NONCOMMUTATIVE TWO-DIMENSIONAL TOPOLOGICAL FIELD THEORIES AND HURWITZ

NUMBERS FOR REAL ALGEBRAIC CURVES

A.ALEXEEVSKI ^{1)}, S.NATANZON ^{2)}

Contents

1. Introduction 2 2. Structure algebra 5 2.1. Definition of a structure algebra 5 2.2. Semisimple structure algebras 8 2.3. Structure algebra of a finite group 12 3. Cuts of stratified surfaces 15 3.1. Stratified surfaces 15 3.2. Cut systems 18 3.3. Basic and simple surfaces 24 3.4. Neighboring complete cut systems 29 4. Two-dimensional topological field theory 34 4.1. Definition of Klein topological field theory 34 4.2. Correlators of Klein topological field theory 38 4.3. From system of correlators to structure algebras 44 4.4. From structure algebra to Klein topological field theory 47 5. Hurwitz topological field theory 52 5.1. Stratified coverings 52 5.2. Cut coverings 57 5.3. Hurwitz topological field theory 62 5.4. Hurwitz numbers 66 References 68

Abstract It is well-known that classical two-dimensional topological field the-

ories are in one-to-one correspondence with Frobenius algebras. We define topological field theories for non necessary oriented surfaces possibly with a boundary and call them Klein topological field theories. We prove that Klein topological field theories bijectively correspond to a more complicated algebraic structure which we call a structure algebra.

This research is partially supported by 1) CRDF grant RM1-2088 2)RFBR-01-01-00739, NSh-1972.2003.1 and INTAS-00-0259.

A structure algebra is a non-commutative analogue of a Frobenius algebra. Semisimple structure algebras are classified in the work. Starting from an arbitrary finite group we construct a structure algebra and prove that it is semisimple one.

We prove that classical Hurwitz numbers of complex algebraic curves and their analogues for real algebraic curves have natural interpretation in terms of structure constants of a structure algebra for symmetric group.

Key words: topological field theory, Frobenius algebra, Hurwitz numbers. MSC: 16W, 57M, 81T

1. Introduction

The notion of a topological field theory (TFT) was introduced in works [35, 4, 33]. TFTs play important role both in pure mathematics and in mathematical physics. In particular, two-dimensional topological field theories are used in string theory and brane theory [23], G-flux and K-theories [25], WZW, G/G and CS theories [13]. In this work we deal with two-dimensional TFTs only.

A TFT is based essentially on a functor Ω ↦ V_{Ω} from a category of topological manifolds with singularities into a category of vector spaces. V_{Ω} depends on singularities of a manifold Ω. If a functor is fixed, than TFT is defined as an invariant choice of a linear form Φ ∈ V_{Ω}^{∗} for each manifold Ω.

Two-dimensional topological field theory was defined first for a category of compact oriented surfaces with boundary contours [4] (’closed’ theories). Contracting these contours to points we can and will consider closed topological field theories as theories based on the category of compact oriented surfaces without boundaries endowed with finitely many special points.

Later was defined so called ’open-closed’ TFT [23, 26]. In our terms open-closed TFTs are based on the category of compact oriented surfaces with finite set of special points lying either in the interior or on the boundary contours of a surface. Formerly, in [19] was outlined the construction of TFT for the category to nonoriented surfaces. In this work (in contrast to previously mentioned ones) partition functions for triangulated surfaces are defined instead of a functor.

Closed 2D TFTs are classified by Frobenius algebras, i.e., associative algebras endowed with invariant nondegenerate scalar products [10, 1, 11, 34]. In [23] is constructed an algebra corresponding to an openclosed TFT. In more explicit form this algebra is described in [26]. In [26] is claimed (without a proof) that this correspondence is a bijection. An algebraisation is outlined also in [19].

In this work we give a definition of a topological field theory based on a category of arbitrary compact surfaces (i.e., nonoriented surfaces, possibly nonorientable, possibly with a boundary) that are endowed with a finite set of special points lying either in the interior or on the boundary. We call any such TFT a Klein topological field theory (KTFT).

We prove that Klein topological field theories correspond bijectively to noncommutative analogues of Frobenius algebras. We call these algebras structure algebras.

A structure algebra is an associative (typically noncommutative) algebra endowed with an invariant scalar product and three features: a decomposition into direct sum of a commutative subalgebra and an ideal (reflecting two kinds of special points, interior and boundary ones), an involutive antiautomorphism (reflecting the change of the local orientation at a special point) and an element U (reflecting nonorientability of a surface). Emphasize that KTFT gives an additional structure even for the category of orientable surfaces; it is an involutive antiautomorphism. This involutive antiautomorphism is missed in closed and open-closed TFTs because fixing the orientation of a surface allows to work without it.

We classify all semisimple structure algebras. In terms of TFTs this is the classification of massive Klein topological field theories.

Starting from any finite group we construct structure algebras. (Only one structure algebra corresponds to symmetric group, but it can happen that several structure algebras correspond to the same group G.) These structure algebras are semisimple.

We apply Klein topological field theory to study (generalized) Hurwitz numbers. More than 100 years ago Hurwitz [18] raised the following problem. Let Ω be a complex algebraic curve of genus g, p_{1}, …, p_{m} ∈ Ω be pairwise distinct points and α_{1}, …, α_{m} be partitions n (a decomposition of number n into unordered sums of positive integers). Denote by S(α_{1}, ..., α_{m}) the set of classes of birational equivalence of morphisms π : P → Ω of order n having critical values p_{1}, …, p_{m} and for i = 1, …, m local degrees of π at points π^{−1}(p_{i}) are summands from α_{i}. The problem is to comput the sum ∑_{π∈S(α_{1},...,α_{m})} \frac{1}{| Aut(π)|}, where | Aut(π)| is the order of the group of automorphisms of a covering π. These sums are called Hurwitz numbers.

The generating function for Hurwitz numbers is the partition function for 2D supersymmetric Yang-Mills theory [7]. These numbers are connected with string theory [16], mirror symmetry [9], theory of singularities [3], matrix models [22] and integrable systems [32]. There exist combinatorial, differential and integral formulas for the simplest of Hurwitz numbers. See, for example, [3, 15, 12].

Analog of Hurwitz numbers for morphisms into fixed real algebraic curve is defined in this work. We call these numbers Hurwitz numbers of a real algebraic curve, or generalized Hurwitz numbers, or just Hurwitz numbers for short. Generalized Hurwitz numbers include classical Hurwitz numbers as a particular case.

Generalized Hurwitz numbers can be defined in topological terms as weighted numbers of ’stratified coverings’ over a compact stratified surface possibly with boundary (see 5.1).

According to [7] Hurwitz numbers of real algebraic curves are important for string theory and supersymmetric Yang-Mills theory.

Hurwitz numbers of real algebraic curves were investigated in [24, 5] in several special cases.

In this work we prove that Hurwitz numbers of real algebraic curves generate a Klein topological field theory. We call this KTFT by Hurwitz topological field theory (HTFT) and prove that its structure algebra is isomorphic to the structure algebra of symmetric group S_{n}. Using this construction we obtain simple algebraic expressions for all Hurwitz numbers. Namely, any Hurwitz number is a sum of products of Hurwitz numbers of three surfaces: a sphere with three special points, a triangle, i.e., a disc with three boundary special points, and a disc with one boundary and one interior special point. Character formulas for classical Hurwitz numbers [9] are essentially particular cases of our expressions.

The paper is organized as follows.

In section 2 we define a structure algebra, classify semisimple structure algebras and construct semisimple structure algebras for any finite group.

In section 3 we prove a geometrical theorem about ’complete cut systems’ of an arbitrary surface with special points. A complete cut system generates the decomposition of a surface. Obtained decompositions are analogs of ’punt’ decompositions of a surfaces without a boundary. We define adjacent complete cut systems and prove that two arbitrary complete cut systems of a surface are connected by a chain of adjacent ones.

In section 4 we define a Klein topological field theory. We construct a correspondence between KTFTs and structure algebras and prove that it is bijective correspondence. The theorem about complete cut systems (section 3) is a base of the proof.

In section 5 we construct a topological field theory (HTFT) from generalized Hurwitz numbers. We prove that its structure algebra is isomophic to the structure algebra of symmetric group and give a formula for Hurwitz numbers. At the end of section 5 we note, that there are G-analogs of generalized Hurwitz numbers for a finite group G. They come from coverings with structure group G instead of S_{n}. The theory of ’G-Hurwitz numbers’ is quit similar to the theory described in section 5.

Acknowledgment. The final of this paper was written during the second author stay at Max-Planck-Institut für Mathematik in Bonn. He would like to thank this institution for its support and hospitality.

2. Structure algebra

2.1. Definition of a structure algebra. All vector spaces (algebras) in the paper are vector spaces (algebras) over complex numbers. Let X be an associative algebra with unit endowed with a nondegenerate invariant scalar product (x, y) : X ⊗ X → ℂ, i.e., (x, y) = (y, x), (xy, z) = (x, yz) and (x, a) = 0 for an arbitrary x ∈ X implies a = 0.

Fix a basis E of X. We denote by α, α′, ⋯ ∈ E elements of the basis. We use the same letters as indices of tensors. Thus, we denote by F_{α′,α′′} the matrix of the scalar product in basis E and we denote by F^{α′,α′′} the tensor of the dual form. Hence, F_{α′,α′′}F^{α′′,α′′′} = δ_{α′}^{α′′′}. As usual, repeating indices mean summation.

Denote by K̂_{X} an element F^{α′,α′′}α′ ⊗α′′ of the tensor product X ⊗X and by K_{X} Casimir element. By definition, K_{X} = F^{α′,α′′}α′α′′ ∈ X. Clearly, elements K̂_{X} and K_{X} have coordinate-independent definitions. There are three operators X → X associated with Casimir element. They are the left multiplication by K_{X}, the right multiplication by K_{X} and operator x ↦ F^{α′,α′′}α′xα′′. The latter we denote by V_{K_{X}}.

By x ↦ x^{∗} we denote an involutive antiautomorphism of X, i.e., (x^{∗})^{∗} = x and (xy)^{∗} = y^{∗}x^{∗}. Denote by K_{X,∗} twisted Casimir element F^{α′,α′′}α′α′′^{∗}. Obviously, it coincides with Casimir element of twisted scalar product (x, y)_{∗} = (x, y^{∗}).

Definition. A structure algebra H = {H = A ∔ B, (x, y), x ↦ x^{∗}, U} is a finite dimensional associative algebra H endowed with

a decomposition H = A ∔ B of H into the direct sum of two vector spaces;

an invariant symmetric bilinear form (x, y) : H ⊗ H → ℂ; an involutive antiautomorphism H → H, denoted by x ↦ x^{∗}; an element U ∈ A,

such that the following axioms hold:

1^{∘} A is a subalgebra belonging to the center of algebra H; algebra A has unit 1_{A} ∈ A and 1_{A} is also the unit of algebra H;

2^{∘} B is a two-sided ideal of H (typically noncommutative); algebra B has a unit 1_{B} ∈ B;

3^{∘} restrictions (x, y)|_{A} and (x, y)|_{B} are nondegenerate scalar products on algebras A and B resp.

4^{∘} (V_{K_{B}}(b_{1}), b_{2}) = (K̂_{A}, b_{1} ⊗ b_{2}) for arbitrary b_{1}, b_{2} ∈ B (this axiom reflects Cardy relation [23]);

5^{∘} an involutive antiautomorphism preserves the decomposition H = A∔B and the form (x, y) on H, i.e., A^{∗} = A, B^{∗} = B, (x^{∗}, y^{∗}) = (x, y);

6^{∘} U^{2} = K_{A,∗};

7^{∘} (U, b) = (K_{B,∗}, b) for any b ∈ B;

8^{∘} (aU)^{∗} = aU for any a ∈ A.

Remark. Forms (x, y)|_{A} and (x, y)|_{B} are nondegenerate and we use them to raise or lower indices in tensors. Bilinear form (x, y) on H is not assumed to be nondegenerate. Linear subspaces A and B are not assumed to be orthogonal. Clearly, if an element a ∈ A is orthogonal to B then aB = 0. Therefore, if A is orthogonal to B then AB = 0.

Fix a structure algebra H = {H = A ∔ B, (x, y), x ↦ x^{∗}, U}. Let a be an element of A. Then formula (ϕ(a), b) = (a, b) defines an element ϕ(a) ∈ B because bilinear form (x, y)|_{B} is nondegenerate.

Lemma 2.1. The mapping ϕ : A → B is a homomorphism of algebra A into the center of the algebra B such that for any a ∈ A, b ∈ B the equality ab = ϕ(a)b holds.

Proof. By invariance of the form (x, y) we have (ab, b′) = (a, bb′) = (ϕ(a), bb′) = (ϕ(a)b, b′). Therefore, ab = ϕ(a)b, which means that ϕ is a homomorphism. By definition, A lies in the center of H. Hence (ab, b′) = (ba, b′) = (b, ab′) = (b, ϕ(a)b′) = (bϕ(a), b′). Therefore, ϕ(a)b = bϕ(a) and ϕ(a) belongs to the center of B. □

We shall reformulate the definition of a structure algebra H = {H = A ∔ B, (x, y), x ↦ x^{∗}, U} in a coordinate form. Fix basis E_{A} of A and E_{B} of B. We denote by α, α_{1}, … the elements of E_{A} and by β, β_{1}, … the elements of E_{B}. Indices of tensors are denoted below by the same letters α, α_{1}, …; β, β_{1}, . . ..

Define tensors:

(1) F_{α_{1},α_{2}} = (α_{1}, α_{2});

(2) F_{β_{1},β_{2}} = (β_{1}, β_{2});

(3) R_{α,β} = (α, β);

(4) S_{α_{1},α_{2},α_{3}} = (α_{1}α_{2}, α_{3});

(5) T_{β_{1},β_{2},β_{3}} = (β_{1}β_{2}, β_{3});

(6) R_{α,β_{1},β_{2}} = (αβ_{1}, β_{2});

(7) I_{α_{1},α_{2}} = (α_{1}^{∗}, α_{2});

(8) I_{β_{1},β_{2}} = (β_{1}^{∗}, β_{2});

(9) D_{α} = (U, α);

(10) J_{α} = (1_{A}, α);

(11) J_{β} = (1_{B}, β).

Tensors (1)-(11) completely define a structure algebra. Indeed, tensors (1)-(3) define the bilinear form, tensors (4)-(6) define the multiplication in algebra H, tensors (7)-(8) define the involutive antiautomorphism, tensor (9) defines the element U and tensors (10), (11) define units 1_{A} ∈ A and 1_{B} ∈ B.

We call tensors (1)-(11) structure constants of structure algebra H in basis E_{A}, E_{B}.

Let us write down exact formulas. Suppose we are given by two linear spaces A and B with basis E_{A} of A, E_{B} of B and by arbitrary tensors F_{α_{1},α_{2}}, …, J_{β} that have the same type as tensors from the left-hand sides of equalities (1) - (11) and are denoted by the same letters.

Assume F_{α_{1},α_{2}} and F_{β_{1},β_{2}} are symmetric and nondegenerate. Using F_{α_{1},α_{2}} and F_{β_{1},β_{2}} and dual tensors we will raise or lower indices of tensors. In the case of asymmetric tensors we always raise the last index.

Let H = A ⊕ B. Define bilinear form on H by formulas (1)-(3). Define a multiplication in H as follows: α_{1}α_{2} = S_{α_{1},α_{2}}^{α}α; β_{1}β_{2} = T_{β_{1},β_{2}}^{β}β; βα = αβ = R_{α,β}^{β′}β′. By definition, A is a subalgebra and B is an ideal of H. Define a linear map x ↦ x^{∗} as follows: α^{∗} = I_{α}^{α′}α′; β^{∗} = I_{β}^{β′}β′. Define elements U ∈ A, 1_{A}, 1_{B} ∈ B by formulas U = D^{α}α, 1_{A} = J^{α}α, 1_{B} = J^{β}β. Denote by S_{α_{1},α_{2},α_{3},α_{4}} the contraction S_{α_{1},α_{2}}^{α′}S_{α′,α_{3},α_{4}} and by T_{β_{1},β_{2},β_{3},β_{4}} the contraction T_{β_{1},β_{2}}^{β′}T_{β′,β_{3},β_{4}}.

Lemma 2.2. A set of data H = {H = A ∔ B, (x, y), x ↦ x^{∗}, U} is a structure algebra if and only if

(1) F_{α_{1},α_{2}}, F_{β_{1},β_{2}} are symmetric nondegenerate tensors;

(2) S_{α_{1},α_{2},α_{3}} and S_{α_{1},α_{2},α_{3},α_{4}} are symmetric tensors;

(3) tensors T_{β_{1},β_{2},β_{3}} and T_{β_{1},β_{2},β_{3},β_{4}} are invariant under cyclic permutations;

(4) R_{α,β_{1},β_{2}} = R_{α}^{β′}T_{β′,β_{1},β_{2}};

(5) R_{β}^{α′}S_{α′,α_{1},α_{2}} = R_{α_{1}}^{β′}R_{α_{2}}^{β′′}T_{β′,β′′,β};

(6) R_{α,β_{1},β_{2}} = R_{α,β_{2},β_{1}};

(7) R_{α′,β_{1}}F^{α′,α′′}R_{α′′,β_{2}} = T_{β_{1},β′}^{β′′}T_{β′′,β_{2}}^{β′}

(8) I_{α_{1}}^{α′}I_{α′,α_{2}} = F_{α_{1},α_{2}}, I_{β_{1}}^{β′}I_{β′,β_{2}} = F_{β_{1},β_{2}}

(9) I_{α_{1},α_{2}} = I_{α_{2},α_{1}}, I_{β_{1},β_{2}} = I_{β_{2},β_{1}}, I_{α}^{α′}R_{α′,β} = I_{β}^{β′}R_{α,β′}

(10) I_{α_{1}}^{α′}I_{α_{2}}^{α′′}I_{α_{3}}^{α′′′}S_{α′′′,α′′,α′} = S_{α_{1},α_{2},α_{3}}, I_{β_{1}}^{β′}I_{β_{2}}^{β′′}I_{β_{3}}^{β′′′}T_{β′′′,β′′,β′} = T_{β_{1},β_{2},β_{3}},

(11) S_{α,α′,α′′}D^{α′}D^{α′′} = S_{α,α′,α′′}I^{α′,α′′};

(12) D^{α′}R_{α′,β} = I^{β′,β′′}T_{β′,β′′,β}

(14) D^{α′}S_{α′,α_{1},α_{2}} = I_{α_{1}}^{α′}D^{α′′}S_{α′,α′′,α_{2}}

(15) J^{α′}S_{α′,α_{1},α_{2}} = F_{α_{1},α_{2}}, J^{α′}R_{α′,β_{1},β_{2}} = F_{β_{1},β_{2}}

(16) J^{β′}T_{β′,β_{1},β_{2}} = F_{β_{1},β_{2}}

Proof. We shall list below the correspondence between relations (1)- (12) and axioms of a structure algebra. All proofs are by direct calculations.

Relation (1) coincides with axiom 3^{∘}.

Relation (2) is equivalent to the claim ’A is an associative commutative algebra endowed with an invariant scalar product’.

Relation (3) is equivalent to the claim ’B is an associative algebra endowed with invariant scalar product’.

Define a map ϕ : A → B by ϕ(α) = R_{α}^{β}β. Clearly (α, β) = (ϕ(α), β). Relation (4) is equivalent to the claim αβ = ϕ(α)β. Relation (5) is equivalent to the claim ’ϕ is a homomorphism of

algebras’.

Relation (6) is equivalent to the claim ϕ(α)β = βϕ(α).

It is easy to show that if ϕ is a homomorphism and its image lies in the center of algebra B then H is an associative algebra and the form (x, y) is invariant and vice versa.

Relation (7) is equivalent to axiom 4^{∘}.

Relation (8) is equivalent to the claim ’x ↦ x^{∗} is an involution’. Relation (9) is equivalent to the claim ’the involution x ↦ x^{∗} pre-

serves the bilinear form (x, y)’.

Relations (10) is equivalent to the claim ’the involution x ↦ x^{∗} preserving the form (x, y) is an antiautomorphism’. Indeed, two relations from (10) are the direct reformulation of this fact for subalgebras A and B. Let us prove that (αβ)^{∗} = β^{∗}α^{∗}. We have (ϕ(α), β) = (α, β). Therefore, (ϕ(α^{∗}), β) = (α^{∗}, β) = (α, β^{∗}). Analogously, (ϕ(α)^{∗}, β) = (ϕ(α), β^{∗}) = (α, β^{∗}). We obtain that ϕ(α^{∗}) = ϕ(α)^{∗}. Hence, (αβ)^{∗} = (ϕ(α)β)^{∗} = β^{∗}ϕ(α)^{∗} = β^{∗}ϕ(α^{∗}) = β^{∗}α^{∗}.

Relation (11) is equivalent to axiom 6^{∘}.

Relation (12) is equivalent to axiom 7^{∘}.

Relation (14) is equivalent to axiom 8^{∘}.

Relations (15), (16) are equivalent to the claims ’1_{A} is a unit of H’ and ’1_{B} is a unit of B’.

□

We call (1) - (16) relations for structure constants of a structure algebra.

2.2. Semisimple structure algebras. A structure algebra H is called semisimple if the algebra H is semisimple.

Let H be a finite dimensional complex semisimple structure algebra. Then both subalgebras A and B are semisimple algebras. Indeed, B is an ideal of H and A is isomorphic to the factor H/B.

By Wedderburn theorem, B is isomorphic to the direct sum of matrix algebras, B = ⊕_{i=1}^{k} M_{n_{i}}. A is commutative semisimple algebra; therefore, A is isomorphic to the direct sum of one-dimensional algebras,

A = \overbrace{ℂ ⊕ ⋯ ⊕ ℂ}^{m}. Clearly, as an abstract algebra H is isomorphic to the direct sum A ⊕ B. Despite this fact, the decomposition H = A ∔ B that is included in the set of data H typically does not coincide with a decomposition into the direct sum of two-sided ideals.

Let us write down formulas for the multiplication A × B → B and for the invariant bilinear form (x, y) : H ⊗ H → ℂ. There is a uniquely defined, up to permutations, complete system e_{1}, … e_{m} of orthogonal idempotents in A and A = ℂe_{1} ⊕ ⋯ ⊕ ℂe_{m}. Orthogonal idempotents are orthogonal with respect to the invariant scalar product. Therefore, we have (e_{i}, e_{j}) = λ_{i}δ_{i,j}, λ_{i} ≠ 0.

Similarly, components M_{n_{i}} of algebra B = ⊕_{i=1}^{k} M_{n_{i}} are orthogonal with respect to any invariant scalar product on B. All invariant scalar products on M_{n_{i}} are proportional to each other. Hence, if X, Y ∈ M_{n_{i}} then (X, Y) = μ_{i} tr(XY), μ_{i} ≠ 0.

Denote by E_{i} the unit matrix of M_{n_{i}}. Clearly, elements {E_{i}|i = 1, …, k} form a basis of the center of algebra B and they are orthogonal idempotents. By lemma 2.1, morphism ϕ : A → B is a homomorphism into the center of B. Therefore, the image of a complete system {e_{i}|i = 1, …, m} of orthogonal idempotents is a system of orthogonal idempotents in the center of B. Hence, ϕ(e_{i}) = ∑_{j∈N_{i}} E_{j}, where N_{i} ⊂ {1, …, k} and if i ≠ i′ then N_{i} ∩ N_{i′} = ∅. Thus,

e_{i}b = be_{i} = (∑_{j∈N_{i}} E_{j})b for any b ∈ B (2.1)

and this formula completely describes the multiplication between elements of A and B. For elementary matrices E_{s,i,j} ∈ M_{n_{s}} we get

(e_{i}, E_{s,i,j}) = (ϕ(e_{i}), E_{j}) = \{ \begin{matrix}μ_{s}δ_{i,j} if s ∈ N_{i} \\ 0 if s ∉ N_{i} \end{matrix} (2.2)

Conversely, let A = ℂ^{m} and B = ⊕_{i=1}^{k} M_{n_{i}} be algebras with invariant scalar products defined by constants λ_{i}, i = 1, …, m and μ_{j}, j = 1, …, k resp. Define a multiplication between A and B by formula 2.1 and the scalar product between A and B by formula 2.2. Then axioms 1^{∘}-3^{∘} of a structure algebra are satisfied. Let us denote such an algebra endowed with an invariant bilinear form by H̃ = A ∔ B.

Lemma 2.3. Cardy axiom 4^{∘} is satisfied in algebra H̃ if and only if (1) for i = 1, …, m a subset N_{i} of the set {1, …, k} is either empty set or contains one element;

(2) {1, …, k} = ⊔_{i}N_{i};

(3) if N_{i} = {j} then λ_{i} = μ_{j}^{2}.

Proof. Let us compute element K̂_{A} and transformation V_{K_{b}}. Evidently, K̂_{A} = ∑_{i=1}^{m} \frac{1}{λ_{i}} e_{i} ⊗ e_{i} and K_{A} = ∑_{i=1}^{m} \frac{1}{λ_{i}} e_{i}. Denote by E_{s,i,j} the elementary matrix E_{i,j} ∈ M_{n_{s}}. Evidently, all matrices E_{s,i,j} form a basis of algebra B. We obtain by direct calculations that K_{B} = ∑_{s=1}^{k} \frac{n_{s}}{μ_{s}} E_{s} and V_{K_{B}}(E_{s,i,j}) = δ_{i,j} \frac{1}{μ_{s}} E_{s}. Therefore, putting b_{1} = E_{s,i,j}, b_{2} = E_{s′,i′,j′} in axiom 4^{∘} we obtain that:

a) left side = δ_{s,s′}δ_{i,j}δ_{i′,j′} \frac{1}{m_{s}} μ_{s} = δ_{s,s′}δ_{i,j}δ_{i′,j′};

b) right side = \{ \begin{matrix}\frac{μ_{s}μ_{s′}}{λ_{l}} δ_{i,j}δi′, j′ if s, s′ ∈ N_{l} \\ 0 if s and s′ do not lie in the same set N_{l} \end{matrix} The proof is completed by comparing these formulas.

□

Assume that axiom 4^{∘} holds for an algebra H̃. Then we can and will reorder components of A and B in such a way that

a) for i = 1, …, k we have ϕ(e_{i}) = E_{i}, where E_{i} is the unit matrix of the component M_{n_{i}};

b) for i = k + 1, …, m we have ϕ(e_{i}) = 0.

Let us check axioms related to an involutive antiauthomorphism x ↦ x^{∗} and an element U for algebra H̃. Suppose x ↦ x^{∗} is an involutive transformation satisfying axiom 5^{∘}.

Obviously, involutive antiautomorphism x ↦ x^{∗} can either fix an idempotent e_{s} or permute two idempotents e_{s′} and e_{s′′}. If the latter occurs then λ_{s′} = λ_{s′′} because this antiautomorphism preserves the scalar product. Thus, the involution of the set {1, …, m} is induced. We denote it by the same sign s ↦ s^{∗}.

Analogously, the involution x ↦ x^{∗} either preserves a component M_{n_{s}} or permutes two components M_{n_{s}} and M_{n′_{s}} of algebra B. If the latter occurs then we have n_{s} = n_{s′} and μ_{s} = μ_{s′}. Thus, the involution of the set {1, …, k} is induced. This involution coincides with the restriction to {1, …, k} of previously defined involution s ↦ s^{∗} of the set {1, …, m} because transformation x ↦ x^{∗} preserves bilinear form and e_{i} is orthogonal to M_{n_{j}} if and only if i ≠ j.

Obviously, the restriction of the transformation x ↦ x^{∗} to A is completely defined by the involution s ↦ s^{∗} of the set {1, …, m}. There is an ambiguity for the restriction of the involutive transformation to subalgebra B. The possibilities are well known. We describe them below without proofs.

If s ≤ k and s^{∗} ≠ s then the restriction of x ↦ x^{∗} to M_{n_{s}} ⊕ M_{n_{s^{∗}}} is conjugated to the transformation {X, Y } → {Y′, X′}, where X′ denotes the transpose of X.Here {X, Y } is an element of M_{n_{s}} ⊕ M_{n_{s^{∗}}}, i.e., two matricies. Thus, changing the basis of B if necessary, we have {X, Y }^{∗} = {Y′, X′}.

If s ≤ k and s^{∗} = s then an involutive antiautomorphism of M_{n_{s}} is conjugated to one of two canonical antiautomorphisms.

The first of them is associated with a symmetric bilinear form and coincides with the transpose of a matrix.

The second of them is associated with a nondegenerate skew-symmetric bilinear form. Hence, it exists only if n_{s} = 2r. Let us identify the set {1, …, n_{s}} with the set ℤ_{2} × {1, …, r}, where ℤ_{2} is the group with two elements {0, 1}. If i = (ϵ, i′) then put ϵ(i) = ϵ and i^{τ} = (ϵ + 1, i′). Define a transformation τ : M_{n_{s}} → M_{n_{s}} by formula (E_{i,j})^{τ} = (−1)^{ϵ(i)+ϵ(j)}E_{j^{τ} ,i^{τ}}. One can check directly that τ is the involutive antiautomorphism preserving the invariant scalar product.

Therefore, if s^{∗} = s then in an appropriate basis we have either E_{s,i,j}^{∗} = E_{s,j,i} or E_{s,i,j}^{∗} = (−1)^{ϵ(i)+ϵ(j)}E_{s,j^{τ} ,i^{τ}}. In order to distinguish these cases let us introduce an invariant ν = ν(s) ∈ ±1 and put ν(s) = 1 in the former case, ν(s) = −1 in the latter case.

Denote by P a set of fixed points of involution s ↦ s^{∗} and put P_{0} = P ∩ {1, …, k}.

As a result of the above considerations, we obtain the following lemma.

Lemma 2.4. Let x ↦ x^{∗} be an involutive antiautomorphisms of algebra H̃. Suppose, x ↦ x^{∗} satisfies axiom 5^{∘}. Then antiautomorphism x ↦ x^{∗} induces

∘ involution s ↦ s^{∗} of set {1, …, m} such that ({1, …, k})^{∗} =

{1, …, k} and μ_{s} = μ_{s^{∗}}, λ_{s} = λ_{s^{∗}};

∘ numbers ν(s) ∈ {±1} for s ∈ P_{0} where P_{0} = {s|s ≤ k, s^{∗} = s}

Antiautomorphisms s ↦ s^{∗} up to inner automorphism of H̃ are classified by pairs (s ↦ s^{∗}; ν(s)).

Let us fix additionally an involutive antiautomorphism satisfying axiom 5^{∘} and check the possibilities of choosing an element U ∈ A satisfying axioms 6^{∘} − 8^{∘}.

Lemma 2.5. An element U ∈ A satisfies axioms 6^{∘} − 8^{∘} if and only if U = ∑_{i∈P_{0}} \frac{ν_{i}}{μ_{i}} e_{i} + ∑_{j∈P \P_{0}} x^{j}e_{j}, where (x^{j})^{2} = \frac{1}{λ_{j}}.

Proof. Let U = ∑_{i=1}^{m} x^{i}e_{i} be an element of A satisfying axioms 6^{∘} − 8^{∘}. Then U^{2} = ∑_{i=1}^{m}(x^{i})^{2}e_{i}. Twisted Casimir element K_{A,∗} is equal to ∑_{i} \frac{1}{λ_{i}} e_{i}e_{i^{∗}} = ∑_{i∈P} \frac{1}{λ_{i}} e_{i}. Therefore, U = ∑_{i∈P} x^{i}e_{i} and (x^{i})^{2} = \frac{1}{λ_{i}}

(axiom 6^{∘}) Evidently, (U, E_{s,i,j}) = \{ \begin{matrix}x_{s}μ_{s}δ_{i,j} if s ∈ P_{0} \\ 0 if s ∉ P_{0} \end{matrix}

Compute element K_{B,∗}. By definition, K_{B,∗} = ∑_{s=1}^{k} ∑_{i,j} \frac{1}{μ_{s}} E_{s,i,j}E_{s,j,i}^{∗}. If s^{∗} ≠ s then E_{s,i,j}E_{s,j,i}^{∗} = 0, hence only summands corresponding to s ∈ P_{0} are nonzero.

If ν_{s} = 1 then ∑_{i,j} \frac{1}{μ_{s}} E_{s,i,j}E_{s,j,i}^{∗} = ∑_{i,j} \frac{1}{μ_{s}} E_{s,i,j}E_{s,i,j} = \frac{1}{μ_{s}} ∑_{i} E_{s,i,i} = \frac{1}{μ_{s}} E_{s} = \frac{1}{μ_{s}} ν_{s}E_{s}.

If ν_{s} = −1 then ∑_{i,j} \frac{1}{μ_{s}} E_{s,i,j}E_{s,j,i}^{∗} = ∑_{i,j} \frac{1}{μ_{s}} E_{s,i,j}(−1)^{ϵ(j)+ϵ(i)}E_{s,i^{∗},j^{∗}} = \frac{1}{μ_{s}} ∑_{i} E_{s,i,i^{∗}}(−1)E_{s,i^{∗},i} = \frac{1}{μ_{s}} ν_{s}E_{s}.

Therefore, K_{B,∗} = ∑_{s∈P_{0}} \frac{1}{μ_{s}} ν_{s}E_{s} and (K_{B,∗}, E_{s,i,j}) = \{ \begin{matrix}ν_{s}δ_{i,j} if s ∈ P_{0} \\ 0 if s ∉ P_{0} \end{matrix}

By axiom 7^{∘} x_{s}μ_{s} = ν_{s} for s ∈ P_{0}. Thus, we prove that U = ∑_{i∈P_{0}} \frac{ν_{i}}{μ_{i}} e_{i} + ∑_{j∈P \P_{0}} x^{j}e_{j}, where (x^{j})^{2} = \frac{1}{λ_{j}}.

Conversely, let U is given by the latter formula. Then reversing our steps, we obtain that it satisfies axioms 6^{∘} and 7^{∘}. By direct calculations we obtain also that U satisfies axioms 8^{∘}. □

Remark. In the case of a semisimple structure algebra axiom 8^{∘} follows from axioms 6^{∘} and 7^{∘}.

Thus, we proved the following theorem.

Theorem 2.1. Let H = {H = A ∔ B, (x, y), x ↦ x^{∗}, U} be a semisimple structure algebra. Then

(1) A = ℂe_{1} ⊕ ⋯ ⊕ ℂe_{m}, B = M_{n_{1}} ⊕ ⋯ ⊕ M_{n_{k}}, k ≤ m and ϕ(e_{i}) = E_{i} for i ≤ k;

(2) (X_{i}, X̃_{i}) = μ_{i} tr(X_{i}X̃_{i}), μ_{i} ≠ 0, for X_{i}, X̃_{i} ∈ M_{n_{i}}, i = 1, …, k, and (X_{i}, X̃_{j}) = 0 for i ≠ j;

(3) (e_{i}, e_{i}) = μ_{i}^{2} for i = 1, …, k; (e_{i}, e_{i}) = λ_{i} for i > k; (e_{i}, e_{j}) = 0 for i ≠ j;

(4) if M_{n_{i}}^{∗} = M_{n_{j}} then e_{i}^{∗} = e_{j};

a) if i ≠ j then n_{i} = n_{j}, μ_{i} = μ_{j} and for X ∈ M_{n_{i}} we have X^{∗} = X′ ∈ M_{n_{j}};

b) if i = j then either X^{∗} = X′ or n_{i} is even and X^{∗} = X^{τ}; put ν_{i} = 1 in the first case and ν_{i} = −1 in the second case;

(5) for i > k we have e_{i}^{∗} = e_{j} and if i ≠ j then λ_{i} = λ_{j};

(6) U = ∑_{i∈P_{0}} \frac{ν_{i}}{μ_{i}} e_{i} + ∑_{j∈P \P_{0}} x^{j}e_{j}, where P = \{x ∈ \{1, …, m\}|e_{i}^{∗} = e_{i}\}, P_{0} = P ∩ \{1, …, k\} and (x^{j})^{2} = \frac{1}{λ_{j}}.

Conversely, if conditions (1)-(6) hold then H is a semisimple structure algebra

Corollary 2.1. If a semisimple associative algebra H, its decomposition H = A ⊕ B and bilinear form (x, y) satisfy axioms 1^{∘} − 4^{∘} then there exists at least one involutive antiautomorphism satisfying axiom 5^{∘}. For each of them there exists 2^{p} > 0, where p = |P \ P_{0}| elements U ∈ A such that H = {H = A ⊕ B, (x, y), x ↦ x^{∗}, U} is a semisimple structure algebra.

Proof is evident.

2.3. Structure algebra of a finite group. In this section we assign a structure algebra H = H(G) to any finite group G. Denote by |Λ| the cardinality of a set Λ. Denote by ℂ[G] the group algebra of group G. Denote by A the center of algebra ℂ[G]. Let us assign an element E_{α} = ∑_{g∈α} g ∈ ℂ[G] to a class α of conjugated elements. It is known, that elements E_{α} form a basis of A.

Denote by M(G) an algebra of all endomorphisms of vector space ℂ[G]. Elements of G form a basis of ℂ[G]). Therefore, an elementary matrix in M(G) corresponds to an ordered pair (g_{1}, g_{2}) of elements of G. Denote an elementary matrix by E_{g_{1},g_{2}}.

Two ordered pairs (g_{1}, g_{2}), (g′_{1}, g′_{2}) of elements of G are called conjugated if and only if there exists an element g ∈ G such that gg_{1}g^{−1} = g′_{1}, gg_{2}g^{−1} = g′_{2}. Let (s_{1}, s_{2}) be an ordered pair of involutive elements of G and β be a class of ordered pairs conjugated to (s_{1}, s_{2}). Define an element E_{β} ∈ M(G) by the formula E_{β} = ∑_{(s′,s′′)∈β} E_{s′,s′′}. Denote by B a linear subspace of M(G) generated by all E_{β}. Obviously elements E_{β} form a basis of B. Action of g ∈ G on group G by conjugation x ↦ gxg^{−1} defines an element of M(G); we denote it by V_{g}. Obviously mapping g ↦ V_{g} defines the representation V : ℂ[G] → M(G).

Lemma 2.6. .

(1) V (g) = ∑_{h∈G} E_{gh,hg}.

(2) V (g)E_{g_{1},g_{2}} = E_{gg_{1}g^{−1},g_{2}}, E_{g_{1},g_{2}}V (g) = E_{g_{1},g^{−1}g_{2}g}

(3) V (g)E_{g_{1},g_{2}}V (g^{−1}) = E_{gg_{1}g^{−1},gg_{2}g^{−1}}

The proof is elementary.

Lemma 2.7. Linear subspace B coincides with the set of matrices X ∈ M(G) such that

a) X is a linear combination of elements E_{s_{1},s_{2}}, where s_{1}, s_{2} are involutive elements of G;

b) X commutes with all V_{g}, g ∈ G.

The proof follows from lemma 2.6. □

Put V_{α} = V (E_{α}). Define the algebra structure on the direct sum H = A ⊕ B. The multiplication of elements of A and elements of B follows from algebra structure on A and B respectively. Define multiplication of elements E_{α} ∈ A and E_{β} ∈ B by formulas: E_{α}E_{β} = V_{α}E_{β}, E_{β}E_{α} = E_{β}V_{α}.

Lemma 2.8. H = A ⊕ B is a semisimple associative algebra, A is a central subalgebra, B is an ideal.

Proof. Associativity of H and properties of the decomposition H = A ⊕ B can be easily checked by direct calculations. A is a semisimple algebra as the center of a group algebra.

Let us proove that B is a semisimple algebra. Denote by S the set of all involutions in G. Clearly, the subspace ℂ[S] ⊂ ℂ[G] is invariant with respect to the representation V : ℂ[G] → M(G). By lemma 2.7, B coincides with the centralizer of V (G) ⊂ M(S). Therefore, B is a semisimple algebra. Hence H is a semisimple algebra. □

There are natural involutive antiautomorphism in both algebras A and B. Namely, denote by ”^{∗}” the linear extension of the involution g ↦ g^{−1}, g ∈ G, to group algebra ℂ[G]. Obviously, E_{α}^{∗} = E_{α^{∗}}, where α^{∗} consists of inverse elements to elements of α and ”^{∗}” is an involutive antiautomorphism of A.

Denote by the same sign ”^{∗}” the transpose of matrices of M(G), it is an involutive antiautomorphism of the algebra. For a class β = [(s_{1}, s_{2})] put β^{∗} = [(s_{2}, s_{1})]. Clearly, E_{β}^{∗} = E_{β^{∗}}. Therefore, B^{∗} = B. So we have defined the involutive transformation of H.

Lemma 2.9. Involutive transformation x ↦ x^{∗} is an antiautomorphism of algebra H.

Proof. We should check that (E_{α}E_{β})^{∗} = E_{β}^{∗}E_{α}^{∗}. We have (E_{α}E_{β})^{∗} = (V_{α}E_{β})^{∗} = E_{β}^{∗}V_{α}^{∗}. By lemma 2.6 V_{g} = ∑_{h∈G} E_{gh,hg}. Therefore, V_{g}^{∗} = ∑_{h∈G} E_{hg,gh} = ∑_{h∈G} E_{g^{−1}h′,h′g^{−1}}, where h′ = ghg. Hence V_{g}^{∗} = V_{g^{−1}} and V_{α}^{∗} = V_{α^{∗}}. □

It is well-known that any invariant symmetric bilinear form (x, y) is uniquely defined by a linear form f(x) such that f(xy − yx) = 0 identically. Bilinear form corresponding to f(x) is defined as (x, y) = f(xy). Define a linear form f on H by formulas: f(E_{α}) = \frac{1}{|G|}δ_{α,1}, f(E_{β}) = \frac{1}{|G|} tr(E_{β}) (recall that E_{β} is an element of matrix algebra M(G)). Clearly, f defines the invariant symmetric bilinear form (x, y) on H.

The restriction (x, y)|_{A} coincides with the restriction to the center of the standard invariant form on ℂ[G] given by formula (g_{1}, g_{2}) = \frac{1}{|G|}δ_{g_{1},g_{2}^{−1}} for g_{1}, g_{2} ∈ G. Therefore, (E_{α_{1}}, E_{α_{2}}) = \frac{1}{ν_{α_{1}}} δ_{α_{1},α_{2}^{∗}}, where ν_{α} =

\frac{|G|}{|α|}. Obviously ν_{α} is equal to the number of elements in the centralizer of any g ∈ α. Note that form (x, y)|_{A} is nondegenerate.

The restriction (x, y)|_{B} coincides with the restriction to B of the standard invariant form on M(G) given by formula (X_{1}, X_{2}) = \frac{1}{|G|} tr(X_{1}X_{2}).

Therefore, (E_{β_{1}}, E_{β_{2}}) = \frac{1}{ν_{β_{1}}} δ_{β_{1},β_{2}^{∗}}, where ν_{β} = \frac{|G|}{|β|}. Obviously ν_{β} is equal to the number of elements in the stabilizer of any element (s_{1}, s_{2}) ∈ β. Clearly, the form (x, y)|_{B} is nondegenerate. Note that scalar products (E_{α}, E_{β}) can be nonzero because by definition (E_{α}, E_{β}) = f(E_{α}E_{β}) = \frac{1}{|G|} tr(V_{α}E_{β}).

Clearly, involution x ↦ x^{∗} preserves this bilinear form. According to notations of previous subsection put F_{α_{1},α_{2}} = (E_{α_{1}}, E_{α_{2}}), F_{β_{1},β_{2}} = (E_{β_{1}}, E_{β_{2}}) and R_{α,β} = (E_{α}, E_{β}).

Lemma 2.10. Cardy axiom 4^{∘} holds for algebra H.

Proof. First, let us compute the left hand side L = (V_{K_{B}}(E_{β_{1}}), E_{β_{2}}) of the identity in axiom 4^{∘}. By definition, V_{K_{B}}(E_{β_{1}}) = F^{β′,β′′}E_{β′}E_{β_{1}}E_{β′′}. We have F^{β′,β′′} = ν_{β′}δ^{β′,β′′^{∗}}. Hence V_{K_{B}}(E_{β_{1}}) = ∑_{β} ν_{β}E_{β}E_{β_{1}}E_{β^{∗}} and L = ∑_{β} ν_{β} \frac{1}{|G|} tr(E_{β}E_{β_{1}}E_{β^{∗}}E_{β_{2}}) = ∑_{β} \frac{1}{|β|} tr(E_{β}E_{β_{1}}E_{β^{∗}}E_{β_{2}}).

Each summand in this formula is a product of elementary matrices E_{x,y}E_{s_{1},s_{2}}E_{u,v}E_{s_{3},s_{4}}, where (x, y) ∈ β, (s_{1}, s_{2}) ∈ β_{1}, (u, v) ∈ β^{∗}, (s_{3}, s_{4}) ∈ β_{2}. If the trace of the summand is nonzero then y = s_{1}, u = s_{2}, v = s_{3}, x = s_{4}. Let this conditions be satisfied. Then the trace is equal to 1 and the pair (s_{4}, s_{1}) is conjugated to the pair (s_{3}, s_{2}). Therefore,

L = ∑_{(s_{1},s_{2},s_{3},s_{4},g|(s_{1},s_{2})∈β_{1},(s_{3},s_{4})∈β_{2},gs_{2}g^{−1}=s_{1},gs_{3}g^{−1}=s_{4},g∈G} \frac{1}{|[(s_{4}, s_{1})]|} \frac{1}{|Z_{G}(s_{4}, s_{1})|} =

= \frac{|\{(s_{1}, s_{2}, s_{3}, s_{4}, g|(s_{1}, s_{2}) ∈ β_{1}, (s_{3}, s_{4}) ∈ β_{2}, g ∈ G, gs_{2}g^{−1} = s_{1}, gs_{3}g^{−1} = s_{4}\}|}{|G|}.

Let us compute the right hand side R = (K̂_{A}, E_{β_{1}} ⊗ E_{β_{2}}) of the identity from axiom 4^{∘}. By definition,

R = ∑_{α′,α′′} F^{α′,α′′}(E_{α′}, E_{β_{1}})(E_{α′′}, E_{β_{2}}) = ∑_{α} ν_{α}(E_{α}, E_{β_{1}}) · (E_{α^{∗}}, E_{β_{2}}) = ∑_{α} ν_{α} \frac{1}{|G|^{2}} tr(V_{α}E_{β_{1}}) tr(V_{α^{∗}}E_{β_{2}}).

By lemma 2.6, we have V_{α} = ∑_{h∈G,g∈α} E_{gh,hg} = ∑_{x∈G,g∈α} E_{x,g^{−1}xg}. Analogously, V_{α^{∗}} = ∑_{y∈G,h∈α} E_{y,hyh^{−1}}.

Hence,

R = ∑_{α} \frac{1}{|G||α|}(∑_{x∈G,g∈α,(s_{1},s_{2})∈β_{1}} tr(E_{x,g^{−1}xg}E_{s_{1},s_{2}}))· ·(∑_{y∈G,h∈α,(s_{3},s_{4})∈β_{2}} tr(E_{y,hyh^{−1}}E_{s_{3},s_{4}})).

Nonzero summands arise from the cases x = s_{2}, g^{−1}xg = s_{1}, y = s_{4}, hyh^{−1} = s_{3}; in this case a summand is equal to \frac{1}{α}.

Therefore,

R = ∑_{α} \frac{1}{|G||α|}|\{s_{1}, s_{2}, s_{3}, s_{4}, g, h|(s_{1}, s_{2}) ∈ β_{1}, (s_{3}, s_{4}) ∈ β_{2}, g ∈ α, h ∈ α, s_{2} = gs_{1}g^{−1}, s_{3} = hs_{4}h^{−1}}|.

Element h runs through the set α of elements conjugated to g. Hence, we can present it as h = zgz^{−1}, where z runs through all elements of group G. Each element h has k presentations of this type, where k = |Z_{G}(g)| is the cardinality of the centralizer of element g. Therefore, R = ∑_{α} \frac{1}{|G||α|} \frac{1}{ν_{α}} |\{s_{1}, s_{2}, s_{3}, s_{4}, g, z|(s_{1}, s_{2}) ∈ β_{1}, (s_{3}, s_{4}) ∈ β_{2}, g ∈ α, z ∈ G, s_{2} = gs_{1}g^{−1}, s_{3} = (zgz^{−1})s_{4}(zg^{−1}z^{−1}\}|. Note that \frac{1}{|G||α|} \frac{1}{ν_{α}} = \frac{1}{|G|^{2}} and therefore, this coefficient can be carried out of summing. Therefore, R = \frac{1}{|G|^{2}} |\{s_{1}, s_{2}, s_{3}, s_{4}, g, z|(s_{1}, s_{2}) ∈ β_{1}, (s_{3}, s_{4}) ∈ β_{2}, g ∈ G, z ∈ G, s_{2} = gs_{1}g^{−1}, s_{3} = (zgz^{−1})s_{4}(zg^{−1}z^{−1}}|.

The equality s_{3} = (zgz^{−1})s_{4}(zg^{−1}z^{−1}) can be rewritten as s′_{3} = gs′_{4}g^{−1}, where s′_{3} = z^{−1}s_{3}z, s′_{4} = z^{−1}s_{4}z. For two fixed z = z_{1} and z = z_{2} the numbers of tuples (s_{1}, s_{2}, s_{3}, s_{4}, g, z) satisfying conditions are equal since the pairs (s_{3}, s_{4}) run through all representatives of the class β_{2}. Therefore, R = \frac{1}{|G|^{2}} |G||\{s_{1}, s_{2}, s_{3}, s_{4}, g, z|(s_{1}, s_{2}) ∈ β_{1}, (s_{3}, s_{4}) ∈ β_{2}, g ∈ G, s_{2} = gs_{1}g^{−1}, s_{3} = gs_{4}g^{−1}}|.

We get that L = R. □

Thus, we have proved all axioms for a structure algebra except those concerning an element U ∈ A. By corollary 2.1, there exists at least one element U ∈ A satisfying axioms 6^{∘} − 8^{∘}. Let us fix it. Therefore, we prove theorem.

Theorem 2.2. For any finite group G the set of data H(G) = (H = A ⊕ B, (x, y), x ↦ x^{∗}, U) is a structure algebra.

Remark. It can be shown that in the case of the symmetric group G = S_{n} the element U is uniquely determined.

3. Cuts of stratified surfaces

3.1. Stratified surfaces. Denote by ∂X a boundary of a topological space X and denote by X^{∘} its interior \overline{X} \ ∂X. We deal below with compact topological manifolds possibly with boundary and call them ’manifolds’ for short.

Examples

(1) A connected one-dimensional manifold is homeomorphic either to a circle or to a segment.

(2) A connected orientable two-dimensional manifold is homeomorphic to a sphere with g handles and s holes. We call it a surface of type (g, s, 1).

(3) A connected nonorientable two-dimensional manifold is homeomorphic either to a projective plane with a handles and s holes or to a Klein bottle with a handles and s holes. We call it a surface of type (g, s, 0), where g = a + 1/2 in the former case and g = a + 1 in the latter case.

Definition. Let Λ be a finite set. The decomposition Ω = ∐_{λ∈Λ} Ω_{λ} of a topological space Ω into the disjoint union of subspaces (strata) Ω_{λ} ⊂ Ω is called a stratification if and only if

(1) each stratum Ω_{λ} is homeomorphic to a connected manifold without a boundary;

(2) each stratum is open in its closure;

(3) the boundary ∂Ω_{λ} of a stratum coincides with the union of strata of less dimension.

We call topological space Ω itself a base of the stratification. Zerodimensional strata are called special points.

Remark.

(1) A stratum is not necessarily homeomorphic to an open ball.

(2) Joining of strata induces a partial ordering of set Λ.

(3) There are several different definitions of a stratified manifold (see [17]). All these definitions are equivalent in dimensions 1 and 2. Fortunately, in this work we deal with dimensions 1 and 2 only. □

A homeomorphism ϕ : Ω → Ω′ of the bases of stratified manifolds Ω = ∐_{λ∈Λ} Ω_{λ} and Ω′ = ∐_{λ∈Λ′} Ω′_{λ} is called an isomorphism of stratifications, if and only if the restriction of ϕ to any stratum of Ω is the homeomorphism with an appropriate stratum of Ω′. Clearly, an isomorphism of stratifications induces the isomorphism of partial ordered sets Λ and Λ′.

Definition. A stratification Ω = ∐_{λ∈Λ} Ω_{λ} of a manifold Ω is called special stratification if and only if all strata of codimension one belong to the boundary of Ω.

Examples (1) A connected specially stratified one-dimensional manifold is iso-

morphic either to a circle with the unique stratum or to a segment with natural stratification (strata are two end points and the open interval).

(2) Let Ω = ∐_{λ∈Λ} Ω_{λ} be a connected specially stratified two-dimensional manifold. We call the set of data G = (g, ε, m, m_{1}, …, m_{s}) a type of Ω. Here (g, s, ε) is the type of surface Ω, m is the number of interior special points, m_{i} is the number of special points on i-th boundary contour. Two types that differ only in the order of m_{i} are considered as equal. It can be easily shown that an isomorphism class connected two-dimensional specially stratified manifolds is uniquely determined by its type.

(3) Let Ω be a stratified surface consisting of connected components Ω_{i}. Denote by G unordered set of types G^{i} of surfaces Ω_{i}. We call G a type of Ω. Clearly, up to isomorphism, Ω is uniquely determined by its type.

Remark. A special stratification of a surface is uniquely defined by a set Ω_{0} of its special points.

We call a two-dimensional specially stratified manifold Ω consisting of finitely many connected components a stratified surface for short. Let Ω consists of c connected components. Denote by G^{i} = (g^{i}, ε^{i}, m^{i}, m_{1}^{i}, …, m_{s}^{i}) (i = 1, …, c) a type of i-th connected component Ω^{i}. Define an invariant μ(Ω) by formula

μ(Ω) = ∑_{i} 2g^{i} + ∑_{i} m^{i} + ∑_{i} s^{i} + \frac{1}{2} ∑_{i} ∑_{j} m_{j}^{i} − 2c

From the definition follows that μ(Ω) is half-integer. Clearly, invariant μ is additive with respect to the disjoint union of stratified surfaces:

μ(Ω_{1} ∐ Ω_{2}) = μ(Ω_{1}) + μ(Ω_{2})

We say that a connected stratified surface Ω is a trivial surface if and only if μ(Ω) ≤ 0. All trivial surfaces can be easily listed.

Lemma 3.1. Any trivial stratified surface is isomorphic to one of surfaces from the following list:

(1) sphere S^{2} without special points (μ = −2);

(2) projective plane ℝP^{2} without special points (μ = −1);

(3) disc D^{2} without special points (μ = −1);

(4) sphere (S^{2}, p) with a unique interior special point p (μ = −1);

(5) disc (D^{2}, q) with a unique boundary special point q and without interior special points (μ = −\frac{1}{2});

(6) sphere (S^{2}, p_{1}, p_{2}) with two interior special points (μ = 0);

(7) projective plane (ℝP^{2}, p) with one interior special point (μ = 0);

(8) torus T^{2} without special points (μ = 0);

(9) Klein bottle Kl without special points (μ = 0); (10) disc (D^{2}, p) with a unique interior special point and without boundary special points (μ = 0);

(11) disc (D^{2}, q_{1}, q_{2}) with two boundary special points and without interior special points (μ = 0);

(12) Möbius band Mb without special points (μ = 0);

(13) cylinder Cyl without special points (μ = 0).

3.2. Cut systems. Let Ω be a stratified surface. A generic not selfintersecting curve γ ⊂ Ω is called a simple cut. This implies that γ does not meet any special point and either is (closed) contour consisting purely of interior points, or is a segment, its end points belong to the boundary of Ω and all interior points of the segment are interior points of the surface.

A set Γ of pairwise nonintersecting simple cuts γ ⊂ Ω is called a cut system of Ω (Fig.1a).

Let Ω, Ω′ be two stratified surfaces endowed with cut systems Γ, Γ′. We say that an isomorphism ϕ : Ω → Ω′ of stratified surfaces is an isomorphism of pairs (Ω, Γ), (Ω′, Γ′) if and only if ϕ(Γ) = Γ′.

Definition. A triple (Ω_{∗}, Γ_{∗}, τ) consisting of ∘ a stratified surface Ω_{∗}; ∘ a subset Γ_{∗} ⊂ ∂Ω_{∗} such that each connected component of Γ_{∗} coincides with the closure of a one-dimensional stratum of Ω_{∗}; ∘ an involutive homeomorphism τ : Γ_{∗} → Γ_{∗} having no fixed

points is called cut surface. In this case pair (Γ_{∗}, τ) is called a gluing system. An isomorphism of cut surfaces (Ω_{∗}, Γ_{∗}, τ), (Ω′_{∗}, Γ′_{∗}, τ′) is an isomor-

phism ϕ : Ω_{∗} → Ω′_{∗} of stratified surfaces such that ϕ(Γ_{∗}) = Γ′_{∗} and τ′ ∘ ϕ = ϕ ∘ τ.

Let (Ω_{∗}, Γ_{∗}, τ) be a cut surface. Gluing points x and τ(x) we obtain a surface Ω and ’gluing topological map’ glue : Ω_{∗} → Ω. Clearly, stratification of Ω_{∗} induces the stratification of Ω and the image Γ = glue(Γ_{∗}) is a cut system of Ω and glue : (Ω_{∗}, Γ_{∗}, τ) → (Ω, Γ) is a functor from the category of cut surfaces to the category of pairs (Ω, Γ) (in both categories morphisms are isomorphisms).

Conversely, if Γ is a cut system of Ω then one can construct cut surface (Ω_{∗}, Γ_{∗}, τ) as follows. Points of Ω_{∗} are points of Ω \ Γ and pairs (x, c), where x ∈ Γ and c is a coorientation of Γ in a neighborhood of point x. The stratification of Ω_{∗} and gluing system (Γ_{∗}, τ) are defined evidently (Fig.1b). Clearly, we constructed a functor cut : (Ω, Γ) → (Ω_{∗}, Γ_{∗}, τ).

Lemma 3.2. Functors glue and cut establish the equivalence of categories of cut surfaces (with morphisms defined as isomorphisms of cut surfaces) and pairs (Ω, Γ) (with morphisms defined as isomorphisms of pairs).

Figure 1. The construction of a contracted cut surface.

Fig.1a. Cut system Γ = {γ_{1}, γ_{2}, γ_{3}} of surface Ω.

Fig.1b. Cut surface (Ω_{∗}, Γ_{∗}, τ) obtained from cut system Γ. Here Γ_{∗} = {γ′_{1∗}, γ′′_{1∗}, γ′_{2∗}, γ′′_{2∗}, γ′_{3∗}, γ′′_{3∗}}. Fig.1c. Contracted cut surface Ω_{#} = Ω/Γ. Special points obtained as contractions of the connected components of Γ_{∗} are marked by ∗.

Proof is skipped.

Let (Ω_{∗}, Γ_{∗}, τ) be a cut surface. Denote by Ω_{#} a surface obtained by contracting each connected component of Γ_{∗} to a point. Clearly, the stratification of Ω_{∗} induces the stratification of Ω_{#}. Special points of this stratification are images of special points of Ω_{∗} and points that are obtained as contracted connected components of Γ_{∗} (Fig.1c). Homeomorphism τ induces the involution of the set of special points of Ω_{#} coming from contracted connected components. We denote this involution by the same letter τ. Clearly, we have constructed a functor contr : (Ω_{∗}, Γ_{∗}, τ) → Ω_{#} from category of cut surfaces to a category of stratified surfaces.

Therefore, we can assign a stratified surface Ω_{#} = contr(cut(Ω, Γ)) to any pair (Ω, Γ), where Γ is a cut system of a stratified surface Ω. We call Ω_{#} a contracted cut surface and denote it by Ω/Γ.

Remark. The set (Ω_{#})_{0} of special points of a contracted cut surface Ω_{#} = Ω/Γ carries an additional structure. Namely, 1) (Ω_{#})_{0} contains the distingushed subset Ω_{0} consisting of special points that are images of special points of Ω; 2) there is fixed involution τ : (Ω_{#})_{0} \ Ω_{0} → (Ω_{#})_{0} \ Ω_{0} (τ may have fixed points).

Let us classify simple cuts of a surface as follows. Denote by γ an arbitrary simple cut of a connected stratified surface Ω of type G = (g, ε, m, m_{1}, …, m_{s}). Two situations may occur. First, contracted cut surface Ω/{γ} is a connected surface. Denote its type by G_{#} = (g_{#}, ε_{#}, m_{#}, m_{#1}, …, m_{#s}). Second, Ω/{γ} consists of two connected components. Then type G_{#} of Ω/{γ} is the set of types of its connected components. Denote types of components by G_{#}^{i} = (g_{#}^{i}, ε_{#}^{i}, m_{#}^{i}, m_{#1}^{i}, …, m_{#s^{i}}^{i}), where i = 1, 2. In this case the image of a special point of Ω in Ω_{#} belongs to one of components. Thus, γ induces a division Ω_{0} = (Ω_{0})_{1} ⊔ (Ω_{0})_{2} of set Ω_{0} of special points of Ω into two subsets.

For a cut system γ consisting of one simple cut we use notations (Ω_{∗}, γ_{∗}, τ) for cut surface obtained by cutting stratified surface Ω along γ.

Lemma 3.3. Let Ω be a stratified surface. Then

(1) any simple cut γ of Ω belongs to just one of classes 1 − 9 from the list below;

(2) the identities between types G_{#} and G that are written in each item of the list holds.

List of classes of simple cuts

A. In classes 1-4 a simple cut γ is supposed to be homeomorphic to a circle

Class 1 consists of separating contours (Fig.2a). A simple cut γ is called a separating contour if and only if γ_{∗} ⊂ Ω_{∗} consists of two contours and surface Ω_{#} consists of two connected components,

G = (g_{#}^{1} + g_{#}^{2}, ε_{#}^{1}ε_{#}^{2}, m_{#}^{1} + m_{#}^{2} − 2, m_{#1}^{1}, …, m_{#s_{1}}^{1}, m_{#1}^{2}, …, m_{#s_{2}}^{2}) Class 2 consists of cuts of a handle (Fig.2b). A simple cut γ is called

a cut of a handle if and only if γ_{∗} ⊂ Ω_{∗} consists of two contours, Ω_{#} is connected surface and ε_{#} = ε,

G = (g_{#} + 1, ε_{#}, m_{#} − 2, m_{#1}, …, m_{#s}).

Class 3 consists of cuts of a neck of Klein bottle (Fig.2c). A simple cut γ is called a cut of a neck of Klein bottle if and only if γ_{∗} ⊂ Ω_{∗} consists of two contours, Ω_{#} is connected surface , ε = 0 and ε_{#} = 1,

G = (g_{#} + 1, 0, m_{#} − 2, m_{#1}, …, m_{#s}).

Class 4 consists of Möbius cuts (Fig.2d). A simple cut γ is called a Möbius cut if and only if γ_{∗} ⊂ Ω_{∗} is connected contour. In this case ε = 0, Ω_{#} is connected surface , ε_{#} is equal either to 0 or to 1,

G = (g_{#} + \frac{1}{2}, 0, m_{#} − 1, m_{#1}, …, m_{#s}).

B. In classes 5-9 a simple cut γ is supposed to be homeomorphic to a segment. Hence, γ_{∗} consists of two disjoint segments.

Class 5 consists of cuts between two holes (Fig.2e). A simple cut γ is called a cut between two holes if and only if end points of γ belong to different boundary contours. In this case Ω_{#} is a connected surface and in an appropriate numeration of boundary contours we obtain the identity G_{#} = (g, ε, m, m_{1} + m_{2} + 2, m_{3}, …, m_{s}).

C. In classes 6-9 both boundary points of a segment γ are supposed to belong to the same boundary contour ω. Denote by ω_{∗} the preimage of ω in cut surface Ω_{∗} and by ω_{#} the image of ω_{∗} in contracted cut surface Ω_{#}.

Class 6 consists of separating segments (Fig.2f). A simple cut γ is called a separating segment if and only if Ω_{#} consists of two connected components. In this case ω_{#} consists of two contours and in an appropriate numeration of boundary contours of Ω_{#} we obtain the identity

G = (g_{#}^{1}+g_{#}^{2}, ε_{#}^{1}ε_{#}^{2}, m_{#}^{1}, +m_{#}^{2}, m_{#1}^{1}+m_{#1}^{2}−2, m_{#2}^{1}, …, m_{#s_{1}}^{1}, m_{#2}^{2}, …, m_{#s_{2}}^{2}). Class 7 consists of cuts of a handle through a hole (Fig.2g). A simple cut γ is called a cut of a handle through a hole, if and only

if Ω_{#} is connected surface, ω_{#} consists of two contours and ε_{#} = ε. In this case in an appropriate numeration of boundary contours of Ω_{#} we obtain the identity

G = (g_{#} + 1, ε_{#}, m_{#}, m_{#1} + m_{#2} − 2, m_{#3}, …, m_{#s}).

Class 8 consists of cuts of a neck of a Klein bottle through a hole (Fig.2h). A simple cut γ is called a cut of a neck of a Klein bottle through a hole if and only if Ω_{#} is connected surface, ω_{#} consists of two contours, ε = 0 and ε_{#} = 1.

In this case in an appropriate numeration of boundary contours of Ω_{#} we obtain the identity

G = (g_{#} + 1, 0, m_{#}, m_{#1} + m_{#2} − 2, m_{#3}, …, m_{#s}).

Class 9 consists of cuts across Möbius band (Fig.2i). A simple cut γ is called a cut across Möbius band if and only if boundary contour ω_{#} is connected contour.

In this case Ω_{#} is a connected surface, ε = 0, ε_{#} equals either to 0 or to 1 and in an appropriate numeration of boundary contours of Ω we obtain the identity

G = (g_{#} + \frac{1}{2}, 0, m_{#}, m_{#1} − 2, m_{#2}, …, m_{#s}).

Lemma 3.4. Let Γ be a cut system of a stratified surface Ω. Then μ(Ω) = μ(Ω_{#}), where Ω_{#} = Ω/Γ is the contracted cut surface.

Proof. The equality can be easily checked for every class of simple cuts. □

Definition. Two cut systems Γ′ and Γ′′ of Ω are called equivalent if there exists an isomorphism ϕ : Ω → Ω of the stratified surfaces conserving all special points, all one-dimensional strata and their orientations, and such that ϕ(Γ′) = Γ′′.

Remark. In this definition the requirement of conserving one-dimensional strata of a stratified surface Ω is essential only for boundary contours with zero, one or two special points. If a boundary contour contains more that two special points then any isomorfism of Ω conserving all Figure 2. Examples of cuts of classes 1-9. Each cut is marked by γ. Fig.2a. A separating contour. Fig.2b. A cut of a handle. Fig.2c. A cut of a neck of Klein bottle. Fig.2d. A Möbius cut. Fig.2e. A cut between two holes. Fig.2f. A separating segment. Fig.2g. A cut of a handle through a hole. Fig.2h. A cut of a neck of a Klein bottle through a hole. Fig.2i. A cut across Möbius band

special points conservses also all one-dimentional strata on the contour and their orientations.

Lemma 3.5. Two simple cuts γ′ and γ′′ of a stratified surface Ω are equivalent if and only if (1) γ′ and γ′′ belong to the same class;

(2) types of contracted cut surfaces Ω′_{#} = Ω/γ′ and Ω′′_{#} = Ω/γ′′ coincide;

(3) if γ′ and γ′′ are separating contours or separating segments the divisions of the set of special points of Ω, induced by γ′ and γ′′, coincide;

(4) if γ′ and γ′′ are segments then their end points belong to the same one-dimensional strata.

Proof. Obviously, if γ′ and γ′′ are equivalent cuts of a stratified surface Ω then conditions (1)–(4) are satisfied.

Conversely, let γ′ and γ′′ be simple cuts satisfying conditions (1)

– (4). Denote by (Ω′_{∗}, γ′_{∗}, τ′_{∗}) and (Ω′′_{∗}, γ′′_{∗}, τ′′_{∗}) cut surfaces obtained by cutting Ω along γ′ and γ′′ resp. Conditions (1) – (4) provide the equality of types of stratified surfaces Ω′_{∗} and Ω′′_{∗}. Therefore, there exists an isomorphism ϕ : Ω′_{∗} → Ω′′_{∗} of stratified surfaces. Moreover, it can be easily shown for every class of simple cuts that one can choose ϕ satisfying the properties:

∘ for any special point r ∈ Ω_{0} of Ω isomorphism ϕ brings the

image r′ of r in Ω′_{∗} to the image r′′ of r in Ω′′_{∗} ∘ ϕ(γ′_{∗}) = γ′′_{∗} ∘ ϕ is the isomorphism of cut surfaces, i.e., ϕ ∘ τ′ = τ′′ ∘ ϕ.

Clearly, ϕ generates an homeomorphism \overline{ϕ} : Ω → Ω such that \overline{ϕ} preserves all special points and \overline{ϕ}(γ′) = γ′′. □

Corollary 3.1. Let Γ′ and Γ′′ be two cut systems of a stratified surface Ω. Suppose Γ′ contains a simple cut γ′ and Γ′′ contains a simple cut γ′′ such that γ′ and γ′′ are of the same class, types of Ω/γ′ and Ω/γ′′ coincide, γ′ and γ′′ induces the same division of the set Ω_{0} (if applicable) and end points of γ′ and γ′′ belong to the same one-dimensional stratum (if applicable). Then there exists a cut system \overline{Γ} such that it is equivalent to Γ′′ and contains cut γ′.

Proof is evident.

Lemma 3.6. Let Γ′ and Γ be two equivalent cut systems of a stratified surface Ω. Fix local orientations in a small neighborhoods of all special points. Then there exists an isomorphism ϕ : Ω → Ω of the stratified surface such that ϕ(Γ′) = Γ, ϕ fixes all special points of Ω and preserves an orientation of a small neighborhood of any special point.

Proof is skipped.

Lemma 3.7. Let Γ′ and Γ′′ be two cut systems of a stratified surface Ω. Suppose both Γ′ and Γ′′ contain the same simple cut γ. If cut systems Γ′ \ γ and Γ′′ \ γ of contracted cut surface Ω/γ are equivalent then Γ′ and Γ′′ are equivalent cut systems of Ω.

Proof. Suppose, cut systems Γ̃′ = Γ′ \ γ and Γ̃′′ = Γ′′ \ γ of Ω/γ are equivalent. Let (Ω_{∗}, γ_{∗}, τ_{∗}) be the cut surface, corresponding to the pair (Ω, γ). Denote by ω_{∗} the image of γ_{∗} in Ω/γ. Clearly, ω_{∗} consists of either two or one special point of contracted cut surface Ω/γ. Denote by U the joint of neighborhoods of points from ω_{∗} such that U ∩ (Γ̃′ ∪ Γ̃′′) = ∅. By lemma 3.6, there exists a homeomorphism ϕ̃: Ω/γ → Ω/γ, that is identical on U and such that ϕ̃(Γ̃′) = Γ̃′′. Thus there exist a homeomorphism ϕ_{∗} : Ω_{∗} → Ω_{∗} such that ϕ_{∗}(Γ̃′) = Γ̃′′ and τ_{∗}ϕ_{∗} = ϕ_{∗}τ_{∗}. Gluing by τ_{∗} gives a homeomorphism ϕ : Ω → Ω such that ϕ(γ) = γ, ϕ(Γ̃′) = Γ̃′′ and ϕ̃ fixes all special points. □

Corollary 3.2. Let Γ′ and Γ′′ be two cut systems of a stratified surface Ω. Suppose Γ^{∘} is a cut system such that Γ^{∘} ⊂ Γ′ and Γ^{∘} ⊂ Γ′′. If cut systems Γ′ \ Γ^{∘} and Γ′ \ Γ^{∘} of Ω/Γ^{∘} are equivalent cut systems of Ω_{#} = Ω/Γ_{0} then Γ′ and Γ′′ are equivalent cut systems of Ω.

3.3. Basic and simple surfaces.

Definition. A stratified surface Ω is called a stable surface if and only if any connected component Ω_{i} of Ω is a nontrivial surface (i.e., μ(Ω_{i}) > 0)

A cut system Γ = {γ_{1}, …, γ_{k}} of a stable stratified surface Ω is calledd a stable cut system if and only if the contracted cut surface Ω_{#} = Ω/Γ is also stable.

A connected stratified surface Ω is called a basic surface if and only if there is no nonempty stable cut systems of Ω.

A cut system Γ of a stable stratified surface Ω is called a complete cut system if and only if any connected component of the contracted cut surface Ω_{#} = Ω/Γ is a basic surface.

Let us formulate several elementary statements. A complete cut system is a stable cut system. Let Γ = {γ_{1}, …, γ_{k}} be a cut system of a stratified surface Ω. Choose a subset Γ′ = {γ_{i_{1}}, …, γ_{i_{l}}} of Γ. Then Γ′ is a cut system of Ω and the image of Γ \ Γ′ in the contracted cut surface Ω_{#} = Ω/Γ′ is a cut system of Ω_{#}. If Γ is a stable (complete) cut system then Γ\Γ′ is a stable (complete) cut system of Ω_{#}. Conversely, a cut system Γ′′ on the contracted cut surface Ω_{#} = Ω/Γ′ can be lifted to a cut system Γ = Γ′ ⊔ \overline{Γ}′′, where \overline{Γ}′′ ⊂ Ω is obtained from Γ′′ in two steps: first, take the preimage of Γ′′ in cut surface Ω_{∗}; second, take image of the preimage under glueing map glue : Ω_{∗} → Ω. If Γ′′ is a stable (complete) cut system then Γ is a stable (complete) cut system of Ω_{#}.

Lemma 3.8. A basic stratified surface Ω is isomorphic to one of the three stratified surfaces:

(1) sphere (S^{2}, p_{1}, p_{2}, p_{3}) with three interior special points p_{1}, p_{2}, p_{3} (μ = 1); (2) triangle, i.e., disc (D^{2}, q_{1}, q_{2}, q_{3}) with three boundary special points q_{1}, q_{2}, q_{3} (μ = \frac{1}{2});

(3) disc (D, p, q) with one interior special point p and one boundary special point q (μ = \frac{1}{2}).

Proof is elementary

Lemma 3.9. For any stable stratified surface Ω there exists a complete cut system of Ω.

Proof. It is sufficient to prove lemma for connected surfaces. If genus g of a stratified surface Ω is not zero, then there exist a cut of a handle or a Möbius cut. Using one of this cuts we reduce the question to a surface with less genus and, by inductive arguments, to a surface of genus zero. After it one can use inductive arguments with respect to the number of special points plus number of boundary contours. For example, if Ω contains at least two interior special points and is not a basic surface then a contour γ separating a disc with two of these points off the rest of surface is a stable cut and contracted cut surface Ω/γ is the disjoint union of a sphere with three interior points and a stable surface with less parameter of induction. We skip elementary details. □

Let us describe isomorphism class of complete cut systems for several stratified surfaces.

Lemma 3.10. Denote by N be the number of isomorphism classes of complete cut systems of a stratified surfaces. Then

(1) N = 3 for a sphere with four interior special points (Fig.3a);

(2) N = 2 for a Klein bottle with one interior special point (Fig.3b);

(4) N = 2 for a disc with four boundary special points (Fig.3c);

(5) N = 2 for a disc with two boundary special points and one interior special point (Fig.3d);

(6) N = 3 for a disc with one boundary special point and two interior special points (Fig.3e);

(7) N = 2 for a Möbius band with one boundary special point (Fig.3f);

(8) N = 2 for a cylinder having one boundary special point on each boundary contour, two boundary special points in total (Fig.3g).

Proof. (1) Let Ω be a sphere with four interior special points. Clearly, a complete cut system Γ of Ω consists of one separating contour that induces the division of Ω_{0} into two pairs of special points. By lemma 3.5 two contours inducing the same division are equivalent. Therefore, there are three nonequivalent complete cut systems Γ_{d} marked by a partition d of the set {1, 2, 3, 4} into two-element subsets.

(2) Let Ω be a Klein bottle with one interior special point. Any complete cut system of Ω contains a simple cut γ such that the genus of Ω/γ is less than 1. Clearly, only two cases are possible:

Figure 3. Non-equivalent complete cut systems on simple stratified surfaces. Different complete cut systems are drawn by diffent line types. Cuts in a cut system are denoted by γ with the same number of accents and are distinguished by lower indeces. Surfaces are numbered according to lemma 3.10. Fig.3b presents Klein bottle cutted by cut γ forming one cut system.

i) γ = γ_{k}, where γ_{k} denotes a cut of the neck of the Klein bottle. Clearly, Γ_{1} = {γ_{k}} is a complete cut system.

ii) γ = γ_{m}, where γ_{m} is a Möbius cut. Clearly, there is a complete cut system Γ_{2} = {γ_{m}, γ′_{m}} consisting of two Möbius cuts and all complete cut systems consisting of two Möbius cuts are equivalent.

(3) Let Ω be a surface of genus g = \frac{3}{2} (i.e., a torus with a hole glued by a Möbius band) having no special points. There are the following stable simple cuts (up to equivalence) of Ω.

∘ Möbius cut γ_{m1} such that Ω/γ_{m1} is a torus with one interior

special point; ∘ Möbius cut γ_{m2} such that Ω/γ_{m2} is a Klein bottle with one

interior special point; ∘ cut of a handle γ_{h}; in this case Ω/γ_{h} is ℝP^{2} with two interior

special points.

Clearly, any complete cut system Γ of Ω contains simple cut γ reducing the genus of Ω by \frac{1}{2} (i.e., genus of Ω/γ is equal to 1). Hence, γ is a Möbius cut.

Suppose, γ = γ_{m1}. Then, evidently, Γ = Γ_{1}, where Γ_{1} = {γ_{m1}, γ_{h}} and all complete cut systems containing γ_{m1} are equivalent.

Suppose, γ = γ_{m2}. Then Γ \ γ is a complete cut system of a Klein bottle Ω/γ having one interior special point. By (2) there are two isomorphism classes of complete cut systems of Ω/γ. Therefore, either Γ = Γ_{2}, where Γ_{2} = {γ_{m2}, γ′_{m2}, γ′′_{m2}} or Γ = Γ_{3}, where Γ_{3} = {γ_{m2}, γ_{h}} (lifting of a cut of the neck of Klein bottle Ω/γ to Ω is a cut of a handle). The latter is not equivalent to γ_{1} because Möbius cuts in them are not equivalent.

(3’) Let Ω be a surface of genus (g = \frac{3}{2}) (i.e., a torus with a hole glued by a Möbius band) with one interior special point. There are the following stable simple cuts (up to equivalence) of Ω.

∘ Möbius cut γ_{m1} such that Ω/γ_{m1} is a torus with two interior

special points; ∘ Möbius cut γ_{m2} such that Ω/γ_{m2} is a Klein bottle with two

interior special points; ∘ separating contour γ_{s1} such that Ω/γ_{s1} consists of two connected components, namely, a torus with one interior special point and ℝP^{2} with two interior special points.

∘ separating contour γ_{s2} such that Ω/γ_{s2} consists of two connected components, namely, a Klein bottle with one interior special point and ℝP^{2} with two interior special points;

∘ cut of a handle γ_{h}; in this case Ω/γ_{h} is ℝP^{2} with three interior special points.

Let Γ be a complete cut system of Ω.

Suppose, Γ contains a separating contour γ_{s1}. Then Ω/γ_{s1} is the disjoint union of a torus with one interior special point and ℝP^{2} with two interior special points. Clearly, there is only one equivalent class of complete cut systems of Ω/γ_{s1}. The lifting Γ_{1} of a complete cut system of Ω/γ_{s1} to Ω consists of the following simple cuts: Γ_{1} = {γ_{m1}, γ_{s1}, γ_{h}}.

Suppose, Γ contains a separating contour γ_{s2}. Then Ω/γ_{s2} is the disjoint union of a a Klein bottle with one interior special point and ℝP^{2} with two interior special points. By (2) there are two equivalency classes of complete cut systems of Ω/γ_{s2}. Their liftings to Ω are as follows: Γ_{2} = {γ_{m2}, γ_{s2}, γ_{h}} and Γ_{3} = {γ_{m2}, γ′_{m2}, γ′′_{m2}, γ_{s2}}.

Suppose Γ does not contain a separating contour and contains a simple cut γ_{m1}. Therefore, Ω/γ_{m1} is a torus with two interior special points. It can be shown that in this case Γ = Γ_{4}, where Γ_{4} = {γ_{m1}, γ_{h}, γ′_{h}}.

Suppose, Γ contains a simple cut γ_{m2}. Therefore, Ω/γ_{m1} is a Klein bottle with two interior special points. It can be shown that in this case Γ = Γ_{5}, where Γ_{5} = {γ_{m2}, γ_{h}, γ′_{h}}.

(4) Let Ω be a disc with four boundary special points. Clearly, any complete cut system of Ω consists of a single simple cut γ, which is separating segment inducing the partition of Ω_{0} into two pairs of consecutive boundary points. Mark boundary special points by 0, 1, 2, 3 going around boundary contour. There are two isomorphism classes of γ, namely, Γ_{1} = {γ_{0,1}}, where γ_{0,1} separates points 0, 1 of 2, 3, and Γ_{2} = {γ_{1,2}}, where γ_{1,2} separates points 1, 2 of 3, 0.

(5) Let Ω be a a disc with two boundary special points and one interior special point. Clearly, any complete cut system of Ω consists of a single simple cut γ, which is separating segment, its end points belong to the same segment of the boundary and γ surrounds the interior special points. Thus, there are two isomorphism classes of complete cut systems of Ω, namely, Γ_{1} = {γ_{l}} and Γ_{2} = {γ_{r}}, where end points of γ_{l} (resp. γ_{r}) belong to the ’left’ (resp. ’right’) segment of the boundary contour.

(6) Let Ω be a a disc with one boundary special point and two interior special points. Clearly, there are three isomorphism classes of complete cut systems. First of them, denoted by Γ_{1}, consists of a single separating contour γ_{o}. Each of two other complete cut systems, Γ_{2} and Γ_{3} consist of two separating segments γ_{s}, γ′_{s}, such that end points of any segment belong to the same segment of the boundary and the segment surrounds one interior special points. The only difference is the order of end points of two segments with respect to a fixed orientation of the boundary contour.

(7) Let Ω be a a Möbius band with one boundary special point. Clearly, there are two isomorphism classes of complete cut systems of Ω. They are as follows.

Γ_{1} = {γ_{m}}, where γ_{m} is a Möbius cut of Ω.

Γ_{2} = {γ_{mh}}, where γ_{mh} is a cut across Möbius band.

(8) Let Ω be a a cylinder having one boundary special point on each boundary contour, two boundary special points in total. Clearly, there are two isomorphism classes of complete cut systems of Ω. They are as follows.

Γ_{1} = {γ_{s}}, where γ_{s} is a separating contour.

Γ_{2} = {γ_{sh}}, where γ_{sh} is a cut between two holes.

□

Definition. Stratified surfaces (1) – (8) from list in lemma 3.10 are called simple stratified surfaces.

3.4. Neighboring complete cut systems. In this subsection we deal with complete cut systems of a fixed stratified surface. It is supposed that any boundary contour of Ω contains at least one special point.

We call a pair of complete cut systems Γ_{1} and Γ_{2} of Ω an adjacent pair if the following condition holds. Denote by Γ_{0} the joint of all simple cuts that belongs to Γ_{1} and Γ_{2} simultaneously and denote by Ω_{#} the contracted cut surface Ω/Γ_{0}. Both Γ_{1} \ Γ_{0} and Γ_{2} \ Γ_{0} give rise to the complete cut systems of Ω_{#}. The condition is that cut systems Γ_{1} \Γ_{0} and Γ_{2} \Γ_{0} belong to the same connected component of Ω_{#} and this component is a simple surface.

We call isomorphism classes C_{1} and C_{2} of complete cut systems an adjacent classes if and only if there exists representatives Γ_{1} ∈ C_{1} and Γ_{2} ∈ C_{2} such that Γ_{1} and Γ_{2} are adjacent complete cut systems.

We call two isomorphism classes C′, C′′ of complete cut systems a neighboring classes if and only if there exists a sequence of isomorphism class C_{1}, …, C_{k} such that C_{1} = C′, C_{k} = C′′ and C_{i} is adjacent to C_{i+1} for i = 1, …, k−1. We call two complete cut systems from neighboring isomorphism classes neighbors.

We call a stable surface an almost simple surface if and only if it is isomorphic to one of the following stratified surfaces:

(1) a disc with two or less interior special points and an arbitrary number of boundary special points;

(2) a cylinder with at most one interior special points and an arbitrary number of boundary special points;

(3) a Möbius band with at most one interior special points and an arbitrary number of boundary special points.

(4) a surface of genus (g = \frac{3}{2}) (i.e., a torus with a hole glued by a Möbius band) having no special points or with one interior special point.

Lemma 3.11. All complete cut systems of an almost simple stratified surface Ω are neighbors.

Proof. If Ω is a disc without interior special points then the statement of lemma can be reformulated in terms of triangulations of a polygon and follows from the connectedness of an associhedron [8].

Let Ω be a disc with one interior special point. Then the statement of lemma in this case follows from the following claims.

1) Any complete cut system of Ω contains a unique simple cut γ such that Ω/γ consists of two connected components and one of components is a disc with one interior and one boundary special points. Let us call a cut of this type a type H cut. Obviously end points of γ belong to the same boundary segment.

2) All complete cut systems of Ω containing fixed cut γ of type H are neighbours.

3) A complete cut system Γ containing γ of type H is adjacent to a complete cut system Γ′ containing a cut γ′ of type H such that its end points lie on the neighbor segment to a segment containing end points of γ.

Let Ω be a disc with two interior special point. In this case the following claims leads to the proof of lemma.

1) A complete cut system Γ contains either two cuts of type H or a contour separating a disc with two interior special points from the rest of surface. We call the last cut a cut of type P.

2) A complete cut system containing a cut of type H is a neighbor of a complete cut system containing a cut of type P.

3) All complete cut systems containing a fixed cut γ of type H are neighbors.

4) A complete cut system Γ containing γ of type H is adjacent to a complete cut system Γ′ containing a cut γ′ of type H such that its end points lie on the neighbor segment to a segment containing end points of γ.

5)Let Ω be a surface of genus g = \frac{3}{2} (i.e., a torus with a hole glued by a Möbius band) having no special points. We will prove that there are just 3 nonequivalent complete cut systems of Ω and all of them are neighbours.

There are the following stable simple cuts (up to equivalence) of Ω.

∘ Möbius cut γ_{m1} such that Ω/γ_{m1} is a torus with one interior

special point; ∘ Möbius cut γ_{m2} such that Ω/γ_{m2} is a Klein bottle with one

interior special point; ∘ cut of a handle γ_{h}; in this case Ω/γ_{h} is ℝP^{2} with two interior

special points. Clearly, any complete cut system Γ of Ω contains simple cut γ re-

ducing the genus of Ω by \frac{1}{2} (i.e., genus of Ω/γ is equal to 1). Hence, γ is a Möbius cut.

Suppose, γ = γ_{m1}. Then, evidently, Γ = Γ_{1}, where Γ_{1} = {γ_{m1}, γ_{h}} and all complete cut systems containing γ_{m1} are equivalent.

Suppose, γ = γ_{m2}. Then Γ \ γ is a complete cut system of a Klein bottle Ω/γ having one interior special point. By (2) there are two isomorphism classes of complete cut systems of Ω/γ. Therefore, either Γ = Γ_{2}, where Γ_{2} = {γ_{m2}, γ′_{m2}, γ′′_{m2}} or Γ = Γ_{3}, where Γ_{3} = {γ_{m2}, γ_{h}} (lifting of a cut of the neck of Klein bottle Ω/γ to Ω is a cut of a handle).

The latter is not equivalent to γ_{1} because Möbius cuts in them are not equivalent.

Finally, it can be easely shown, that Γ_{2} is a neighbor of Γ_{3} and Γ_{1} is a neighbor of Γ_{3}. Indeed, both cut systems in a pair have equivalent simple cut.

(5’) Let Ω be a surface of genus (g = \frac{3}{2}) (i.e., a torus with a hole glued by a Möbius band) with one interior special point.

We will prove that there are just 5 nonequivalent complete cut systems of Ω and all of them are neighbours.

There are the following stable simple cuts (up to equivalence) of Ω.

∘ Möbius cut γ_{m1} such that Ω/γ_{m1} is a torus with two interior

special points; ∘ Möbius cut γ_{m2} such that Ω/γ_{m2} is a Klein bottle with two

interior special points; ∘ separating contour γ_{s1} such that Ω/γ_{s1} consists of two connected components, namely, a torus with one interior special point and ℝP^{2} with two interior special points.

∘ separating contour γ_{s2} such that Ω/γ_{s2} consists of two connected components, namely, a Klein bottle with one interior special point and ℝP^{2} with two interior special points;

∘ cut of a handle γ_{h}; in this case Ω/γ_{h} is ℝP^{2} with three interior

special points. Let Γ be a complete cut system of Ω. Suppose, Γ contains a separating contour γ_{s1}. Then Ω/γ_{s1} is the

disjoint union of a torus with one interior special point and ℝP^{2} with two interior special points. Clearly, there is only one equivalent class of complete cut systems of Ω/γ_{s1}. The lifting Γ_{1} of a complete cut system of Ω/γ_{s1} to Ω consists of the following simple cuts: Γ_{1} = {γ_{m1}, γ_{s1}, γ_{h}}.

Suppose, Γ contains a separating contour γ_{s2}. Then Ω/γ_{s2} is the disjoint union of a a Klein bottle with one interior special point and ℝP^{2} with two interior special points. By (2) there are two equivalency classes of complete cut systems of Ω/γ_{s2}. Their liftings to Ω are as follows: Γ_{2} = {γ_{m2}, γ_{s2}, γ_{h}} and Γ_{3} = {γ_{m2}, γ′_{m2}, γ′′_{m2}, γ_{s2}}.

Suppose Γ does not contain a separating contour and contains a simple cut γ_{m1}. Therefore, Ω/γ_{m1} is a torus with two interior special points. It can be shown that in this case Γ = Γ_{4}, where Γ_{4} = {γ_{m1}, γ_{h}, γ′_{h}}.

Suppose, Γ contains a simple cut γ_{m2}. Therefore, Ω/γ_{m1} is a Klein bottle with two interior special points. It can be shown that in this case Γ = Γ_{5}, where Γ_{5} = {γ_{m2}, γ_{h}, γ′_{h}}.

Like in the case (5), it can be easely shown, that Γ_{1} is a neighbor of Γ_{2}, Γ_{1} is a neighbor of Γ_{4} and Γ_{2}, Γ_{3}, Γ_{5} are neighbors. Indeed, both cut systems in each pair have equivalent simple cut.

6) For the rest of almost simple surfaces the proof is by similar arguments, we skip them here.

□

A pair of complete cut systems is said to be parallel if they contain at least one common simple cut. A pair of isomorphism classes of complete cut systems is said to be parallel if there exists parallel representatives of the classes.

Lemma 3.12. All complete cut systems of an orientable surface Ω without a boundary are neighbors.

Proof. We use inductive arguments with respect to genus g of the surface Ω. The statement of lemma is known for the sphere [21]. If g > 0 then any complete cut system Γ contains a cut of a handle. By lemmas 3.5 and corollary 3.1, any pair of isomorphism classes of complete cut systems is parallel. The proof follows from lemma 3.7 and the inductive arguments. □

A simple cut γ of Ω is called projective if it is a Möbius cut and Ω/γ is a nonorientable surface.

Lemma 3.13. All complete cut systems of a nonorientable surface Ω without a boundary are neighbors.

Proof. We use inductive arguments with respect to genus g of the surface Ω. Let Γ be a complete cut system of Ω.

If g = \frac{1}{2} then any complete cut system of Ω contains a Möbius cut. By lemma 3.12 and 3.7, we obtain that all complete cut systems are neighbors. Similar arguments are valid for g = 1. Using a slightly more complicated reasoning one can prove lemma for g = \frac{3}{2}.

Let g > \frac{3}{2} and lemma is proven for smaller genera. It can be shown, that any complete cut system Γ contains a simple cut γ such that genus g_{#} of Ω/γ is less than the genus of Ω and Ω/γ is nonorientable. By inductive arguments, Γ \ γ is equivalent to a complete cut system containing a projective cut. By lemma 3.7, Γ is equivalent to a complete cut system containing a projective cut. Therefore, all isomorphism classes of complete cut systems of Ω are parallel. The proof of lemma is completed by inductive arguments using lemma 3.7. □

A stable cut γ of a connected stratified surface Ω is called normal if it is either a contour or a separating segment such that one of the connected components of a contracted cut surface is homeomorphic to a disk without interior special points (no restrictions for boundary special points). If Ω is disconnected stratified surface then γ is said normal if it is a normal cut of a connected component of Ω.

Clearly, any simple cut of a stratified surface without boundary is normal.

We call a simple cut, that is not normal cut, a special cut.

Definition. A complete cut system Γ is called normal cut system if Γ consists of normal simple cuts only.

For example, any complete cut system of a stratified surface without boundary is normal.

Clearly, if Γ is a normal cut system of Ω and Γ_{0} ⊂ Γ is a subsystem of cuts then Γ \ Γ_{0} is normal cut system of Ω/Γ_{0}.

Lemma 3.14. Any stable surface admits a normal complete cut system.

Proof. The proof is by inductive arguments with respect to the genus and the number of boundary contours. The step of induction is from Ω to Ω/γ where γ is a normal cut. □

Lemma 3.15. All normal cut systems of a stratified surface Ω are neighbors.

Proof. If Ω is a surfaces without a boundary then the lemma follows from lemmas 3.12 and 3.13.

If Ω is a disk without interior special points then the lemma follows from lemma 3.11.

In order to prove the lemma for an arbitrary Ω we use inductive arguments with respect to the number of boundary contours. Let Γ′ and Γ′′ be normal cut systems of Ω. Let ω be a boundary contour of Ω. It can be easily shown that any normal cut system contains a separating oval γ with the following properties. The connected component of Ω/γ that contains ω is homeomorphic to a disc with one interior special point (coming from γ). The homotopy class of γ does not depend on the choice of a normal cut system. Therefore, all isomorphism classes of normal cut systems of Ω are parallel. The proof of the lemma is completed by inductive arguments. □

Lemma 3.16. Let Ω be a stable surface and γ be a stable special cut of Ω. Suppose, Ω is not a simple or almost simple surface. Then there exists a normal cut γ′ homeomorphic to an oval such that {γ, γ′} is a stable cut system of Ω.

Proof. If contracted cut surface Ω_{#} = Ω/γ admits a stable simple cut homeomorphic to an oval then the claim of lemma is obviously true. Suppose, Ω_{#} does not admit a simple cut with this property. Then any connected component of Ω_{#} is either a basic surface or a disk with no more than one interior special point. It can be proven by exhaustion that Ω is either a simple or almost simple surface. □

Let Γ be a complete cut system of Ω. Denote by Γ_{n} the subset of all normal cuts from Γ. Γ is called an almost normal if one connected component of Ω/Γ_{n} is either a simple or an almost simple surface and all other connected components are basic surfaces.

By lemmas 3.11 and 3.14, we obtain the following statement.

Lemma 3.17. Any almost normal cut system has a neighbor that is a normal cut system.

Lemma 3.18. Any special stable cut γ can be included in an almost normal complete cut system.

Proof. Use inductive arguments with respect to genus, number of boundary contours and number of interior special points.

If Ω is either a simple or an almost simple surfaces then the claim of lemma is trivial.

Otherwise choose a normal simple cut γ′ provided by lemma 3.16. By inductive hypothesis, there exists an almost normal cut system Γ_{#} of the contracted cut surface Ω/γ′ such that γ ⊂ Ω_{#}. Clearly, Γ_{#} ⊔ γ′ is an almost simple cut system of Ω. □

Theorem 3.1. Let Ω be an arbitrary stable stratified surface. Then all complete cut systems of Ω are neighbors.

Proof. By lemma 3.15, it is sufficient to prove that any complete cut system Γ is the neighbor of a normal cut system. Use inductive arguments with respect to the number of special cuts. Suppose, Γ contains n > 0 special cuts. Let γ ∈ Γ be a special cut. By lemma 3.18, there exists an almost normal cut system Γ′ such that γ ⊂ Ω′.

By inductive hypothesis, complete cut systems Γ \ γ and Γ′ \ γ of the contracted cut surface Ω/γ are neighbors. Therefore, Γ and Γ′ are neighbors. By lemma 3.17, Γ′ is the neighbor of a normal cut system. Hence, Ω is the neighbor of a normal cut system. □

4. Two-dimensional topological field theory

4.1. Definition of Klein topological field theory. First, we shall fix a tensor category of surfaces (a ’basic’ category) and a functor from it to the tensor category of vector spaces.

A set of local orientations. For any special point r ∈ Q fix an orientation o_{r} of its small neighborhood. Denote the set of these local orientations by O. Let Ω be a connected stratified surface. A set O of local orientations is called admissible if and only if either Ω is orientable surface and all local orientations are induced by an orientation of Ω (Fig.4a) or Ω is a nonorientable surface and local orientations at all special points from any boundary contour ω_{i} are compatible with one of orientations of ω_{i} (Fig.4b). Thus, there are two admissible sets of local orientations in the former case and there are 2^{s}, where s is a number of connected components of ∂Ω, admissible sets of local orientations in the latter case.

A set of local orientations for disconnected Ω is said admissible if and only if it is admissible for each connected component of Ω.

Lemma 4.1. Let (Ω′, O′) and (Ω′′, O′′) be two pairs consisting of stratified surfaces and admissible sets of local orientations at special points. If stratified surfaces Ω′ and Ω′′ are isomorphic then there exists an isomorphism ϕ : Ω′ → Ω′′ such that ϕ(O′) = O′′.

Figure 4. Admissible set of local orientations of (4a)

oriented surface; (4b) non-oriented surface.

Proof follows from standard properties of surfaces.

Let Ω be a stratified surface. Denote by Ω_{a} the set of all interior special points and by Ω_{b} the set of all boundary special points. Put also Ω_{0} = Ω_{a} ⊔ Ω_{b}. Fix a local orientation o_{r} in a small neighborhood of any special point r ∈ Ω_{0} and denote by O the set of all these local orientations. Pairs (Ω, O) are objects of the basic category. Morphisms are any combinations of morphisms of types 1)-4).

1) Isomorphism ϕ : (Ω, O) → (Ω′, O′). By definition, ϕ is an isomorphism ϕ : Ω → Ω′ of stratified surfaces compatible with local orientations at special points.

2) Changing of local orientations ψ : (Ω, O) → (Ω, O′). Thus, there is one such morphism for any pair (O, O′) of sets of local orientations on a stratified surface Ω.

3) Adding a special point ξ : (Ω, O) → (Ω′, O′). Morphism ξ depends on a point r ∈ Ω\Ω_{0} endowed with a local orientation o_{r}. By definition, topological surface Ω′ coincides with surface Ω. The stratification of Ω′ is defined as a refinement of the stratification of Ω by additional point r. Therefore, Ω′_{0} = Ω_{0} ⊔ {r}. Local orientation o_{r} completes the set of local orientations, i.e., O′ = O ⊔ {o_{r}}.

4) Cutting η : (Ω, O) → (Ω_{#}, O_{#}). Morphism η depends on a cut system Γ endowed with orientations of all simple cuts γ ∈ Γ. Ω_{#} is defined as contracted cut surface Ω/Γ. Stratified surface Ω_{#} inherits special points of Ω and local orientations at any of them. A special point r ∈ (Ω_{#})_{0} \ Ω_{0} appears as contracted connected component of a cut contour γ_{∗} in cut manifold Ω_{∗}. In obvious way the orientation of simple cut γ induces the local orientation at r. The set of all these local orientations at all points of (Ω_{#})_{0} form the set O_{#}.

Define ’tensor product’ θ : (Ω′, O′) × (Ω′′, O′′) → (Ω, O) of two pairs (Ω′, O′) and (Ω′′, O′′) as their disjoint union (Ω, O) = (Ω′⊔Ω′′, O′⊔O′′).

We shall define below a functor (Ω, O) → V (Ω, O) from the basic category of surfaces to the category of vector spaces.

Let {X_{m}|m ∈ M} be a finite set of n = |M| vector spaces X_{m}. The action of symmetric group S_{n} on the set {1, …, n} induces its action on the linear space (⊕_{σ:{1,...,n}⇆M}X_{σ(1)} ⊗ ⋯ ⊗ X_{σ(n)}), an element s ∈ S_{n} takes X_{σ(1)}⊗⋯⊗X_{σ(n)} to X_{σ(s(1))}⊗⋯⊗X_{σ(s(n))}. Denote by ⊗_{m∈M}X_{m} the subspace of all invariants of this action.

Vector space ⊗_{m∈M}X_{m} is canonically isomorphic to a tensor product of all X_{m} in any fixed order, the isomorphism is a projection of vector space ⊗_{m∈M}X_{m} to the summand that is equal to the tensor product of X_{m} in a given order. Assume that all X_{m} are equal to a fixed vector space X. Then any bijection M ↔ M′ of sets induces the isomorphism ⊗_{m∈M}X_{m} ↔ ⊗_{m′∈M′}X_{m′}.

Let A and B be finite dimensional vector spaces over complex numbers endowed with involutive transformations A → A and B → B, which we denote by x ↦ x^{∗} (x ∈ A) and y ↦ y∗ (y ∈ B) resp.

Let (Ω, O) be a pair consisting of a stratified surface Ω and a set O of local orientations at its special points. Assign a copy A_{p} of a vector space A to any point p ∈ Ω_{a} and a copy B_{q} of a vector space B to any point q ∈ Ω_{b}. Elements of A_{p} (resp. B_{q}) are called interior (resp. boundary) fields. Put V_{Ω} = (⊗_{p∈Ω_{a}}A_{p}) ⊗ (⊗_{q∈Ω_{b}}B_{q}).

For any morphism of pairs (Ω, O) → (Ω′, O′) define a morphism of vector spaces V_{Ω} → V_{Ω′} as follows.

1) An isomorphism ϕ : (Ω, O) → (Ω′, O′) induces the isomorphism ϕ_{∗} : V_{Ω} → V_{Ω′} because ϕ generates the bijections Ω_{a} ⇆ Ω′_{a} and Ω_{b} ⇆ Ω′_{b} of sets of special points.

2) For a changing of local orientations ψ : (Ω, O) → (Ω, O′) define a linear map ψ_{∗} : V_{Ω} → V_{Ω} as (⊗_{r∈Ω_{0}}ψ_{r}), where for any r ∈ Ω_{0}

ψ_{r}(x) = \{ \begin{matrix}x & if o_{r} = o′_{r} \\ x^{∗} & if o_{r} = −o′_{r} \end{matrix}

3) In order to define a morphism ξ_{∗} : V_{Ω} → V_{Ω′} for adding a special point ξ : (Ω, O) → (Ω′, O′) we need to fix elements 1_{A} ∈ A and 1_{B} ∈ B. These notations of elements are motivated in the frame of a Klein topological field theory, where they are ’trivial fields’.

In this case there exists a canonical isomorphism V_{Ω′} = V_{Ω} ⊗ X, where X is equal either to A (if adding special point r belongs to the interior of Ω) or to B (if adding special point r belongs to the boundary of Ω). For x ∈ V_{Ω} put ξ_{∗}(x) = x⊗1_{X}, where 1_{X} is either 1_{A} or 1_{B} resp.

4) In order to define a morphism η_{∗} : V_{Ω} → V_{Ω_{#}} for any cutting morphism η : (Ω, O) → (Ω_{#}, O_{#}) we need to fix elements K̂_{A,∗} ∈ A⊗A, K̂_{B,∗} ∈ B ⊗ B and U ∈ A. (Notations of them will be clear from the sequel.)

Evidently, it is sufficient to define η_{∗} for an arbitrary oriented simple cut γ ⊂ Ω. In this case we have a canonical isomorphism V_{Ω_{#}} = V_{Ω}⊗X, where

X = \{ \begin{matrix}A ⊗ A & if γ is an contour and is not Möbius cut \\ B ⊗ B & if γ is a segment \\ A & if γ is a Möbius cut \end{matrix}

For x ∈ V_{Ω} put η_{∗}(x) = x ⊗ z, where z is either K̂_{A,∗}, or K̂_{B,∗}, or U resp.

Finally, for a ’tensor product’ θ : (Ω′, O′)×(Ω′′, O′′) → (Ω′ ⊔Ω′′, O′ ⊔ O′′) there is evident canonical linear map θ_{∗} : V_{Ω′} ⊗ V_{Ω′′} → V_{Ω′⊔Ω′′}

We say that a set of data T = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {Φ_{(Ω,O)}}), where A and B are linear spaces, x ↦ x^{∗} and y ↦ y^{∗} are involutive linear transformations of A and B resp., {Φ_{(Ω,O)}} is a system of linear forms Φ_{(Ω,O)} : V_{Ω} → ℂ, is a Klein topological field theory if the following axioms are satisfied.

1^{∘} Topological invariance.

For any isomorphism of pairs ϕ : (Ω, O) → (Ω′, O′) the following identity holds

Φ_{(Ω,O)}(x) = Φ_{(Ω′,O′)}(ϕ_{∗}(x).)

2^{∘} Invariance of a change of local orientations.

For any change of local orientations ψ : (Ω, O) → (Ω, O′) the following identity holds

Φ_{(Ω,O)}(x) = Φ_{(Ω,O′)}(ψ_{∗}(x)).

3^{∘} Nondegeneracy.

Define first a bilinear form (x, x′)_{A} on vector space A. Namely, let (Ω, O) be a pair, where Ω is a sphere with 2 interior special points p, p′ and the set O = {o_{p}, o_{p′}} is such that local orientations o_{p}, o_{p′} induce the same global orientations of the sphere. Put (x, x′)_{A} = Φ_{(Ω,O)}(x_{p} ⊗ x′_{p′}), where x_{p} and x′_{p′} are images of x ∈ A and x′ ∈ A in A_{p} and A_{p′} resp. The correctness of this definition follows from axioms 1^{∘} and 2^{∘}. Evidently, (x, x′)_{A} is a symmetric bilinear form.

Similarly, define a bilinear form (y, y′)_{B} on vector space B using a disc with 2 boundary special points q, q′ instead of a sphere with two interior special points p, p′. As in the previous case local orientations o_{q}, o_{q′} must induce the same global orientations of the disc. Evidently, (y, y′)_{B} is a symmetric bilinear form.

Axiom is that both forms (x, x′)_{A} and (y, y′)_{B} are nondegenerate. 4^{∘} Invariance of adding unit fields.

Axioms 1^{∘}−3^{∘} allows us to choose elements 1_{A} ∈ A, 1_{B} ∈ B. Indeed, let Ω be a sphere with one special point p and o_{p} be a local orientation at p. Then linear form Φ_{(Ω,{o_{p}})} is an element of the vector space dual to A. Identify dual vector space with A by means of nondegenerate bilinear form (x, y)_{A}. Thus, we obtain an element of A, which we denote by 1_{A}. Analogously, starting from a disc with one boundary special point we obtain an element of B, which we denote by 1_{B}. An element 1_{A} (resp. 1_{B}) is called trivial interior (resp. boundary) field. We shall use just these elements in the definition of morphisms ξ_{∗} of type 3).

Axiom is that for any adding of special point ξ : (Ω, O) → (Ω′, O′) the following identity holds

Φ_{(Ω,O)}(x) = Φ_{(Ω′,O′)}(ξ_{∗}(x)).

5^{∘} Cut invariance.

Axioms 1^{∘} − 3^{∘} allows us to choose elements K̂_{A,∗} ∈ A × A, K̂_{B,∗} ∈ B ×B and U ∈ A. Indeed, any nondegenerate bilinear form on a vector space X canonically defines an element K̂_{X} ∈ X ⊗ X. Taking forms (x, x′)_{A,∗} = (x, x′^{∗})_{A} and (y, y′)_{B,∗} = (y, y′^{∗})_{B} we obtain elements K̂_{A,∗} and K̂_{B,∗}.

Linear form Φ_{(Ω,O)} for a projective plane Ω with one interior special point is an element of the vector space dual to A. We denote by U the image of this element in A under the isomorphism induced by nondegenerate bilinear form (x, x′)_{A}.

We shall use just these elements for morphisms η_{∗} of type 4).

Axiom is that for any cut system Γ endowed with orientations of all simple cuts the following identity is required

Φ_{(Ω,O)}(x) = Φ_{(Ω_{#},O_{#})}(η_{∗}(x))

(here η_{∗} depends on Γ, see 4) above)

6^{∘} Multiplicativity.

Axiom is that the product θ : (Ω′, O′)×(Ω′′, O′′) → (Ω′⊔Ω′′, O′⊔O′′) of any two pairs (Ω′, O′) and (Ω′′, O′′) the following identity holds

Φ_{(Ω′,O′)}(x_{1})Φ_{(Ω′′,O′′)}(x_{2}) = Φ_{(Ω′⊔Ω′′,O′⊔O′′)}(θ_{∗}(x_{1} ⊗ x_{2}))

The example of a Klein topological field theory is given in section 5.

4.2. Correlators of Klein topological field theory. Axioms of a topological field theory can be reformulated in terms of so called ’systems of correlators’. In terms of correlators the dependence of a linear form Φ_{(Ω,O)} on an individual stratified surface Ω endowed with a set of local orientations O is reduced to the dependence on a topological type G = (g, ε, m_{1}, …, m_{s}) of a surface.

In this section we present the reformulation of a Klein topological field theory in terms of correlators. Let A and B be two linear spaces, x ↦ x^{∗} and y ↦ y^{∗} be fixed involutive transformations of A and B resp.

Denote by G = (g, ε, m, m_{1}, …, m_{s}) a data that are the topological type of a connected stratified surface.

We assume throughout this section that ∘ m_{i} > 0 for i = 1, …, s (m may be zero; s may be zero) ∘ m + ∑ m_{i} > 0 ∘ numbers m_{i} are given in ascending ordered m_{1} ≤ m_{2} ≤ ⋯ ≤

m_{s}.

Denote by V_{G} tensor product A^{⊗m} ⊗ B^{⊗(m_{1}+···+m_{s})} of vector spaces.

For any G fix one linear function f_{G} : V_{G} → ℂ. According to the style of notations used in field theories we denote corresponding polylinear function as follows:

⟨x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{(g,ε)} := f_{G}(z),

where z = x_{1} ⊗ ⋯ ⊗ x_{m} ⊗ y_{1}^{1} ⊗ ⋯ ⊗ y_{m_{1}}^{1} ⊗ ⋯ ⊗ y_{1}^{s} ⊗ ⋯ ⊗ y_{m_{s}}^{s} ∈ V_{G}.

Note that vector spaces V_{G} and V_{G′} coincide for some G ≠ G′ but corresponding polylinear functions ’remember’ all data from G = (g, ε, m, m_{1}, …, m_{s}) due to bracketing of arguments. We call a bracket (y_{1}^{j}, …, y_{m_{j}}^{j}) a block of arguments.

We use capital letters X, X_{1} etc. to denote subsets consisting of several x_{i} and several blocks of arguments. For example, ⟨x_{1}, X⟩_{g,ε}, where X = (x_{2}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})) denotes a function ⟨x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{g,ε}.

We write down ⟨X⟩_{G} instead of ⟨X⟩_{g,ε} in order to emphasize explicitly the dependence on a type G.

We denote by X_{a} the set of all x_{i} in a correlator and we denote by X_{b} the set of all blocks of arguments. Thus, we write down

⟨X⟩_{G} = ⟨X_{a}, X_{b}⟩_{G} = ⟨x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{g,ε} We use capital letters Y, Y_{1}, etc. to denote sequential sets of argu-

ments in a block. For example, ⟨X, (y_{1}^{s}, Y)⟩_{g,ε}, where

X = (x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s−1}, …, y_{m_{s−1}}^{s−1})) and Y = (y_{2}^{s}, …, y_{m_{s}}^{s}), denotes polylinear function

⟨x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{g,ε}.

For an arbitrary Y = (y_{1}, …, y_{k}) put Y^{∗} = (y_{k}^{∗}, …, y_{1}^{∗}).

Choose a basis E_{A} of A and E_{B} of B and denote basic elements by α, α′, ⋯ ∈ E_{A} and β, β′, ⋯ ∈ E_{B}. Then we can present f_{G} as a tensor

⟨α_{1}, …, α_{m}, (β_{1}^{1}, …, β_{m_{1}}^{1}), …, (β_{1}^{s}, …, β_{m_{s}}^{s})⟩_{(g,ε)}

Let C = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {⟨X⟩_{G}}) be a set of data consisting of ∘ linear spaces A and B, ∘ involutive linear transformations x ↦ x^{∗} (x ∈ A) and y ↦ y^{∗}

(y ∈ B) of A and B resp., ∘ polylinear function ⟨X⟩_{G} for any topological type G = (g, ε, m, m_{1}, …, m_{s})

of a stratified surface .

Definition. A set of data C = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {⟨X⟩_{G}}) is called a system of correlators and any function ⟨X⟩_{G} is called a correlator if and only if the following axioms hold.

(i) Any correlator ⟨x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{g,ε} is invariant with respect to a permutation of x_{i}, a permutation of blocks (y_{1}^{i}, …, y_{m_{i}}^{i}) of equal size, a cyclic permutations of arguments inside each block (y_{1}^{i}, …, y_{m_{i}}^{i}). (ii) If ε = 1 then a correlator is invariant with respect to replacing of x_{i} by x_{i}^{∗} and block Y^{j} = (y_{1}^{j}, …, y_{m_{j}}^{j}) by (Y^{j})^{∗} for all i = 1, …, m, j = 1, …, s simultaneously.

If ε = 0 then a correlator is invariant with respect to replacing of x_{i} by x_{i}^{∗} for any fixed i and with respect to replacing a block Y^{j} = (y_{1}^{j}, …, y_{m_{j}}^{j}) by (Y^{j})^{∗} for any fixed j. (iii) Bilinear maps ⟨x_{1}, x_{2}⟩_{0,1} and ⟨(y_{1}, y_{2})⟩_{0,1} are nondegenerate bilinear forms on A and B resp.; below we denote them by (x_{1}, x_{2})_{A} and (y_{1}, y_{2})_{B} resp. (iv) Fix a basis E_{A} = {α} of A and a basis E_{B} = {β} of B. Denote by F^{α′,α′′} the tensor dual to the tensor F_{α′,α′′} of bilinear form (α′, α′′), by F^{β′,β′′} the tensor dual to the tensor F_{β′,β′′} := (β′, β′′) and by D^{α} the tensor obtained by lifting the index of tensor D_{α} = ⟨α⟩_{\frac{1}{2} ,0}. Axiom is that for arbitrary sets X, X_{1}, X_{2}, Y_{1}, Y_{2} of arguments denoted according to the conventions described above the relations (1) − (9) hold.

(1) ⟨X_{1}, X_{2}⟩_{g_{1}+g_{2},ε_{1}ε_{2}} = ∑_{α′,α′′∈E_{A}}⟨X_{1}, α′⟩_{g_{1},ε_{1}}F^{α′,α′′}⟨α′′, X_{2}⟩_{g_{2},ε_{2}}

(2) ⟨X⟩_{g+1,ε} = ∑_{α′,α′′∈E_{A}}⟨X, α′, α′′⟩_{g,ε}F^{α′,α′′}

(3) ⟨X⟩_{g+1,0} = ∑_{α′,α′′∈E_{A}}⟨X, α′, α′′^{∗}⟩_{g,1}F^{α′,α′′}

(4) ⟨X⟩_{g+ \frac{1}{2} ;0} = ∑_{α′∈E_{A}}⟨X, α′⟩_{g,ε}D^{α′}

(5) ⟨X, (Y_{1}), (Y_{2})⟩_{g,ε} = ∑_{β′,β′′∈E_{B}}⟨X, (Y_{1}, β′, Y_{2}, β′′)⟩_{g,ε}F^{β′,β′′}

(6) ⟨X_{1}, X_{2}, (Y_{1}, Y_{2})⟩_{g_{1}+g_{2},ε_{1}ε_{2}} = ∑_{β′,β′′∈E_{B}}⟨X_{1}, (Y_{1}, β′)⟩_{g_{1},ε_{1}}F^{β′,β′′}⟨X_{2}, (Y_{2}, β′′)⟩_{g_{2},ε_{2}}

(7) ⟨X, (Y_{1}, Y_{2})⟩_{g+1,ε} = ∑_{β′,β′′∈E_{B}}⟨X, (Y_{1}, β′), (Y_{2}, β′′)⟩_{g,ε}F^{β′,β′′}

(8) ⟨X, (Y_{1}, Y_{2})⟩_{g+1,0} = ∑_{β′,β′′∈E_{B}}⟨X, (Y_{1}, β′), (Y_{2}^{∗}, β′′^{∗})⟩_{g,1}F^{β′,β′′}

(9) ⟨X, (Y_{1}, Y_{2})⟩_{g+ \frac{1}{2} ,0} = ∑_{β′,β′′∈E_{B}}⟨X, (Y_{1}, β′, Y_{2}^{∗}, β′′^{∗})⟩_{g,ε}F^{β′,β′′}

Remark. We pay no attention to the order in which blocks of arguments are written in a correlator because due to (i) of the definition of systems of correlators all orderings of blocks in ascending order of their size give the same function.

Remark. We shall define a correlator for a type \overline{G} of disconnected surfaces. First, fix any ordering of all types G of connected stratified surfaces (use for example the lexicographic ordering). A type \overline{G} of disconnected surfaces is a set \overline{G} = (G_{1}, …, G_{k}) of types G_{i} = (g^{(i)}, ε^{(i)}, m^{(i)}, m_{1}^{(i)}, …, m_{s_{i}}^{(i)}) of connected stratified surface. We require types G_{i} to be numbered according the ordering.

Put V\overline{G} = A^{⊗(∑ m^{(i)})} ⊗ B^{⊗(∑ m_{j}^{(i)})}. Define polylinear function ⟨X⟩\overline{G} on V\overline{G} as a product of correlators

⟨X⟩\overline{G} = ⟨X^{(1)}⟩_{G_{1}} … ⟨X^{(k)}⟩_{G_{k}}

Here X = (X_{a}^{(1)}, …, X_{a}^{(k)}, X_{b}^{(1)}, …, X_{b}^{(k)}) and X^{(i)} = (X_{a}^{(i)}, X_{b}^{(i)}). A type of an arbitrary stratified surface possibly disconnected will

be denoted below by G.

Fix a basis E_{A} of A and a basis E_{B} of B and use nondegenerate tensors F_{α′,α′′} = (α′, α′′)_{A}, F_{β′,β′′} = (β′, β′′)_{B} and dual to lower and raise indices.

Denote by 1_{A} the element ⟨α′⟩_{0,1}F^{α′,α′′}α′′ ∈ A. Denote by 1_{B} the element ⟨β′⟩_{0,1}F^{β′,β′′}β′′ ∈ B. Denote by U the element ⟨α′⟩_{\frac{1}{2} ,0}F^{α′,α′′}α′′ ∈ A.

Lemma 4.2. The following identities hold for any acceptable sets of arguments X, Y.

(1) ⟨1_{A}, X⟩_{g,ε} = ⟨X⟩_{g,ε}

(2) ⟨X, (1_{B}, Y)⟩_{g,ε} = ⟨X, (Y)⟩_{g,ε}

(3) ⟨U_{A}, X⟩_{g,ε} = ⟨X⟩_{g+ \frac{1}{2} ,0}

Proof. (1) Use identity (1) from the definition 4.2 and put in it X_{1} = ∅, g_{1} = 0, ε_{1} = 1, X_{2} = X, g_{2} = g, ε_{2} = ε.

(2) Use identity (6) from the definition 4.2 and put in it X_{1} = ∅, Y_{1} = ∅, g_{1} = 0, ε_{1} = 1, X_{2} = X, Y_{2} = Y, g_{2} = g, ε_{2} = ε.

(3) Use identity (1) from the definition and put in it X_{1} = ∅, g_{1} = \frac{1}{2}, ε_{1} = 0, X_{2} = X, g_{2} = g, ε_{2} = ε. □

Remark. Let γ be an unstable simple cut of a stratified surface Ω. Then one of the connected components of contracted cut surface Ω_{#} is a trivial surface. This component is one of the surfaces (4), (5), (6), (7), (11) from the list of trivial surfaces given in lemma 3.1 due to the assumption that any component of boundary of Ω contains at least one special point. In cases (4), (5), (7) simple cut γ induces one of the relations from lemma 4.2. In cases (6), (11) simple cut γ induces trivial relation, following from the definition of bilinear forms (x′, x′′)_{A} and (y′, y′′)_{B}.

We shall assign a system of correlators to a Klein topological field theory.

Let T = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {Φ_{(Ω,O)}}) be a Klein topological field theory. For any type G = (g, ε, m, m_{1}, …, m_{s}) choose a connected stratified surface Ω of type G and an admissible set of local orientations O. Note that for any boundary contour ω_{i} admissible set of local orientations generates a cyclic ordering of special points from ω_{i}.

Denote by M the set of multipliers of the tensor product V_{G} = A^{⊗m} ⊗ B^{⊗(m_{1}+···+m_{s})}. A bijection N : M ↔ Ω_{0} between set M and set Ω_{0} of special points on a stratified surface Ω endowed with an admissible set of local orientations O is called an admissible bijection if

∘ multipliers A correspond to interior special points;

∘ first m_{1} multipliers B correspond to special points from boundary contour ω_{1} such that |ω_{1}| = m_{1}, next m_{2} multipliers B correspond to special points from boundary contour ω_{2} such that |ω_{2}| = m_{2} etc.;

∘ the linear order of special points on any ω_{i} induced by the order of multipliers B in V_{G} is compatible with the cyclic order induced by O.

Fix an admissible bijection N : M ↔ Ω_{0}. Denote by ϕ_{N} : V_{G} → V_{(Ω,O)} the isomorphism of vector spaces induced by this bijection.

For any element z = x_{1}⊗⋯⊗x_{m}⊗y_{1}^{1}⊗⋯⊗y_{m_{1}}^{1}⊗⋯⊗y_{1}^{s}⊗⋯⊗y_{m_{s}}^{s} ∈ V_{G} put

⟨x_{1}, …, x_{m}(y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{G} = Φ_{(Ω,O)}(ϕ_{N} (z)).

Thus, we obtain a linear form ⟨X⟩_{G} for any topological type G of a stratified surface.

Lemma 4.3. Linear form ⟨X⟩_{g,ε} does not depend on the choice of a stratified surface Ω, an admissible set of local orientations O and an admissible bijection N.

Proof. Lemma follows immediately from topological invariance of Klein field theory. □

Denote by C(T ) the set of data (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {⟨X⟩_{G}}).

Lemma 4.4. The set of data C(T ) is a system of correlators.

Proof. Identities (i) - (iii) follows from appropriate axioms of a Klein topological field theory.

We will pay more attention to identity (iv). Let γ be a simple cut of a stratified surface Ω of type G = (g, ε, m, m_{1}, …, m_{s}) with m_{i} > 0. Then contracted cut surface Ω_{#} = Ω/γ is either a connected surface or consists of two connected components (Ω_{#})_{1}, (Ω_{#})_{2}. Denote by G_{#} the type of Ω_{#}. Thus, G_{#} = ((G_{#})_{1}, (G_{#})_{2}) in the latter case.

Cut invariance of a Klein topological field theory leads to the relation between correlators ⟨X⟩_{G} and ⟨X⟩_{G_{#}}.In order to establish this relation we need to fix

admissible sets of local orientations O on Ω and O′_{#} on Ω_{#}; admissible bijections N : M ↔ Ω_{0} and N_{#} : M_{#} ↔ (Ω_{#})_{0}. (Note that a cut morphism η : (Ω, O) → (Ω_{#}, O_{#}) may bring an

admissible set of local orientations O to nonadmissible set of local orientations O_{#}.)

We have the chain of morphisms

V_{G} ⟶^{ϕ_{N}} V_{(Ω,O)} ⟶^{η_{∗}} V_{Ω_{#}} ⟶^{ϕ_{N′}^{−1}}→ V_{Ω′_{#}} → V_{G_{#}}.

Using cut invariance of linear forms Φ_{(Ω,O)} and the definition of ⟨X⟩_{g,ε} we obtain the identity, namely, tensor ⟨X⟩_{G} is equal to a contraction of tensor ⟨X_{#}⟩_{G_{#}}.

We did an appropriate choice of admissible sets of local orientations O and O′ separately for all 9 classes of simple cuts. The resulting relations for correlators just form a list of relations in the definition of a sysem of correlators.

□

Conversely, assume we are given a system of correlators C = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, ⟨X⟩_{G}}).

First, define a linear form Φ_{(Ω,O)} for stratified surfaces of type G = (g, ε, m, m_{1}, …, m_{s}) such that m_{i} > 0 for i = 1, …, s and an admissible set of local orientations O. For this purpose use an admissible bijection N : M → Ω_{0} and invert the above consideration. The function Φ_{(Ω,O)} is well defined because the correlators are symmetric.

Second, using axioms of a Klein topological field theory extend the definition of linear forms Φ_{(Ω,O)} to all pairs (Ω, O), i.e., to arbitrary sets of local orientations O (using axiom 2^{∘}) and to surfaces with boundaries having no special points (using axiom 4^{∘}). It can be easily checked that constructed set of data T (C) = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {Φ_{(Ω,O)}}) satisfies all axioms of a Klein topological field theory. From definitions it follows that the correspondence C ⇝ T (C) is inverse to the correspondence T ⇝ C(T ). Thus

Theorem 4.1. Correspondence T ⇝ C(T ) is a bijection between Klein topological field theories and systems of correlators.

4.3. From system of correlators to structure algebras. Fix a system of correlators C = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {⟨X⟩_{G}}). In this subsection we consider correlators as tensors in basis E_{A} and E_{B} of vector spaces A and B resp. Thus, in notation ⟨X⟩_{G} of a correlator we assume that X is a set of basic elements α_{i} ∈ E_{A}, β_{i}^{j} ∈ E_{B}. An index with star like α^{∗} means the contraction with the matrix I_{α′}^{α′′} of involutive transformation x ↦ x^{∗}. For example, ⟨α^{∗}, X⟩_{(g,ε)} = I_{α}^{α′}⟨α′, X⟩_{(g,ε)}

We deal here with correlators corresponding to a type G of not necessarily connected stratified surface. It was shown above that any simple cut γ of a stratified surface Ω of type G generates an identity with tensor ⟨X⟩_{G} in the left hand side and a contraction of tensor ⟨X_{#}⟩_{G_{#}} in the right hand side. Here G_{#} is the type of the contracted cut surface Ω_{#} = Ω/γ. The explicit formulas for these identities corresponding to simple cuts of all classes are those that are listed in the definition of systems of correlators.

Identities for correlators that differ only by symmetries (i)-(ii), we consider as equal. Accepting this agreement we may claim, that the identity corresponding to a simple cut γ ⊂ Ω does not depend on the choice of Ω and an admissible set of local orientations O. The identities corresponding to isomorphic pairs (Ω, γ) and (Ω′, γ′) are equal.

We denote this identity as follows:

⟨X⟩_{G} = Contr_{(Ω,γ)}(⟨X_{#}⟩_{G_{#}}).

Here X_{#} is tensor of a type encoded by the type G_{#} of contracted cut surface Ω_{#} = Ω/γ, namely, if G_{#} = (g_{#}, ε_{#}, m_{#}, m_{#1}, …, m_{#s_{#}}) then X_{#} has m_{#} first indices from A and ∑ m_{#i} remaining indices from B. Right hand side expression is a contraction of this tensor by means of tensors F^{α′,α′′}, F^{β′,β′′}, D^{α} and possibly by their contractions with matrices of star involutions I_{α′}^{α}, I_{β′}^{β}. The explicit form of the contraction is defined by a pair (Ω, γ) and is encoded in the notation of operator Contr_{(Ω,γ)}(…). Indices of X_{#} that are not involved into contraction are just indices of tensor X. The type of X is encoded by G.

A cut system Γ = {γ_{1}, …, γ_{k}} generates a chain of identities and finally we obtain the identity as follows:

⟨X⟩_{G} = Contr_{(Ω,γ_{1})}(Contr_{(Ω/{γ_{1}},γ_{2})}(… Contr_{(Ω/{γ_{1},...,γ_{k−1}},γ_{k})}(⟨X_{#}⟩_{G_{#}}) …)),

where G_{#} is topological type of contracted cut surface Ω_{#} = Ω/Γ. Clearly, the result does not depend on the ordering of simple cuts. We denote this identity as

⟨X⟩_{G} = Contr_{(Ω,Γ)}(⟨X_{#}⟩_{G_{#}}).

Isomorphic pairs (Ω, Γ) and (Ω′, Γ′) generate equal identities. Note that operator Contr_{(Ω,Γ)}(…) can be applied to any tensor \overline{X}_{#} of type corresponding to the type G_{#} of the contracted cut surface Ω/Γ.

The following identities are evident.

Lemma 4.5. (1) If Γ^{∘} ⊂ Γ is a subsystem of a cut system Γ then Contr_{(Ω,Γ)}(\overline{X}_{#}) = Contr_{(Ω,Γ^{∘})}(Contr_{(Ω/Γ^{∘},Γ\Γ^{∘})}(\overline{X}_{#})).

(2) Let Γ′ be a cut system on Ω′, Γ_{1} and Γ_{2} be two cut systems on Ω^{∘} and \overline{X}_{#}^{∘} (resp. \overline{X}′_{#}) be a tensor of type corresponding to the type G_{#}^{∘} (resp. G′_{#}) of the contracted cut surface Ω′/Γ′ (resp.Ω^{∘}/Γ^{∘}). If Contr_{(Ω^{∘},Γ_{1})}(\overline{X}_{#}) = Contr_{(Ω^{∘},Γ_{2})}(\overline{X}_{#}) then Contr_{(Ω^{∘}⊔Ω′,Γ^{∘}⊔Γ_{1})}(\overline{X}_{#}^{∘}·\overline{X}′_{#}) = Contr_{(Ω^{∘}⊔Ω′,Γ^{∘}⊔Γ_{2})}(\overline{X}_{#}^{∘} · \overline{X}′_{#}).

Correlators corresponding to types G of trivial surfaces are called trivial correlators. There are five of them; they were used above for definitions of bilinear forms (α′, α′′)_{A}, (β′, β′′)_{B} and elements 1_{A} ∈ A, 1_{B} ∈ B, U ∈ A.

Correlators corresponding to types G of basic surfaces are called basic correlators. Therefore, there are three basic correlators

⟨x_{1}, x_{2}, x_{3}⟩_{0,1}, ⟨(y_{1}, y_{2}, y_{3})⟩_{0,1}, ⟨x, (y)⟩_{0,1}.

Lemma 4.6. The correlator ⟨X⟩_{G} for the type G of any connected stable stratified surface Ω is equal to a contraction of a product of basic correlator.

Proof. Indeed, take a stratified surface Ω of type G and a complete cut system Γ of Ω. Then we get the identity ⟨X⟩_{G} = Contr_{(Ω,Γ)}(⟨X_{#}⟩_{G_{#}}). By definition, connected components of Ω_{#} are basic surfaces. Therefore, correlator ⟨X_{#}⟩_{G_{#}} is a product of basic correlators. □

Let Γ′ and Γ′′ be two complete cut systems of a stable connected stratified surface Ω. Then we have two identities ⟨X⟩_{G} = Contr_{(Ω,Γ′)}(⟨X′_{#}⟩_{G′_{#}}) and ⟨X⟩_{G} = Contr_{(Ω,Γ′′)}(⟨X′′_{#}⟩_{G′′_{#}}). Hence

Contr_{(Ω,Γ′)}(⟨X′_{#}⟩_{G′_{#}}) = Contr_{(Ω,Γ′′)}(⟨X′′_{#}⟩_{G′′_{#}}),

and we obtain the relation between basic correlators. Obviously, replacing Γ′ or Γ′′ by an equivalent complete cut system brings this relation to the equal one. Moreover, if two triples (Ω_{1}, Γ′_{1}, Γ′′_{1}) and (Ω_{2}, Γ′_{2}, Γ′′_{2}) are equivalent, i.e., there exists an isomorphism ϕ : Ω_{1} → Ω_{2} such that ϕ(Γ′_{1}) = Γ′_{2}, ϕ(Γ′′_{1}) = Γ′′_{2} (or ϕ(Γ′_{1}) = Γ′′_{2}, ϕ(Γ′′_{1}) = Γ′_{2}), then corresponding relations are equal.

We call the relations corresponding to nonequivalent pairs of complete cut systems on simple surfaces defining relations. The relations corresponding to unstable cuts of simple and trivial surfaces we call defining relations too.

Using the list of simple surfaces and the classification of complete cut systems on them (see lemma 3.10) we obtain the following list of defining relations between basic correlators. In the list we omit relations that are consequence of given ones. The enumeration of relations corresponds to the enumeration of simple surfaces. In order to simplify formulas we use nonbasic correlators ⟨α, α′, α′′, α′′′⟩_{0,1} and ⟨(β, β′, β′′, β′′′)⟩_{0,1} because they have clear expression via basic correlators.

Defining relations corresponding to complete cut systems of simple surfaces

(1) ⟨α_{1}, α_{2}, α′⟩_{0,1}F^{α′,α′′}⟨α′′, α_{3}, α_{4}⟩_{0,1} = = ⟨α_{1}, α_{3}, α′⟩_{0,1}F^{α′,α′′}⟨α′′, α_{2}, α_{4}⟩_{0,1}

(2) ⟨α, α′, α′′^{∗}⟩_{0,1}F^{α′,α′′} = ⟨α, α′, α′′⟩_{0,1}D^{α′}D^{α′′}

(4) ⟨(β_{1}, β_{2}, β′)⟩_{0,1}F^{β′,β′′}⟨(β′′, β_{3}, β_{4})⟩_{0,1} = = ⟨(β_{2}, β_{3}, β′)⟩_{0,1}F^{β′,β′′}⟨(β′′, β_{4}, β_{1})⟩_{0,1}

(5) ⟨(β_{1}, β_{2}, β′)⟩_{0,1}F^{β′,β′′}⟨α, (β′′)⟩_{0,1} = ⟨(β_{2}, β_{1}, β′)⟩_{0,1}F^{β′,β′′}⟨α, (β′′)⟩_{0,1}

(6) ⟨α_{1}, α_{2}, α′⟩_{0,1}F^{α′,α′′}⟨α′′, (β)⟩_{0,1} = = ⟨(β, β′, β′′)⟩_{0,1}F^{β′,β′′′}⟨α_{1}, (β′′′)⟩_{0,1}F^{β′′,β^{(iv)}}⟨α_{2}, (β^{(iv)})⟩_{0,1}

(7) ⟨α′, (β)⟩_{0,1}D^{α′} = ⟨(β, β′, β′′^{∗})⟩_{0,1}F^{β′,β′′}

(8) ⟨α′, (β_{1})⟩_{0,1}F^{α′,α′′}⟨α′′, (β_{2})⟩_{0,1} = ⟨(β_{1}, β′, β_{2}, β′′)⟩_{0,1}F^{β′,β′′}

Defining relations completely describe properties of systems of correlators. Precise statements are formulated and proved in sequel subsection using structure algebras.

Denote trivial and basic correlators as follows.

(1) F_{α_{1},α_{2}} = ⟨α_{1}, α_{2}⟩_{0,1}

(2) F_{β_{1},β_{2}} = ⟨(β_{1}, β_{2})⟩_{0,1};

(3) R_{α,β} = ⟨α, (β)⟩_{0,1};

(4) S_{α_{1},α_{2},α_{3}} = ⟨α_{1}, α_{2}, α_{3}⟩_{0,1};

(5) T_{β_{1},β_{2},β_{3}} = ⟨(β_{1}, β_{2}, β_{3})⟩_{0,1};

(6) R_{α,β_{1},β_{2}} = ⟨α, (β_{1}, β_{2})⟩_{0,1};

(7) I_{α_{1},α_{2}} = ⟨α_{1}^{∗}, α_{2}⟩_{0,1};

(8) I_{β_{1},β_{2}} = ⟨(β_{1}^{∗}, β_{2})⟩_{0,1};

(9) D_{α} = ⟨α⟩_{\frac{1}{2} ,0};

(10) J_{α} = ⟨α⟩_{0,1};

(11) J_{β} = ⟨(β)⟩_{0,1}.

Theorem 4.2. Tensors (1)-(11) are structure constants of a structure algebra, which we denote by H(C).

Proof. By lemma 2.2 it is sufficient to verify relations (1)–(16) for these tensors. These relations either follow from the symmetries of correlators or coincide with defining relations for basic correlators (1)-(8).

□

4.4. From structure algebra to Klein topological field theory. Let H = (H = A ⊕ B, (x, y), x ↦ x^{∗}, U ∈ A) be a structure algebra. We shall construct a Klein topological field theory T (H) = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {Φ_{(Ω,O)}}) with the same vector spaces A and B and involutive transformations x ↦ x^{∗} and y ↦ y^{∗} which are restrictions of involutive antiautomorphism of algebra H to its summands.

First, define linear forms Φ_{(Ω,O)} : V_{Ω} → ℂ for trivial and basic surfaces Ω. Choose basis E_{A} of vector space A and E_{B} of vector space B. It is sufficient to define a linear form on basic elements (⊗_{p∈Ω_{a}}α_{p})⊗(⊗_{q∈Ω_{b}}β_{q}) of vector space V_{Ω}. Here Ω_{a} (resp. Ω_{b}) is the set of all interior (resp. boundary) special points of Ω, α_{p} ∈ E_{A}, β_{q} ∈ E_{B}.

For example, let Ω be isomorphic to a disc with three boundary special points q_{1}, q_{2}, q_{3}. Choose an admissible set O of local orientations and fix an admissible ordering N of special points. In this case O and N are uniquely defined by fixing an orientation of the disc and choosing a first special point on its boundary. The order of special points defines the isomorphism V_{Ω} ≂ B_{1} ⊗ B_{2} ⊗ B_{3}, where B_{i} is a copy of vector space B and i means that the isomorphism brings a vector β_{i} assigned to i-th special point q_{i} to this multiplier (i = 1, 2, 3). For basic element ⊗_{i=1}^{3}β_{i} ∈ V_{Ω} put Φ_{(Ω,O)}(⊗_{i=1}^{3}β_{i}) = T_{β_{1},β_{2},β_{3}}, where tensor T_{β_{1},β_{2},β_{3}} belongs to structure constants of structure algebra H (see subsection 2.1). For a nonadmissible set O of local orientations define Φ_{(Ω,O)} by invariance of changing of local orientations (axiom 2^{∘} from the definition of Klein topological field theory). Properties of structure constants of a structure algebra (see lemma 2.2) provide the correctness of the definition of Φ_{(Ω,O)}, its topological invariance and invariance with respect to changing of local orientations.

Analogous reasoning leads to correct definitions of Φ_{(Ω,O)} for all trivial and basic surfaces. Key formulas for a surface Ω, an admissible set O of local orientations and an admissible numbering N of special points are as follows. Right-hand sides of these formulas are expressed via structure constants of H.

Trivial surfaces

(1) Ω is a sphere S^{2} without special points: V_{Ω} = ℂ, Φ_{(Ω,O)}(1) = J^{α}J_{α}.

(2) Ω is a projective plane ℝP^{2} without special points: V_{Ω} = ℂ, Φ_{(Ω,O)}(1) = J_{α}D^{α}.

(3) Ω is a disc D^{2} without special points: V_{Ω} = ℂ, Φ_{(Ω,O)}(1) = J^{β}J_{β}.

(4) Ω is a sphere S^{2} with an interior special point p: V_{Ω} = A, Φ_{(Ω,O)}(α) = J_{α}.

(5) Ω is a disc D^{2} with a boundary special point q: V_{Ω} = B, Φ_{(Ω,O)}(β) = J_{β}.

(6) Ω is a sphere S^{2} with two interior special points p_{1}, p_{2}: V_{Ω} ≂ A ⊗ A, Φ_{(Ω,O)}(α_{1} ⊗ α_{2}) = F_{α_{1},α_{2}}.

(7) Ω is a projective plane ℝP^{2} with an interior special point p: V_{Ω} = A, Φ_{(Ω,O)}(α) = D_{α}.

(8) Ω is a torus T^{2} without special points: V_{Ω} = ℂ, Φ_{(Ω,O)}(1) = F_{α′,α′′}F^{α′,α′′} = dim A.

(9) Ω is a Klein bottle Kl without special points: V_{Ω} = ℂ, Φ_{(Ω,O)}(1) = I_{α′,α′′}F^{α′,α′′} = tr(x ↦ x^{∗}).

(10) Ω is a disc D^{2} with an interior special point p: V_{Ω} = A, Φ_{(Ω,O)}(α) = J^{β}R_{α,β}.

(11) Ω is a disc D^{2} with two boundary special points q_{1}, q_{2}: V_{Ω} ≂ B ⊗ B, Φ_{(Ω,O)}(β_{1} ⊗ β_{2}) = F_{β_{1},β_{2}}.

(12) Ω is a Möbius band Mb without special points: V_{Ω} = ℂ, Φ_{(Ω,O)}(1) = J^{β}D^{α}R_{α,β}.

(13) Ω is a cylinder Cyl without special points: V_{Ω} = ℂ, Φ_{(Ω,O)}(1) = J^{β′}R_{α′,β′}F^{α′,α′′}J^{β′′}R_{α′′,β′′}.

Basic surfaces

(1) Ω is a sphere S^{2} with three interior special points p_{1}, p_{2}, p_{3}: V_{Ω} ≂ A ⊗ A ⊗ A, Φ_{(Ω,O)}(α_{1} ⊗ α_{2} ⊗ α_{3}) = S_{α_{1},α_{2},α_{3}}.

(2) Ω is a disc D^{2} with three boundary special points q_{1}, q_{2}, q_{3}: V_{Ω} ≂ B ⊗ B ⊗ B, Φ_{(Ω,O)}(β_{1} ⊗ β_{2} ⊗ β_{3}) = T_{β_{1},β_{2},β_{3}}.

(3) Ω is a disc D^{2} with a boundary special point q and an interior special point p: V_{Ω} = A ⊗ B, Φ_{(Ω,O)}(α ⊗ β) = R_{α,β}.

Multiplicativity (axiom 6^{∘} from the definition of Klein topological field theory) allows us to define correctly linear forms Φ_{(Ω,O)} also for stratified surfaces Ω that are disjoint union of several trivial and basic surfaces.

Lemma 4.7. The already defined linear forms Φ_{(Ω,O)} for trivial and basic surfaces and their disjoint unions satisfy topological invariance axiom, invariance of a change of local orientations axiom, nondegeneracy axiom, invariance of adding trivial field axiom (when applicable), cut invariance axiom (for cuts of trivial and basic surfaces) and multiplicativity axiom.

Proof. Topological invariance and invariance of a change of local orientations follows from the definitions of functions Φ_{(Ω,O)} as it was shown above for a disc with three boundary special points. Nondegeneracy follows from nondegeneracy of bilinear forms (x′, x′′)|_{A} and (y′, y′′)|_{B} in algebra H. Invariance of adding a trivial field and cut invariance for trivial and basic surfaces are checked for all such surfaces. Any verification turns into a simple identity for structure constants of H. We skip the details. Multiplicativity follows from the definition of Φ_{(Ω,O)} for disjoint unions of trivial and basic surfaces. □

Define Φ_{(Ω,O)} for a connected stable surface Ω. Choose a complete cut system Γ of Ω and orientations of all simple cuts γ ⊂ Γ. Denote by η : (Ω, O) → (Ω_{#}, O_{#}) cut morphism corresponding to Γ (see subsection 4.1). By definition of a complete cut system, all connected components of contracted cut surface Ω_{#} are basic surfaces. Therefore, function Φ_{(Ω_{#},O_{#})} is already defined. Put Φ_{(Ω,O,Γ)}(x) = Φ_{(Ω_{#},O_{#})}(η_{∗}(x)). We have to prove that this function does not depend on the choice of Γ.

Lemma 4.8. Let Γ and Γ′ be equivalent complete cut systems endowed with orientations of all simple cuts of a stable surface Ω. Then Φ_{(Ω,O,Γ)}(x) = Φ_{(Ω,O,Γ′)}(x). Particularly, Φ_{(Ω,O,Γ)}(x) does not depend on a choice of orientations of simple cuts.

Proof. Suppose, we change an orientation of a simple cut γ ⊂ Γ of a complete cut system Γ. Denote the same complete cut system with changed orientation of the cut by Γ′. We have two cut morphisms η_{Γ} : (Ω, O) → (Ω_{#}, O_{#}) and η_{Γ′} : (Ω, O) → (Ω_{#}, O′_{#}) and changing local orientations morphism ψ : (Ω_{#}, O_{#}) → (Ω_{#}, O′_{#}). Vector space V_{Ω_{#}} is isomorphic to V_{Ω} ⊗ X, where vector space X depends on Γ. By definition, linear map η_{∗} : V_{Ω} → V_{Ω_{#}} brings element x ∈ V_{Ω} to x ⊗ z, where z is a tensor product of several elements K_{A,∗} ∈ A, K_{B,∗} ∈ B and U ∈ A (see subsection 2.1). It can be easily checked that these elements of structure algebra are invariants of involution x ↦ x^{∗}. Therefore, (η_{Γ′})_{∗}(x) = (η_{Γ})_{∗}(ψ_{∗}(x)) = (η_{Γ})_{∗}(x) and Φ_{(Ω,O,Γ)}(x) = Φ_{(Ω,O,Γ′)}(x).

The equality Φ_{(Ω,O,Γ)} = Φ_{(Ω,O,Γ′)} for different but equivalent complete cut systems Γ and Γ′ follows from the topological invariance for trivial and basic surfaces and their disjoint unions applied to linear forms Φ_{(Ω/Γ,O_{#})} and Φ_{(Ω/Γ′,O′_{#})}. □

Lemma 4.9. If Ω is a simple stratified surface then Φ_{(Ω,O,Γ)} = Φ_{(Ω,O,Γ′)} for any pair Γ, Γ′ of complete cut systems of Ω.

Proof. Consider separately simple surfaces (1) - (8) (see lemma 3.10). For example, let Ω be a disc with two interior special points p_{1}, p_{2} and one boundary special point q. By lemma 4.8, it is sufficient to consider only nonequivalent complete cut systems Γ and Γ′. There are two nonequivalent complete cut systems Γ and Γ′ of Ω. One of them, Γ, consists of one separating contour γ and Ω/Γ is isomorphic to the disjoint union of a sphere with three interior special points p_{1}, p_{2}, p′ and a disc with one interior special point p′′ and one boundary special point q. Another one, Γ′, consists of two separating segments cut γ_{1}, γ_{2} and Ω/Γ′ is isomorphic to the disjoint union of three components:

∘ a disc with interior special point p_{1} and boundary special point q′_{1}, ∘ a disc with interior special point p_{2} and boundary special point q′_{2}, ∘ a disc with three boundary special points q, q′′_{1}, q′′_{2}. From definitions of Φ_{(Ω,O,Γ)} and Φ_{(Ω,O,Γ′)} we obtain that the equality

Φ_{(Ω,O,Γ)} = Φ_{(Ω,O,Γ′)} is equivalent to the equality

R_{α_{1},β′_{1}}F^{β′_{1},β′′_{1}}R_{α_{2},β′_{2}}F^{β′_{2},β′′_{2}}T_{β′β′′β} = S_{α_{1}α_{2}α′}F^{α′,α′′}R_{α′′,β}

for structure constants. The latter coincides with relation (5) for structure constants (see lemma 2.2) of structure algebra H.

Analogous reasoning are applicable in all cases. Equalities for structure constants derived from equalities Φ_{(Ω,O,Γ)} = Φ_{(Ω,O,Γ′)} for complete cut systems of simple surfaces are just relations for structure constants of a structure algebra. We omit the details. □

It follows from the lemma that formula Φ_{(Ω,O)} = Φ_{(Ω,O,Γ)} correctly defines a function Φ_{(Ω,O)} for all simple surfaces. Now, we can define by multiplicativity linear form Φ_{(Ω,O)} for any surface that is the disjoint union of trivial, basic and simple surfaces. Topological invariance for these forms and invariance of change of local orientations can be easily checked.

Lemma 4.10. Let Γ and Γ′ be adjacent complete cut systems of a stable stratified surface Ω. Then Φ_{(Ω,O,Γ)} = Φ_{(Ω,O,Γ′)}

Proof. Let Γ and Γ′ be two nonequivalent adjacent complete cut systems of Ω. By lemma 4.8, the replacement of Γ by equivalent cut system leads to the same linear form. Hence, we may assume that the set Γ^{∘} of cuts common to Γ and Γ′ has the following property. Contracted cut surface Ω/Γ^{∘} is the disjoint union of a simple surface Ω^{∘} and several basic surfaces and cut systems \overline{Γ} = Γ \ Γ^{∘}, \overline{Γ′} = Γ′ \ Γ^{∘} belong to Ω^{∘}; therefore \overline{Γ} and \overline{Γ′} are complete cut systems of simple surface Ω^{∘}.

We have that linear forms Φ_{(Ω/Γ^{∘},O_{#}^{∘})} are already correctly defined for any set O_{#}^{∘} of local orientations. From transitivity property of cut morphisms we obtain that Φ_{(Ω,O,Γ)}(x) = Φ_{(Ω/Γ^{∘},O_{#}^{∘})}(η^{∘}(x)) and Φ_{(Ω,O,Γ′)}(x) = Φ_{(Ω/Γ^{∘},O_{#}^{∘})}(η^{∘}(x)), where η^{∘} : V_{Ω} → V_{Ω/Γ^{∘}} is linear map induced by cut morphism η^{∘} : (Ω, O) → (Ω/Γ^{∘}, O_{#}^{∘}). Therefore, Φ_{(Ω,O,Γ)} = Φ_{(Ω,O,Γ′)}. □

By theorem 3.1 any pair of complete cut systems of Ω is connected by a chain of adjacent complete cut systems. Thus, formula Φ_{(Ω,O)} = Φ_{(Ω,O,Γ)} correctly defines function Φ_{(Ω,O)} and we define linear forms Φ_{(Ω,O)} for all connected stratified surfaces. Define Φ_{(Ω,O}) for disconnected surfaces by multiplicativity.

Theorem 4.3. A tuple of data T (H) = (A, x ↦ x^{∗}, B, y ↦ y^{∗}, {Φ_{(Ω,O)}}) is a Klein topological field theory.

Proof. Let us verify axioms of a Klein topological field theory.

1^{∘} It is sufficient to check topological invariance for connected stable surfaces that are not trivial or basic surfaces. Let ϕ : (Ω, O) → (Ω′, O′) be an isomorphism. Choose a complete cut system Γ of Ω and denote by Γ′ complete cut system ϕ(Γ) of Ω′. The isomorphism ϕ induces the isomorphism ϕ_{#} : (Ω_{#}, O_{#}) → (Ω′_{#}, O′_{#}) of contracted cut surfaces Ω_{#} = Ω/Γ and Ω′_{#} = Ω′/Γ′. The topological invariance with respect of ϕ follows from the topological invariance of ϕ_{#}. The latter was proven in lemma 4.7.

2^{∘} Invariance of changing of local orientations is verified analogously to 1^{∘}.

3^{∘} Nondegeneracy follows from nondegeneracy of bilinear forms (x′, x′′)|_{A} and (y′, y′′)_{B}.

4^{∘} Invariance of adding trivial field follows from the same statement proved for basic and trivial surfaces (lemma 4.7). Indeed, let Ω be a stable surface and r be a nonspecial point. Choose complete cut system Γ that does not meet r. Then the image of r in contracted cut surface belongs to a connected component Ω_{#i}, which is basic surface. Invariance of adding new special point r follows from the same statement for its image in Ω_{#i}.

5^{∘} It is sufficient to check cut invariance for a simple cut γ of a connected stable surface Ω that is not trivial or basic surface.

Suppose that γ is a stable cut. Then it can be included in a complete cut system Γ of Ω. By properties of cut morphisms (η_{Γ})_{∗} = (η_{Γ\γ})_{∗} ∘ (η_{γ})_{∗}, where (η_{Γ})_{∗} : V_{Ω} → V_{Ω/Γ}, (η_{Γ\γ})_{∗} : V_{Ω/γ} → V_{Ω/Γ}, (η_{γ})_{∗} : V_{Ω} → V_{Ω/γ} are linear maps corresponding to cut morphisms. Therefore, Φ_{(Ω,O)}(x) = Φ_{(Ω/γ,O_{#})}(η_{γ})_{∗}(x)).

Suppose that γ is trivial (unstable) cut. Then there exists a complete cut system Γ such that Γ ∩ γ = ∅. Hence, the image of γ in contracted cut surface is a simple cut of a connected component Ω_{#i}, which is basic surface. The invariance with respect to cut γ follows from the invariance with respect of its image in the basic surface, which was proved in lemma 4.7.

6^{∘} Multiplicativity is clear because linear forms Φ_{(Ω,O)} for disconnected surfaces are defined using this axiom.

□

Theorem 4.4. (Main) The correspondences T ⇝ C(T ) ⇝ HC(T )) and H ⇝ T (H) are reciprocal bijections between Klein topological field theories T = {A, x ↦ x^{∗}, B, y ↦ y^{∗}, Φ_{(Ω,O)}} and isomorphism classes of structure algebras H = {H = A ∔ B, (x, y), x ↦ x^{∗}, U}. The correlators for Klein topological theory T (H) have the following expressions in terms of structure algebra H:

⟨x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{g,1} =
= (x_{1} … x_{m}(y_{1}^{1} … y_{m_{1}}^{1})V_{K_{B}}(y_{1}^{2} … y_{m_{1}}^{2}) … V_{K_{B}}(y_{1}^{s} … y_{m_{s}}^{s}), K_{A}^{g})

and

⟨x_{1}, …, x_{m}, (y_{1}^{1}, …, y_{m_{1}}^{1}), …, (y_{1}^{s}, …, y_{m_{s}}^{s})⟩_{g,0} =
= (x_{1} … x_{m}(y_{1}^{1} … y_{m_{1}}^{1})V_{K_{B}}(y_{1}^{2} … y_{m_{1}}^{2}) … V_{K_{B}}(y_{1}^{s} … y_{m_{s}}^{s}), U^{2g})

Proof. The first statement follows from the exact constructions of the correspondences T ⇝ C(T ), C ⇝ H(C), H ⇝ T (H) and from theorems 4.1, 4.2, 4.3. The second statement follows from a representation ⟨X⟩_{G} = Contr_{(Ω,Γ)}(⟨X_{#}⟩_{G_{#}}), where Γ is a specially chosen complete cut system. Namely, Γ contains a system of cuts Γ^{∘} between holes 1st and 2nd, 2nd and 3d, etc. and Γ \ Γ^{∘} is a normal cut system of Ω/Γ^{∘}. □

Let H be a Klein topological field theory. One can restrict basic category of surfaces to the category of oriented surfaces with local orientations at all special points induced by a global orientation of a surface. It can be shown that this restriction is equivalent to an open-closed topological field theory as it is defined in [23], [26].

It is convenient to discuss the correspondence between variants of TFTs in terms of algebras. Note, that both involutive antiautomorphism x ↦ x^{∗}, responsible for changes of local orientations, and element U, responsible for gluing a hole by Möbius band, are not in use in oriented case.

The following claim is clear from Main theorem and results of section 2.

Corollary 4.1. Any massive (corresponding to a semisimple algebra) open-closed topological field theory can be extended to a Klein topological field theory. Equivalent classes of extensions are in the bijection with pairs, consisting of an involutive antiautomorphism x ↦ x^{∗} and an element U satisfying axioms of a structure algebra.

Number of extensions of an open-closet TFT to KTFT can be easely computed by results of subsection 2.2

5. Hurwitz topological field theory

5.1. Stratified coverings. Fix a stratified topological space Ω = ∐_{λ∈Λ} Ω_{λ}. Let π : P → Ω be a continuous epimorphic map of a topological space P onto Ω. Denote by P_{λ} the preimage of a stratum Ω_{λ} and by π_{λ} : P_{λ} → Ω_{λ} the restriction of π to P_{λ}. Analogously, for an arbitrary subset X ∈ Ω denote by π_{X} : P_{X} → X the restriction of π to P_{X} = π^{−1}(X).

Definition. A continuous map π : P → Ω is called a covering over the stratification (or a stratified covering) if and only if for any stratum λ the restriction π_{λ} : P_{λ} → Ω_{λ} is a local homeomorphism, i.e., a covering.

A stratified covering π : P → Ω is called special coverings if and only if

(1) Ω is a specially stratified manifold (i.e., any codimension 1 stratum belongs to the boundary of Ω);

(2) P is a manifold, possibly with a boundary.

In this work we deal with specially stratified coverings over surfaces and call them coverings for short.

A covering is called n-sheeted covering if and only if the full preimage of any point of a stratum of maximal dimension consists of n points. n-sheeted coverings over disconnected base can (and sometimes, will) be considered. In general, it is not assumed that a cover space P is connected.

An isomorphism of two coverings π_{1} : P_{1} → Ω and π_{2} : P_{2} → Ω over the same base is a homeomorphism ϕ : P_{1} → P_{2} such that π_{2} ∘ ϕ = π_{1}. By Cov_{n}(Ω) we denote the category of n-sheeted stratified coverings over Ω with morphisms being isomorphisms of coverings. By Cov_{n}(Ω) is denoted the set of isomorphism classes in category Cov_{n}(Ω). Clearly, Cov_{n}(Ω) is a finite set.

Lemma 5.1. An isomorphism ϕ : Ω → Ω′ of stratified manifolds induces the bijection between sets Cov_{n}(Ω) and Cov_{n}(Ω′).

Indeed, the bijection is induced by morphism π ↦ ϕ ∘ π.

Definition. Let π : P → Ω and π′ : P′ → Ω′ be two coverings over stratified manifolds Ω and Ω′. A pair of homeomorphisms ϕ̂ : P → P′ and ϕ : Ω → Ω′ is called a topological equivalence of coverings if and only if π′ ∘ ϕ̂ = ϕ ∘ π. A class of topologically equivalent coverings is called a topological type of coverings.

Obviously, two isomorphic coverings over a stratified manifold Ω are of the same topological type. The converse is not true: there exist nonisomorphic coverings over a surface that are of the same topological type.

Let Aut Ω be the group of all homeomorphisms of Ω, that preserve the stratification of Ω (a permutation of strata is allowed). The group Aut Ω acts on the set Cov(Ω): if ϕ ∈ Aut Ω and π ∈ Cov(Ω) then ϕ(π) = ϕ ∘ π. Clearly, two coverings π, π′ ∈ Cov(Ω) are of the same topological type if and only if they lie in the same orbit of the group Aut(Ω).

Note, that one can correctly assign a topological type of a covering π : P → Ω to a topological type of Ω. In contrast, the isomorphism class [π] of covering π can be associated with just manifold Ω but not with its topological type because one cannot choose an isomorphism between two different (but isomorphic) bases Ω and Ω′ canonically.

For an oriented base Ω we define an oriented topological type of a covering with respect to homeomorphisms preserving the orientation.

There exist only two, up to isomorphism, compact connected special stratified one-dimensional manifolds, a circle S^{1} and a closed segment I. It is convenient to consider also one noncompact manifold, namely, a ray R (with two obvious strata).

5.1.1. Coverings over a circle. A covering π : P → S^{1} over a circle has no special points, i.e., it is a nonramified covering. Isomorphism classes of n-sheeted coverings over S^{1} are in one-to-one correspondence with unordered partitions of the number n into a sum n = n_{1} + ⋯ + n_{s}. If π : P → S^{1} is a covering then s is the number of connected components of P and n_{i} is the degree of the restriction of π to i-th connected component of P.

Equivalently, the isomorphism class of a covering over S^{1} can be presented 1) as a Young diagram of order n; 2) as a conjugacy class of a symmetric group S_{n}. We denote by α a partition of n, as well, as corresponding Young diagram of order n and a conjugacy class of S_{n}. Group Aut S^{1} acts trivially on the set Cov_{n}(S^{1}) of isomorphism classes of coverings. Therefore, topological type of a covering can be also presented by the partition α.

5.1.2. Coverings over a ray. Let π : P → R be an n-sheeted covering over a ray. Denote by M the set of connected components of the preimage of an open interval R^{∘} = R − ∂R. Obviously, |M| = n and the restriction of π to any component is a homeomorphism. Any point p′ of the preimage of the end-point p ∈ R belong to the closure of either just one or just two components of M. Therefore, a special covering defines an involutive element s of group Aut M of all bijections M → M. Group Aut M is isomorphic to symmetric group S_{n}; there is no canonical isomorphism, therefore, involution s ∈ M correctly defines only conjugasy class of its image in S_{n}.

Evidently, isomorphism classes of coverings over the ray R are in one-to-one correspondence with conjugacy classes of involutions in S_{n}. On other hand, conjugacy classes of involutions are in one-to-one correspondence with decompositions of n into the sum n = k + 2l.

Group Aut R acts trivially on the set Cov_{n}(R) of isomorphism classes of coverings. Therefore, topological type of a covering can be also presented as a conjugacy class of an involutive permutation.

5.1.3. Coverings over a segment. (Fig.5)

Formal description of isomorphism classes of n-sheeted coverings over an interval I is as follows. Denote by p_{1} and p_{2} end points of I. Let π : P → I be an n-sheeted covering. Denote by M the set of connected components of the preimage of I^{∘} in P. Obviously, #M = n and the restriction of π to any component is a homeomorphism.

A point p_{i} (i = 1, 2) defines an involutive permutation of set M, which we denote by s_{i}. Thus, to each special covering π we assign the ordered pair (s_{1}, s_{2}) of involutions s_{1}, s_{2} ∈ Aut M ≈ S_{n}. Two pairs (s_{1}, s_{2}) and (s′_{1}, s′_{2}) are called conjugated if there exists g ∈ S_{n} such that s′_{1} = gs_{1}g^{−1} and s′_{2} = gs_{2}g^{−1}.

Figure 5. Stratified covering π over segment I.

Lemma 5.2. There exists a natural one-to-one correspondence between set Cov_{n}(I) of isomorphism classes of n-sheeted coverings over an interval I and the set of conjugacy classes of pairs of involutions in S_{n}.

Proof is skipped.

5.1.4. Analogue of Young diagram for pairs of involutions. The analogue of Young diagram can be defined for an ordered pair of involutions. Let D_{∞} = ⟨Σ_{1}, σ_{2}|σ_{1}^{2} = σ_{2}^{2} = 1⟩ be the infinite dihedral group and (s_{1}, s_{2}) be an ordered pair of involutive permutations of the symmetric group S_{n} acting by permutations of the set M = {1, …, n}. Then the correspondence σ_{i} ↦ s_{i} can be uniquely extended to the representation ρ of D_{∞} by permutations of the set M. Thus, M can be decomposed into the orbits M_{1}, …, M_{l} of D_{∞}. Any orbit can be considered as a transitive representation of D_{∞}.

Let M_{i} be an orbit of D_{∞}. Then it can be of one of four types: • (type 1) σ_{1} and σ_{2} has no fixed points in M_{i};

• (type 2) σ_{1} has one fixed point and σ_{2} has one fixed point;

• (type 3) σ_{1} has two fixed point and σ_{2} has no fixed point;

• (type 4) σ_{1} has no fixed points and σ_{2} has two fixed points. An orbit of type 1 consists of even number 2n_{i} of elements and any

two orbits of type 1 of 2n_{i} elements are equivalent as representations of D_{∞}. An orbit of type 2 consists of odd number 2n_{i} − 1 of elements and any two orbits of type 2 of 2n_{i} − 1 elements are equivalent as representations of D_{∞}. An orbit of type 3 consists of even number 2n_{i} of elements and any two orbits of type 3 of 2n_{i} elements are equivalent as representations of D_{∞}. An orbit of type 4 consists of even number 2n_{i} of elements and any two orbits of type 4 of 2n_{i} elements are equivalent as representations of D_{∞}.

Thus, to each pair (s_{1}, s_{2}) we can assign a decomposition of n into the sum of natural numbers; the sum is separated into four blocks as follows: n = (2n_{1}^{1} + ⋯ + 2n_{s_{1}}^{1}) + ((2n_{1}^{2} − 1) + ⋯ + (2n_{s_{2}}^{2} − 1)) + (2n_{1}^{3} + ⋯ + 2n_{s_{3}}^{3}) + (2n_{1}^{4} + ⋯ + 2n_{s_{4}}^{4}) Blocks correspond to types of orbits, summands in a block are unordered. We denote this decomposition by a single letter β and call it a dihedral decomposition of n.

Lemma 5.3. Let (s_{1}, s_{2}) and (s′_{1}, s′_{2}) be two ordered pairs of involutive elements of S_{n} and β, β′ be corresponding dihedral decompositions of n. Then (s_{1}, s_{2}) is conjugated to (s′_{1}, s′_{2}) if and only if β = β′.

Proof is skipped.

Thus, dihedral decompositions of n are in bijection with conjugacy classes of ordered pairs of involutions. Obviously, one can present β also as the set of four Young diagrams with easily derived special properties. We call this presentation a dihedral Young diagram of a pair (s_{1}, s_{2}). In sequel we consider terms ’a dihedral decomposition of n’ and ’a dihedral Young diagram of order n’ as synonyms. Let β be the dihedral Young diagram of a pair (s_{1}, s_{2}). Denote by β^{∗} a dihedral Young diagram of the pair (s_{2}, s_{1}). The operation β ↦ β^{∗} defines an involutive automorphism of the set Cov_{n}(I), which we denote by the same sign ”^{∗}”.

Lemma 5.4. Dihedral Young diagram β^{∗} can be obtained from β by replacing blocks of summands of types 3 and 4.

Proof is elementary.

Assign to each conjugacy class [(s_{1}, s_{2})] conjugacy classes of involutions [s_{1}] and [s_{2}] and denote them by ι_{1}(β) and ι_{2}(β) respectively.

A conjugacy classes of a pair [(s, s)], s ∈ S_{n}, is called a trivial class; corresponding dihedral Young diagram is said to be trivial. Evidently, they are in one-to-one correspondence with conjugacy classes of involutive elements in S_{n}. The dihedral Young diagram β of the class [(s, s)] is n = (2 + ⋯ + 2) + (1 + ⋯ + 1) (blocks of summands of types 3 and 4 are empty).

Fix an orientation of interval I. Then homeomorphisms of I preserving the orientation fixes points p_{1} and p_{2}. These homeomorphisms generate the index two subgroup Aut^{+} I of automorphism group Aut I. Evidently, Aut^{+} I acts trivially on the set Cov(I). Thus, oriented topological types of special coverings over a segment I are in one-to-one correspondence with dihedral Young diagrams.

Any element from Aut I \Aut^{+} I acts on the set Cov(I) as (s_{1}, s_{2}) ↦ (s_{2}, s_{1}). Thus, coverings with dihedral Young diagrams β and β^{∗} are of the same topological type. Therefore, topological type and isomorphism class of coverings do not coincide for the segment.

5.1.5. Dimension 2. (Fig.6)

Let π : P → Ω be a covering over a stratified surface Ω = ∐_{λ∈Λ} Ω_{λ} and p ∈ Ω be any point of base. Topological invariants of the restriction π_{U} : P_{U} → U of the covering to a sufficiently small neighborhood U of p depends only on a stratum containing p. More precisely, if p belongs to the generic stratum, then π_{U} is trivial covering, i.e., π_{U} mappings n copies of U onto U. Thus, n is a unique invariant at a generic point.

If p is an interior special point, then one can choose U isomorphic to an open disc. In this case π_{U} is defined, up to topological equivalence, by the topological type of its restriction to the boundary ∂U (it is a circle) [20]. Thus, a Young diagram describes the covering in the neighborhood of p.

If p belongs to a one-dimensional boundary stratum E, then it is convenient to choose U homeomorphic to the direct product of a neighborhood U_{E} of p in E and a ray R [29]. Clearly, topological type of the covering π_{U} is uniquely defined by the topological type of its restriction to the ray R. Therefore, the covering in the neighborhood of p ∈ E is described by a conjugacy class of an involutive element of S_{n}.

If q is a boundary special point, then it is convenient to choose a neighborhood U homeomorphic to an open cone C^{∘}(I) over a closed interval I. Clearly, isomorphism classes of coverings over U are in one-to-one correspondence with isomorphism classes of coverings over I. As it was mentioned above, topological type of a covering over I can include more than one isomorphism classes of coverings over I. In order to control isomorphism classes of coverings over U, let us fix an admissible set of local orientations at special points of the surface. Hence, to each boundary special point q we can assign an isomorphism class of coverings over the segment, i.e., dihedral Young diagram. This diagram describes the covering in the neighborhood of q.

Clearly, local invariants of a covering coincide for all points from the same stratum. Thus, to each stratum of Ω we assigned a combinatorial invariant.

5.2. Cut coverings. Recall that a cut surface Ω_{∗} = (Ω_{∗}, Γ_{∗}, τ) is a triple, consisting of

a stratified surface Ω_{∗},

a set Γ_{∗} ⊂ Ω_{∗} such that Γ_{∗} coincides with the joint of several closed pairwise nonintersecting one-dimensional boundary strata,

an involutive homeomorphism τ : Γ_{∗} → Γ_{∗} having no fixed points (see section 3.2).

Fix a cut surface Ω_{∗} = (Ω_{∗}, Γ_{∗}, τ).

Figure 6. Stratified covering π over two-dimensional stratified surface Ω.

Let π_{∗} : P_{∗} → Ω_{∗} be an n-sheeted covering. Denote by Γ̂_{∗} the preimage of Γ_{∗} in P_{∗}. Suppose that the following condition holds:

(∗)  Γ̂_{∗} ⊂ ∂P_{∗}

In sequel we deal with coverings π_{∗} satisfying this condition.

Denote by ξ : Γ̂_{∗} → Γ_{∗} the restriction of π_{∗} to Γ̂_{∗}. Due to (∗) the mapping ξ : Γ̂_{∗} → Γ_{∗} is an n-sheeted covering over Γ_{∗}.

An involutive homeomorphism τ̂ : Γ̂_{∗} → Γ̂_{∗} is called an admissible involution if π_{∗} ∘ τ̂ = τ ∘ π_{∗}.

Definition. Let π_{∗} : P_{∗} → Ω_{∗} be a covering over a cut surface Ω_{∗}. Suppose condition (∗) is satisfied. Let τ̂ : Γ̂_{∗} → Γ̂_{∗} be an admissible involution. Then pair (π_{∗}, τ̂) is called a cut covering.

An isomorphism of two cut coverings (π_{∗}, τ̂) and (π′_{∗}, τ̂′) is an isomorphism ϕ : P_{∗} → P′_{∗} of coverings such that τ̂′ ∘ ϕ = ϕ ∘ τ̂.

Denote by CutCov(Ω_{∗}) the category of cut coverings over a cut surface Ω_{∗} = (Ω_{∗}, Γ_{∗}, τ) with morphisms being isomorphisms of cut coverings; denote by CutCov(Ω_{∗}) the set of isomorphism classes of cut coverings. In sequel, we shall use the same difference in fonts in order to distinguish a category and the set of isomorphism classes of objects in it and in other cases.

Clearly, CutCov(Ω_{∗}) is a finite set.

Let π : P → Ω be a covering over stratified surface Ω, Γ ⊂ Ω be a cut system, (Ω_{∗}, Γ_{∗}, τ) be a cut surface obtained by cutting Ω along Γ and glue : Ω_{∗} → Ω be gluing map.

Clearly, the preimage Γ̂ = π^{−1}(Γ) is a cut system of P. Denote by (P_{∗}, Γ̂_{∗}, τ̂) the cut surface obtained by cutting P along Γ̂ and denote by ĝlue the glueing map P_{∗} → P. Obviously, there is a natural covering π_{∗} : P_{∗} → Ω_{∗} such that the following diagram is commutative.

P_{∗} \underrightarrow{ĝlue} P
π_{∗}↓  ^{π}↓
Ω_{∗} ⟶_{glue} Ω

Evidently, covering π_{∗} satisfies (∗) and the correspondence π ↦ π_{∗} is a functor cut : Cov(Ω) → Cov(Ω_{∗}). This functor generates a map cut : Cov(Ω) → Cov(Ω_{∗}) of isomorphism classes of coverings. Functor cut is a superposition of two functors

(1) Cov(Ω) → CutCov(Ω_{∗}) and

(2) CutCov(Ω_{∗}) → Cov(Ω_{∗}),

where (1) is induced by cutting along Γ and (2) is just ”forgetting” τ̂.

Lemma 5.5. Functor Cov(Ω) → CutCov(Ω_{∗}) induces the equivalence of categories.

Proof. One can easily construct an inverse functor CutCov(Ω_{∗}) → Cov(Ω). It ’glues’ a cut covering (π_{∗}, τ̂) into a covering π : P_{∗}/τ̂ → Ω. □

Let (Ω_{∗}, Γ_{∗}, τ) be a cut surface. Denote by Cov(Ω_{∗}, Γ_{∗}) the subcategory of Cov(Ω_{∗}), that consists of coverings π_{∗} : P_{∗} → Ω_{∗} over Ω_{∗} satisfying condition (∗).

Fix a covering π_{∗} ∈ Cov(Ω_{∗}, Γ_{∗}); denote by [π_{∗}] its class. Denote by ξ the restriction π_{∗}|Γ̂_{∗}.

Clearly, an automorphism of covering π_{∗} induces the automorphism of covering ξ. Hence, we obtain a homomorphism ϕ : Aut π_{∗} → Aut ξ. The preimage of a generic point of Γ_{∗} consists of n elements where n is the degree of π_{∗}. Therefore, ϕ is a monomorphism. We shall consider group Aut π_{∗} as a subgroup of Aut ξ.

Denote by T (π_{∗}) the set of all admissible involutions τ̂: Γ̂_{∗} → Γ̂_{∗}. Clearly, the composition τ̂∘τ̂′ of two admissible involutions is an au-

tomorphism of covering ξ and vice versa, the composition of an admissible involution and an automorphism of ξ is an admissible involution. Hence, the number of admissible involutions |T (π_{∗})| coincides with the order of group Aut ξ.

Group Aut ξ acts on T (π_{∗}) as follows: g(τ̂)(x) = g ∘τ̂∘g^{−1}(x), where g ∈ Aut ξ, τ̂ ∈ T (π_{∗}), x ∈ Γ̂_{∗}.

Denote by CutCov(Ω_{∗})_{[π_{∗}]} the preimage of [π_{∗}] under the map CutCov(Ω_{∗}) → Cov(Ω_{∗}) (coming from functor (2) above).

Lemma 5.6. There exists a bijection between the set CutCov(Ω)_{[π_{∗}]} and the set of orbits T (π_{∗})/ Aut(π_{∗}).

Proof. Choose a representative π_{∗} ∈ [π_{∗}]. Any class from CutCov(Ω_{∗})_{[π_{∗}]} contains a representative (π_{∗}, τ̂) that includes just the covering π_{∗}. Clearly, cut coverings (π_{∗}, τ̂) and (π_{∗}, τ̂′) with the same π_{∗} are isomorphic, if and only if τ̂ and τ̂′ belong to the same orbit of Aut π_{∗}. □

Evidently, isomorphic coverings have isomorphic automorphism groups. Therefore, we can assign the isomorphism class of groups Aut[π] to an isomorphism class of coverings [π]. Analogously, to [π_{∗}] we assign a class of isomorpic sets of admissible involutions T ([π]).

By lemma 5.5 there is a bijection between sets CutCov(Ω_{∗}) and Cov(Ω). Denote by Cov(Ω)_{[π_{∗}]} the image of CutCov(Ω_{∗})_{[π_{∗}]} in Cov(Ω).

Theorem 5.1. Let Γ be a cut system of a stratified surface Ω. Denote by (Ω_{∗}, Γ_{∗}, τ) a cut surface obtained by cutting Ω along Γ. Let [π_{∗}] be an arbitrary class of isomorphic coverings π_{∗} : P_{∗} → Ω_{∗} satisfying (∗). Then

∑_{[π]∈Cov(Ω)_{[π∗]}} \frac{1}{| Aut[π]|} = |T ([π_{∗}])| · \frac{1}{| Aut[π_{∗}]|}.

Proof. Clearly, the stabilizer of τ̂ ∈ T (π_{∗}) in group Aut π_{∗} coincides with the group of all automorphisms of cut covering (π_{∗}, τ̂). By lemma 5.5, there is the bijection between cut coverings over cut surface Ω_{∗} and coverings over Ω. Let π ∈ Cov(Ω) corresponds to cut covering (π_{∗}, τ̂). Therefore, the stabilizer of τ̂ in Aut π_{∗} is isomorphic to Aut π. Denote by Δ([π]) the orbit in T ([π_{∗}]) corresponding to [π] (see lemma 5.6). Hence |Δ([π])| = \frac{| Aut[π_{∗}]|}{| Aut[π]|}.

In order to complete the proof substitute the latter equality into the equality |T (π_{∗})| = ∑_{[π]∈Cov(Ω)_{[π∗]}} |Δ(π)| which reflects the decomposition of T (π_{∗}) into orbits of Aut π_{∗}. The claim of theorem coincides with obtained equality up to algebraic transformations. □

Remark. Theorem 5.1 can be easily generalized to stratified coverings in any dimension.

Let Γ be a cut system of a stratified surface Ω. Denote by (Ω_{∗}, Γ_{∗}, τ) a cut surface obtained by cutting Ω along Γ and denote by Ω_{#} the contracted cut surface.

Let π : P → Ω be a stratified covering. Denote by (π_{∗}, τ̂) the cut covering corresponding to π. By definition, π_{∗} ∈ Cov(Ω_{∗}, Γ_{∗}). Clearly, covering π_{∗} : P_{∗} → Ω_{∗} induces the covering π_{#} : P_{#} → Ω_{#} where P_{#} is the surface obtained by contracting to a point each connected component of Γ_{∗}.

Lemma 5.7. The correspondence π_{∗} → π_{#} induces the bijection between sets Cov(Ω_{∗}, Γ_{∗}) and Cov(Ω_{#}) of classes of equivalent coverings.

Proof is skipped.

Let Γ be a cut system of a stratified surface Ω. The composition of maps Cov(Ω) → CutCov(Ω_{∗}) → Cov(Ω_{∗}, Γ_{∗}) → Cov(Ω_{#}) defines the map Cov(Ω) → Cov(Ω_{#}). By theorem 5.1 and lemma 5.7, there is a relation between number \frac{1}{| Aut[π_{#}]|} where [π_{#}] ∈ Cov(Ω_{#}) and numbers \frac{1}{| Aut[π]|} for all preimages [π] ∈ Cov(Ω) of [π_{#}]. We shall rewrite them below in slightly different form for simple cuts of all classes.

Let γ be a simple cut of Ω. Thus, in our notations Γ = {γ}. We will write γ, γ_{∗}, etc. instead of Γ, Γ_{∗} etc.

Clearly, there are three possibilities for γ_{∗}: (1) γ_{∗} consists of two ovals; (2) γ_{∗} consists of two segments; (3) γ_{∗} consists of one oval.

Denote by r′, r′′ two special points of Ω_{#} coming from components of γ_{∗} in cases (1) and (2) and by r the analogous point in case (3).

We have to fix local orientations at points r′, r′′ or at r. It is required that these local orientations can be completed to an admissible set of local orientations at all special points of Ω_{#}. In any case there are local orientations at r′, r′′ or r (case (3)) induced by an orientation of simple cut γ. Note, that these local orientations may not satisfy the requirement. We correct them by the following rule. If γ is a simple cut of class 1, 2, 5, 6, 7 than change the local orientation at point r′′ by opposite one. Fix obtained local orientations and denote them by ≀′, ≀′′ or ≀ (case (3)). It can be checked that these orientations satisfy the requirement.

Let π be a covering over Ω and γ′_{∗} be a connected component of γ_{∗}. There is an orientation of γ′_{∗} that is compatible with fixed local orientation at point r′ ∈ Ω_{#} coming from the contraction of γ′_{∗}. Denote by ξ′ the restriction of covering π_{∗} to the preimage π_{∗}^{−1}(γ′_{∗}). Thus, we have that ξ′ is defined, up to isomorphism, either by a Young diagram or by a dihedral Young diagram.

Therefore, we obtain a local invariant of covering π_{#} : P_{#} → Ω_{#} at this point as either a Young diagram or a dihedral Young diagram.

Let π_{#} be an arbitrary covering over Ω_{#}. Denote by α′, α′′ the Young diagram describing local invariants of π_{#} at points r′, r′′ resp. in case (1). Analogously, denote by β′, β′′ dihedral Young diagrams in case (2) and by α Young diagram in case (3).

Denote by Cov(Ω)_{[π_{#}]} the preimage of [π_{#}] ∈ Ω_{#} in Cov(Ω).

Put α^{∗} = α for a Young diagramm α. We use α^{∗} below in order to obtain forumas that will be correct also for coverings with structure group G different from S_{n}.

Corollary 5.1. Let γ be a simple cut of a stratified surface Ω and π_{#} be a covering over contracted cut surface Ω_{#}. Then the following identities hold:

(1) If γ belongs to one of classes 1, 2 then

∑_{[π]∈Cov(Ω)_{[π_{#}]}} \frac{1}{| Aut[π]|} = δ_{α′,α′′^{∗}}| Aut α′| · \frac{1}{| Aut[π_{#}]|}

(2) If γ belongs to class 3 then

∑_{[π]∈Cov(Ω)_{[π_{#}]}} \frac{1}{| Aut[π]|} = δ_{α′,α′′}| Aut α′| · \frac{1}{| Aut[π_{#}]|}

(3) If γ belongs to one of classes 5,6,7 then

∑_{[π]∈Cov(Ω)_{[π_{#}]}} \frac{1}{| Aut[π]|} = δ_{β′,β′′^{∗}}| Aut β′| · \frac{1}{| Aut[π_{#}]|}

(4) If γ belongs to one of classes 8,9 then

∑_{[π]∈Cov(Ω)_{[π_{#}]}} \frac{1}{| Aut[π]|} = δ_{β′,β′′}| Aut β′| · \frac{1}{| Aut[π_{#}]|}

(5) If γ belongs to class 4 then

∑_{[π]∈Cov(Ω)_{[π_{#}]}} \frac{1}{| Aut[π]|} = d_{α} · \frac{1}{| Aut[π_{#}]|}

where d_{α} is a number of involutions in symmetric group S_{n} having no fixed points and commuting with an element g ∈ S_{n} that belongs to conjugated class corresponding to Young diagram α.

Proof follows from theorem 5.1 and lemma 5.7.

5.3. Hurwitz topological field theory. Denote by A = A_{n} the set of Young diagrams of order n and denote by A a vector space of formal linear combinations of Young diagrams. Denote by B = B_{n} the set of dihedral Young diagrams of order n and denote by B a vector space of formal linear combinations of dihedral Young diagrams. Define an involutive linear transformation ^{∗} : A → A as identical map. Define an involutive linear transformation ^{∗} : B → B as linear continuation of the map β ↦ β^{∗} for dihedral Young diagram β.

Let (Ω, O) be a pair consisting of a stratified surface and a set of local orientations O at special points of Ω. Denote by Ω_{a} the set of interior special points of Ω and by Ω_{b} the set of boundary special points.

Let α : Ω_{a} → A and β : Ω_{b} → B be two maps. Denote by α_{p} (resp., β_{q}) the image of an interior special point p (resp., boundary special point q) in set A (resp.,B). Denote by Cov(Ω, {α_{p}}, {β_{q}}) the set of isomorphism classes of coverings having a local invariant α_{p} at each point p ∈ Ω_{a} and a local invariant β_{q} at each point q ∈ Ω_{b}.

We shall define linear function H_{(Ω,O)} : V_{(Ω,O)} → ℂ, where V_{(Ω,O)} = (⊗_{p∈Ω_{a}}A_{p}) ⊗ (⊗_{q∈Ω_{b}}B_{q}).

Assume first that O is an admissible set of local orientations. Then for element (⊗_{p∈Ω_{a}}α_{p}) ⊗ (⊗_{q∈Ω_{b}}β_{q}) ∈ V_{(Ω,O)} put

H_{(Ω,O)}(⊗_{p∈Ω_{a}}α_{p}) ⊗ (⊗_{q∈Ω_{b}}β_{q})) = ∑_{[π]∈Cov(Ω,\{α_{p}\},\{β_{q}\})} \frac{1}{| Aut[π]|}.

Elements (⊗_{p∈Ω_{a}}α_{p}) ⊗ (⊗_{q∈Ω_{b}}β_{q}) form a basis of V_{(Ω,O)}, hence we can expand H_{(Ω,O)} by linearity. For nonadmissible O define linear form H_{(Ω,O)} in such a way that axiom 2^{∘} of Klein field theory is satisfied. This rule correctly defines H_{(Ω,O)} in all cases because ∑_{[π]∈Cov(Ω,\{α_{p}\},\{β_{q}\})} \frac{1}{| Aut π|} does not depend on the choice of an admissible set of local orientations.

Theorem 5.2. The set of data H = {A,^{∗}: A → A, B,^{∗}: B → B, H_{(Ω,O)}} is a Klein topological field theory. We call it Hurwitz topological field theory of degree n.

Proof. Evidently, axioms 1^{∘}, 2^{∘}, 3^{∘} and 4^{∘} are satisfied. Bilinear forms can be easely computed. We obtain equalities: (α′, α′′)_{A} = δ_{α′,α′′^{∗}} \frac{1}{| Aut α′|} and (β′, β′′)_{B} = δ_{β′,β′′^{∗}} \frac{1}{| Aut β′|}. Element U ∈ A is equal to ∑_{α} d_{α}(α, α)_{A} where numbers d_{α} are defined in corollary 5.1. Cleary, there are the following identities for tensors corresponding to bilinear forms and U: F^{α′,α′′} = δ_{α′,α′′^{∗}}| Aut α′|, F^{β′,β′′} = δ_{β′,β′′^{∗}}| Aut β′|, D^{α} = d_{α}.

Axiom 5^{∘} essentially follows from theorem 5.1 and corollary 5.1.

For example, let us verify the axiom for γ of class 5.

Let Ω be a stratified surface, O be a set of local orientations at special points and and γ be a simple cut of Ω of class 5, i.e., a cut between two holes. Denote by Ω_{#} the contracted cut surface.

The set (Ω_{#})_{0} of all special points of Ω_{#} consists of the image of set Ω_{0} and two additional boundary points q′, q′′ coming from connected components of γ_{∗}. Choose a set of local orientations O such that the following conditions are satisfied. First, O is an admissible set of local orientations. Second, induced set of local orientations O_{#} at special points of Ω_{#} coincides with an admissible set of local orientations O_{#adm} in all points except q′′. Clearly, for any Ω and γ of class 5 it is possible to choose O satisfying these conditions. (Note, that there is no admissible set of local orientations O such that O_{#} is also admissible set of local orientations.)

Fix an interior primary field α_{p} at each interior special point p of Ω and a dihedral primary field β_{q} at each boundary special point q of Ω. Denote by x = (⊗_{p∈Ω_{a}}α_{p}) ⊗ (⊗_{q∈Ω_{b}}β_{q})) the element of vector space V_{(Ω,O)}. Then, by definition, element η_{∗}(x) ∈ V_{(Ω,O)} is equal to y = ∑_{β′,β′′} F^{β′β′′}(⊗_{p∈Ω_{a}}α_{p}) ⊗ (⊗_{q∈Ω_{b}}β_{q}) ⊗ β′ ⊗ β′′^{∗}.

Sustitute the values F^{β′β′′}: y = ∑_{β} | Aut β|(⊗_{p∈Ω_{a}}α_{p})⊗(⊗_{q∈Ω_{b}}β_{q}))⊗ β ⊗ β.

Introduce notation:

R = H_{(Ω_{#},O_{#})}(η_{∗}(x))

Substitute y = η_{∗}(x):

R = ∑_{β} | Aut β|H_{(Ω_{#},O_{#})}((⊗_{p∈Ω_{a}}α_{p}) ⊗ (⊗_{q∈Ω_{b}}β_{q}) ⊗ β ⊗ β)

By (already established) invariance of change of local orientations, we obtain:

R = ∑_{β} | Aut β|H_{(Ω_{#},O_{#adm})}((⊗_{p∈Ω_{a}}α_{p}) ⊗ (⊗_{q∈Ω_{b}}β_{q}) ⊗ β ⊗ β^{∗})

By definition of H_{(Ω_{#},O_{#adm})},

R = ∑_{β} | Aut β| ∑_{[π_{#}]∈Cov(Ω_{#},\{α_{p}\},\{\{β_{q}\},β,β^{∗}\})} \frac{1}{| Aut π_{#}|}

By theorem 5.1 and corollary 5.1 we have

R = ∑_{β} | Aut β| ∑_{[π_{#}]∈Cov(Ω_{#}),\{α_{p}\},\{\{β_{q}\},β,β^{∗}\}) \frac{1}{| Aut β|}} ∑_{[π]∈Cov(Ω)_{[π∗]}} \frac{1}{| Aut π|} =

= ∑_{β} ∑_{[π_{#}]∈Cov(Ω_{#}),\{α_{p}\},\{\{β_{q}\},β,β^{∗}\})} ∑_{[π]∈Cov(Ω)_{[π∗]}} \frac{1}{| Aut π|} =

= ∑_{[π]∈Cov(Ω)} \frac{1}{| Aut π|} = H_{(Ω,O)}(x).

For other class of simple cuts axiom 5^{∘} can be verified quit similar. Axiom 6^{∘} is satisfied because the sum ∑_{[π]∈Cov(Ω,\{α_{p}\},\{β_{q}\})} \frac{1}{| Aut[π]|} is

multiplicative with respect to disjoint union of surfaces.

□

Structure algebra of Hurwitz topological field theory is a vector space H = A ⊕ B with multiplication constructed by tensors from C(H).

Let H̄ = Ā ⊕ B̄ be a structure algebra of symmetric group. Note that Young diagram α corresponds to a class of conjugated elements of S_{n}. This class we denote by the same sign α. Define a linear map ϕ : A → Ā as ϕ(α) = E_{α}, where E_{α} = ∑_{g∈α} g is an element of the center of group algebra ℂ[S_{n}]. Analogously, dihedral Young diagram β corresponds to a conjugacy class of ordered pairs of involution in S_{n}. Put ϕ(β) = E_{β}, where E_{β} = ∑_{(s_{1},s_{2})∈β} E_{s_{1},s_{2}} and E_{s_{1},s_{2}} ∈ M(S_{n}) is an elementary matrix.

Theorem 5.3. The linear map ϕ is an isomorphism between the structure algebra of Hurwitz topological field theory of degree n and the structure algebra of symmetric group S_{n}.

Proof. Obviously, ϕ defines an isomorphism of linear spaces H and H̄. Moreover, ϕ preserves scalar products on A and B and commutes with ’^{∗}’ involutions.

In order to prove that ϕ is an isomorphism of structure algebras it is sufficient to check the following equalities:

(1) S_{α_{1},α_{2},α_{3}} = S_{E_{α_{1}},E_{α_{2}},E_{α_{3}}},

(2) T_{β_{1},β_{2},β_{3}} = T_{E_{β_{1}},E_{β_{2}},E_{β_{3}}},

(3) R_{α,β} = R_{E_{α},E_{β}}.

Proof of (1). After Hurwitz, it is known that computation of weighted numbers of isomorphism classes of coverings over two-dimensional sphere is equivalent to the solution of factorization problem in S_{n} (see [14]).

Therefore, S_{α_{1},α_{2},α_{3}} = |({(a_{1}, a_{2}, a_{3})|a_{i} ∈ α_{i}, a_{1}a_{2}a_{3} = 1}/S_{n})|, where g ∈ S_{n} acts as (a_{1}, a_{2}, a_{3}) ↦ (ga_{1}g^{−1}, ga_{2}g^{−1}, ga_{3}g^{−1}). By direct calculations, we obtain that E_{α_{1}}E_{α_{2}} = S_{α_{1},α_{2}}^{α_{3}}E_{α_{3}}, where S_{α_{1},α_{2}}^{α_{3}} = S_{α_{1},α_{2},α_{3}}| Aut α_{3}|. Hence, S_{α_{1},α_{2},α_{3}} = S_{E_{α_{1}},E_{α_{2}},E_{α_{3}}}.

Proof of (2). By definition T_{β_{1},β_{2},β_{3}} is a number of equivalent classes of coverings over triangle Δ with local invariants at vertices q_{i} equal to β_{i} (i = 1, 2, 3).

Let π : P → Δ be a stratified covering. Evidently, preimage of Δ^{∘} consists of n connected components and each one if homeomorphic to Δ^{∘}. An edge e_{i,j} of Δ defines a involutive permutation s_{i,j} of these preimages. Therefore, the covering is uniquely defined by triple of involutions (s_{0,1}, s_{1,2}, s_{2,0}) of symmetric group S_{n} up to equivalence (s_{0,1}, s_{1,2}, s_{2,0}) ∼ (gs_{0,1}g^{−1}, gs_{1,2}g^{−1}, gs_{2,0}g^{−1}). Conversely an equivalent class [(s_{0,1}, s_{1,2}, s_{2,0})] of triple of involutions defines an equivalent class of coverings over Δ. Local invariant at vertex q_{i} is a dihedral Young diagram β_{i} corresponding to a conjugacy class [(s_{i−1,i}, s_{i,i+1})] (where i − 1 and i + 1 are taken modulo 3) of the pair of involutions.

Hence, T_{β_{1},β_{2},β_{3}} = |({(s_{0,1}, s_{1,2}, s_{2,0})|s_{i,i+1} ∈ S_{n}, s_{i,i+1}^{2} = 1, [(s_{i−1,i}, s_{i,i+1})] = β_{i}}/ ∼)|. By direct calculations we obtain that E_{β_{1}}E_{β_{2}} = T_{β_{1},β_{2},β_{3}}| Aut β_{3}|E_{β_{3}^{∗}}. Therefore, T_{β_{1},β_{2},β_{3}} = T_{E_{β_{1}},E_{β_{2}},E_{β_{3}}}

Proof of (3). By definition R_{α,β} is the number of equivalent classes of coverings over disc (D, p, q) with local invariant α at interior special point p and local invariant β at boundary special point q.

Let π : P → D be a stratified covering. Let us connect points p and q by a segment δ and denote by D^{∘} the set of all interior point except δ. Evidently, the preimage of D^{∘} consists of n connected components and each one is homeomorphic to D^{∘}.

Segment δ defines a permutation a of these preimages and the boundary circle defines an involutive permutation s of these preimages. (If X is a connected component of the preimage of D^{∘} then Y = s(X) is another component such that \overline{X} ∩\overline{Y} is equal to a connected component of the preimage of a boundary without point q; if there no such component then s(X) = X). Therefore, the covering is uniquely defined by a pair (a, s) up to equivalence (a, s) ∼ (gag^{−1}, gsg^{−1}).

Conversely an equivalent class [(a, s)] defines an equivalent class of coverings over D. Local invariant at vertex p is Young diagram α = [a] and local invariant at vertex q is dihedral Young diagram β corresponding to a pair of involutions (s, asa^{−1}). Hence, R_{α,β} = |({(a, s)|a, s ∈ S_{n}, a ∈ α, (s, asa^{−1}) ∈ β}/ ∼)|. By direct calculations we obtain that (E_{α}, E_{β}) = R_{α,β} (see subsection 2.3)

□

5.4. Hurwitz numbers. Classical Hurwitz numbers are weighted numbers of algebraic maps f : P → Q to Riemann sphere Q = S^{2} having prescribed branchings at finite number of fixed points z_{1}, …, z_{m} ∈ Q. Young diagramms α_{1}, …, α_{m} are fixed. It is required that z_{1}, …, z_{m} are critical values of f and the branching of f at z_{i} is described by Young diagramm α_{i}.

We use term ’generalized Hurwitz numbers’ or ’Hurwitz numbers’ for short for morphisms of both complex and real algebraic curves, an arbitrary base Q and an arbitrary local types of branchings. Definitions are given below.

A real algebraic curve is a pair (P, τ), where P is a complex algebraic curve, i.e., a compact Riemann surface, and τ : P → P is an antiholomorphic involution [2, 30]. Involution τ is called complex conjugation. Fixed points of τ are called real points of the real curve (P, τ). An algebraic map f : (P, τ) → (Q, ω) of real algebraic curves is defined as a holomorphic map f : P → Q such that fτ = ωf.

Algebraic maps f : (P, τ) → (Q, ω) of a fixed degree n to a fixed real algebraic curve form a category Map(Q, ω); a morphism ϕ : f → f′, where f : (P, τ) → (Q, ω) and f′ : (P′, τ′) → (Q, ω) are two algebraic maps to (Q, ω), is defined as a holomorphic map ϕ : P′ → P such that ϕτ′ = τϕ and f′ = fϕ.

Any complex algebraic curve P_{c} generates real algebraic curve (P, τ). Namely, P = P_{c} ∐ P̄_{c} where P̄_{c} is P_{c} endowed with conjugated complex structure and τ is just permutation of components P_{c} and P̄_{c}. Thus, complex algebraic curves can be considered as a particular case of real algebraic curves.

Denote by Ω the factor-space Q/τ. It is known that Ω carries the structure of Klein surface [2, 27]. This implies that Ω is nooriented, possibly nonorientable, possibly with boundary, surface endowed with an atlas such that any transaction function is dianalitic, i.e., either holomorphic or antiholomorphic function . Morphisms of Klein surfaces are defined as dianalitic functions. According to [2] the category of real algebraic curves is isomorphic to the category of Klein surfaces.

Fix a real curve (Q, ω) and put Ω = Q/τ. Any morphism f : (P, τ) → (Q, ω) of real algebraic curves generates the dianalytic map π(f) : P/τ → Ω. Denote by p_{1}, …, p_{m} ∈ Ω critical values of π(f) that are interior points, and denote by q_{1}^{j}, …, q_{m_{j}}^{j} critical values of π(f) that belong to j-th boundary contour of Ω. Clearly, points p_{i}, q_{i}^{j} come from critical values z_{0}, …, z_{t} ∈ Q of f: an interior point p_{i} is the image of two points z_{k}, z_{l}, a boundary point q_{i}^{j} is the image of one point z_{r} and z_{r} is a fixed point of τ. Points p_{i}, q_{i}^{j} induce the stratification of Ω. Forgetting dianalitic structures on P/τ and Ω we obtain that π(f) is a stratified covering, i.e., an element of Cov(Ω).

Fix the stratification of Ω with special points p_{i}, q_{i}^{j}. According to [28] forgetting dianalitic structures induces the bijection between the set of isomorphism classes of dianalitic maps to Ω, such that all critical values are special points of the stratification, and the set Cov(Ω) of isomorphism classes of stratified covering over Ω. Thus, up to isomorphisms, algebraic maps to (Q, τ) can be identified with stratified coverings over Ω.

The latter fact allows us to define Hurwitz numbers for real algebraic curves in topological terms. Namely, define Hurwitz numbers for Ω as

Hurw_{n}(Ω, \{α_{p}\}, \{β_{q}\}) = ∑_{[π]∈Cov(Ω,\{α_{p}\},\{β_{q}\})} \frac{1}{| Aut π|}

Here α_{p} (resp., β_{q}) is a Young diagram (resp., a dihedral Young diagram) of degree n assigned to an iterior special point p (resp., to a boundary special point q) of Ω.

By previous considerations, these Hurwitz numbers are weighted numbers of classes of alrebraic maps with prescribed branchings to real algebraic curve (Q, τ).

For G = (0, 1) our definition gives weighted numbers of meromorphic functions with fixed critical values of fixed topological types. They includes numbers that were first introduced by Hurwitz [18].

By the results of previous subsection, Hurwitz numbers are equal to correlators for Hurwitz topological field theory. Thus, from theorem 5.3 and theorem 4.4 we get the following theorem.

Theorem 5.4. Let Ω be a stratified surface of type G. Fix an admissible set of local orientations at special points of Ω.Then (1)

Hurw_{n}(Ω, {α_{p}}, {β_{q}}) = ⟨α_{1}, …, α_{m}, (β_{1}^{1}, …, β_{m_{1}}^{1}), …, (β_{1}^{s}, …, β_{m_{s}}^{s})⟩_{G}

(2) If G = (g, 1, m, m_{1}, …, m_{s}) then

Hurw_{n}(Ω, {α_{p}}, {β_{q}}) =

= (α_{1} … α_{m}(β_{1}^{1} … β_{m_{1}}^{1})V_{K_{B}}(β_{1}^{2} … β_{m_{1}}^{2}) … V_{K_{B}}(β_{1}^{s} … β_{m_{s}}^{s}), K_{A}^{g})

(3) If G = (g, 0, m, m_{1}, …, m_{s}) then

Hurw_{n}(Ω, {α_{p}}, {β_{q}}) =

= (α_{1} … α_{m}(β_{1}^{1} … β_{m_{1}}^{1})V_{K_{B}}(β_{1}^{2} … β_{m_{1}}^{2}) … V_{K_{B}}(β_{1}^{s} … β_{m_{s}}^{s}), U^{2g}) In (2) and (3) right hand sides are expressions in the structure alge-

bra associated with symmetric group S_{n} (see subsection 2.3).

Remark.

Let G be a finite group and Ω be a stratified surface. Denote by Cov_{G}(Ω) the caregory of stratified coverings with structure group G (G-coverings). A definition of a G-covering in the case of stratified surfaces possibly with boundary is a generalization of a standard one and should be explained.

Fix a generic point x ∈ Ω. Let δ ⊂ Ω be a path with ends at x. Path δ is called generic if it does not cross any special point and cross the boundary of Ω in finitely many points.

Let π : P → Ω be a stratified covering and x̂ ∈ P be one of preimages of x. Lift δ to δ̂ ⊂ P using the following rule of lifting through a point ŷ such that π(y) ∈ δ ∩ ∂Ω: if the covering π_{U} : Û → U of a neighborhood Û of point ŷ over U = π(Û) is two-sheeted covering then δ̂ must go from one sheet of Û to another sheet through point ŷ.

This rule allows to define a fundamental group, a monodromy group etc. for stratified surfaces and stratified covering. All theory of fundamental group is consistent for this generalization. Note that these definitions are valid for all dimensions in the case of specially stratified manifolds and specially stratified coverings (see definition 5.1). (Recall that in this work we deal with specially stratified surfaces and specially stratified coverings only).

Fix an action of group G on the set of preimages of x. A covering π is said to be G-covering if (generalized) monodromy group at x is contained in G. All definitions and considerations of this section, particularly, the definition of Hurwitz numbers, can be applied to G-coverings without changes. Thus, we can construct ’G-Hurwitz topological field theory’. Structure algebra of it is isomorphic to the structure algebra of group G. G-Hurwitz numbers have just the same representation as in theorem 5.4; these formulas coincide with correlators for G-Hurwitz topological field theory.

G-Hurwits numbers for oriented surfaces without a boundary were considered in [9].

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A.Alexeevski Moscow State University Moscow 119899 aba@belozersky.msu.ru

S.Natanzon Moscow State University Moscow 119899 Independent universyty of Moscow Institute Theoretical and Experimental Physics natanzon@mccme.ru