Edge contraction on dual ribbon graphs and 2D TQFT
EEDGE CONTRACTION ON DUAL RIBBON GRAPHS AND 2D TQFT
OLIVIA DUMITRESCU AND MOTOHICO MULASE
Abstract.
We present a new set of axioms for 2D TQFT formulated on the category of cell graphswith edge-contraction operations as morphisms. We construct a functor from this category to theendofunctor category consisting of Frobenius algebras. Edge-contraction operations correspondto natural transformations of endofunctors, which are compatible with the Frobenius algebrastructure. Given a Frobenius algebra A , every cell graph determines an element of the symmetrictensor algebra defined over the dual space A ∗ . We show that the edge-contraction axioms makethis assignment depending only on the topological type of the cell graph, but not on the graphitself. Thus the functor generates the TQFT corresponding to A . Contents
1. Introduction 12. Frobenius algebras 53. 2D TQFT 74. The edge-contraction axioms 115. Category of cell graphs and Frobenius ECO functors 17References 211.
Introduction
The purpose of the present paper is to give a new set of axioms for two-dimensional topo-logical quantum field theory (2D TQFT) formulated in terms of dual ribbon graphs. Thekey relations between ribbon graphs are edge-contraction operations , which correspondto the degenerations in the moduli space M g,n of stable curves of genus g with n labeledpoints that create a rational component with 3 special points. The structure of Frobeniusalgebra is naturally encoded in the category of dual ribbon graphs, where edge-contractionoperations form morphisms and represent multiplication and comultiplication operations.As Grothendieck impressively presents in [14], it is a beautiful and simple yet very surpris-ing idea that a graph drawn on a compact topological surface gives an algebraic structureto the surface. When a positive real number is assigned to each edge as its length, a uniquecomplex structure of the surface is determined. This association leads to a combinatorialmodel for the moduli space M g,n of smooth algebraic curves of genus g with n markedpoints [15, 21, 24, 26, 27]. By identifying these graphs as Feynman diagrams of [29] ap-pearing in the asymptotic expansion of a particular matrix integral, and by giving a graphdescription of tautological cotangent classes on M g,n , Kontsevich [17] shows that Witten’sgenerating function [30] of intersection numbers of these classes satisfies the KdV equations.Kontsevich’s argument is based on his discovery that an weighted sum of these intersectionnumbers is proportional to the Euclidean volume of the combinatorial model of M g,n . Mathematics Subject Classification.
Primary: 14N35, 81T45, 14N10; Secondary: 53D37, 05A15.
Key words and phrases.
Topological quantum field theory; Frobenius algebras; ribbon graphs; cell graphs. a r X i v : . [ m a t h . AG ] S e p O. DUMITRESCU AND M. MULASE
The Euclidean volume of M g,n depends on the choice of the perimeter length of each faceof the graph drawn on a surface. Kontsevich used the Laplace transform of the volume as afunction of the perimeter length to obtain a set of relations among intersection numbers ofdifferent values of ( g, n ). These relations are equivalent to the conjectured KdV equations.Recall that if each edge has an integer length, then the resulting Riemann surface by theStrebel correspondence [27] is an algebraic curve defined over Q [3, 21]. Thus a systematiccounting of curves defined over Q gives an approximation of the Euclidean volume of Kontse-vich by lattice point counting. Since these lattice points naturally correspond to the graphsthemselves, the intersection numbers in question can be obtained by graph enumeration, af-ter taking the limit as the mesh length approaches to 0. Now we note that edge-contractionoperations give an effective tool for graph enumeration problems. Then one can ask: whatinformation do the edge-contraction operations tell us about the intersection numbers? We found in [9, 12, 22] that the Laplace transform of the counting formula obtained bythe edge-contraction operations on graphs is exactly the Virasoro constraint conditions of[6] for the intersection numbers. Indeed it gives the most fundamental example of topologicalrecursion of [13].Euclidean volume is naturally approximated by lattice point counting. It can be alsoapproximated as a limit of hyperbolic volume. The latter idea applied to moduli spaces ofhyperbolic surfaces gives the same Virasoro constraint conditions, as beautifully describedin the work of Mirzakhani [19, 20]. Mirzakhani’s technique of symplectic and hyperbolicgeometry can be naturally extended to character varieties of surface groups. Yet there areno Virasoro constraints for this type of moduli spaces. We ask: what do edge-contractionoperations give us for the character varieties?
This is our motivation of the current paper. Instead of discussing the application of ourresult to character varieties, which will be carried out elsewhere, we focus in this paper ourdiscovery of the relation between edge-contraction operations and 2D TQFT.A TQFT of dimension d is a symmetric monoidal functor Z from the monoidal cat-egory of ( d − d -dimensionaloriented cobordism forming morphisms among ( d − K [2, 25]. Since there is only one compact manifold in dimension 1, a 2D TQFT is associatedwith a unique vector space A = Z ( S ), and the Atiyah-Segal axioms of TQFT makes A acommutative Frobenius algebra. It has been established that 2D TQFTs are classified byfinite-dimensional Frobenius algebras [1, 5]. We ask the following question, in the reversedirection: Question 1.1.
Suppose we are given a finite-dimensional commutative Frobenius algebra.What is the combinatorial realization of the algebra structure that leads to the corresponding2D TQFT?
The answer we propose in this paper is the category of dual ribbon graphs , with edge-contraction operations as morphisms. This category does not carry the information ofa specific Frobenius algebra. In our forthcoming paper, we will show that our categorygenerates all Frobenius objects among any given monoidal category.For a given Frobenius algebra A and a ribbon graph γ g,n with n vertices drawn on atopological surface of genus g , we assign a multilinear map γ g,n : A ⊗ n −→ K. The edge-contraction axioms of Section 4 determine the behavior of this map under thechange of ribbon graphs via edge contractions. Theorem 4.7, our main result of this paper,
DGE CONTRACTION AND 2D TQFT 3 exhibits a surprising statement that the map γ g,n depends only on g and n , and is indepen-dent of the choice of the graph γ g,n . We then evaluate γ g,n for each v ⊗ · · · ⊗ v n ∈ A ⊗ n andprove that this map indeed defines the TQFT corresponding to A .A ribbon graph (also called as a dessin d’enfant, fatgraph, embedded graph, or a map)is a graph with an assignment of a cyclic order of half-edges incident at each vertex. Thecyclic order induces the ribbon structure to the graph, and it becomes the 1-skeleton of thecell-decomposition of a compact oriented topological surface of genus, say g , by attachingoriented open discs to the graph. Let n be the number of the discs attached. We call thisribbon graph of type ( g, n ).
123 4
Figure 1.1.
Top Row: A cyclic order of half-edges at a vertex induces a localribbon structure to a graph. Second Row: Globally, a ribbon graph is the 1-skeletonof a cell-decomposition of a compact oriented surface. Third Row: A ribbon graphis thus a graph drawn on a compact oriented surface.
An assignment of a positive real number to each edge of a ribbon graph determinesa concrete holomorphic coordinate system of the topological surface of genus g with n labeled marked points [21], thus making it a Riemann surface. This construction gives theidentification of the space of ribbon graphs of type ( g, n ) with positive edge lengths assigned,and the space M g,n × R n + , as an orbifold. The operation of edge-contraction of an edgeconnecting two distinct vertices then defines the boundary operator, which introduces thestructure of orbi-cell complex on M g,n × R n + . Each ribbon graph determines the stratumof this cell complex, whose dimension is the number of edges of the graph.Since the ribbon graphs we need for the consideration of TQFT have labeled vertices butno labels for faces, we use the terminology cell graph of type ( g, n ) for a ribbon graph ofgenus g with n labeled vertices. A cell graph of type ( g, n ) is the dual of a ribbon graph ofthe same type ( g, n ). The set of all cell graphs of type ( g, n ) is denoted by Γ g,n .Ribbon graphs naturally form orbi-cell complex. Their dual cell graphs naturally form acategory CG , as we shall define in Section 5. We then consider functors ω : CG −→ F un ( C /K, C /K ) , where ( C , ⊗ , K ) is a monoidal category with the unit object K , and F un ( C /K, C /K ) isthe endofunctor category over the category of K -objects of C . Each cell graph correspondsto an endofunctor, and edge-contraction operations among them correspond to naturaltransformations. Our consideration can be generalized to the cohomological field theory ofKontsevich-Manin [18]. After this generalization, we can construct a functor that gives aclassification of 2D TQFT. Since we need more preparation, these topics will be discussedin our forthcoming paper. O. DUMITRESCU AND M. MULASE
Edge-contraction operations also provide an effective method for graph enumeration prob-lems. It has been noted in [12] that the Laplace transform of edge-contraction operationson many counting problems corresponds to the topological recursion of [13]. In a separatepaper [11], we give the construction of the mirror B-models corresponding to the simpleand orbifold Hurwitz numbers, by using only the edge-contraction operations. In general,enumerative geometry problems, such as computation of Gromov-Witten type invariants,are solved by studying a corresponding problem on the mirror dual side. The effectivenessof the mirror problem relies on the technique of complex analysis. The question is: How dowe find the mirror of a given enumerative problem? In [11], we give an answer to this ques-tion for a class of graph enumeration problems that are equivalent to counting of orbifoldHurwitz numbers. The key is again the same edge-contraction operations. The base case,or the case for the “moduli space” M , , of the edge contraction in the counting problemidentifies the mirror dual object, and a universal mechanism of complex analysis, known asthe topological recursion of [13], solves the B-model side of the counting problem. Thesolution is a collection of generating functions of the original problem for all genera.The edge-contraction operation causes the degeneration of P with one marked point p into two P ’s with one marked point on each, connected by a P with 3 special points, twoof which are nodal points and the third one representing the original marked point p . Interms of graph enumeration, the P with 3 special points does not play any role. So webreak the original vertex into two vertices, and separate the graph into two disjoint pieces(Figure 1.2). p pL Figure 1.2.
The edge-contraction operation on a loop is a degeneration process.The graph on the left is a connected cell graph of type (0 , L changes it to the one on the right. Here, a P with one marked point p degenerates into two P ’s with one marked point on each, connected by a P with3 special points. Once we have our formulation of 2D TQFT and topological recursion in terms of edge-contraction operations, we can consider a TQFT-valued topological recursion. An immedi-ate example is the Gromov-Witten theory of the classifying space BG of a finite group G .In our forthcoming paper, we will show that a straightforward generalization of the topo-logical recursion for differential forms with values in tensor products of a Frobenius algebraautomatically splits into the product of the usual scalar-valued solution to the topologicalrecursion and a 2D TQFT. Therefore, topological recursion implies TQFT. Here, we re-mark the similarity between the topological recursion and the comultiplication operationin a Frobenius algebra. Indeed, the topological recursion itself can be regarded as a comul-tiplication formula for an infinite-dimensional analogue of the Frobenius algebra (Vertexalgebras, or conformal field theory).The authors have noticed that the topological recursion appears as the Laplace transformof edge-contraction operations in [12]. The geometric nature of the topological recursionwas further investigated in [7, 8, 10], where it was placed in the context of Hitchin spectral DGE CONTRACTION AND 2D TQFT 5 curves for the first time, and the relation to quantum curves was discovered. The presentpaper is the authors’ first step toward identifying the topological recursion in an algebraicand categorical setting. We note that Hitchin moduli spaces are diffeomorphic to charactervarieties of a surface group. The TQFT point of view of our current paper in the context ofthese character varieties, in particular, their Hodge structures, will be discussed elsewhere.The paper is organized as follows. We start with a quick review of Frobenius algebras,for the purpose of setting notations, in Section 2. We then recall two-dimensional TQFTin Section 3. In Sections 4, we give our formulation of 2D TQFT in terms of the edge-contraction axioms of cell graphs. A categorical formulation of our axioms is given inSection 5. 2.
Frobenius algebras
In this paper, we are concerned with finite-dimensional, unital, commutative Frobeniusalgebras defined over a field K . In this section we review the necessary account of Frobeniusalgebra and set notations.Let A be a finite-dimensional, unital, associative, and commutative algebra over a field K . A non-degenerate bilinear form η : A ⊗ A −→ K is a Frobenius form if(2.1) η (cid:0) v , m ( v , v ) (cid:1) = η (cid:0) m ( v , v ) , v (cid:1) , v , v , v ∈ A, where m : A ⊗ A −→ A is the multiplication. We denote by(2.2) λ : A ∼ −→ A ∗ , (cid:104) λ ( u ) , v (cid:105) = η ( u, v ) , the canonical isomorphism of the algebra A and its dual. We assume that η is a symmetricbilinear form. Let ∈ A denote the multiplicative identity. Then it defines a counit , or a trace , by(2.3) (cid:15) : A −→ K, (cid:15) ( v ) = η ( , v ) . The canonical isomorphism λ introduces a unique cocommutative and coassociative coalge-bra structure in A by the following commutative diagram.(2.4) A δ −−−−→ A ⊗ A λ (cid:121) (cid:121) λ ⊗ λ A ∗ −−−−→ m ∗ A ∗ ⊗ A ∗ It is often convenient to use a basis for calculations. Let (cid:104) e , e , . . . , e r (cid:105) be a K -basis for A . In terms of this basis, the bilinear form η is identified with a symmetric matrix, and itsinverse is written as follows:(2.5) η = [ η ij ] , η ij := η ( e i , e j ) , η − = [ η ij ] . The comultiplication is then written as δ ( v ) = (cid:88) i,j,a,b η (cid:0) v, m ( e i , e j ) (cid:1) η ia η jb e a ⊗ e b . From now on, if there is no confusion, we denote simply by m ( u, v ) = uv . The symmetricFrobenius form and the commutativity of the multiplication makes(2.6) η (cid:0) e i · · · e i j , e i j +1 · · · e n (cid:1) = (cid:15) ( e i · · · e i n ) , ≤ j < n, completely symmetric with respect to permutations of the indices. O. DUMITRESCU AND M. MULASE
The following is a standard formula for a non-degenerate bilinear form:(2.7) v = (cid:88) a,b η ( v, e a ) η ab e b . It immediately follows that
Lemma 2.1.
The following diagram commutes: (2.8) A ⊗ A ⊗ A m ⊗ id (cid:38) (cid:38) A ⊗ A id ⊗ δ (cid:56) (cid:56) m (cid:47) (cid:47) δ ⊗ id (cid:38) (cid:38) A δ (cid:47) (cid:47) A ⊗ A.A ⊗ A ⊗ A id ⊗ m (cid:56) (cid:56) Or equivalently, for every v , v in A, we have δ ( v v ) = ( id ⊗ m ) (cid:0) δ ( v ) , v (cid:1) = ( m ⊗ id ) (cid:0) v , δ ( v ) (cid:1) . Proof.
Noticing the commutativity and cocommutativity of A , we have δ ( v v ) = (cid:88) i,j,a,b η ( v v , e i e j ) η ia η jb e a ⊗ e b = (cid:88) i,j,a,b η ( v e i , v e j ) η ia η jb e a ⊗ e b = (cid:88) i,j,a,b,c,d η ( v e i , e c ) η cd η ( e d , v e j ) η ia η jb e a ⊗ e b = (cid:88) i,j,a,b,c,d η ( v , e i e c ) η cd η ia η ( e d v , e j ) η jb e a ⊗ e b = (cid:88) i,a,c,d η ( v , e i e c ) η cd η ia e a ⊗ ( e d v )= ( id ⊗ m ) (cid:0) δ ( v ) , v ) . (cid:3) In the lemma above we consider the composition δ ◦ m . The other order of operationsplays an essential role in 2D TQFT. Definition 2.2 (Euler element) . The
Euler element of a Frobenius algebra A is definedby(2.9) e := m ◦ δ ( ) . In terms of basis, the Euler element is given by(2.10) e = (cid:88) a,b η ab e a e b . Another application of (2.7) is the following formula that relates the multiplication andcomultiplication.(2.11) ( λ ( v ) ⊗ id ) δ ( v ) = v v . DGE CONTRACTION AND 2D TQFT 7
This is because( λ ( v ) ⊗ id ) δ ( v ) = (cid:88) a,b,k,(cid:96) ( λ ( v ) ⊗ id ) η ( v , e k e (cid:96) ) η ka η (cid:96)b e a ⊗ e b = (cid:88) a,b,k,(cid:96) η ( v e (cid:96) , e k ) η ka η ( v , e a ) η (cid:96)b e b = (cid:88) b,(cid:96) η ( v , v e (cid:96) ) η (cid:96)b e b = v v .
2D TQFT
The axiomatic formulation of conformal and topological quantum field theories was es-tablished in 1980s. We refer to Atiyah [2] and Segal [25]. We consider only two-dimensionaltopological quantum field theories in this paper. Again for the purpose of setting notations,we provide a brief review of the subject in this section. We refer to fundamental literature,such as [16, 28], for more detail of 2D TQFT.A 2D TQFT is a symmetric monoidal functor Z from the cobordism category of orientedsurfaces (a surface being a cobordism of its boundary circles) to the monoidal category offinite-dimensional vector spaces over a fixed field K with the operation of tensor products.The Atiyah-Segal TQFT axioms automatically make the vector space(3.1) Z ( S ) = A a unital commutative Frobenius algebra over K .Let Σ g,n be an oriented surface of finite topological type ( g, n ), i.e., a surface obtainedby removing n disjoint open discs from a compact oriented two-dimensional topologicalmanifold of genus g . The boundary components are labeled by indices 1 , . . . , n . We alwaysgive the induced orientation at each boundary circle. The TQFT then assigns to such asurface a multilinear map(3.2) Ω g,n def = Z (Σ g,n ) : A ⊗ n −→ K. If we change the orientation at the i -th boundary, then the i -th factor of the tensor productis changed to the dual space A ∗ . Therefore, if we have k boundary circles with inducedorientation and (cid:96) circles with opposite orientation, then we have a multi-linear mapΩ g,k, ¯ (cid:96) : A ⊗ k −→ A ⊗ (cid:96) . The sewing axiom of Atiyah [2] requires thatΩ g ,(cid:96), ¯ n ◦ Ω g ,k, ¯ (cid:96) = Ω g + g + (cid:96) − ,k, ¯ n : A ⊗ k −→ A ⊗ n . Figure 3.1.
A 2D TQFT can be also obtained as a special case of a CohFT of [18].
O. DUMITRESCU AND M. MULASE
Definition 3.1 (Cohomological Field Theory) . We denote by M g,n the moduli space ofstable curves of genus g ≥ n ≥ g − n >
0. Let(3.3) π : M g,n +1 −→ M g,n be the forgetful morphism of the last marked point, and gl : M g − ,n +2 −→ M g,n (3.4) gl : M g ,n +1 × M g ,n +1 −→ M g + g ,n + n (3.5)the gluing morphisms that give boundary strata of the moduli space. An assignment(3.6) Ω g,n : A ⊗ n −→ H ∗ ( M g,n , K )is a CohFT if the following axioms hold: CohFT 0: Ω g,n is S n -invariant, i.e., symmetric, and Ω , ( , v , v ) = η ( v , v ) . CohFT 1: Ω g,n +1 ( v , . . . , v n , ) = π ∗ Ω g,n ( v , . . . , v n ) . CohFT 2: gl ∗ Ω g,n ( v , . . . , v n ) = (cid:88) a,b Ω g − ,n +2 ( v , . . . , v n , e a , e b ) η ab . CohFT 3: gl ∗ Ω g + g , | I | + | J | ( v I , v J ) = (cid:88) a,b Ω g , | I | +1 ( v I , e a )Ω g , | J | +1 ( v J , e b ) η ab , where I (cid:116) J = { , . . . , n } .If a CohFT takes values in H ( M g,n , K ) = K , then it is a 2D TQFT. In what follows,we only consider CohFT with values in H ( M g,n , K ). Remark 3.2.
The forgetful morphism makes sense for a stable pointed curve, but it doesnot exist for a topological surface with boundary in the same way. Certainly we cannot just forget a boundary. For a TQFT, eliminating a boundary corresponds to capping a disc. Inalgebraic geometry language, it is the same as gluing a component of g = 0 and n = 1.Since H ( M g,n , K ) = K is not affected by the morphism (3.3)-(3.5), the equationΩ g,n ( , v , . . . , v n ) = Ω g,n − ( v , . . . , v n )is identified with CohFT 3 for g = 0 and J = ∅ , if we define(3.7) Ω , ( v ) := (cid:15) ( v ) = η ( , v ) , even though M , does not exist. We then haveΩ g,n ( v , . . . , v n ) = (cid:88) a,b Ω g,n +1 ( v , . . . , v n , e a ) η ( , e b ) η ab = Ω g,n +1 ( v , . . . , v n , )by (2.7). In other words, the isomorphism of the degree 0 cohomologies(3.8) π ∗ : H ( M g,n , K ) −→ H ( M g,n +1 , K )is replaced by its left inverse(3.9) σ ∗ i : H ( M g,n +1 , K ) −→ H ( M g,n , K ) , where(3.10) σ i : M g,n −→ M g,n +1 is one of the n tautological sections. Of course this consideration does not apply for CohFT. DGE CONTRACTION AND 2D TQFT 9
Remark 3.3.
In the same spirit, although M , does not exist either, we can define (3.11) Ω , ( v , v ) := η ( v , v )so that we exhaust all cases appearing in the Atiyah-Segal axioms for 2D TQFT. In partic-ular, for g = 0 and J = { n } , we haveΩ g,n ( v , . . . , v n ) = Ω g,n v , . . . , v n − , (cid:88) a,b η ( v n , e b ) η ab e a = (cid:88) a,b Ω g,n ( v , . . . , v n − , e a )Ω , ( v n , e b ) η ab . Thus Ω , ( v , v ) functions as the identity operator of the Atiyah-Segal axiom [2]. Remark 3.4.
A marked point p i of a stable curve Σ ∈ M g,n is an insertion point for thecotangent class ψ i = c ( L i ), where L i is the pull-back of the relative canonical sheaf on theuniversal curve π : M g,n +1 −→ M g,n by the i -th tautological section σ i : M g,n −→ M g,n +1 .If we cut a small disc around p i ∈ Σ, then the orientation induced on the boundary circleis consistent with the orientation of the unit circle in T ∗ p i Σ. This orientation is opposite tothe orientation that is naturally induced on T p i Σ. In general, if V is an oriented real vectorspace of dimension n , then V ∗ naturally acquires the opposite orientation with respect tothe dual basis if n ≡ , g, n ) is oriented according to the induced orientation, then this is an input circleto which we assign an element of A . If a boundary circle is oppositely oriented, then it is an output circle and Σ produces an output element at this boundary. Thus if Σ has an inputcircle and Σ an output circle, then we can sew the two surfaces together along the circleto form a connected sum Σ , where the output from Σ is placed as input for Σ . Proposition 3.5.
The genus values of a 2D TQFT is given by (3.12) Ω ,n ( v , . . . , v n ) = (cid:15) ( v · · · v n ) , provided that we define(3.13) Ω , ( v , v , v ) := (cid:15) ( v v v ) . Proof.
This is a direct consequence of CohFT 3 and (2.7). (cid:3)
One of the original motivations of TQFT [2, 25] is to identify the topological invariant Z (Σ) of a closed manifold Σ. In our current setting, it is defined as(3.14) Z (Σ g ) := (cid:15) (cid:0) λ − (Ω g, ) (cid:1) for a closed oriented surface Σ g of genus g . Here, Ω g, : A −→ K is an element of A ∗ , and λ : A ∼ −→ A ∗ is the canonical isomorphism. Proposition 3.6.
The topological invariant Z (Σ g ) of (3.14) is given by (3.15) Z (Σ g ) = (cid:15) ( e g ) , where e g ∈ A represents the g -th power of the Euler element of (2.9) . Lemma 3.7.
We have (3.16) e := m ◦ δ (1) = λ − (Ω , ) . Proof.
This follows fromΩ , ( v ) = (cid:88) a,b Ω , ( v, e a , e b ) η ab = (cid:88) a,b η ( v, e a e b ) η ab = η ( v, e )for every v ∈ A . (cid:3) Proof of Proposition 3.6.
Since the starting case g = 1 follows from the above Lemma, weprove the formula by induction, which goes as follows:Ω g, ( v ) = (cid:88) a,b Ω g − , ( v, e a , e b ) η ab = (cid:88) i,j,a,b Ω , ( v, e a , e b , e i )Ω g − , ( e j ) η ab η ij = (cid:88) i,j,a,b η ( ve a e b , e i )Ω g − , ( e j ) η ab η ij = (cid:88) i,j η ( v e , e i )Ω g − , ( e j ) η ij = Ω g − , ( v e )= Ω , ( v e g − )= η ( v e g − , e ) = η ( v, e g ) . (cid:3) A closed genus g surface is obtained by sewing g genus 1 pieces with one output boundariesto a genus 0 surface with g input boundaries. Since the Euler element is the output of thegenus 1 surface with one boundary, we obtain the same result Z (Σ g ) = Ω ,g ( g (cid:122) (cid:125)(cid:124) (cid:123) e , . . . , e ) . Finally we have the following:
Theorem 3.8.
The value or the 2D TQFT is given by (3.17) Ω g,n ( v , . . . , v n ) = (cid:15) ( v · · · v n e g ) . Proof.
The argument is the same as the proof of Proposition 3.6:Ω g,n ( v , . . . , v n ) = Ω ,n ( v e g − , v , . . . , v n )= (cid:88) a,b Ω ,n +2 ( v e g − , v , . . . , v n , e a , e b ) η ab = (cid:15) ( v · · · v n e g ) . (cid:3) Example 3.9.
Let G be a finite group. The center of the complex group algebra Z C [ G ] isa semi-simple Frobenius algebra over C . For every conjugacy class c of G , the sum of groupelements in c , v ( C ) := (cid:88) u ∈ C u ∈ C [ G ] , is central and defines an element of Z C [ G ]. Although we do not discuss it any further here,the corresponding TQFT is equivalent to counting problems of character varieties of thefundamental group of n -punctured topological surface of genus g into G . DGE CONTRACTION AND 2D TQFT 11 The edge-contraction axioms
In this section we give a formulation of 2D TQFTs based on the edge-contraction opera-tions on cell graphs and a new set of axioms. The main theorem of this section, Theorem 4.7,motivates our construction of the category of cell graphs and the Frobenius ECO functorin Section 5.
Definition 4.1 (Cell graphs) . A connected cell graph of topological type ( g, n ) is the1-skeleton (the union of 0-cells and 1-cells) of a cell-decomposition of a connected compactoriented topological surface of genus g with n labeled 0-cells. We call a 0-cell a vertex , a1-cell an edge , and a 2-cell a face , of a cell graph. Remark 4.2.
The dual of a cell graph is usually referred to as a ribbon graph , or a dessind’enfant of Grothendieck. A ribbon graph is a graph with cyclic order assigned to incidenthalf-edges at each vertex. Such assignments induce a cyclic order of half-edges at eachvertex of the dual graph. Thus a cell graph itself is a ribbon graph. We note that verticesof a cell graph are labeled, which corresponds to the usual face labeling of a ribbon graph.
Remark 4.3.
We identify two cell graphs if there is a homeomorphism of the surfaces thatbrings one cell-decomposition to the other, keeping the labeling of 0-cells. The only possibleautomorphisms of a cell graph come from cyclic rotations of half-edges at each vertex.We denote by Γ g,n the set of connected cell graphs of type ( g, n ) with labeled vertices.
Definition 4.4 (Edge-contraction axioms) . The edge-contraction axioms are the fol-lowing set of rules for the assignment(4.1) Ω : Γ g,n −→ ( A ∗ ) ⊗ n of a multilinear map Ω( γ ) : A ⊗ n −→ K to each cell graph γ ∈ Γ g,n . We consider Ω( γ ) an n -variable function Ω( γ )( v , . . . , v n ),where we assign v i ∈ A to the i -th vertex of γ . • ECA 0 : For the simplest cell graph γ = • ∈ Γ , that consists of only one vertexwithout any edges, we define(4.2) Ω( • )( v ) = (cid:15) ( v ) , v ∈ A. • ECA 1 : Suppose there is an edge E connecting the i -th vertex and the j -th vertexfor i < j in γ ∈ Γ g,n . Let γ (cid:48) ∈ Γ g,n − denote the cell graph obtained by contracting E . Then(4.3) Ω( γ )( v , . . . , v n ) = Ω( γ (cid:48) )( v , . . . , v i − , v i v j , v i +1 , . . . , (cid:98) v j , . . . , v n ) , where (cid:98) v j means we omit the j -th variable v j at the j -th vertex, which no longerexists in γ (cid:48) . Figure 4.1.
The edge-contraction operation that shrinks a straight edge connect-ing Vertex i and Vertex j . • ECA 2 : Suppose there is a loop L in γ ∈ Γ g,n at the i -th vertex. Let γ (cid:48) denote thepossibly disconnected graph obtained by contracting L and separating the vertex totwo distinct vertices labeled by i and i (cid:48) . For the purpose of labeling all vertices, weassign an ordering i − < i < i (cid:48) < i + 1. Figure 4.2.
The edge-contraction operation that shrinks a loop attached Vertex i . If γ (cid:48) is connected, then it is in Γ g − ,n +1 . We call L a loop of a handle . We thenimpose(4.4) Ω( γ )( v , . . . , v n ) = Ω( γ (cid:48) )( v , . . . , v i − , δ ( v i ) , v i +1 , . . . , v n ) , where the outcome of the comultiplication δ ( v i ) is placed in the i -th and i (cid:48) -th slots.If γ (cid:48) is disconnected, then write γ (cid:48) = ( γ , γ ) ∈ Γ g , | I | +1 × Γ g , | J | +1 , where(4.5) (cid:40) g = g + g I (cid:116) J = { , . . . , (cid:98) i, . . . , n } . In this case L is a separating loop . Here, vertices labeled by I belong to the con-nected component of genus g , and those labeled by J on the other component. Let( I − , i, I + ) (reps. ( J − , i, J + )) be reordering of I (cid:116) { i } (resp. J (cid:116) { i } ) in the increasingorder. We impose(4.6) Ω( γ )( v , . . . , v n ) = (cid:88) a,b,k,(cid:96) η ( v i , e k e (cid:96) ) η ka η (cid:96)b Ω( γ )( v I − , e a , v I + )Ω( γ )( v J − , e b , v J + ) , which is similar to (4.4), just the comultiplication δ ( v i ) is written in terms of thebasis. Here, cocommutativity of A is assumed in this formula. Remark 4.5.
We do not assume the permutation symmetry of Ω( γ )( v , . . . , v n ). Thecumbersome notation of the axioms is due to keeping track of the ordering of indices. Remark 4.6.
Let us define m ( γ ) = 2 g − n for γ ∈ Γ g,n . The edge-contraction operationsare reduction of m ( γ ) exactly by 1. Indeed, for ECA 1, we have m ( γ (cid:48) ) = 2 g − n −
1) = m ( γ ) − . ECA 2 applied to a loop of a handle produces m ( γ (cid:48) ) = 2( g − − n + 1) = m ( γ ) − . For a separating loop, we have2 g − | I | + 1+) 2 g − | J | + 12 g + 2 g − | I | + | J | + 2 = 2 g − n − . This reduction is used in the proof of the following theorem.
DGE CONTRACTION AND 2D TQFT 13
Theorem 4.7 (Graph independence) . As the consequence of the edge-contraction axioms,every connected cell graph γ ∈ Γ g,n gives rise to the same map (4.7) Ω( γ ) : A ⊗ n (cid:51) v ⊗ · · · ⊗ v n (cid:55)−→ (cid:15) ( v · · · v n e g ) ∈ K, where e is the Euler element of (2.9) . In particular, Ω( γ )( v , . . . , v n ) is symmetric withrespect to permutations of indices. Corollary 4.8 (ECA implies TQFT) . Define Ω g,n ( v , . . . , v n ) = Ω( γ )( v , . . . , v n ) for any γ ∈ Γ g,n . Then { Ω g,n } is the 2D TQFT associated with the Frobenius algebra A . Every 2DTQFT is obtained in this way, hence the two descriptions of 2D TQFT are equivalent.Proof of Corollary 4.8 assuming Theorem 4.7. Since both ECAs and 2D TQFT give theunique value Ω( γ )( v , . . . , v n ) = (cid:15) ( v · · · v n e g ) = Ω g,n ( v , . . . , v n )for all ( g, n ) from (3.17), we see that the two sets of axioms are equivalent, and also thatthe edge-contraction axioms produce evert 2D TQFT. (cid:3) To illustrate the graph independence, let us first examine three simple cases.
Lemma 4.9 (Edge-removal lemma) . Let γ ∈ Γ g,n . (1) Suppose there is a disc-bounding loop L in γ (the graph on the left of Figure 4.3).Let γ (cid:48) ∈ Γ g,n be the graph obtained by removing L from γ . (2) Suppose there are two edges E and E between two distinct vertices Vertex i andVertex j , i < j , that bound a disc (the middle graph of Figure 4.3). Let γ (cid:48) ∈ Γ g,n bethe graph obtained by removing E . (3) Suppose two loops, L and L , are attached to the i -th vertex (the graph on the rightof Figure 4.3). If they are homotopic, then let γ (cid:48) ∈ Γ g,n be the graph obtained byremoving L from γ .In each of the above cases, we have (4.8) Ω( γ )( v , . . . , v n ) = Ω( γ (cid:48) )( v , . . . , v n ) . Figure 4.3.
Proof. (1) Contracting a disc-bounding loop attached to the i -th vertex creates ( γ , γ (cid:48) ) ∈ Γ , × Γ g,n , where γ consists of only one vertex and no edges. Then ECA 2 readsΩ( γ )( v , . . . , v n ) = (cid:88) a,b,k,(cid:96) η ( v i , e k e (cid:96) ) η ka η (cid:96)b γ ( e a )Ω( γ (cid:48) )( v , . . . , v i − , e b , v i +1 . . . , v n )= (cid:88) a,b,k,(cid:96) η ( v i , e k e (cid:96) ) η ka η (cid:96)b η (1 , e a )Ω( γ (cid:48) )( v , . . . , v i − , e b , v i +1 . . . , v n )= (cid:88) b,k,(cid:96) η ( v i , e k e (cid:96) ) δ k η (cid:96)b Ω( γ (cid:48) )( v , . . . , v i − , e b , v i +1 . . . , v n ) = (cid:88) b,(cid:96) η ( v i , e (cid:96) ) η (cid:96)b Ω( γ (cid:48) )( v , . . . , v i − , e b , v i +1 . . . , v n )= Ω( γ (cid:48) )( v , . . . , v i − , v i , v i +1 . . . , v n ) . (2) Contracting Edge E makes E a disc-bounding loop at Vertex i . We can remove itby (1). Note that the new Vertex i is assigned with v i v j . Restoring E makes the graphexactly the one obtained by removing E from γ . Thus (4.8) holds.(3) Contracting Loop L makes L a disc-bounding loop. Hence we can remove it by (1).Then restoring L creates a graph obtained from γ by removing L . Thus (4.8) holds. (cid:3) Remark 4.10.
The three cases treated above correspond to eliminating degree 1 and 2vertices from the ribbon graph dual to the cell graph. In combinatorial moduli theory, wenormally consider ribbon graphs that have no vertices of degree less than 3 [21].
Definition 4.11 (Reduced graph) . We call a cell graph reduced if it does not have anydisc-bounding loops or disc-bounding bigons. In other words, the dual ribbon graph of areduced cell graph has no vertices of degree 1 or 2.We can see from Lemma 4.9 (1) that every γ , ∈ Γ , gives rise to the same map(4.9) Ω( γ , )( v ) = (cid:15) ( v ) . Likewise, Lemma 4.9 (1) and (2) show that every γ , ∈ Γ , gives the same mapΩ( γ , )( v , v ) = η ( v , v ) . This is because we can remove all edges and loops but one that connects the two vertices,and from ECA 1, the value of the assignment is (cid:15) ( v v ). Proof of Theorem 4.7.
We use the induction on m = 2 g − n . The base case is m = − g, n ) = (0 , g, n ) with 2 g − n < m . Now let γ ∈ Γ g,n be a cell graph of type ( g, n ) such that2 n − n = m .Choose an arbitrary straight edge of γ that connects two distinct vertices, say Vertex i and Vertex j , i < j . By contracting this edge, we obtain by ECA 1,Ω( γ )( v , . . . , v n ) = Ω( γ g,n − )( v , . . . , v i − , v i v j , v i +1 . . . , (cid:98) v j , . . . , v n ) = (cid:15) ( v . . . v n e g ) . If we have chosen an arbitrary loop attached to Vertex i , then its contraction by ECA 2gives two cases, depending on whether the loop is a loop of a handle, or a separating loop.For the first case, by appealing to (2.7) and (2.10), we obtainΩ( γ )( v , . . . , v n ) = (cid:88) a,b,k,(cid:96) η ( v i , e k e (cid:96) ) η ka η (cid:96)b Ω( γ g − ,n +1 )( v , . . . , v i − , e a , e b , v i +1 , . . . , v n )= (cid:88) a,b,k,(cid:96) η ( v i e k , e (cid:96) ) η ka η (cid:96)b Ω( γ g − ,n +1 )( v , . . . , v i − , e a , e b , v i +1 , . . . , v n )= (cid:88) a,k η ka Ω( γ g − ,n +1 )( v , . . . , v i − , e a , v i e k , v i +1 , . . . , v n )= (cid:88) a,k η ka (cid:15) ( v · · · v n e g − e a e b )= (cid:15) ( v · · · v n e g ) . DGE CONTRACTION AND 2D TQFT 15
For the case of a separating loop, again by appealing to (2.7), we haveΩ( γ )( v , . . . , v n ) = (cid:88) a,b,k,(cid:96) η ( v i , e k e (cid:96) ) η ka η (cid:96)b Ω( γ g , | I | +1 ) (cid:0) v I − , e a , v I + (cid:1) Ω( γ g , | J | +1 ) (cid:0) v J − , e b , v J + (cid:1) = (cid:88) a,b,k,(cid:96) η ( v i , e k e (cid:96) ) η ka η (cid:96)b (cid:15) (cid:32) e a (cid:89) c ∈ I v c e g (cid:33) (cid:15) (cid:32) e b (cid:89) d ∈ J v d e g (cid:33) = (cid:88) a,b,k,(cid:96) η ( v i e k , e (cid:96) ) η ka η (cid:96)b η (cid:32)(cid:89) c ∈ I v c , e a e g (cid:33) (cid:15) (cid:32) e b (cid:89) d ∈ J v d e g (cid:33) = (cid:88) a,k η ka η (cid:32)(cid:89) c ∈ I v c e g , e a (cid:33) (cid:15) (cid:32) v i e k (cid:89) d ∈ J v d e g (cid:33) = (cid:15) (cid:32) v i (cid:89) c ∈ I v c e g (cid:89) d ∈ J v d e g (cid:33) = (cid:15) ( v · · · v n e g + g ) . Therefore, no matter how we apply ECA 1 or ECA 2, we always obtain the same result.This completes the proof. (cid:3)
Remark 4.12.
There is a different proof of the graph independence theorem, using atopological idea of deforming graphs similar to the one used in [23].As we see, the key reason for the graph independence of Theorem 4.7 is the property ofthe Frobenius algebra A that we have, namely, commutativity, cocommutativity, associativ-ity, coassociativity, and the Frobenius relation (2.1). These properties are manifest in thefollowing graph operations. Although the next proposition is an easy consequence of Theo-rem 4.7, we derive it directly from the ECAs so that we can see how the algebraic structureof the Frobenius algebra is encoded into the TQFT. Indeed, the graph-independence theo-rem also follows from Proposition 4.13. This fact motivates us to introduce the category ofcell graphs and the Frobenius ECO functor in the next section. Proposition 4.13 (Commutativity of Edge Contractions) . Let γ ∈ Γ g,n . (1) Suppose Vertex i is connected to two distinct vertices Vertex j and Vertex k by twoedges, E j and E k . The graph we obtain, denoted as γ (cid:48) ∈ Γ g,n − , by first contracting E j and then contracting E k , is the same as contracting the edges in the oppositeorder. The two different orders of the application of ECA 1 then gives the sameanswer. For example, if i < j < k , then we have (4.10) Ω( γ )( v , . . . , v n ) = Ω( γ (cid:48) )( v , . . . , v i − , v i v j v k , v i +1 , . . . , (cid:98) v j , . . . , (cid:98) v k , . . . , v n ) . (2) Suppose two loops L and L are connected to Vertex i . Then the contraction of thetwo loops in different orders gives the same result. (3) Suppose a loop L and a straight edge E are attached to Vertex i , where E connectsto Vertex j , i (cid:54) = j . Then contracting L first and followed by contracting E , gives thesame result as we contract L and E in the other way around.Proof. (1) There are three possible cases: i < j < k , j < i < k , and j < k < i . In eachcase, the result is replacing v i by v i v j v k , and removing two vertices. The associativity andcommutativity of the multiplication of A make the result of different contractions the same. (2) There are two cases here: After the contraction of one of the loops, (a) the other loopremans to be a loop, or (b) becomes an edge connecting the two vertices created by thecontraction of the first loop.In the first case (a), the contraction of the two loops makes Vertex i in γ into threedifferent vertices i , i , i of the resulting graph γ (cid:48) , which may be disconnected. The loopcontractions in the two different orders produce triple tensor products(1 ⊗ δ ) δ ( v i ) = ( δ ⊗ δ ( v i ) , which are equal by the coassociativity A ⊗ A ⊗ δ (cid:38) (cid:38) A δ (cid:60) (cid:60) δ (cid:34) (cid:34) A ⊗ A ⊗ A.A ⊗ A δ ⊗ (cid:56) (cid:56) For (b), the contraction of the loops in either order will produce m ◦ δ ( v i ) on the same i -thslot of the same graph γ (cid:48) ∈ Γ g − ,n .(3) This amounts to proving the equation δ ( v i v j ) = (1 ⊗ m ) (cid:0) δ ( v i ) , v j (cid:1) = ( m ⊗ (cid:0) v j , δ ( v i ) (cid:1) , which is Lemma 2.1. (cid:3) Remark 4.14.
If we have a system of subsets Γ (cid:48) g,n ⊂ Γ g,n for all ( g, n ) that is closed underthe edge-contraction operations, then all statements of this section still hold by replacingΓ g,n by Γ (cid:48) g,n . Remark 4.15.
Chen [4] proved the graph independence for a special case of A = Z C [ S ],the center of the group algebra for symmetric group S , by direct computation. This resultled the authors to find a general proof of Theorem 4.7.The edge-contraction operations are associated with gluing morphisms of M g,n that aredifferent from those in (3.4) and (3.5). ECA 1 of (4.3) is associated with(4.11) α : M , × M g,n − −→ M g,n . The handle cutting case of ECA 2 of (4.4) is associated with(4.12) β : M , × M g − ,n +1 −→ M g,n , and the separating loop contraction with(4.13) β : M , × M g , | I | +1 × M g , | J | +1 −→ M g + g , | I | + | J | +1 . Although there are no cell graph operations that are directly associated with the forgetfulmorphism π and the gluing maps gl and gl , there is an operation on cell graphs similarto the connected sum of topological surfaces. Definition 4.16 (Connected sum of cell graphs) . Let γ (cid:48) be a cell graph with the followingconditions.(1) There is a vertex q in γ (cid:48) of degree d .(2) There are d distinct edges incident to q . In particular, none of them is a loop.(3) There are exactly d faces in γ (cid:48) incident to q . DGE CONTRACTION AND 2D TQFT 17
Given an arbitrary cell graph γ with a degree d vertex p , we can create a new cell graph γ ( p,q ) γ (cid:48) , which we call the connected sum of γ and γ (cid:48) . The procedure is the following. Welabel all half-edges incident to p with { , , . . . , d } according to the cyclic order of the cellgraph γ at p . We also label all edges incident to q in γ (cid:48) with { , , . . . , d } , but this timeopposite to the cyclic oder given to γ (cid:48) at q . Cut a small disc around p and q , and connectall half-edges according to the labeling. The result is a cell graph γ ( p,q ) γ (cid:48) . Remark 4.17.
The connected sum construction can be applied to two distinct vertices p and q of the same graph, provided that these vertices satisfy the required conditions. Remark 4.18.
The total number of vertices decreases by 2 in the connected sum. There-fore, two 1-vertex graphs cannot be connected by this construction.The connected sum construction provides the inverse of the edge-contraction operationsas the following diagrams show. It is also clear from these figures that the edge-contractionoperations are degeneration of curves producing a rational curve with three special points,as indicated in Introduction.
123 4 5 67 1 23456 7
EEp qLL
123 4 5 67 1 23456 7
EEp qLL
Figure 4.4.
The connected sum of a cell graph with a particular type (0 ,
3) cellgraph gives the inverse of the edge-contraction operation on E that connects twodistinct vertices. The connected sum with the (0 ,
3) piece has to be done so thatthe edges incidents on each side of E match the original graph.
123 4 5 67 1 23456 7
EEp qLL
123 4 5 67 1 23456 7
EEp qLL
Figure 4.5.
The edge-contraction operation on a loop L is the inverse of twoconnected sum operations, with a type (0 ,
3) piece in the middle. Category of cell graphs and Frobenius ECO functors
In the previous section, we started from a Frobenius algebra A and constructed the cor-responding TQFT through edge-contraction axioms. The key step is the assignment of thelinear map Ω( γ ) : A ⊗ n −→ K to each cell graph γ ∈ Γ g,n . As we have noticed, edge-contraction operations encode the structure of a Frobenius algebra. These considerationssuggest that cell graphs are functors, and edge-contraction operations are natural transfor-mations. In this section, we define the category of cell graphs, and define Frobenius ECOfunctors, which make edge-contraction operations correspond to natural transformations.Let ( C , ⊗ , K ) be a monoidal category with a bifunctor ⊗ : C × C −→ C and its left andright identity object K ∈ Ob ( C ). The example we keep in mind is the monoidal category( Vect , ⊗ , K ) of vector spaces defined over a field K with the vector space tensor product operation. Fore brevity, we call the bifunctor ⊗ just as a tensor product. A K -object in C is a pair ( V, f : V −→ K ) consisting of an object V and a morphism f : V −→ K . Wedenote by C /K the category of K -objects in C . A K -morphism h : ( V , f ) −→ ( V , f ) is amorphism h : V −→ V in C that satisfies the commutativity(5.1) V f −−−−→ K h (cid:121) (cid:13)(cid:13)(cid:13) V −−−−→ f K. We note that every morphism h : V −→ V in C yields a new object ( V , f ) from a given( V , f ) as in (5.1). This is the pull-back object. The category C /K itself is a monoidalcategory with respect to the tensor product, and the final object ( K, id K : K −→ K ) of C /K as its identity object.We denote by F un ( C /K, C /K ) the endofunctor category , consisting of monoidal func-tors α : C /K −→ C /K as its objects. Let α and β be two endofunctors, and τ a natural transformation betweenthem. Natural transformations form morphisms in the endofunctor category. V h (cid:15) (cid:15) f (cid:32) (cid:32) α ( V ) α ( h ) (cid:15) (cid:15) α ( f ) (cid:34) (cid:34) τ (cid:47) (cid:47) β ( V ) (cid:15) (cid:15) β ( f ) (cid:34) (cid:34) K K τ (cid:47) (cid:47) KW g (cid:62) (cid:62) α ( W ) α ( g ) (cid:60) (cid:60) τ (cid:47) (cid:47) β ( W ) β ( g ) (cid:60) (cid:60) The final object of F un ( C /K, C /K ) is the functor(5.2) φ : ( V, f : V −→ K ) −→ ( K, id K : K −→ K )which assigns the final object of the codomain C /K to everything in the domain C /K . Withrespect to the tensor product and the above functor φ as its identity object, the endofunctorcategory F un ( C /K, C /K ) is again a monoidal category. Definition 5.1 (Subcategory generated by V ) . For every choice of an object V of C , wedefine a category of K -objects T ( V • ) /K as the full subcategory of C /K whose objects are( V ⊗ n , f : V ⊗ n −→ K ), n = 0 , , , . . . . We call T ( V • ) /K the subcategory generated by V in C /K . Definition 5.2 (Monoidal category of cell graphs) . The finite coproduct (or cocartesian) monoidal category of cell graphs CG is defined as follows. • The set of objects Ob ( CG ) consists of a finite disjoint union of cell graphs. • The coproduct in CG is the disjoin union, and the coidentity object is the emptygraph.The set of morphism Hom( γ , γ ) from a cell graph γ to γ consists of equivalence classes ofsequences of edge-contraction operations and graph automorphisms. For brevity of notation,if E is an edge connecting two distinct vertices of γ , then we simply denote by E itself as DGE CONTRACTION AND 2D TQFT 19 the edge-contraction operation shrinking E , as in Figure 4.1. If L is a loop in γ , then wedenote by L the edge-contraction operation of Figure 4.2. Let (cid:93) Hom( γ , γ ) = (cid:26) composition of a sequence of edge-contractionsand graph automorphisms that changes γ to γ (cid:27) . This is the set of words consisting of edge-contraction operations and graph automorphismsthat change γ to γ when operated consecutively. If there is no such operations, thenwe define (cid:93) Hom( γ , γ ) to be the empty set. The morphism set Hom( γ , γ ) is the set ofequivalence classes of (cid:93) Hom( γ , γ ). The equivalence relation in the extended morphism setis generated by the following cases of equivalences.(1) Suppose γ has a non-trivial automorphism σ . Then for every edge E of γ , E and σ ( E ) are equivalent.(2) Suppose Vertex i of γ ∈ Γ g,n is connected to two distinct vertices Vertex j andVertex k by two edges, E j and E k . The graph we obtain, denoted as γ ∈ Γ g,n − , byfirst contracting E j and then contracting E k , is the same as contracting the edgesin the opposite order. The two words E E and E E are equivalent.(3) Suppose two loops L and L of γ are connected to Vertex i . Then the contractionoperations of the two loops in different orders give the same result. The two words L L and L L are equivalent.(4) Suppose a loop L and a straight edge E in γ are attached to Vertex i , where E connects to Vertex j , i (cid:54) = j . Then contracting L first and followed by contracting E , gives the same result as we contract L and E in the other way around. The twowords EL and LE are equivalent.(5) Suppose γ has two edges (including loops) E and E that have no common vertices,and γ is obtained by contracting them. Then E E is equivalent to E E .(6) Suppose two edges E and E are both incident to two distinct vertices. Then E E is equivalent to E E . Example 5.3.
A few simple examples of morphisms are given below.Hom( • E • E • , • • ) = { E , E } , Hom( • E • E • , • ) = { E E } , Hom (cid:18) E •(cid:13)• E , •(cid:13) (cid:19) = { E } = { σ ( E ) } = { E } , Hom (cid:18) E •(cid:13)• E , • • (cid:19) = { E E } . The cell graph on the left of the third and fourth lines has an automorphism σ thatinterchanges E and E . Thus as the edge-contraction operation, E = E ◦ σ = σ ( E ). Remark 5.4. If γ ∈ Γ g,n , then Hom( γ, γ ) = { id γ } .We have seen in the last section that when we have made a choice of a unital commutativeFrobenius algebra A , a cell graph γ ∈ Γ g,n defines a multilinear map Ω A ( γ ) : A ⊗ n −→ K subject to edge-contraction axioms. For a different Frobenius algebra B , we have a differentmultilinear map Ω B ( γ ) : B ⊗ n −→ K , subject to the same axioms. These two maps areunrelated, unless we have a Frobenius algebra homomorphism h : A −→ B . Theorem 4.7 tells us that we have a K -morphism of (5.1) which induces Ω A ( γ ) as the pull-back of Ω B ( γ ). A h (cid:15) (cid:15) A ⊗ n (cid:15) (cid:15) Ω A ( γ ) (cid:47) (cid:47) K (cid:15) (cid:15) B B ⊗ n Ω B ( γ ) (cid:47) (cid:47) K This consideration suggests that Ω( γ ) is a functor defined on the category of Frobeniusalgebras. But since we are encoding the Frobenius algebra structure into the category ofcell graphs, the extra choice of Frobenius algebras is redundant.We are thus led to the following definition. Definition 5.5 (Frobenius ECO functor) . An Frobenius ECO functor is a monoidalfunctor(5.3) ω : CG −→ F un ( C /K, C /K )satisfying the following conditions. • The graph γ = • of (4.2) of type (0 ,
1) consisting of only one vertex and no edgescorresponds to the identity endofunctor:(5.4) • −→ ( id : C /K −→ C /K ) . • A graph γ ∈ Γ g,n of type ( g, n ) corresponds to a functor(5.5) γ (cid:55)−→ (cid:2) ( V, f : V −→ K ) −→ ( V ⊗ n , ω V ( γ ) : V ⊗ n −→ K ) (cid:3) . The Frobenius ECO functor assigns to each edge-contraction operation a natural transfor-mation of endofunctors C /K −→ C /K . Remark 5.6.
The unique construction of the Frobenius ECO functor for (
Vect , ⊗ , K )requires us to generalize our categorical setting to include CohFT of Kontsevich-Manin[18]. Then we will be able to show that this unique functor actually generates all Frobeniusobjects of (
Vect , ⊗ , K ). This topic will be treated in our forthcoming paper.Let us consider the monoidal (not full) subcategory A ⊂ ( Vect , ⊗ , K ) consisting ofcommutative Frobenius algebras. Theorem 5.7 (Construction of 2D TQFTs) . There is a canonical Frobenius ECO functor (5.6) Ω :
CG −→ F un ( A /K, A /K ) . When we start with a Frobenius algebra A , this functor generates a network of multilinearmaps Ω A ( γ ) : A ⊗ n −→ K for all cell graphs γ ∈ Γ g,n for all values of ( g, n ) . This is the 2D TQFT corresponding tothe Frobenius algebra A .Proof. This follows from the graph independence of Theorem 4.7. (cid:3)
Acknowledgement.
The authors are grateful to the American Institute of Mathematics inCalifornia, the Banff International Research Station, the Institute for Mathematical Sciencesat the National University of Singapore, Kobe University, Leibniz Universit¨at Hannover, theLorentz Center for Mathematical Sciences, Leiden, Max-Planck-Institut f¨ur Mathematik inBonn, Mathematisches Forschungsinstitut Oberwolfach, and Institut Henri Poincar´e, fortheir hospitality and financial support during the authors’ stay for collaboration related
DGE CONTRACTION AND 2D TQFT 21 to the subjects of this paper. They thank Ruian Chen, Maxim Kontsevich and ShintaroYanagida for valuable discussions. They also thank the referee for useful comments in im-proving the manuscript. O.D. thanks the Perimeter Institute for Theoretical Physics, andM.M. thanks the Hong Kong University of Science and Technology and the Simons Centerfor Geometry and Physics, for financial support and hospitality. During the preparationof this work, the research of O.D. has been supported by GRK 1463
Analysis, Geometry,and String Theory at Leibniz Universit¨at Hannover, and Max-Planck-Institut f¨ur Math-ematik, Bonn. The research of M.M. has been supported by NSF grants DMS-1104734,DMS-1309298, DMS-1619760, DMS-1642515, and NSF-RNMS: Geometric Structures AndRepresentation Varieties (GEAR Network, DMS-1107452, 1107263, 1107367).
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E-mail address : [email protected] Simion Stoilow Institute of Mathematics, Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest,RomaniaM. Mulase: Department of Mathematics, University of California, Davis, CA 95616–8633
E-mail address : [email protected]@math.ucdavis.edu