 2 with one critical value with the ramification profile ∆.
Formulas (22) and (14) allow us to give an independent proof of the fact that Hurwitz numbers satisfy the axioms of topological field theory: (22). Consider the set of partitions ∆ i , i = 1, . . . , f 1 + f 2 of the same weight d. We have the handle cut relation (Fig 3): (that is the manifistation of (3) ), and we have surface cut relation (Fig 2): (that is can be either cuts given by (3) and (5)), or the Moebius cut (6).
For instance, we get that the Hurwitz numbers of the projective plane can be obtained from the Hurwitz numbers of the Riemann sphere, while the Hurwitz numbers of the torus and the Klein bottle can be obtained from the Hurwitz numbers of the projective plane.
Jucys-Murphy elements. Jucys-Murphy elements serves to relate Hurwitz numbers to classical and also to the simplest quantum integrable systems (see Subsection 4.1 below). First applications of Jucys-Murphy elements to the description of integrable systems (namely, to the so-called KP and to the so-called TL hierarchies of integrable equations) were found in [32], [37] and in most clarified way in [40]. Let us consider the sums of transpositions as follows: where G is a symmetric function of the arguments, and c 1 , . . . , c d is the set of the contents of all d nodes of λ. (The content of the node of Young diagram λ with coordinates (i, j) is defined as j − i [61]). In other words, we get where We are going to explain the following: A) Integrable hierarchies generate correlators of the following type: The Kadomtsev-Petviashvili (KP) hierarchy [82] generates (see Proposition 5) The two-component version of the KP hierarchy (and the relativistic Toda lattice hierarchy) generates The BKP hierarchy [47] generates (see Proposition 6) Here G is any (symmetric) function defined by the choice of the solution of the equations of the hierarchies. B) The insertion of G can be equivalently described with help of the so-called completed cycles. This link will be discussed (see Lemma 1).
C) The insertion of G can be obtained as a result of the action of certain vertex operators on generating functions for Hurwitz numbers. Sometimes (see Subsection 4.2) the generating function can be interpreted as the so-called tau function of the integrable hierarchy. In this case this action is known as the action of the additional symmetries of classical integrable systems studied in [86]. Then the vertex operators can also be considerd treated as evolutionary operators for certain simple quantum models, see Subsection 4.1.
Finally, at the end of this subsection, we would like to mention the excellent book [59], which discusses many related topics.

The generating function for Hurwitz numbers.
Important applications of the Hurwitz numbers are related to the corresponding generating functions for 1-and 2-Hurwitz numbers. The (disconnected) 1-Hurwitz number h • m,∆ is the notation for the Hurwitz The generating function for the 1-Hurwitz numbers depends on the infinite number of formal variables p 1 , p 2 , . . . . We associate a Young diagram ∆ with rows of length ∆ 1 , . . . , ∆ k with the monomial p ∆ = p ∆1 , . . . , p ∆ k . The generating function for the 1-Hurwitz numbers is defined as This function has a number of remarkable properties that have been discovered relatively recently. The first one is the relationship where This relationship was first discovered in [35] by purely combinatorial methods. But it also has a geometric explanation [68], [69]. Consider a covering ϕ : Σ → S 2 of the type (∆, Γ 1 , . . . , Γ m ). Let q, p ∈ S 2 be the critical points of the covering ϕ related to the Young diagrams ∆ and Γ m , respectively. Connect the points q and p with a line l without self-intersections. The preimage of ϕ −1 (l) consists of d − 1 connected components, exactly one of which, sayl, contains the critical pointp with the critical value p. The ends of the componentl are the pre-imagesq 1 andq 2 of q. Now we move the point p along the line l in the direction of the point q, respectively, continuously changing the covering ϕ. As a result, we get a covering ϕ of the type (∆ , Γ 1 , . . . , Γ m−1 ). Let's see which Young diagram ∆ can serve this. Letq 1 =q 2 and c be the branching order of the covering ϕ at this pointq =q 1 =q 2 . The orders of the critical points other thanq, will not change if the cover ϕ is deformed into a cover ϕ . As a result of deformation, the pointq is splitted into 2 points with branch orders a and b, where a + b = c. Thus, the monomial p ∆ becomes a monomial p a p b ∂p∆ ∂pc . Suppose that the critical pointsq 1 andq 2 do not coincide and the orders of their branching are a and b, respectively. Then, as before, in the process of deformation of covering ϕ into a covering ϕ , the orders of critical points other thanq 1 andq 2 will not change. As a result of the deformation, the pointsq 1 and q 2 will be transferred to one critical point of order c = a+b. Thus, the monomial p ∆ becomes a monomial p c We defined the Hurwitz numbers, as the weighted number (2) of coverings ϕ : P → Σ where the surface P is not necessarily connected. Such Hurwitz numbers are often called disconnected Hurwitz numbers. If in this definition we consider only connected surfaces P , then the resulting number is called connected Hurwitz number. Let us denote h • m,∆ the connected version of 1-Hurwitz number h • m,∆ . It was proved that its generating function is a solution to the KP hierarchy. By simple combinatorial methods it is easy to prove that Therefore (34) generates

Gaussian integrals over sets of complex matrices
On this subject there is an extensive literature, for instance see [1-3, 96, 97]. We will consider integrals over N × N complex matrices Z 1 , . . . , Z n where the measure is defined as where the integration domain is C N 2 × · · · × C N 2 and where c n N is the normalization constant defined via dΩ(Z 1 , . . . , Z n ) = 1.
The Wick rule. Let us recall the following property of the Gaussian integrals: From (c is the normalization constant) it follows that d! that enters the right hand side can be interpreted as the number of ways to split the product z · · · zz · · ·z into the pairwise products zz. This funny observation is very useful and is known in physics as the Wick rule. The Wick rule is associated with the interpretations of matrix models in many problems of physics, for example, in the papers on statistical physics [43] quantum gravity [16] and also in [14], [13] which precede our work.
In what further, it will be applied to the product of matrix entries (Z i ) a,b and its complex conjugate Z i a,b , i = 1, . . . , n, a, b = 1, . . . , N . Then, the thanks to the measure dΩ, the integral of a monomial in matrix entries either vanishes, or is equal to a power of N .
The integrands we will consider in Sections 3.1,4 are polynomials in entries of matrices (Z i A i ) and (Ẑ iÂi ), i = 1, . . . , n (in certain cases integrands can be formal series in polynomials) where matrices A i andÂ i are not necessarily related while eachẐ i is Hermitian conjugate Z i .

Geometrical and combinatorial definition of Hurwitz numbers via graphs
Let Σ be a connected compact orientable surface without boundary of the Euler characteristic e. We fix on Σ two nonempty sets of points. The points of the first set {c 1 , . . . , c f } will be called capitals, and the points of the second set {w 1 , . . . , w v } will be called watch towers. To each capital c j we assign a Young diagram ∆ j = [∆ j 1 , . . . , ∆ j (j) ] with rows ∆ j 1 ≥ ∆ j 2 , . . . , ≥ ∆ j (j) > 0 from the set Υ d of all Young diagrams of weight d.
Consider a graph Γ with vertices {w 1 , . . . , w v } on Σ. We require that the edges of the graph do not intersect at interior points, and that the complement to the edges disintegrates into connected, simply connected domains P j c j (j = 1, . . . , f). (Such a partition always exists, except when Σ is a sphere and f = v = 1.) We denote the number of edges of Γ by n, the Euler characteristic of Σ is e = f + v − n. We will call the domains P j basic polygons . The boundary of each basic polygon consists of sides generated by the edges of the graph and vertices generated by the vertices of the graph. Thus, an edge of the graph generates either two different side of the same basic polygon or two sides of two different basic polygons. A vertex w of the graph (the watchtower) generates k vertices of basic polygons, where k is the number of basic polygons with vertex w. Fix one of vertices v of each basic polygon P .
Consider a d -sheeted branched covering of the basic polygon P j , corresponding to the Young diagram ∆ j having a single critical value c j . This covering consists of (j) ∆ j i -sheeted cyclic covers ϕ j i : P j i → P j . The preimages of the sides of the basic polygon P j are called sides of the polygon P j i . The order |Aut(ϕ j )| of the automorphism group of the covering Aut(ϕ j ) is equal to the order of the automorphism group of the Young diagram ∆ j i.e. |Aut(∆ j )| = m j (1)! . . . m j (∆ j 1 )!∆ j 1 ∆ j 2 , . . . , ∆ j (j) , where m j (r) is the number of rows of length r in the Young diagram ∆ j .
We divide the sides of the set of polygons {P j i } into pairs so that the images of the both sides of a pair coincide under the action of the maps ϕ j i and belong to the closure of only one basic polygon P j if and only if each side of the pair does not belong to the polygon P k i where k = j. Glue the sides of one pair so that the images of glued points coincide under the action of the covers ϕ j i . We call such gluing systems admissible. The number of different ways of gluing of the preimages of an edge of Γ is d!. Therefore, the total number of admissible gluing systems equals (d!) n . Each admissible gluing system ξ generates a branched covering ϕ(ξ) : Σ(ξ) → Σ. The surface Σ(ξ) that is glued from polygons P j i is compact, orientable, but possibly not connected. Critical values of covering ϕ(ξ) lie in the set {c 1 , . . . , c f } ∪ {w 1 , . . . , w v }, and the topological type of the critical value c j is equal to ∆ j . Topological type ∆ k of critical values w k depends on the gluing system. Automorphisms of the coverings ϕ j transfer admissible gluing into admissible ones, preserving the topological type of the glued covers. Now consider an arbitrary covering ϕ :Σ → Σ, the complement Σ 0 = Σ \ Γ and the preimagê Σ 0 = ϕ −1 (Σ 0 ). Adding the preimage ϕ −1 (Γ) implements the admissible gluing system of the surfacê Σ 0 to get the surfaceΣ. Thus, an admissible gluing system allows you to get any branched cover from any equivalence class of coverings of the surface Σ with critical values {c 1 , . . . , c f } ∪ {w 1 , . . . , w v } of topological type ∆ j at points c j . Denote by Φ v (∆ 1 , . . . , ∆ f ) the set of equivalence classes of all such coverings.
The correspondence ξ → ϕ(ξ) generates a mapping Ψ of the set Ξ(∆ 1 , . . . , ∆ f ) of all admissible gluing of the sides of the polygons P j i on the set Φ v (∆ 1 , . . . , ∆ f ). The mapping Ψ is constant on the orbits of the action of the group Aut = j Aut(∆ j ). In addition, the kernel of the action of the group Aut on the set Ξ(∆ 1 , . . . , ∆ f ) coincides with automorphisms of the coverings ϕ(ξ). Thus, In order to find a specific Hurwitz number H Σ (∆ 1 , . . . , ∆ f , ∆ 1 , . . . , ∆ v ) we assign N × N matrices to the sides and corners of the basic polygons.
The matrix assigned to a side l of a basic polygon P is denoted by Z l,P . Thus, the edge l separating the basic polygons P , P corresponds to the matrices Z l,P and Z l,P . The case P = P requires a separate discussion. In this case, the matrices Z l,P and Z l,P correspond to the sides of the same basic polygon and these sides are identified on the surface Σ. In all cases, we choose the matrices Z l,P and Z l,P Hermitian conjugate. We get n pairs of Hermitian conjugate complex matrices, this collection we denote {Z}.
At the vertex of the basic polygon P is the watchtower w. Denote by A w,P the matrix assigned to this vertex of the basic polygon. We will call these matrices source matrices. One gets a set of matrices A w,P1 , . . . , A w,P k related to vertex w, where P 1 , . . . , P k are basic polygons adjacent to the watchtower w. The collection of 2n source matrices we denote {A}.
The matrices Z l1,P , . . . , Z ls,P and A w 1 ,P , . . . , A w s ,P correspond to the basic polygon P with the capital c. We assume that the indexes of the sides and vertices are ordered counterclockwise, with the vertex w i lying between sides l i and l i+1 . Denote by M ({ZA} P ) = Z l1,P A w 1 ,P . . . Z ls,P A w s ,P the product of the matrices corresponding to polygon border bypass, we call this product monodromy around capital c. Here {ZA} P denotes the set of all pairwise products Z li,P A w i ,P , where the matrix Z li,P assigned to the side l i belonging to the country P , and the matrix A w i ,P is assigned to the vertex w i , which is the endpoint of the side l i (remember that the sides are oriented). In what follows we will write M Below {ZA} is the collection of all {ZA} P related to the set of all polygons P 1 , . . . , P f . Put Each edge of the graph corresponds to a pair of Hermitian conjugate matrices. Their union belongs to the set Z, consisting of n pairs of Hermitian conjugate matrices, where n is the number of edges of the graph Γ. The right side of the formula (38) contains d sets of matrices {Z}. Moreover, to each edge l of the graph Γ corresponds to d pairs of Hermitian conjugate matrices (Z l,1 ,Ẑ l,1 , . . . , Z l,d ,Ẑ l,d ) ∈ Z d . We denote their matrix elements Z l,i αβ ,Ẑ l,i αβ . Put U l;s,t α,β = Z l,s αβẐ l,t βα . Now on the set Z d we define the measure dΩ, assuming that the integral of a monomial product of entries of matrices from Z is equal to N −d if it is a product of monomials of the form U l;s,t α,β with respect to all edges of the graph Γ. The integral of the remaining monomials is assumed to be 0. (This measure is exactly the mesaure introduced in Subsection 2.6). Put where {A} is the collection of all source matrices and where {∆} is the set of Young diagrams ∆ 1 , . . . , ∆ f of the weight d.
We now consider an arbitrary watchtower w v and the matrices A wv,P1 , . . . , A wv,P k associated with it. We assume that the indexes of the basic polygons P i correspond to the clockwise rounds around the watchtower w v . Denote by A wv = A wv,P1 . . . A wv,P k the matrix product. We will call this product monodromy around the watchtower w v .
As we can see, I(Γ, {A}, {∆}) depends on the eigenvalues of special products of the original matrices, namely, on the eigenvalues of the watchtower monodromies A w1 , . . . , A wv .
Proof. If the integral of the monomial of matrix elements is not equal to 0, then its part, generated by the matrices Z andẐ is generated by monomials U l;s,t α = Z l,s αβẐ l,t βα . These monomials disappear after integration and contribute giving the factor N −nd . Grouping the remainder of the monomial by watchtowers, we obtain a monomial polynomial of the form where We now describe the expression (39) in terms of the coverings of the surface Σ. Consider the set of {l} edges of the graph Γ and its complement Σ 0 = Σ \ {l}. It splits into basic polygons. Consider the coveringΣ 0 → Σ 0 with the critical value in the capitals {c j } of basic polygons of topological type {∆ j }. The integration of the monomial U l;s,t α,β glues together the sides s and t covering the edge l. Thus, the integral of the monomial generates a coveringΣ → Σ of degree d and a monomial of a polynomial of the form (39). Denote by K the set of coverings constructed in this way. Let ∆ be the Young diagram of a basic polygon P . Consider the restrictionP → P of the coverings ofΣ 0 → Σ 0 on the preimage of the basic polygon P . Group Aut(∆) acts on the cover ofP → P by automorphisms. This action changes the gluing system and therefore acts on the set K, preserving the equivalence class of the covering. Denote by Φ the set of equivalence classes of coverings from K.
Suppose that all covers ϕ ∈ K have the trivial automorphism group. Then the set K splits into equivalence classes. The automorphism of the covering ϕ is realized by replacing the gluing system preserving the equivalence class of the covering. Thus, in the general case, the formula (40) gives the sum Summation over all monomials gives the statement of the theorem. tr (ZAZAZA) tr ẐÂẐÂẐÂ dΩ = where the summation over repeating indexes is implied.
The gluing of sheets is the same as the selection the non-vanishing types of monomials which means the pairing the entries of Z andẐ. The pairings (i) of the first Z from the left to the firstẐ from the left in the integrand, then (ii) the second Z from the left to the secondẐ and at last, (iii) the third Z from the left to the thirdẐ is related, respectively, to the equalities thus, one can get rid of the subscripts with the hats. One can record the way of gluing as 1 → 1, 2 → 2, 3 → 3. As one can see the integral of such monomials results in tr AÂ 3 . There are two other ways of gluing where one obtains the same answer. One is the case where we glue the first Z to the thirdẐ, the second to the first and the third to the second, let us present the gluing of the sheets as 1 → 3, 2 → 1, 3 → 2, that is we put Next, we glue the first Z to the secondẐ, the second to the third and the third to the first, let us present it as 1 → 2, 2 → 3, 3 → 1, that is we put Thus, these three ways of gluing result in∆ = (3).
One can see that the both ways of gluing 1 → 1, 2 → 3, 3 → 2 and 1 → 2, 2 → 1, 3 → 3 result in the same answer which is tr AÂ 3 , which is related to∆ = (1, 1, 1). The last way of gluing 1 → 3, 2 → 2, 3 → 1: , which is related to the case∆ = (3). In the end, we obtain The monodromies related to the 4-gon and to the single vertex are respectively equal to where Z i ,Ẑ i is the pair of Hermitian conjugated matrices, i = 1, 2, and where A i ,Â i , i = 1, 2 are four source matrices.
where the summation over repeating subscripts a i j , b i j ,â i j ,b i j is implied, i = 1, 2 and j = 1, 2. One gets where one should equate all indexes marked with hats to indexes without hats, as well as summing over duplicate indexes.
We have 4 ways of the equating of indexes (or, which is the same, of gluing the sheets) which are equations of type a i j =b i k and b i j =â i k where only the subscripts j and k can be different. The first one is thus, one can get rid of subscripts with hats. The summation of monomials over repeating subscripts of the matrix entries yields the term 1 , thus it is related to∆ = (1, 1). Three different ways of gluing: yields the same result. It means that The purely geometric consideration of this answer is as follows. Find the weighted number of 2-sheet coverings ϕ (the Hurwitz number) of the torus without branching. Fix a basis A, B of the fundamental group of the torus T . The number of the connected components (circuits) of the inverse images ϕ −1 (A) and ϕ −1 (B) is 1 or 2. This gives 4 non-isomorphic covers (one unconnected and 3 connected). The automorphism group of each of the covers is Z 2 . Thus, the Hurwitz number is 2.
One gets the 4 sets of equations identical to the previous case, however, now the result is which is related to∆ = (2).

Non-orientable case. Integrals and the topological theory
Hurwitz numbers for non-orientable surfaces were studied in [64], [45], [7], [8]. In the context of integrable systems it was done in [79], [80], [17]. Now we relate it to matrix integrals. Consider where D(∆)is given by (24), and where sums range over all partitions. We use (41) for the description of (6) and we use (42) in relation with (5) and (4), see Figure 3, as follows: The interpretation of the equality in the Theorem can be as follows. Consider the original orientable surface Σ and the drawn graph Γ of Subsection 3.1 and remove h pairs of faces with capitals (c 2i , c 2i−1 ) , i = 1, . . . , 2h and seal each pair of holes with a handle. Further, in addition, remove m faces with capitals c i , i = 2h + 1, . . . , 2h + m and seal each one with a Moebius strip. We obtainΣ.
Proof. Proof follows from the axioms of topological theories presented by relations (25) and (27) and from the explicit form of (41) and of (42) and from Theorem 3.

Hurwitz numbers and quantum and classical integrable models
In this section we show the links of Hurwitz numbers with certain quantum integrable systems.

Dubrovin's commuting quantum Hamiltonians, Okounkov's completed cycles, Jucys-Murphy elements, classical integrable systems
Vertex operators of Kyoto school. Consider one-parametric series of differential operators in infinitely many variables p = (p 1 , p 2 , . . . ): which depends on the parameter z ∈ S 1 and acts on formal series in p 1 , p 2 , . . . . Properly ordered exponentials of θ(z) and properly defined products of such exponentials are known in physics as vertex operators 1 . As it was explained in the works of Kyoto school, the vertex operators play an important role in the theory of classical integrability. Now, let us consider the vertex operator that depends on two parameters z and q = e y : : 1 At first, the vertex operators were considered in the work of Pogrebkov and Sushko in [94] in their studies of quantum Thirring model. Dots :: A :: denote the so-called bosonic normal ordering of A, that means that all differential operators ∂ ∂pm , m > 0 called in physics "ellimination operators" which are present in A are moved to the right, while all "creation operators", which are all p m , m ≥ 0, are moved to the left.
This vertex operator acts on functions of p 1 , p 2 , . . . as a shift operator in infinitely many variables. In classical soliton theory such vertex operator acts on the so-called tau functions [44] where this action means the adding of the so-called soliton to a solution to Kadomtsev-Petviashvili equation [82]. In [86], [39] it was used to construct symmetries of the KP hierarchy sometimes calledŴ 1+∞ symmetries.
The important fact pointed out in [84] (see also [70], [69]) is that H 3 is the cut-and-join operator (35), introduced in [35]. (c) For any λ, Schur functions s λ (p) are eigenfunctions of these Hamiltonians, or, the same, the Schur functions are eigenstates of quantum dispersionless KdV equations: The vacuum vector is given by s 0 = 1.
The proof of this statement is contained in [23] (see also [25]). (The last formula was known much earlier from the boson-fermion correspondence 2 .) Remark 1. Hamiltonians {Ĥm} belong to the Cartan subalgebra ofŴ1+∞ and the Schur functions are eigenfunctions of the whole set of these operators. There exits more general algebra of commuting operators which does not belong toŴ1+∞ and whose eigenfunctions are the Schur functions, see [68], [69].  (45), that means ∂ t (∂ tm θ z ) = ∂ tm (∂ t θ z ). The expansion of the right hand side of (46) in y = log q gives rise to the higher dispersionless quantum KdV equations.
From Proposition 3 it follows Corollary 2.
[84] For any Young diagram λ, the Schur function s λ is the eigenfunction of the operator Formula (48) was used by Okounkov to generate the so-called completed cycles, see [84].
Proof follows from (29) and from the relation between the eigenvalues e λ (q) and the so-called quantum contents q i−j of the nodes of Young diagram λ with coordinates i, j: The last equality is obtained by the re-summation of the left hand side. Let us introduce the following Hurwitz numbers where J i are Jusys-Murphy elements, see Subsection 2.4. Let us mention that the expansion of the left side in the small parameter y = log q generates Hurwitz numbers defined on the completed cycles.

Proposition 4. We have
Proof follows from Cauchy-Littlewood identity from Corollary 2 and from Lemma 1.

Hurwitz numbers (52) generated by tau functions.
Let us consider the commuting set of quantum HamiltoniansĤ(q),Ĥ(q 2 ),Ĥ(q 3 ), . . . which generate the following evolution flows The flow parameters t(q) = (t 1 (q), t 2 (q), . . . ) can be treated as the higher times for soliton lattice solutions related to the Hurwitz problem. In this subsection we shall use the notation t m instead of the notation t m (q) and the notation t instead of t(q).
We refer the reader to the important notion of tau function introduced by Kyoto school, see [44] for a review of the topic. There are only few cases where Hurwitz numbers (52) are generated by tau functions and we are going to select them. Let us note that the tau functions we need is are the tau functions of the hypergeometric family described in [89] (the so-called 2KP case) and in [90] (the so-called BKP case). As functions of the variables p 2KP tau functions are written as series in the Schur functions ( [98], [100] and also in [58], [88] and the specification we need in is written down in [89]. These tau functions solve a pair KP hierarchy: one KP hierarchy with respect to the higher times p = (p 1 , p 2 , . . . ) and other KP hierarchy with respect to the higher times t = (t 1 , t 2 , . . . ). Below we also need the BKP analogue of KP series written down in [90]. As functions in the variables t solutions below presented in [89] and in [90] may be called soliton lattices in the R ∞ space with coordinates t. Lemma 2. Let t = (t 1 , t 2 , . . . ), k ∈ Z is the infinite set of parameters and p(0, q) = (p 1 (0, q), p 2 (0, q), . . . ) is the special set given by p m (0, q) = 1 1−q m . Then = e m>0 1 m tmq mkĤ (q m ) · e m>0 1 m pm 1−q m is a KP tau function (the soliton lattice one) with respect to the parameters t which plays the role of the KP higher times and the parameters p and k define initial configuration of solitons. It is also a KP tau function of hypergeometric family with respect to the parameters p which, now, play the role of the KP higher times while the parameters t and k are given constants. It is also a tau function of the two-dimensional Toda lattice hierarchy with respect to the sets t, p and the discrete variable k. The last statement means that φ k (p, t) = log τ k (p, t) − log τ k−1 (p, t) solves Toda lattice equation The proof is basically contained in [89].
The proof is basically contained in [90].
where sum ranges over all partitions Remark 2. Simirlaly, we can show that ∂ solves KP equation with respect to the variables t * 1 , t * 2 , t * 3 and according to [103] it also solves the so-called Burges equation. Also < e is the tau function of the BKP hierarchy. We will not explain it in details, let us point out that in this case, one needs an additional triangle transform of from the set of the KP (respectively, of the BKP) higher times to the set t * 1 , t * 2 , . . . . This can be derived from the relation (29) and the results of [80].

2D Yang-Mills quantum theory
If we slightly modify the integrand in Theorem 3, then the integral obtained will lead to another interesting problem. This is the problem of calculating the correlation functions of two-dimensional quantum gauge theories in the case of both orientable and non-orientable surfaces. Quantum two-dimensional gauge theories (also known as 2D Yang-Mills (2DYM) models) were analyzed in the works of A. Migdal and his students [67], [95], and in more detail in Witten's work [102], in which the non-orientable case was also considered. The following answer for the partition function Z Σ of the two-dimensional Yang-Mills theory with the unitary gauge group was obtained [102]: Here ρ is the coupling constant of the theory and for the case G = SU (N ) one has is the dimension of the representation λ, e is the Euler characteristic of the surface Σ (in [67] only the case Σ = S 2 was studied). The following answer for the correlation function of the pair of the so-called Wilson loops around points z 1 , z 2 ∈ Σ was obtained Z Σ (ρ; Θ 1 , Θ 2 ) = e z 1 Adz e z 2 Adz The meaning of the left side of (59) is explained in [67], s λ on the right side denotes the Schur function and Θ i ∈ SU (N ) denotes the class of the Wilson loop e z i Adz , i = 1, 2, A is the so-called gauge field on Σ, which is a flat connection on Σ without z 1 , z 2 . In [89] it was noted that the right-hand side coincides with a certain hypergeometric tau function of the Toda lattice [58], [87], [88].
The answer for the correlation function of k Wilson loops of 2DYM theory on a surface Σ with an Euler characteristic e = 2−2g (or, that is the same, the Yang-Mills path integral over all flat connections on Σ\{z i } where the holonomies around k marked points belong to the conjugacy classes Θ 1 , . . . , Θ k ) is [102]: Our claim is to present the same right hand side of ( [102]) with the help of Gaussian integral over complex matrices and relate it to the counting of Hurwitz numbers.
Consider the following series over Young diagrams: τ (X 1 , X 2 , t, a) := where a ∈ C, and X 1 , X 2 , Y ∈ GL N (C). According to [88], (64) is the tau function of the Toda lattice (TL) hierarchy introduced in [44]. The sets tr(X m i ), m = 1, 2, . . . , i = 1, 2 play the role of the higher times of the TL hierarchy. According to [90], (65) is the tau function of the so-called BKP hierarchy introduced in [47]. The set tr(Y m ), m = 1, 2, . . . plays the role of the set of the higher times of the BKP hierarchy.
Relation to Hurwitz numbers. The comparance with Theorem 4 and also with the help of (11) and character map relation (61) and with the fact that dimλ is equal to dim U λ times a polynomial in N shows that right hand sides of (66), (60) generates a specific series in Hurwitz numbers. We will do it similarly it was done in [91], or in [40]. Namely, we need where φ λ (k) := ∆ (∆)=d−k ϕ λ (∆), k = 0, . . . , d − 1 The corollary of (67) is where eachφ λ (k) is built of a collection {φ λ (i) , i > 0}: e(e − 1) · · · (e − l + 1) Here µ = (µ 1 , . . . , µ l ) is a partition which may be written alternatively [61] as µ = (1 m1 2 m2 3 m3 · · · ) where m i is the number of times a number i occurs in the partition of |µ| = k. Thus the set of all non-vanishing m ja , a = 1, . . . l , (l ≤ l) defines the partition µ of length (µ) = l a=1 m ja = l and of weight |µ| = l a=1 j a m ja = k. Then the order of the automorphism group of the partition µ is |aut µ| := m j1 ! · · · m j l ! We seeφ λ (k; 1) = φ λ (k). Note that φ λ (1) = ϕ λ (2, 1 |λ|−2 ) that is the normalized character evaluated on the cycle class of transpositions. The profile (2, 1 |λ|−2 ) is related to the so-called simple branch point. (68) is called the length of permutation with cycle structure λ and will be denoted by * (λ) (also called the colength of the partition λ). The colength enters the well-known Riemann-Hurwitz formula which relates the Euler characteristic e of a base surface to the Euler characteristic e of it's d-branched cover:

Remark 3. The quantity d − (λ) which is used in the definition
where the sum is over all branch points zi , i = 1, 2, . . . and where the ramification profiles are given by partitions ∆ i , i = 1, 2, . . . .

Let us introduce
According to (68) this degree is equal to the colength of ramification profiles, and due to Remark 3 it defines the Euler characteristic e of the covering surfaces. From (70) we get degφ λ (i; e) = i In the notations of Theorem 3 we obtain The coefficient in the square brackets is a polynomial function in normalized characters ϕ λ of degree m + i (see (71)). Therefore, thanks to (22) this is a linear combination of Hurwitz numbers, where the Euler characteristics of the base surface and its cover are equal, respectively, to e and to e i,m = de − k v=1 * (∆ (v) ) − m − i. The difference with (22) comes from the coefficient in square brackets where (φ(1)) m describes m extra simple branch points, whileφ(i, e) generates additional branch points in number, not more than i, with weights given by (70).