Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions
aa r X i v : . [ m a t h . C O ] J a n Discrete Dirac Operators, CriticalEmbeddings and Ihara-Selberg Functions
Martin Loebl ∗ Dep. of Applied MathematicsMalostranske n. 25, 118 00 Praha 1Charles University, Czech republic [email protected]
Petr Somberg † Mathematical InstituteSokolovska 83, 180 00 Praha 8Charles University, Czech republic [email protected]
Submitted: ???, 2013; Accepted: ???, 2014; Published: XXMathematics Subject Classifications: 05C50, 82B20, 57M15
Abstract
The aim of the paper is to formulate a discrete analogue of the claim made byAlvarez-Gaume et al., ([1]), realizing the partition function of the free fermion on aclosed Riemann surface of genus g as a linear combination of 2 g Pfaffians of Diracoperators. Let G = ( V, E ) be a finite graph embedded in a closed Riemann surface X of genus g , x e the collection of independent variables associated with each edge e of G (collected in one vector variable x ) and Σ the set of all 2 g Spin-structureson X . We introduce 2 g rotations rot s and (2 | E | × | E | ) matrices ∆( s )( x ), s ∈ Σ,of the transitions between the oriented edges of G determined by rotations rot s .We show that the generating function for the even subsets of edges of G , i.e., theIsing partition function, is a linear combination of the square roots of 2 g Ihara-Selberg functions I (∆( s )( x )) also called Feynman functions. By a result of Foata–Zeilberger holds I (∆( s )( x )) = det( I − ∆ ′ ( s )( x )), where ∆ ′ ( s )( x ) is obtained from∆( s )( x ) by replacing some entries by 0. Thus each Feynman function is computablein polynomial time. We suggest that in the case of critical embedding of a bipartitegraph G , the Feynman functions provide suitable discrete analogues for the Pfaffiansof discrete Dirac operators. Keywords: discrete conformal structure, critical embedding, Ihara-Selberg func-tion, discrete Dirac operator, Ising partition function ∗ Partially supported by the Czech Science Foundation under the contract number P202-13-21988S. † Supported by the grant GACR P201/12/G028. the electronic journal of combinatorics (2009), Introduction
It is well known, cf. [19], [13] and references therein, how to formulate the notion of acritical embedding of a finite graph in a closed Riemann surface in such a way that onecan read the critical values of independent variables (i.e., the edge weights or couplingconstants in the physical terminology) attached to the edges of the Dimer and the Isingproblems on G out of the collection of angles of the embedding termed discrete conformalstructure. It is rather attractive task to study whether some of the properties associatedto the notion of criticality in statistical physics may also be derived from the collectionof geometric data attached to critical embeddings.The main theme of the present paper is the formulation of certain discrete analogueof the claim made by Alvarez-Gaume et all, see [1], that the partition function of freefermion on a closed Riemann surface of genus g is a linear combination of 2 g Pfaffians ofDirac operators. The theory of free fermion is generally accepted to be closely related tothe criticality of both the Dimer and the Ising problems on graphs.The Dimer problem and its determinant type solution are of considerable continualinterest, see e.g., [13], [4]. In [3], the authors study the critical Ising model by reducingit via the determinant type method to the Dimer model. The case of planar graphs iswell-understood in this setting, [13]. However, as proved in [4] and explained in detail inthe Appendix, Subsection 6.3, if one wants to obtain a discrete analogue of the claim in [1]for the Dimer model in a surface of a positive genus, one has to take into account globalrestrictions on the graph embedded into the Riemann surface. In particular, the conditionson Kasteleyn flatness in Corollary 32 need to be satisfied. For the sake of completeness,the combinatorial approach used to describe the determinant type reduction of the criticalDimer model on Riemann surfaces of positive genus is briefly overviewed in the Appendix.The present paper investigates the question whether, using the geometric data pro-vided by a critical embedding of a graph G in a Riemann surface X and the weights of theedges of G in such a way that the matrix of these weights captures the basic propertiesof the discrete Dirac operator, the partition function of the Dimer and Ising models maybe evaluated explicitly.In particular, we propose to overcome the previously mentioned limitations and ob-stacles for the determinant-type reasoning, and replace the determinants by so calledIhara-Selberg functions of the graph G . This in turn allows to rewrite the generatingfunction of the even subsets of edges of G (the Ising model partition function associatedto the graph G ) as a linear combination of 2 g square roots of Ihara-Selberg functionswhich we call Feynman functions. We build the Feynman functions in a combinatorialway in order to capture the emergence of the rotations rot s for Spin structures s in theanalysis of the Ising partition function, where the realization of rot s exploits the spaceof quadratic forms on H ( X, F ). This approach makes use of the original treatment ofthe Ising partition function via Ihara-Selberg functions by Sherman, see [22], [15]. Ourbasic notion of a g − graph G g allows rather convenient treatment of a planar model forthe embedding of the graph G in a closed Riemann surface of genus g . The g − graphshave been successfully used recently in [17] in a related situation. Moreover, the planar the electronic journal of combinatorics (2009), odel may improve the insight into the analysis of the limiting processes, which has notbeen done yet. The rotations are treated in a different way by Johnson [11]. In the present subsection we review the notion of critical embedding associated to discreteconformal structure on a graph embedded in a Riemann surface, [19].Let us consider a triple (
X, G, ϕ ), where X is a closed Riemann surface, G = ( V, E )a graph and ϕ : G ֒ → X an embedding. The image of ϕ defines a CW decomposition of X such that X \ ϕ ( G ) is a disjoint union of open faces. A useful local description of X is given by local charts { ϕ j : U j → C | S j U j = X } j covering X , equipped with the flatmetric and finite number of conical singularities at { P j } j constrained by Gauss-Bonnetformula.An embedding of a graph G in a surface X induces the dual graph G ∗ embedding. Weregard G ∗ as an abstract graph with natural embedding into X , so that each vertex of G ∗ lies on the face of the embedding of G it represents. The central notion related to thecouple G, G ∗ is that of the diamond graph - for simultaneous embedding of G and G ∗ , thediamond graph G + has the vertex set equal to V ( G ) ∪ V ( G ∗ ) and the edges connectingthe end-vertices of each dual pair of edges e, e ∗ into a facial cycle F ( e ) of G + , which is a4 − gon called a diamond (see Figure 1). e be ∗ w Figure 1: Diamond F ( e ) of edge e .An embedding ϕ of G + is critical if each of its faces F ( e ) , e ∈ E , is a rhombus, i.e.,the following conditions hold true with respect to the induced conformal class of metricsin a given local chart on ϕ ( X ):1. The diagonals of each rhombus are perpendicular,2. The lengths of sides of all rhombi are the same. the electronic journal of combinatorics (2009), otice that the first property is independent of the choice of local chart, because thetransition maps ϕ i ◦ ϕ − j are conformal and so angle preserving, while the second conditiondepends on the choice of a representative metric in the conformal class. In the present subsection we discuss the relationship between the Dimer and the Isingpartition functions with emphasis on their combinatorial description.Let us associate an independent variable x e with each edge e ∈ E of G . A subset E ′ ⊂ E of edges is called perfect matching or dimer arrangement, if the induced graph( V, E ′ ) has each vertex of degree one. Let P ( G ) denote the set of all perfect matchingsof G . We define the dimer partition function of G by P ( G, x ) = X M ∈P ( G ) Y e ∈ M x e , (1)where x = ( x e ) e ∈ E is the vector of edge weights.A subset E ′ ⊂ E of edges is called even if the induced graph ( V, E ′ ) has each vertexof even degree. We denote by E ( G ) the set of even subsets of edges of G .The generating function of the even sets of edges of G is defined by E ( G, x ) = X E ′ ∈E ( G ) Y e ∈ E ′ x e . (2)It is well known (see, e.g., [17]) that E ( G, x ) is equivalent to the Ising partition functionon G defined by Z Ising G ( β ) := Z Ising G ( x ) (cid:12)(cid:12)(cid:12) x e := e βJ e ∀ e ∈ E , (3)where J e ( e ∈ E ) are the weights (coupling constants) associated with edges of the graph G , β the scale (inverse temperature) and Z Ising G ( x ) = X σ : V →{ , − } Y e = { u,v }∈ E x σ ( u ) σ ( v ) e . (4) We shall close the first section by a brief formulation and overview of the main result inthe article.We denote by Σ the set of Spin structures on a Riemann surface X . Following [15],[11], [17], we associate the rotation to each Spin structure. This enables us to define(2 | E | × | E | ) matrices ∆( s )( x ), s ∈ Σ, in a way that each ∆( s )( x ) is the transition matrixbetween the oriented edges determined by the rotation rot s corresponding to the Spinstructure s . The main result of our article is the electronic journal of combinatorics (2009), heorem 1. Let G be a graph embedded in a closed Riemann surface X of genus g andlet Σ be the set of Spin structures on X . Then E ( G, x ) is a linear combination of g Feynman functions F (∆( s )( x )) for s ∈ Σ and ∆( s ) the transition matrix of G . This isan Arf invariant formula and each F (∆( s )( x )) is computable in polynomial time.In addition, let G be a bipartite graph critically embedded in the Riemann surface X of genus g . Then for each Spin structure s ∈ Σ the transition matrix ∆( s ) may beconstructed out of the matrix of the discrete Dirac operator associated to s ∈ Σ . The structure of our paper is organized as follows. In Section 2 we introduce our maintool, the discrete Ihara-Selberg functions. We further construct a useful combinatorialmodel of closed Riemann surface X of genus g , which enables to define 4 g rotations ofprime reduced cycles on a graph critically embedded in X . In Section 3 we introducequadratic forms, relate them to rotations and define the Feynman functions. In the endof this section we prove the first part of Theorem 1, namely the Arf-invariant formula.Section 4 treats the topic of discrete Dirac operators, and concludes the proof of Theorem1. In the Appendix we summarize, for the reader’s convenience, the Pfaffian method andits limitations. Acknowledgements
The authors would like to thank David Cimasoni for extensive discussions about discreteDirac operators and Gregor Masbaum for extensive discussions and suggestions aboutrotations and Spin structures.
In this section we assume that the graph G is embedded in a closed Riemann surface X ofgenus g . We suggest to consider the Ising partition function E ( G, x ) instead of the Dimerpartition function P ( G, x ), and the square root of certain Ihara-Selberg functions on thegraph G (which we call the Feynman functions) instead of the determinant. Similar,but much less advanced structure can be found in e.g., [23], under the notion of Isingpreholomorphic observable. Let G = ( V, E ) be a graph. For e ∈ E we denote by o e an orientation of e , and o − e thereversed directed edge to o e . As above, let x = ( x e ) e ∈ E be the formal variables associatedwith edges of G . If o is any orientation of the edge e , we associate the new variable x o with it and always let x o = x e .We consider an equivalence relation on the set of finite-length sequences ( z , . . . , z n )satisfying z = z n : any such sequence is equivalent with each of its cyclic shifts. Theequivalence classes will be called circular sequences . the electronic journal of combinatorics (2009), circular sequence p = v , o , v , o , ..., o n , v n +1 with v n +1 = v and o i = ( v i , v i +1 ) iscalled a prime reduced cycle if the following conditions are satisfied: o i ∈ { o e , o − e : e ∈ E } , o i = o − i +1 and ( o , ..., o n ) = Z m for some sequence Z and a natural number m >
1. Wesay that the ordered pair ( o i , o i +1 ) is a transition of p at v i +1 , and ( o n , o ) is a transitionof p at v . We denote by p − the prime reduced cycle which is the inverse of p . Definition 2.
Let G = ( V, E ) be a graph and assume that the vertex set E is linearlyordered. Let M be 2 | E | × | E | matrix with entries m ( o, o ′ ), o, o ′ ∈ { o e , o − e : e ∈ E } ,where we think of m ( o, o ′ ) as the weight of the transition between directed edges o, o ′ of G . If p is a prime reduced cycle then we set M ( p ) = Q ( o,o ′ ) a transition of p m ( o, o ′ ).We denote the set of prime reduced cycles of G by G . The Ihara-Selberg functionassociated to G is defined by I ( M ) = Y γ ∈G (1 − M ( γ )) , (5)where the infinite product is determined by the formal power series Y γ ∈G (1 − M ( γ )) = X F ( − |F| Y γ ∈F M ( γ ) (6)and the sum is over all finite subsets F of G .A theorem of Foata–Zeilberger (see [8]), generalizing the seminal theorem of Bass (see[2]), states: Theorem 3.
We have I ( M ) = det( I − M ′ ) , (7) where M ′ is the matrix obtained from M by letting m ′ ( o, o ′ ) = 0 if o ′ = o − and m ′ ( o, o ′ ) = m ( o, o ′ ) otherwise. g In this section we restrict ourselves to the following standard representation of closedRiemann surface X of genus g : we regard X as a regular 4 g − gon R (called the basepolygon) in the plane with sides denoted anti-clockwise by z , . . . , z g , and the pairs ofsides z i , z − i +2 and z i +1 , z − i +3 , i = 1 , , . . . , g −
1) + 1, identified. This defines a flat metricon X with one conical singularity of angle 2 π (2 g − X which meet the boundary of R transversely. Moreover, to simplify the arguments,we only consider embeddings of graphs where each edge is represented by a straight lineon X . The general embeddings may be treated analogously.We now describe how an embedding of a graph in a Riemann surface can be used tomake its planar drawing of a special kind, cf. [15], [17]. the electronic journal of combinatorics (2009), efinition 4. The highway surface S g consists of the base polygon R and the bridges R , . . . , R g , where1. Each odd bridge R i − is a rectangle with vertices x ( i, , . . . , x ( i,
4) numbered anti-clockwise. The bridge is glued to R so that its edge [ x ( i, , x ( i, z i − , z i − ] and the edge [ x ( i, , x ( i, z i − , z i − ].2. Each even bridge R i is a rectangle with vertices y ( i, , . . . , y ( i,
4) numbered anti-clockwise. It is glued with R so that its edge [ y ( i, , y ( i, z i − , z i − ] and the edge [ y ( i, , y ( i, z i − , z i − ] (the indexes are always considered modulo 4 g .)There is an orientation-preserving immersion Φ of S g into the complex plane which isinjective except that for each i = 1 , . . . g , the images of the bridges R i and R i − intersectin a square (see Figure 2).Now assume the graph G is piece-wise linearly embedded into a closed Riemann surface X of genus g . We realize X as the union of S g and an additional disk R ∞ glued to theboundary of S g . By an isotopy of the embedding we may assume that G does not meetthe disk R ∞ and all vertices of G lie in the interior of R . We may also assume thatthe intersection of G with any of the rectangular bridges R i consists of disjoint straightlines connecting the two sides of R i glued to the base polygon R . The composition ofthe embedding of G into S g with the immersion Φ yields a drawing ϕ of G in the plane,where each edge of G is represented by a piece-wise linear curve (see Figure 2). A planardrawing of G obtained in this way will be called a g − graph and denoted by G g . Observethat double points of a g − graph can only come from the intersection of the images ofbridges under the immersion Φ of S g into the plane. Thus every double point of a specialdrawing lies in one of the squares Φ( R i ) ∩ Φ( R i − ). Definition 5.
Let G be embedded in S g and let e be an edge of G . By definition, theembedding of e intersects each bridge R i in disjoint straight lines. The number of theselines is denoted by r i ( e ). For a set A of edges of G we denote r ( A ) be the vector of length2 g defined by r ( A ) i = P e ∈ A r i ( e ). Definition 6.
Let p be a prime reduced cycle of G . Then p g denotes the image of p in G g .Clearly, p p g gives bijective correspondence between the prime reduced cycles of G and the prime reduced cycles of G g . Let G be a graph embedded in a closed Riemann surface X of genus g and let G g be its g − graph. We recall that each edge of G in G g is represented by a piece-wise linear curve.Let p be a prime reduced cycle of G and let p g the corresponding prime reduced cycle of G g . We shall introduce 4 g rotations corresponding to p g . the electronic journal of combinatorics (2009), bdae f Figure 2: Immersions of edges { a, b } , { c, d } and { e, f } which cross a side of R .First of all, we denote by 0 the 0 − vector of length 2 g and in analogy with the usualdefinition of the rotation of a regular closed curve in the plane we set rot ( p g ) = P t y ( t )(mod 2), where we sum over all transitions of the linear components of p g . If thetransition t consists in passing from directed segment e to directed segment e ′ then y ( t ) = z ( t )(2 π ) − , where z ( t ) is the angle of the transition. The angle z ( t ) is neg-ative if the transition is clockwise, and z ( t ) is positive if the transition is anti-clockwise(see Figure 3).Let F be the field with two elements. Then we define rot s ( p g ) for each arithmeticvector s ∈ F g in the following way: if the transition t consists in passing from the directedsegment e to the directed segment e ′ and e belongs to the immersion of bridge R i with s i = 1, we set y s ( t ) = y ( t ) + 1 and y s ( t ) = y ( t ) otherwise. Let rot s ( p g ) = P t y s ( t ). Wehave rot s ( p g ) = sr ( p ) + rot ( p g ) (mod 2), where sr denotes the scalar product of vectors s , r . Observe that for each s , ( − rot s ( p g ) = ( − rot s (( p g ) − ) . Example 7.
Let G be a toroidal graph embedded in the highway surface S , and let p be a prime reduced cycle of G intersecting each of the two bridges R i , i = 1 , p has exactly one self-intersection and | rot ( p ) | = 0. Consequently, rot ( p ) = rot ( p ) = 0 and rot ( p ) = rot ( p ) = 1. Definition 8.
We introduce the equivalence relation on the set of prime reduced cyclesof G : we say that p is equivalent to p if p = ( p ) − . The set of equivalence classesfor this equivalence relation is denoted by [ G ]. Analogously, we introduce the equivalencerelation on the set of prime reduced cycles of G g : we say that p g is equivalent to p g if p g = ( p g ) − and the set of equivalence classes is denoted by [ G ] g .The previous considerations allow to introduce the functions F ( G, x, s ) = X [ F ] g ( − | [ F ] g | Y [ γ ] ∈ [ F ] g ( − rot s ( γ ) Y e ∈ γ x e , (8) the electronic journal of combinatorics (2009), π/ − π/ F ] g of [ G ] g . In the proof of the first part ofTheorem 1 in subsection 3.4 we observe that the functions F ( G, x, s ) are the Feynmanfunctions introduced in Definition 16.The following theorem appears in [15] or the book [16]. We assign to s ∈ F g the signaccording to sign( s ) = ( − P i =1 , ,... g − s i s i +1 . Theorem 9.
Let G be a graph with each vertex of degree equal to or , embedded intoa closed Riemann surface X of genus g . Then E ( G, x ) = 2 − g X s ∈ F g sign( s ) F ( G, x, s ) . (9)We remark that Theorem 9 can be easily extended to general graphs and will be usedin the proof of Arf invariant formula (see Subsection 3.4). Theorem 10.
Let G be a graph embedded into a closed Riemann surface X of genus g .Then E ( G, x ) = 2 − g X s ∈ F g sign( s ) F ( G, x, s ) . (10) Proof.
We will construct a graph G ′ along with its embedding into X so that the degreeof each vertex of G ′ is equal to 2 or 4, such that there are two subsets Z, O of E ( G ′ ) anda bijection f : E ( G ′ ) \ ( Z ∪ O ) → E ( G ) inducing E ( G, x ) = E ( G ′ , z ) | z e := x f ( e ) if e/ ∈ Z ∪ O ; z e :=0 if e ∈ Z ; z e :=1 if e ∈ O and, for each s ∈ F g , F ( G, x, s ) = F ( G ′ , z, s ) | z e := x f ( e ) if e/ ∈ Z ∪ O ; z e :=0 if e ∈ Z ; z e :=1 if e ∈ O . Theorem 10 then follows from Theorem 9. We construct graph G ′ in two steps. Let OD denote the set of the vertices of G of an odd degree. We start with O = Z = ∅ . Step 1. If OD = ∅ then it is a standard observation of the graph theory that G has aset of edge-disjoint paths so that each vertex of OD is an end-vertex of exactly one of the the electronic journal of combinatorics (2009), aths, and all the end-vertices of the paths are among the elements of OD . In particular, | OD | is even. Let P denote the set of the edges of these paths. We construct graph G from G by adding, for each edge { u, v } ∈ P , a path of length 3 with end-vertices u, v (seeFigure 4). We further let Z be the set of all the edges of G \ G . Note that G has alldegrees even. vu Figure 4:
Step 2. If v is a vertex of G of an even degree bigger than 4 then we modify G bysplitting the degree of v by introducing the new splitting edge; the operation can be readoff the Figure 5. We repeat this step until the resulting graph G ′ = ( V, E ′ ) has all degreesequal to 2 or 4. Finally we let f be the tautological injection of E into E ′ . a ab c def b c d ef Figure 5: O := O ∪ { the new splitting edge } It is straightforward to realize that all assumptions of the construction are satisfiedafter application of finite number of these steps.In order to associate rotations to quadratic forms, we first need to study self-intersectionsof the prime reduced cycles which traverse each edge of the graph at most once.
In the present section we relate the rotations of prime reduced cycles to quadratic formsand moreover, we introduce the Feynman functions and prove a part of Theorem 1: theArf invariant formula. the electronic journal of combinatorics (2009), .1 Self-intersections of prime reduced cycles Let p be a prime reduced cycle of G which traverses each edge of G at most once. Foreach vertex v of G , let p ( v ) denote the set of the directed edges of p incident with v , andlet P ( p, v ) denote the partition of p ( v ) into pairs which correspond to the transitions of p at v . If a prime reduced cycle p traverses each edge of G at most once then the transitionsof p at v are well described by the directed chord diagram diag ( p, v ) (see Figure 6): Definition 11.
Let p be a prime reduced cycle of G which traverses each edge of G atmost once. The directed chord diagram diag ( p, v ) is obtained by taking the cyclic orderingof the edges of G incident with v and induced from the embedding of G in X , and byintroducing the directed chord ( e, e ′ ) for each class of P ( p, v ) consisting of an orientationof e followed by an orientation of e ′ .We define the number of self-intersections of p as the number of the pairs of intersectingchords of diag ( p, v ), v ∈ V . va bcd a bcd Figure 6: Directed chord diagram diag ( p, v ); p = ac . . . db . . . a We need to extend Definition 11 to the number of self-intersections of general primereduced cycles, i.e., the prime reduced cycles that can go through an edge more thanonce. This can be done for instance in the following way. We define the infinite graph˜ G by replacing each edge e by an infinite sequence of edges e , · · · , e i , · · · with the sameend-vertex as e . We embed ˜ G in X so that we ’thicken’ the embedding of each edge e of G , and embed the edges e , · · · , e i , · · · to this thickened part of e so that they arepiece-wise linear and internally disjoint. Next, for each prime reduced cycle p of G whosecircular sequence of directed edges is ( a . . . , a k ), a j being an orientation of edge e ( j ) of G (possibly e ( j ) = e ( l ) for j = l ), we define prime reduced cycle ˜ p in ˜ G by replacing each a j by the same orientation of e ( j ) j . It is important that the prime reduced cycle ˜ p uses eachedge of ˜ G at most once. We thus define the number of the self-intersections of a primereduced cycle p of G as the number of the self-intersections of the prime reduced cycle ˜ p of ˜ G (see Definition 11). The number of the self-intersections of a prime reduced cycle p g of G g is defined analogously.We note that for the prime reduced cycles of G containing each edge of G at mostonce this is consistent with Definition 11. We also note that the following basic propertyis satisfied. the electronic journal of combinatorics (2009), bservation 12. Let p be a prime reduced cycle of G . Then we have (mod 2)1 + the number of the self-intersections of p g = rot ( p g ) . A subset of edges E ′ ⊂ E of a graph G embedded in a closed Riemann surface X of genus g is called even if each degree of the graph ( V, E ′ ) is even. The set of even subsets of G is denoted by E ( G ). Now we pass to the associated notion of quadratic form, cf. [17].Let H := H ( X, F ) be the first homology group of X with coefficients in the field F .The construction of the highway surface gives us canonical basis of H := H ( X, F ), a , b , . . . , a g , b g , where a i corresponds to the class of the bridge R i − and b i corresponds to the class of R i , i = 1 , . . . , g . Each element of H is represented by the coordinate vector in F g , andtwo even subsets of edges A, B belong to the same class in H if and only if r ( A ) = r ( B )(mod 2) (see Definition 5 for the definition of r ).We recall that H carries a non-degenerate skew-symmetric bilinear form called in-tersection form and denoted by ’ · ’. In the basis chosen above, it is determined by a i · a j = b i · b j = 0 and a i · b j = δ ij for all i, j = 1 , . . . , g . Definition 13.
A quadratic form on ( H, · ) associated to the bilinear form ’ · ’ is a function q : H → F fulfilling q ( x + y ) = q ( x ) + q ( y ) + x · y for all ( x, y ∈ H ).We denote the set of quadratic forms on H by Q . Each quadratic form is given by itsvalues on the basis a , b , . . . , a g , b g , and so the cardinality of Q is 4 g . Let us denote by q the quadratic form whose value on each of the basis vectors a i , b i is zero. For each z ∈ H let q z : H → F be defined by q z ( x ) = q ( x ) + zx . Then q z is a quadratic form and eachquadratic form equals q z for some z ∈ H .If x ∈ H is a vector in F g then we observe sign( x ) = ( − q ( x ) , where sign wasintroduced before the statement of Theorem 9.We now recall the definition of the Arf invariant of a quadratic form q ∈ Q . Let N = 2 g − (2 g + 1) and N = 2 g − (2 g − q has either N timesvalue 0 or N times value 0. In the first case we let Arf ( q ) = 0, and in the second case Arf ( q ) = 1.The relevance of the Arf invariant in our considerations comes from the following fact,established as Lemma 2.10 in [17]: for each x ∈ H − g X q ∈ Q ( − Arf ( q ) ( − q ( x ) = 1 . We will also use the following formula, established in the proof of Lemma 2.10., [17]:for each z ∈ H holds Arf ( q z ) = q ( z ) . (11) the electronic journal of combinatorics (2009), otice that both quadratic forms q z and rotations rot z are parametrized by vectors z ∈ F g . Also recall that for a Riemann surface X the Spin structures are given by theequivalence classes of square roots of the canonical bundle, and these classes correspondto the F -valued first cohomology group H ( X, F ). Because X is assumed to be smooth,the Poincar´e duality implies H ( X, F ) ≃ H ( X, F ).There is a natural bijection between the set Σ of the Spin structures of X and the setof quadratic forms on the F − valued first homology classes of X (see [11]). From now onwe will consider the rotations rot s indexed by Spin structures on X .The usefulness of quadratic forms in studying rotations rot s was suggested to us byG. Masbaum, who also suggested Theorem 14 below. Theorem 14.
Let p be a prime reduced cycle of G and let s ∈ Σ . Then rot s ( p g ) = 1 + the number of the self-intersections of p + q s ( p ) (mod 2) , (12) where p g is the realization of p in the g − graph G g .Proof. By the definition of rot s it suffices to prove the statement for s = 0. We have(mod 2) rot ( p g ) = 1 + the number of the self-intersections of p g =1 + the number of the self-intersections of p + g X i =1 r ( p ) i − r ( p ) i =1 + the number of the self-intersections of p + q ( p ) , where the first equality follows by Observation 12 and the last one is a consequence ofthe definition q . Definition 15.
Let G be a graph embedded in a closed Riemann surface X of genus g . Let E o = ∪{ o e , o − e : e ∈ E } , hence | E o | = 2 | E | . To each Spin structure s ∈ Σ weassociate ∆( s )( x ), the | E o | × | E o | -matrix with entries d s ( o, o ′ ) = ( − y s ( o,o ′ ) ( − κ ( o ) x ′ o ,where1. y s ( o, o ′ ) were introduced in the beginning of the section 2.3,2. κ ( o ) = 0 if o is contained in the interior of R ,3. κ ( o ) = − / o outside R in G g is orientedoppositely to the anti-clockwise orientation of the boundary of R (as directed edge( a, b ) in Figure 2),4. κ ( o ) = 3 / o outside R in G g is oriented inagreement with the anti-clockwise orientation of the boundary of R (as directededge ( a, b ) in Figure 2). the electronic journal of combinatorics (2009), e further define ∆ ′ ( s )( x ) by declaring d ′ s ( o, o ′ ) = 0 if o ′ = o − and d ′ s ( o, o ′ ) = d s ( o, o ′ )otherwise in ∆( s )( x ).We are going to introduce the Feynman functions , see Definition 16 below.
Definition 16.
Let s ∈ Σ be a Spin structure on X . We define F (∆( s )( x )) := X [ F ] Y [ γ ] ∈ [ F ] ( − q s ( γ )+ the number of self-intersections of γ Y e ∈ γ x e . (13)We recall that [ F ] denotes the collection of all the finite sets of the equivalence classes ofthe reduced prime cycles of G . The linear combination appearing in Theorem 1 is the Arf invariant formula, E ( G, x ) = 2 − g X s ∈ Σ ( − Arf( q s ) F (∆( s )( x )) . (14) Proof. (of the first part of Theorem 1, equation 14.) We have F (∆( s )( x )) = X [ F ] ( − | [ F ] | Y γ ∈F ( − q s ( γ )+ the number of the self-intersections of γ Y e ∈ γ x e = X [ F ] ( − | [ F ] | Y γ ∈F ( − rot s ( γ g ) Y e ∈ γ x e = F ( G, x, s ) , where the second equality follows from Theorem 14.Both F (∆( s )( x )) and I (∆( s )( x )) are written as an infinite product (see Definition2). The factors of I (∆( s )( x )) are parametrized by the prime reduced cycles, while thefactors of F (∆( s )( x )) are parametrized by the equivalence classes of prime reduced cycles.Hence each factor of F (∆( s )( x )) appears twice as factor of I (∆( s )( x )). Consequently, F (∆( s )( x )) is the square root of I (∆( s )( x )). By Theorem 3, I (∆( s )( x )) = det( I − ∆ ′ ( s )( x )) and each F (∆( s )( x )) is thus computable in polynomial time.Now the Arf-invariant formula follows from Theorem 10 and a simple observation donein equation 11 mentioned in the end of subsection 3.2: for each s , sign ( s ) = ( − Arf( q s ) . So far we considered arbitrary embeddings of a graph G into a Riemann surface X . Now,let G = ( W, B, E ) be a bipartite graph and the embedding critical. The vertices of W are called white and the vertices of B are called black. We also assume in this subsectionthat the conical singularities are not located at the vertices of G . the electronic journal of combinatorics (2009), e recall the notation E o = ∪{ o e , o − e : e ∈ E } . Let W o be the subset of E o consistingof the edges directed from its black vertices to its white vertices. Analogously, let B o = E o \ W o be the set of edges directed from its white vertex to its black vertex. The keyconstruction has the following structure: Definition 17.
Let T G be the directed transition graph of the orientations of edges of G ,i.e. V ( T G ) = E o and ( o, o ′ ) ∈ E ( T G ) if the head of o is the tail of o ′ . Observation 18.
The graph T G is a directed bipartite graph, T G = ( W o , B o , E ( T G )) , andthe matrix ∆( s )( x ) is a weighted adjacency matrix of T G . The next observation follows directly from Definition 15.
Observation 19.
Let w o ∈ W o be an orientation of the edge e entering the vertex w ∈ W . Let b o be a directed edge leaving w and entering vertex b ∈ B . Then the entry ∆( s )( x )( w o , b o ) equals γ ( s, w o ) e iα ( w o ,b o ) x b o , where α ( w o , b o ) = 1 / z ( w o , b o ) is half of theangle of the transition from w o to b o ; γ ( w o ) equals a complex number depending only on s and w o . Let l ( e ∗ ) denote the length of the dual edge e ∗ of an edge e of G . Definition 20.
We denote by ∆ ( s ) the matrix obtained by taking the square of eachentry of ∆( s )( x ) and substituting the vector of the lengths of the dual edges for thesquares of the variables: ( x e ) := l ( e ∗ ) , e ∈ E o . Corollary 21. (of Observation 19) If w , w are two directed edges of E o entering thesame vertex then the row of ∆ ( s ) indexed by w is a complex multiple of the row of ∆ ( s ) indexed by w . The discrete Dirac operator D ( s ) of a bipartite graph G = ( V, E ), V = W ∪ B ,corresponding to a Spin structure s ∈ Σ, is defined as the weighted adjacency ( | V | ×| V | ) − matrix; [13] contains the definition for the planar graphs and [4] defines half of (or,the chiral part of) D ( s ), which uniquely determines D ( s ). We observe in this subsectionthat the matrix ∆ ( s ) is closely related to D ( s ).To that aim we fix the coordinate chart on the Riemann surface such that a blackvertex v ∈ B lies at 0 ∈ C in this local chart. The directed edges emanating from a whitevertex and terminating at v are denoted e v , . . . , e vk . Similarly, we fix an edge e vw emanatingfrom v and terminating at some white vertex w ∈ W . The ( vw )-matrix element of theDirac operator is D ( vw ) = l (( e vw ) ∗ ) e iα , where α is the angle between the edge e vw and thereal axis of the local coordinate chart. Note that the change in local coordinate chartdoes not preserve the real axis, but does preserve the angle α . On the other hand the( e vi , e vw )-entry of our matrix ∆ ( s ) is equal to l (( e vw ) ∗ ) e iα i , where α i is the angle between e vi , e vw measured anticlockwise, see Figure 3. In particular, in the local coordinate chartin which the edge e vi lies on the real axis, we have α = α i . Equivalently, the matrixcoefficients in a given row of our matrix ∆ ( s ) indexed by directed edge e are (constant)multiples of the row in the Dirac matrix corresponding to the terminal vertex of e . Thesame applies when one starts with the white vertex instead of the black one. the electronic journal of combinatorics (2009), he basic observation about the matrices ∆ ( s ) and D ( s ) can be formulated in thefollowing way. Corollary 22.
Let w o ∈ W o be an orientation of the edge e entering the vertex w ∈ W .Let b o be a directed edge leaving w and entering vertex b ∈ B . Then ∆ ( s )( w o , b o ) = c ( w o ) D ( s )( w, b ) , where c ( w o ) is a complex number depending only on w o . The last part of Theorem 1 is contained in the next Corollary.
Corollary 23.
Each matrix ∆ ( s ) , s ∈ Σ a Spin structure, may be obtained from thediscrete Dirac operator D ( s ) , whose chiral part is defined in [4], by finite number ofoperations:1. Adding identical copy of a row,2. Multiplying a row by a complex number,3. Adding | E | − | V | zero-entries to each row.Remark . The properties that graph G is bipartite and critically embedded, and thecondition that the conic singularities of the metric are not located at the vertices of G , areneeded in [4] to show that constant functions belong to the kernel of the discrete Diracoperators.Each Feynman function can be rewritten as the alternating sum of the traces of skew-symmetric powers of matrices ∆ ′ ( s )( x ), which is an approach closely related to the originof analytic torsion on Riemann surfaces. These considerations may lead to a realizationof the analytic torsion as a continuous limit of the Feynman functions. The present paper is a contribution towards mathematical understanding of fermionicquantum field theory. In particular, we indirectly connect generating functions for theenumeration of even subsets of edges of finite graphs embedded in closed Riemann surfaceswith an expression for partition function of free fermion by means of Dirac operators. Thecrucial role is played by the discrete Ihara-Selberg functions. However, the question oftheir limiting behavior remains elusive.
For the sake of completeness we briefly review the Pfaffian method and indicate its globalrestrictions in order to obtain a discrete analogue of the result by Alvarez-Gaume at all. the electronic journal of combinatorics (2009), .1 Kasteleyn orientations The first step in this theory is the reduction of E ( G, x ) to P ( G δ , x ), where G δ is obtainedfrom G by local changes which do not affect genus of the Riemann surface X as a targetfor embedding of G . This construction (see e.g., [17]) is not relevant here, so we omit itand concentrate on the aspects of the Dimer partition function P ( G, x ).Assume the vertices of G are numbered from 1 to n . An orientation of G is induced byprescribing one of the two possible directions to each edge of G . If D is an orientation of G , we denote by A ( G, D ) the skew-symmetric adjacency matrix of D defined as follows:the diagonal entries of A ( G, D ) are zero, and the off-diagonal entries are A ( G, D ) ij = X e : i → j ± x e , where the sum is taken over all edges e connecting vertices i and j , and the sign in frontof x e is 1 if e is oriented from i to j in the orientation D , and − A ( G, D ) counts perfect matchings P ( G ) of the graph G withsigns: Pfaf A ( G, D ) = X M ∈P ( G ) sign( M, D ) Y e ∈ M x e , where sign( M, D ) = ±
1. We use this as the definition of the sign of a perfect matching M with respect to the orientation D . The following statement is the basic result in thefield: Theorem 25 (Kasteleyn [12], Galluccio-Loebl [9], Tesler [24], Cimasoni-Reshetikhin [6]) . If G embeds into a Riemann surface of genus g , then there exist g orientations D i ( i = 1 , . . . , g ) of G such that the perfect matching polynomial P ( G, x ) can be expressedas a linear combination of the Pfaffian polynomials Pfaf A ( G, D i )( x ) . Moreover, the ori-entations D i are Kasteleyn orientations, i.e., they satisfy the following property: if F isa facial cycle of G then F has an odd number of edges oriented in D i in agreement withthe clockwise traversal of F . The formula of Theorem 25 is called the Arf-invariant formula, as it is based on theproperty of the Arf invariant for quadratic forms in characteristic two. As far as weknow, the relationship with the Arf invariant was first observed in [6]. In [14], Kuper-berg introduced a generalization of the notion of Kasteleyn orientations, called Kasteleynflatness.
We recall that a graph G = ( V, E ) is called bipartite if the set of the vertices V may bepartitioned into two sets W, B so | e ∩ W | = | e ∩ B | = 1 for each e ∈ E . In this and thenext one subsection we restrict ourselves to finite bipartite graphs G = ( W, B, E ), where
W, B are the two edge-less sets of vertices of G and V = W ∪ B . We call the vertices in W white and the vertices in B black and assume that G has at least one perfect matching. the electronic journal of combinatorics (2009), n particular, we restrict to the case of equal cardinalities | W | = | B | . By a cycle in G wemean a subset of edges C ⊂ E ( G ) which form a cycle. The cycle C can be decorated byone of the two possible orientations, i.e. one of the two possible ways of going around C .By an oriented cycle we mean a cycle decorated with an orientation. In this subsectionwe also use O to denote the orientation of G in which each edge is oriented from its blackvertex to its white vertex. Definition 26.
Let G be given with the weights w ( e ) , e ∈ E ( G ). Let C be an orientedcycle of G . We define c ( C ) = ( − | C | / Q e ∈ C + w ( e ) Q e ∈ C − w ( e )as the Kasteleyn curvature of C . Here C + denotes the subset of edges of C whose orien-tation inherited from C coincides with their orientation in O , and C − = C \ C + . Definition 27.
Let G be a weighted graph, w ( e ) , e ∈ E ( G ), embedded in X . We saythat G is Kasteleyn flat if c ( F ) = 1 for each face F of the embedding, arbitrarily oriented.Let G = ( W, B, E ) be a bipartite graph equipped with the weights w ( e ) , e ∈ E ( G ). Letus fix a linear ordering on the set B ∪ W such that the elements of B precede the elementsof W . This allows to introduce D ( w ) as ( B ∪ W ) × ( B ∪ W ) skew-symmetric matrixdefined by D ( w ) uv = w ( uv ) x e if u ∈ B and e = uv ∈ E ( G ) resp. D ( w ) uv = − w ( uv ) x e if u ∈ W and e = uv ∈ E ( G ), D ( w ) uv = 0 otherwise. We further denote by D B ( w ) the( B × W ) − block of D ( w ). For M a perfect matching of G we denote by t ( M ) the coefficientof Q e ∈ M x e in det( D B ( w )). Note that t ( M ) = sign( M ) Q e ∈ M w ( e ), where sign( M ) is thesign of M relative to the fixed linear ordering on the set of vertices B ∪ W .The significance of flatness of the Kasteleyn curvature rests on the following observa-tion (see [14]). Proposition 28.
Let C be a cycle of G embedded in X so that C is a symmetric differenceof two perfect matchings M, N and the orientation of C coincides with the orientation ofthe edges of M in O . Then t ( M ) t ( N ) = c ( C ) . This result leads to
Proposition 29. If G is embedded in the plane and its weight-function is Kasteleyn flatthen t ( M ) is constant. We are able to prove a stronger statement. Let G be embedded in a Riemann surface X of genus g . We say that the weight-function w is simple Kasteleyn flat if it is Kasteleynflat and moreover w ( e ) ∈ { , − } for each e ∈ E . The function w may be viewed asassigning orientation O ( w ) obtained from orientation O by reversing the orientation ofall edges with negative weight. If w is simple Kasteleyn flat the orientation D ( w ) is aKasteleyn orientation, i.e. each face has an odd number of edges oriented clockwise. Thenext definition is motivated by [6]. the electronic journal of combinatorics (2009), efinition 30. We say that two weight-functions w, w ′ are equivalent if w can be ob-tained from w ′ by finite number of vertex multiplications, where a vertex multiplicationconsists in choosing a vertex v and a complex number c = 0 together with multiplicationby c the weights of all edges incident with v . Proposition 31.
Let G be a graph embedded in a closed Riemann surface X of genus g and let w be a Kasteleyn flat weight function, satisfying in addition c ( C ) ∈ { , − } foreach cycle C . Then w is equivalent to a simple Kasteleyn flat weight-function w ′ , i.e., toa Kasteleyn orientation.Proof. We may assume that G is connected. Let T = ( V ( G ) , E ′ ), E ′ ⊂ E ( G ) be aspanning tree of G . We can clearly perform vertex multiplications (in G ) in a way theresulting weight function w ′ satisfies w ′ ( e ) = 1 for each e ∈ E ′ . Let e ∈ E \ E ′ . Thennecessarily w ′ ( e ) ∈ { , − } since e forms a cycle (say, denoted by C ) with a subset of E ′ and c ( C ) has values in { , − } .Proposition 31 along with Theorem 25 immediately imply Corollary 32.
Let G be a graph embedded in a closed Riemann surface X of genus g and w be a Kasteleyn flat weight function satisfying c ( C ) ∈ { , − } for each cycle C . Then P ( G, x ) is a linear combination of g determinants of matrices obtained from D B ( w ) bymultiplying some entries by − . The computation of the Dimer partition function P ( G, x ) for a critically embedded bipar-tite graph G is based on the following strategy (see [13], [4]): define the weight function w using discrete geometric information contained in the data of the critical embedding,and use this information to prove that the kernel of the corresponding matrix D ( w ) hasproperties desired for a discrete analogue of the Dirac operator. Next, prove that w isKasteleyn flat and insert it into Corollary 32.This approach works for planar bipartite graphs as well ([13]), since in this case eachKasteleyn flat weight function satisfies c ( C ) ∈ { , − } for each cycle C , see [14]. On theother hand, the assumptions of Corollary 32 are quite restrictive for non-planar surfaces.By Proposition 31, they are equivalent to w ( e ) ∈ { , − } for each edge e ∈ E . Moreover,it is shown in [4] that the theory of Kasteleyn flatness can not go beyond Corollary 32,see genus 1 example in [4], subsection 4.4. Examples. References [1] L. Alvarez-Gaume, J.B. Bost, G. Moore and C. Vafa. Bosonization on higher genusRiemann surfaces,
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