aa r X i v : . [ m a t h . AG ] F e b FORMULAE FOR CALCULATING HURWITZ NUMBERS
JARED ONGARO
Abstract.
In this paper, we aim to provide an accessible survey to various formulae for calculatingsingle Hurwitz numbers. Single Hurwitz numbers count certain classes of meromorphic functions oncomplex algebraic curves and have a rich geometric structure behind them which has attracted manymathematicians and physicists. Formulation of the enumeration problem is purely of topologicalnature, but with connections to several modern areas of mathematics and physics. Introduction
The number of non-equivalent branched coverings with a given set of branch points and branchedprofile is called the
Hurwitz number . The question of determining the Hurwitz number for agiven branch profile is called the
Hurwitz enumeration problem . Hurwitz numbers count thebranched coverings between complex projective curves with specified branch profile. Branchedcoverings were first described in the famous paper [24] by Riemann who developed the idea ofrepresenting nonsingular curves as branched coverings of P in order to study their moduli. How-ever, systematic investigation of branched coverings was initiated by Hurwitz in [15, 16] more thanthirty years later. Hurwitz numbers can be computed explicitly for non-complicated branched pro-files thanks to the nice combinatorial interpretations they posses that they can be interpreted interms of factorization permutations as first observed by Adolf Hurwitz in [15, 16].Hurwitz numbers have a rich geometric structure behind them which has attracted many mathe-maticians and physicists. The effect is that formulae for computing Hurwitz numbers arise fromdifferent branches of mathematics starting from algebraic geometry, combinatorics, representationof symmetric groups, tropical geometry. Explicit answers to the Hurwitz enumeration problem are usually difficult to obtain. One important case when this problem has a rather explicit answer,is when at most one branch point has an arbitrary branch type while all the others are simple. Incase of Y = P , we usually suppose that the degenerate branch point is at ∞ ∈ P and we call itspreimages f − ( ∞ ) poles. In other words, we are in the situation where all the branch points in C are simple, i.e. correspond to transpositions while the permutation at infinity can be described bysome partition µ = ( µ , . . . , µ n ) .Hurwitz enumeration problem is an old but still active research problem due to its connections toseveral modern areas of mathematics and physics. Below we walk you through the journey fromTopology to Physics in presenting various formulae and connections in calculating single Hurwitznumber. Mathematics Subject Classification.
Acknowledgment.
This paper was written during my fellowship at Department of Mathematics,University of stockholm with funding from International Science Programme, Uppsala University,Sweden. My advisor Boris Shapiro always has been invaluable after introducing me to the subject.I also want to thank S. Shadrin, J. Bergstr¨om, O. Bergvall and Bal´azs Szendr´oi for being availablefor many discussions. Many thanks to M. Shapiro for explaining nagging details of the ELSVformula. 2.
Partitions and Irreducible representations
We work over the field of complex numbers. The cardinality of a set S will be denoted by | S | . Alldefinitions and results on the symmetric group represented below are classical, and can be foundin most standard texts. Definition 2.1. A partition of a positive integer d , is a finite, weakly decreasing sequence ofpositive integers µ = ( µ , µ , . . . , µ n ) called parts of µ such that µ + µ + . . . + µ n = d. Denote a partition by µ ⊢ d and refer to d as the size of µ . The number n of parts of µ is called length of µ and is denoted by ℓ ( µ ) . Example 2.2.
There are integer partitions of d = 4 , namely (4) , (3 , , (2 , , (2 , , , (1 , , , . Denote the set consisting of the first d positive integers { , , . . . , d } by [ d ] . Let i be an integer inthe set { , , . . . , d } , the multiplicity of i in µ which we shall denote by m i ( µ ) is the numberof parts µ j equaling i . We often use exponents to indicate repeated parts, whence a partition µ can be written multiplicatively as µ = 1 m ( µ ) · m ( µ ) . . . k m k ( µ ) with | µ | = P ki =1 im i ( µ ) . Forinstance, the partition (2 , ,
1) = 1 · . The number of permutations of the parts of µ is thequantity | Aut( µ ) | = k Y i =1 m i ( µ )! . We can also represent partitions pictorially using Young diagrams.
Definition 2.3.
A Young diagram is an array of left and top-justified boxes, such that the rowsizes are weakly decreasing. The Young diagram corresponding to µ = ( µ , µ , . . . , µ n ) is theone that has n rows, and µ i boxes in the i th row.For instance, the Young diagrams corresponding to the above mentioned partitions of are givenbelow. ( 4 ) (3 ,
1) (2 ,
2) (2 , ,
1) (1 , , , The conjugate of the Young tableau λ is the reflection of the tableau λ along the main diagonal.This is also a standard Young tableau. We will write λ t to denote the conjugate partition of λ .Conjugate of = ORMULAE FOR CALCULATING HURWITZ NUMBERS 3
Let S d be the group of all permutations on [ d ] , we make the convention that permutations aremultiplied from right to left. A permutation α ∈ S d is a cycle of length k or a k − cycle if thereexist numbers i , i , . . . , i k ∈ [ d ] such that α ( i ) = i , α ( i ) = i , . . . , α ( i k ) = i . Thus, we can write α in the form ( i , i , . . . , i k ) . A cycle of length two is called a transposition .If we fix σ ∈ S d , then σ can be uniquely decomposed into a product of disjoint cycles. The sumof the cycle lengths of σ is equal to d , so the lenghts form a partition of d. The cycle type of σ is an expression of the form m · m . . . d m d , where the m i is the number of i − cycles in σ. We denote the set of all elements conjugate to σ inthe symmetric group S d by C σ , that is C σ = { πσπ − : π ∈ S d } . Recall that two permutations are conjugate if and only if they have the same cycle type. Eachconjugacy class of S d corresponds to a partition of d and we can use the combinatorial propertiesof these partitions to explicitly construct the irreducible representations S λ , from which we cancompute the irreducible characters.Young tableaux and symmetric functions [18] provide not only a straight-forward way of construct-ing irreducible representations of S d , but also an explicit formula for computing the correspondingcharacters. Denote by χ λ ( C ) the character of S λ on the conjugacy class C. Since a conjugacyclass C of an element in S d consists of all permutations of the same cycle type, we use the notation χ λµ to represent the character of S λ at the conjugacy class of the cycle type µ. It can be shown thatthe dimension of the irreducible representation corresponding to λ is given by the hook formula f λ = d ! Q ( i,j ) ∈ λ h ij . The hooklength h ij is the number of boxes directly to the right and directly below ( i, j ) includingthe box ( i, j ) . In particular, h ij = λ i − j + λ tj − i + 1 . For instance, if λ = (3 , , the hooklength h (2 , is . Example 2.4.
The degree of the irreducible representation of S corresponding to partition λ =(3 , ⊢ is the number of standard tableaux which can be calculated as f = 4!4 · · · . Let µ = ( µ , µ , . . . , µ n ) ⊢ d and consider the independent formal variables x = ( x , x , . . . , x m ) . The power sum function p µ ( x ) is defined as p µ ( x ) = n Y i =1 ( x µ i + . . . + x µ i m ) . Theorem 2.5 (Frobenius Character Formula) . Let λ = ( λ , λ , . . . , λ m ) and the partition µ = ( µ , µ , . . . , µ n ) ⊢ d . The character χ λµ is equal to the coefficient of Q ni =1 x λ i + m − ii in ORMULAE FOR CALCULATING HURWITZ NUMBERS 4 ∆( x ) p µ ( x ) where ∆( x ) is the Vandermonde determinant Y i Hurwitz Numbers and Hurwitz Spaces Let X be a complex nonsingular curve of genus g (Note that we impose further conditions on X ,see for example in [21, 22] ). A single Hurwitz numbers h g,µ enumerate Hurwitz covering arisingfrom meromorphic functions on X . A Hurwitz covering of type ( g, µ ) is a meromorphic function f : X → C on X with labelled poles { p , . . . , p n } given by a divisor µ p + . . . + µ n p n , andexcept for these poles, the holomorphic -form of f has simple zeros on X \{ p i , . . . , p n } withdistinct critical values of f . A meromorphic function f gives a finite morphism to the complexprojective line P whose degree d by definition is the degree of the morphism f : X −→ P .Observe, we are call a meromorphic function a covering because if we remove the critical values of f (including ∞ ) from P , then on this open set f becomes a topological covering. More precisely,let the branched locus B = { z , . . . , z w , ∞} denote the set of distinct critical values of f . Then f : X \ f − ( B ) → P \ B is a topological covering of degree d .A holomorphic map f : X −→ P is called a meromorphic function . Thus, given a meromorphicfunction f , for ∞ ∈ P we have the polar divisor f − ( ∞ ) = µ p + . . . + µ n p n , where p , . . . , p n are distinct points on X and µ = ( µ , . . . , µ n ) ⊢ d the branch type of f at a point ∞ gives apartition of d . For instance, the branch type for a simple branch point is (2) or (2 , , . . . , . ORMULAE FOR CALCULATING HURWITZ NUMBERS 5 C , z C , w C , z C , w C , zw = z w = z w = z fX P Figure 1. Local picture of a Hurwitz covering of degree . Example 3.1. Let X be the cubic curve in P defined by y z = x ( x + z )( x − z ) , where [ x, y, z ] are homogenous coordinates in P as discussed in [23]. Let f be the linear projection of X from p = [0 , , ∈ P \ X onto P . It defines a -sheeted branched covering of P . All the branchpoints namely; [0 , , [1 , , [ − , and [1 , ∈ P are simple implying that the meromorphic function on the linear projection of X to P from apoint p = [0 , , is a simple branch covering.The set of all branch points B is called the branching locus of f . In this way, every non-constantmeromorphic function on a curve X is a branched covering . In Hurwitz covering, we consider thecase where branch type at ∞ is given by the partition µ = ( µ , . . . , µ n ) ⊢ d and there are exactly w = 2 g − n + d simple branch points by Riemann-Hurwitz formula. The basic problem isthen the enumeration of such maps f : X −→ P for a given g and d for a prescribed branch typeover each branch point of f . Definition 3.2. Two Hurwitz covering f : X −→ P , covering with poles at { p , . . . , p n } and f : X −→ P with poles at { q , . . . , q n } are called equivalent if there exists an isomorphism φ : X ∼ −→ X such that φ ( p i ) = q i , for all i = 1 , . . . , n and the diagram X X P φ ∼ f f commutes.If X = X , p i = q i , and f = f = φ , then φ is an automorphism of a Hurwitz covering f .Thus, we count a Hurwitz covering with the automorphism factor | Aut( f ) | to compensate for therelabeling. If φ : X → X is just a homeomorphism, we say f and f have the same topologicaltype .Throughtout we let the integers d ≥ , g ≥ and a partition µ = ( µ , . . . , µ n ) ⊢ d . ORMULAE FOR CALCULATING HURWITZ NUMBERS 6 Topological Definition of Hurwitz Numbers.Definition 3.3. The connected single Hurwitz number is h g,µ = X [ f ] | Aut( f ) | , (3.1)where the sum runs all topological equivalence classes of Hurwitz coverings of type ( g, µ ) for g ≥ and a partition µ = ( µ , . . . , µ n ) ⊢ d for connected complex nonsingular curves X of genus g .Its convenient to define the disconnected Hurwitz numbers h • g,µ but without the condition that thecovering surface be connected. as the connected Hurwitz numbers since we can be computed fromthe disconnected Hurwitz via the inclusion-exclusion formula. Definition 3.4. The disconnected single Hurwitz number of type ( g, µ ) for g ≥ and a partition µ = ( µ , . . . , µ n ) ⊢ d is h • g,µ = X [ f ] | Aut( f ) | , (3.2)where the sum runs all topological equivalence classes of Hurwitz coverings f : X −→ P forpossibly a disconnected complex nonsingular curves X of genus g .3.2. Geometric Formulation of Single Hurwitz Numbers. Fix g ≥ and a partition µ ⊢ d on branched coverings f : X → P and the number w of branch points , then equivalence classesof branched coverings form a moduli space called single Hurwitz space of type ( g, µ ) Hurwitzcoverings denoted by H g,µ = ( f : X −→ P (cid:12)(cid:12)(cid:12)(cid:12) µ ⊢ d , X has genus g and f is aHurwitz covering of type ( g, µ ) ) (cid:30) ∼ . (3.3) H g,µ possess the structure of an irreducible smooth algebraic variety (see § 21 of [1] or [10]) ofdimension equal to w = 2 g + 2 d − . The fundamental group of the configuration space of w branch points in P acts on the fibers of H g,µ and the orbits of this action are in one to onecorrespondence with the connected components of H g,µ . Furthermore, H g,µ comes with a naturalfinite ´etale covering Φ : H g,µ −→ Sym w P \ ∆( f : X −→ P ) branch locus of f } (3.4)where Sym w P is the space of unordered w − tuples of points in P and ∆ is the discriminanthypersurface corresponding to sets of cardinality less than w . The morphism Φ is called the branching morphism and its degree is the single Hurwitz number h g,µ = Φ − ( B ) for a fixedbranch locus B .3.3. Group Theoretic Formulation of Hurwitz Numbers. To every meromorphic function f : X −→ P of degree d we can associate its monodromy data and we obtain a formulation ofsingle Hurwitz numbers in terms of counting sequences of factorization of a permutations. Definition 3.5. Fix σ ∈ S d , a sequence ( a b ) , ( a b ) , . . . , ( a n b n ) such that the product ( a b )( a b ) . . . ( a m b m ) = σ is called a transposition factorization of σ of length m . ORMULAE FOR CALCULATING HURWITZ NUMBERS 7 The factorization is not unique, for instance (123) = (12)(13) = (13)(23) . However, the numberof transpositions in the factorization depends on the cycle type of the permutation σ rather thanthe permutation itself. Definition 3.6. Let µ ⊢ d for d ≥ . Consider an ordered sequence of permutations ( τ , . . . , τ w , σ ) ∈ ( S d ) w +1 where w = 2 g − n + d . The single Hurwitz number h g,µ is given by h g,µ = d ! × number of ordered w -tuples ( τ , . . . , τ w ) ∈ ( S d ) w (3.5)such that:(1) (cid:0) τ , . . . , τ w (cid:1) are transpositions in S d ,(2) the product τ ◦ · · · ◦ τ w = σ in S d whose cycle type is µ .(3) The subgroup h τ , . . . , τ w i ⊂ S d acts transitively on the set { , , . . . , d } . Observe that the third condition is equivalent to requiring that the covering surface be connected.So we can define the disconnected Hurwitz numbers by relaxing the third condition: Definition 3.7. The disconnected single Hurwitz number is h • g,µ = number of (cid:26) ( τ , . . . , τ w ) ∈ ( S d ) w (cid:12)(cid:12)(cid:12)(cid:12) τ i are transpositions with τ ◦ · · · ◦ τ w = σ ∈ S d (cid:27) (cid:30) d ! (3.6)such that σ in S d has cycle type is µ .In this form the problem was for the first time formulated by A. Hurwitz.3.4. The Hurwitz Formula. In several specific cases A. Hurwitz calculated h g,µ using purelycombinatorial methods in 1891 and in terms of irreducible characters of S n in 1902. In [15] healso observed that the calculation h g,µ is a purely group-theoretic problem, but its solution iscomplicated for arbitrary g and d . On page 17 of [15], Hurwitz found answers for calculating thedegree of the map (3.4) for small d ≤ and any g ≥ . Namely, h g, = 12 ,h g, = 13! (3 g +3 − ,h g, = 14! (2 g +4 − g +5 − ,h g, = 10 g +8 − g +8 288 + 5 g +8 − g +8 72 + 3 g +8 18 + 2 g +8 − ,h g, = 15 g +10 · (360) − g +10 g +10 · (72) − g +10 · (24) + 6 g +10 · (36) − g +10 360 ++ 4 g +10 − · g +10 − · g +10 + 7271152 . (3.7) ORMULAE FOR CALCULATING HURWITZ NUMBERS 8 For instance, it is immediate to enumerate all degree single Hurwitz numbers for all g ≥ . s s s s s s s ∞ · · · wX P f branch type(partition) C o m pu t i n g t h e s i n g l e H u r w i t z nu m b e r s A. Hurwitz(1891)Example for d = 3 for all g ≥ h g, = h g, = (3 g +3 − ———————— h g, = 3 g Figure 2. Associated to counting permutations up to conjugation in S d Example 3.8. Let µ = (2 , ⊢ and g ≥ . To compute, h g, , all we need, is to countsequences of w = 2 g + 4 transpositions with the above properties. That is, we need to countsequences of w transpositions which generate a transitive subgroup of S d whose product is identity.Notice that we are free to choose g + 3 elements of the sequence as the last of them is determinedby the requirement that the product must be identity as the product of g + 3 transpositions asthe same parity as one transposition in S . Also, to avoid disconnected coverings we have to avoidalways choosing the same transpositions g + 3 times. Thus, we immediately find the number ofsimple branched coverings of degree is h g, = 3 g +3 − for all g ≥ as found by A. Hurwitz in Equation (3.7)Similary, h g, the number for non-isomorphic branched coverings of degree over P with onecomplicated branch point can easily be calculated. Example 3.9. Indeed we establish that the single Hurwitz number h g, = 3 g as follows.Notice that for complicated branch point we can choose freely any -cycle in S . The -cycleguarantee that we generate S . Then we are free to choose cycle for the next g + 1 simple branchpoints, the last is uniquely determined by the fact that the multiplication is identity. So we get · g +1 elements of S . We divide by to account for relabelling of the sheets of the branchedcoverings. ORMULAE FOR CALCULATING HURWITZ NUMBERS 9 Minimal Transposition Factorisation. For genus g = 0 , the single Hurwitz number h ,µ is equivalent to counting factorizations of a permutation σ ∈ S d of cycle type µ ⊢ d into a productof transpositions of minimal length divided by d ! , a result known and published by Hurwitz. Definition 3.10. Let σ ∈ S d be a fixed permutation of length m . The sequence ( τ , . . . , τ n ) iscalled a minimal transitive factorization of σ into transpositions if the following conditionsare satisfied:(1) Product cycle type condition: τ . . . τ n = σ, (2) Minimality condition: n := m + d − ,(3) Transitivity condition: The graph G σ is connected, where G µ is the graph correspondingto factorization σ into a product of n transpositions.Note that, one needs at least d − transpositions to build a cycle of length d . Then n ≥ d − . Example 3.11. (1) If µ = (2) ⊢ and m = 1 , the only transposition is (12) = (21) . Therefore h , = 12 · . This example also shows that Hurwitz numbers can be rational and not always a positiveinteger.(2) If µ = (3) ⊢ , m = 2 there exist transposition factorizations of the three-cycle (123) = (12)(13) := (23)(21) := (31)(32) and we have · three-cycles in S corresponding to connected trees. Thus h , = 16 (3 · 2) = 1 . (3) If µ = (2 , ⊢ and m = 3 we have triples of transpositions but of the triplesconsists of coinciding transpositions and thus the corresponding covering surface is notconnected. This implies that the single Hurwitz number h , = 16 (3 − 3) = 4 . Now, since for µ = ( d ) the graph G µ is a tree, assuming bijective results [19] the correspondingHurwitz number follows immediately from Cayley’s formula of 1860 for enumeration of trees.(Observe, the Cayley formula in the language of transpositions, is attributed to the Hungarianmathematician D´enes [5]). Theorem 3.12 (D´enes) . There exist d d − transposition factorization of an d -cycle into d − distinct transpositions.In the case m = 2 , V.I. Arnol’d [2] found the corresponding Hurwitz number by using the notionof complex trigonometric polynomials. Theorem 3.13 (Arnol’d ) . For a partition µ = ( µ , µ ) ⊢ d the number of distinct minimaltransitive transposition factorizations of σ whose cycle type equals µ is µ µ µ µ ( µ + µ − µ − µ − (3.8)Still another case was settled not that long ago by two physicists M. Crescimanno and W. Taylor. ORMULAE FOR CALCULATING HURWITZ NUMBERS 10 Theorem 3.14 (Crescimano-Taylor) . If m = d means µ = (1 d ) i.e. the factorization of theidentity, then the number of distinct minimal transitive factorizations into transpositions (2 d − d d − (3.9)was discovered in [4], who asked apparently asked the combinatorialist Richard Stanley who con-sulted Goulden-Jackson about the result. Finally, Goulden-Jackson also independently [13, 12]discovered and proved the Hurwitz formula in its complete generality.4. Hurwitz numbers and the symmetric groups Let C [ S d ] be the group algebra of S d . The group algebra C [ S d ] has is d ! dimensional over C .For each partition µ of d , denote by C µ ∈ C [ S d ] the basis elements in S d , i.e. the sum of allpermutations in S d of cyclic type µ . We will denote by C e the class C d = C (1 , ,..., of theidentity permutation, which is the unit of the algebra C [ S d ] , and C for the sum of C (2 , ,..., ofall transpositions. Proposition 4.1. Let S d denote the symmetric group of permutations of d elements. For eachpartition µ of d , the sum of all elements in S d of cyclic type µ span the center of C [ S d ] . Example 4.2. The center of the group algebra C [ S d ] is spanned by the three elements C e = ,C =(12) + (23) + (13) ,C (3) =(123) + (132) . The disconnected simple Hurwitz numbers possess the following natural interpretation. Theorem 4.3. The product of the class C µ with the wth power of the class C . Then h • g,µ = 1 d ! [ C e ] C µ C w (4.1)where [ C e ] C µ C w is the coefficient of C e in the product C µ ◦ C w . Example 4.4. Given d = 3 , g = 1 and µ = (3) and w = 2 g − ℓ ( µ )+ d = 2 · − we have C (3) C = ((123) + (132)) ((12) + (23) + (13))=54 e + 27 ((123) + (132))=54 C e + 27 C (3) . Thus h • g, = h g, = 13! [ C e ] C (3) C = 546 = 9 . Note that in this case, the connected and disconnected cases the calculations are identical as thespecial branch point has exactly one part i.e. (3) ⊢ . ORMULAE FOR CALCULATING HURWITZ NUMBERS 11 Burnside character formula. Calculating Hurwitz numbers is multiplication problem inconjugacy class basis on the center of the group algebra C [ S d ] . Recall that both the conjugacyclasses and irreducible representations of S d are in one to one correspondance with partitions of d . Using Burnside formula in eqrefburnform, we obtain a closed formula by involving a basis formirreducible representation of S d Theorem 4.5 (Burnside character formula) . Let ρ be an irreducible representation of S d . Denotethe character of ρ by χ λ . The single Hurwitz numbers h • g,µ = X λ ⊢ d (cid:18) dim λ | λ | ! (cid:19) f λ ( C µ ) f λ ( C ) w (4.2)where, f λ ( C µ i ) = | C µ i | dim λ χ λ ( µ i ) . This connection was already known to A. Hurwitz in 1902 and has provided a rich interplay betweengeometry and combinatorics for a long time.4.2. Hurwitz Monodromy Group. Recall that the single Hurwitz number corresponding to afixed branching data is given by the degree of the covering map Φ : H g,µ −→ Sym w P \ ∆ (4.3)It is an unsolved problem to determine the image, that is the monodromy group for branchingmorphism as described in (3.4) called the Hurwitz monodromy group . However, in specialcases see [6] a good description can be obtained. This cases include the Hurwitz spaces H g,µ . Theimage of the fundamental group π (Sym w P \ ∆ w ) to the symmetric group S h g,µ (where h g,µ is single hurwitz number) is the Hurwitz monodromy group. Directly from the Hurwitz formulaein (3.7) we have an intuitive indication, that the Hurwitz monodromy groups are less than the fullsymmetric group at least for the first nontrivial cases d = 3 and , but nothing much we cansay for d > from the shape of the formulae seen earlier. Thus for d = 3 or , the Hurwitzmonodromy groups can be anticipated to have a structure which heavily reflects the geometricalstructure of F and F vector spaces.Indeed, as given in (3.7) the single Hurwitz numbers h g, , h g, , for degree and consists ofthe factors n − and n − . Recall that n − is the number of points in the n − dimensionalprojective space over a field with elements and n − is the number of points in a n dimensionalprojective space over a field with elements.In fact, one way to compute the single Hurwitz numbers h g, and h g, via a bijection betweentranspositions and elements of finite fields F and F respectively. Example 4.6. To compute the h g, for degree Hurwitz coverings, we establish a bijectionbetween transpositions t . . . , t w in S specifying a covering curve X with w = 2 g + 4 branch ORMULAE FOR CALCULATING HURWITZ NUMBERS 12 points and the projective space of dimension w − over F . This is easily obtained. Up toconjugation we can assume t = (1 , and consider the assignment µ : (12) , (23) . (4.4)Let f ∗ ((12)) t . . . t w ) = ( µ ( t ) , . . . , µ ( t w − )) we define the map f from the projective pointsvia f ( X ) = f ∗ (( t ) , . . . , µ ( t w − )) As an example, if (12)(13)(12)(23)(13)) represent X , f ( X ) = (1 , , , . One then can easilyshow the map f is well defined from the requirement that the product of the transpositions mustbe identity, moreover its a bijection. Thus, h g, = 12 (3 w − − . which is the number of points in the projective space of dimension w − over a field F with threeelements. 5. Hurwitz numbers in terms monodromy graphs We can now compute single Hurwitz numbers in terms of monodromy graphs. This is motivatedfrom the definition of single Hurwitz numbers as equivalent to counting permutation factorizationsinto transpositions, we have another algebraic definition of single Hurwitz numbers via enumerationof graphs. The presentation, we follow the presentation in [3]The core behind the derivation of this special case is the fact that multiplication of permutation by atransposition τ = ( ab ) can be easily understood; it either cuts or joins cycles of the permutation.Namely, if σ ∈ S d has m cycles then the product τ ◦ σ has either(1) Cut : m − cycles if a and b are in different cycles of σ. (2) Join : m + 1 cycles if a and b are in same cycle of σ. m m + m m m m m m m + m m Figure 3. Effects on multiplicities of a cycle type of σ ∈ S d in the composition τ m ◦ σ . Example 5.1. The multiplication of permutation (12345) ∈ S on the left by (14) gives (15)(234) . In other words, cuts it into two cycles. On the other hand, multiplication of thepermutation (15)(234) on the left by (14) joins the two cycles together.We associate a graph called a monodromy graphs which project to the segment [ ∞ , . . . , w ] withlabeled edges as follows: Definition 5.2. Constructing the monodromy graph Γ , project to the segment [0 , w ] marked andlabelled ∞ , . . . , w as follows: ORMULAE FOR CALCULATING HURWITZ NUMBERS 13 1) Start with n small strands over ∞ decorated by weights labels µ , . . . , µ n .2) Over the point create a three-valent vertex by either joining two strands or splitting onewith weight strictly greater than . Join: If a join, label the new strand with the sum of the weights of the edges joined. Cut: If a cut, label the two new strands in all possible (positive) ways adding to the weightof the split edge,3) In each case, consider a unique representative for any isomorphism class of labeled graphs.4) Repeat (5.2) and (5.2) for all successive integers up to w ,5) Retain all connected graphs that terminate with weight (2) or (2 , , . . . , ⊢ d over w .We obtain a connected graph Γ of genus g with a map to [0 , w ] . We call Γ together with the mapthe monodromy graph of type ( g, µ ) corresponding to ( σ, τ , . . . , τ w ) . Definition 5.3. Given a monodromy graph, a balanced fork is a tripod with weights n, n, n such that the vertices of weight n lie over ∞ or w . A wiener consists of a strand of weight n splitting into two strands of weight n and then re-joining ∞ wnn n n nn n nn n Figure 4. Local structure for balanced left pointing forks, wiener and balancedright pointing forksThe map Γ → [0 , w ] can be viewed as a tropical cover of degree d , where the edges adjacent tovertices over ∞ yield the profile µ . The balancing condition for monodromy graphs , comes from theobservation that by definition a monodromy graph is a combinatorial type of a tropical morphismsee [3] for more details. Definition 5.4. An isomorphism of monodromy graphs Γ → [0 , w ] and Γ → [0 , w ] of type ( g, µ ) is a graph isomorphism f : Γ → Γ , such that Γ Γ [0 , w ] f (5.1)commutes. ORMULAE FOR CALCULATING HURWITZ NUMBERS 14 Table 1. Illustration of Hurwitz numbers using Monodromy graphs for degree . Graph type Q ω ( e ) Aut (Γ) Contribution P ∞ w f = 1 · · · · P ∞ w f = 1 · · · · P ∞ w f = 1 · · · · P ∞ w f = 1 · · · · P ∞ w f = 3 · · · · ORMULAE FOR CALCULATING HURWITZ NUMBERS 15 Definition 5.5. Let B µ denote the set of all isomorphism classes of monodromy graphs. Thenthe single Hurwitz number h g,µ = X [Γ ∈ B µ ] | Aut(Γ) | , (5.2)is the number of monodromy graphs Γ in B µ divided by | Aut(Γ) | .We simplify the definition 5.6 above to give the Cavalieri-Johnson-Markwig formula in [3]. Proposition 5.6. The Hurwitz number h g,µ is computed as a weighted sum over monodromygraphs. h g,µ = X [Γ ∈ B µ ] | Aut(Γ) | Y ω ( e ) (5.3)where we take the product of all the interior edge weights ω ( e ) . The the automorphism group Aut(Γ) involve factors of / coming from the balanced forks and wieners of Γ . Example 5.7. If g = 1 , d = 3 and µ = (2 , ⊢ , we will show that h , = 40 . That is there is a family of non-isomorphic cubics over points in P . Observe that in this case g = 1 , d = 3 , then by Riemann-Hurwitz formula give us w = 2 g − ℓ ( µ ) + d = 2 · − . Note that in the computation of the total contribution, graphs which have a vertical symmetrywill yield another representative. Thus we multiply the factor Q ω ( e ) Aut (Γ) by to factor for thiscontribution. Refer to Table 1 for the specific monodromy graphs. h , =6 + 2 + 12 + 2 + 18=40 . Single Hurwitz numbers turn out to be closely related to the intersection theory on the modulispace of stable curves. We formulate remarkable ELSV formula [7, 8] following a result of E kedahl- L ando- S hapiro- V ainshtein. It provides a strong connection between geometry of moduli spacesand the Hurwitz numbers. In practice it is a very difficult to use but it remains one of the moststriking results related to Hurwitz enumeration problem.6. The ELSV Formula Recall that the Hurwitz number h g,µ is the number of branched coverings of degree d from smoothcurves of genus g to P with one branch point (usually taken to be ∞ ∈ P ) of branched type µ = ( µ , . . . , µ n ) and w = d + n + 2 g − other simple branch points. Theorem 6.1 (The ELSV formula) . Suppose that g, n are integers ( g ≥ , n ≥ such that g − n > . Let µ = ( µ , . . . , µ n ) ⊢ d and Aut( µ ) denote the automorphism group of thepartition µ . Then, h g,µ = w ! | Aut( µ ) | n Y i =1 µ µ i i µ i ! Z M g,n − λ + . . . + ( − g λ g (1 − µ ψ ) . . . (1 − µ n ψ n ) (6.1) ORMULAE FOR CALCULATING HURWITZ NUMBERS 16 where ψ i = c (L i ) ∈ H i ( M g,n , Q ) is the first Chern class of the contagent line bundle L i −→M g,n and λ j = c j ( E ) ∈ H j ( M g,n , Q ) is the j th Chern class of the Hodge bundle E −→ M g,n − µ i ψ i = 1 + µ ψ + . . . + . . . µ ii ψ ii + . . . (Observe that the above expansion terminates because ψ i ∈ H i ( M g,n , Q ) is nilpotent.)Notice that the ELSV formula is a polynomial in the variables µ , . . . , µ n . This fact is stated inthe Golden-Jackson polynomiality conjecture [11] which this formula settles. Remark . The ELSV formula is not applicable to coverings of genus with and markedpoints since the stability condition g − n > is violated. However, the ELSV formularemains true for these two cases as well Z M , − µ ψ ) = 1 µ , and Z M , − µ ψ )(1 − µ ψ ) = 1 µ + µ . (6.2)Apart from the easy combinatorial factor, the ELSV formula involves the integrals of the form Z M g,n ψ m . . . ψ m n λ k . . . λ k g g , (6.3)called the Hodge integrals which can be reduced to other integrals only involving the ψ -classes.The latter integral are called descendant integrals [9]. The explicit evaluation of these integralsor computation of the intersection numbers is a difficult task. On the other hand, we can see thatusing the ELSV formula (6 . makes it possible to calculate the intersection numbers on M g,n once the single Hurwitz numbers are known.6.1. Hurwitz Formula via the ELSV formula. Although, the ELSV formula (6.1) is hard touse, there is a couple of very well-known cases. These cases are related to Witten conjecture [25]now known as the Kontsevich’s theorem [17]which gives a recursive relation for Hodge integralsinvolving ψ classes only. In return some of Hodge integrals can be evaluated recursively throughstring equation and the KdV hierarchy. In particular, we can recover the following well-knowncases. Theorem 6.3 ( Hurwitz Formula [15]) . The single Hurwitz Number formula h ,µ : in is given by, h ,µ = ( n + d − | Aut ( µ ) | n Y i =1 µ µ i i µ i ! d n − (6.4)where n + d − is the number of simple branch points. ORMULAE FOR CALCULATING HURWITZ NUMBERS 17 Proof. By the ELSV formula and string equation h ,µ = ( d + n − | Aut ( µ ) | n Y i =1 µ µ i i µ i ! Z M ,n − µ ψ ) . . . (1 − µ n ψ n )= ( d + n − | Aut ( µ ) | n Y i =1 µ µ i i µ i ! X m + ... + m n = n − h τ m . . . τ m n i · µ m . . . µ m n n = ( d + n − | Aut ( µ ) | n Y i =1 µ µ i i µ i ! X m + ... + m n = n − ( n − m ! . . . m n ! · µ m . . . µ m n n = ( d + n − | Aut ( µ ) | n Y i =1 µ µ i i µ i ! d n − . (cid:3) Moreover, we can recover the classical formulas of Denes, Arnol’d and Crescimano-Taylor, cf. (3.12),(3.8) and (3.9) respectively: Corollary 6.4 (Polynomial case) . If µ = ( d ) then h ,µ = ( d − d d d ! d − = d d − . Corollary 6.5 (Rational case) . If g = 0 and µ = (1 d ) then h ,µ = (2 d − d ! d d − . Corollary 6.6 (Arnol’d Case) . If g = 0 and µ = ( µ , µ ) ⊢ d then h ,µ ,µ = µ µ µ ! · µ µ µ ! · ( µ + µ − . Another well-known case with an explicit generating formula occur in the computation of genus Hurwitz numbers h ,µ . The details can be found in [14]. There has been some progress incalculation of more generalized Hurwitz numbers.7. Single Hurwitz Numbers and Fock space operator techniques Characters of the symmetric group can be easily expressed in the infinite wedge space, and so we canuse Fock space techniques to study the Hurwitz numbers. Fock space techniques arose in physics,and were introduced into Hurwitz theory in [20]. Translating Hurwitz enumeration problem intoa question of operators on the Fock space gives access to the structure of generating functions forenumerative invariants. Let V be a vector space over C with a basis indexed by half-integers V = M i ∈ Z + C · i. Write Z +1 / for the positive half integers, and Z − / for the negative half integers. A state S is asubset of half integers S = { s < s < . . . } ⊂ Z + such that both S \ Z − / and Z − / \ S arefinite. The fermionic Fock space is the vector space ^ ∞ V = M C v S ORMULAE FOR CALCULATING HURWITZ NUMBERS 18 with a basis { v S } spanned by all formal symbols in the wedge product v S = s ∧ s ∧ · · · . Denote by ( . , . ) the unique inner product on V ∞ V for which our basis { v S } is orthonormal.Observe that, the wedge product is associative, bilinear, and anticommutative, that is a ∧ b = − b ∧ a for any half integers a, b . Assign an integer c to each semi-infinite wedge v S of V ∞ V called the charge of S defined by c = | S ∩ Z − / | − | S ∩ Z +1 / | . Let V ∞ c V denotes the subspace generated by semi-infinite wedges of charge c . Then the Fockspace is decomposable by the charge: ^ ∞ V = M c ∈ Z ^ ∞ c V. We will mostly be concerned with the charge zero subspace of Fock space V ∞ V ⊂ V ∞ V – whichis the subspace spanned by all basis elements with charge . Remarkably, the charge zero subspacehas a basis indexed by integer partitions λ = ( λ , . . . , λ ℓ ( λ ) ) of all integers P : v S := v λ = λ − ∧ λ − ∧ λ − ∧ · · · λ i − i + 12 · · · . Note that the state S is given by S = { λ − , λ − , · · · λ i − i + · · · } for some uniqueinteger partition λ .7.1. Representation of a state in a Maya diagram. A useful way to represent a state or rathera basis element of the Fock Space is through a Maya diagram: a sequence of circles at centered at Z + on the real line, with the positive entries going to the left and the negative entries to theright. A black bead is placed at each position i where the corresponding vector s i appears in thewedge of the entries of S . For example, the partition λ = (4 , , , corresponds to the state S = { , , − , − , − , · · · } , the Maya diagram and wedge product below. 112 92 72 52 32 12 − − − − − v λ = v (4 , , , = ∧ ∧ − ∧ − ∧ − ∧ − ∧ · · · This notation is a gateway to an intuitive bijection between partitions P and basis elements of the -charge subspace V ∞ V . To see this, draw a partitions rotated π/ radians counterclockwise andscaled up by a factor of √ , so that each segment of the border path of λ is centered above a halfinteger on the x -axis, with origin above the square . Placing a black bead for every line segmentin λ in the direction (1 , (an upstep) above each half integer s ∈ S . For instance, the partition λ = (4 , , , corresponds to the Maya diagram shown. ORMULAE FOR CALCULATING HURWITZ NUMBERS 19 112 92 72 52 32 12 − − − − − The charge state corresponds to the empty partition represented by a special vector v ∅ calledthe vacuum vector , v ∅ = − ∧ − ∧ − ∧ − ∧ − · · · . Namely, the Maya diagram for the vacuum vector has a black bead for every negative half-integer.7.2. Operators on the fermionic Fock spaces. We now define interesting natural operatorsamong Fock spaces by their action on the basis element.I). Basic operators on Fock spaces V ∞ c V for any charge c .(a) The wedging operator ψ k : V ∞ c V −→ V ∞ c +1 V indexed by the half integers isdefined by ψ k ( v S ) = k ∧ v S = , k ∈ S ± v S ∪{ k } , k S. The sign is obtained by applying the anticommutativity of the wedge product until thesequence is decreasing. For example, ψ ( v (3 , ) = ∧ (cid:16) ∧ − ∧ − ∧ − ∧ · · · (cid:17) = − (cid:16) ∧ ∧ − ∧ − ∧ − ∧ · · · (cid:17) = − v (2 , , . (b) The contracting operator ψ ∗ k : V ∞ c +1 V −→ V ∞ c V is the adjoint of ψ k with respectto our inner product given by ψ ∗ k ( v S ) = k ∧ v S = ± v S ∪{ k } , k ∈ S , k S. These operators satisfy the anti-commutation relations: ψ i ψ ∗ j + ψ ∗ j ψ i = δ ij , ψ i ψ j + ψ j ψ i = 0 , ψ ∗ i ψ ∗ j + ψ ∗ j ψ ∗ i = 0 The operator ψ k increases the charge by , and ψ ∗ k reduces the charge by , we can applythem in sequence and preserve charge which yield interesting operators on V ∞ V . ORMULAE FOR CALCULATING HURWITZ NUMBERS 20 II). To keep track of the convergence of infinite sums of products of the wedging and contractionoperators, we define the normally ordered products of ψ i and ψ ∗ i by : ψ i ψ ∗ j := ( ψ i ψ ∗ j , j > − ψ ∗ j ψ i , j < . (a) The operator E i,j :=: ψ i ψ ∗ j : is attempting to move a bead from position j to position i if it is possible. The normal ordering puts a minus sign if j is negative.If i < j , the result of applying E i,j to a vector v λ , where λ is some partition, is toremove a ribbon to λ if possible and a sign is added according to the parity of the heightof the rim; if the ribbon cannot be removed, the result is zero. For example, E − , (cid:0) v (4 , , (cid:1) = − v (2 , , . This is illustrated in the following diagram below. · · · 92 72 52 32 12 − − − − − · · · · · · · · · If i > j , the result of applying E i,j to a vector v λ is to add a ribbon to λ if possible,and to add a sign according to the parity of the height of the rim. If the addition of theribbon is not possible, the result is zero.The case when i = j , anti-commutation relations and the normal ordering of the producttakes effect.(b) The bosonic operator α n . These operators are constructed from the fermionic oper-ators as follows. α n := X k ∈ Z + E k − n,k If n > , the operator attempts to remove ribbons of length n from the Mya diagramof λ in v λ . If there are multiple ribbons that can be removed, the operator returns thesum of all contributions, weighted by this sign. The sign is ( − h − , where h is theheight h of the ribbon removed and it is defined as the number of rows it occupies.For example, it follows that α v (5 , , = v (3 , , − V (5 , , − V (5 , . This is be illustratedusing Maya diagrams below ORMULAE FOR CALCULATING HURWITZ NUMBERS 21 · · · · · · +ve · · · · · ·· · · · · · -ve · · · · · ·· · · · · · -ve · · · · · · If n < , the operator acts in a similar way It attempts to move downsteps n places tothe right, which graphically corresponds to adding a ribbon of length n . The sign canbe calculated in the same way from the height of the added ribbon.For example, consider the effect of α − on v (3 , . We depict the action in the followingdiagrams: ORMULAE FOR CALCULATING HURWITZ NUMBERS 22 · · · · · · +ve · · · · · ·· · · · · · -ve · · · · · ·· · · · · · +ve · · · · · · It follows that α − v (3 , = v (6 , − v (3 , , − v (3 , , , , .(c) The adjoint of this operator α n can be found from the adjoint of the ψ k operators asfollows: α ∗ n = (cid:16) X k ∈ Z + ψ k ψ ∗ k − n (cid:17) ∗ = X k ∈ Z + ψ k − n ψ ∗ k = X k ∈ Z + ψ k ψ ∗ k + n = α − n ORMULAE FOR CALCULATING HURWITZ NUMBERS 23 (d) The operator, E n ( z ) –a weighted version of the α n . We denote by ζ ( z ) the function e z/ − e − z/ E n ( z ) := X k ∈ Z + e z ( k − E k − n,k + δ n, ζ ( z ) This opertor obeys the commutation relation below [ E a ( z ) , E b ( w )] = ζ ( aw − bz ) E a + b ( z + w ) (e) The content operator F , defined as F := X k ∈ Z + k E k,k . We define the vacuum expectation value of an operator P to be hP i := h |P | i , where h | is thedual of | i with respect to the inner product. The single Hurwitz numbers admit the following animmediate expression in the infinite wedge space: Theorem 7.1. The disconnected single Hurwitz numbers can be computed as expectation valuein the infinite wedge given by h • g,µ = | C µ | d ! D e α F m n Y i =1 α − µ i E . Generating functions of Hurwitz Numbers. In this section, we want to obtain the gen-erating series for the single Hurwitz numbers, giving a recursion for a single Hurwitz number interms of single Hurwitz numbers of lower genera. In the generating function, we consider bothconnected and disconnected coverings.Let p , p , p , . . . be formal commuting variables and set p = ( p , p , p , . . . ) for µ = ( µ , . . . , µ n ) ⊢ d and also p µ = p µ · · · p µ n . Now we introduce the generating functions for connected and dis-connected single Hurwitz numbers as H( t, p) = X g ≥ d,n ≥ X l ( µ )= nµ ⊢ d h g,µ p µ t w w ! (7.1) H • ( t, p) = X g ≥ d,n ≥ X l ( µ )= nµ ⊢ d h • g,µ p µ t w w ! , (7.2)where in each case the summation is over all partitions of length n and w = 2 g − d + n is thenumber of simple branch points. The p = p , p , p , · · · are parameters that encodes the cycletype of σ The parameter t counts the number of simple branch points. Since w and µ recover thegenus g , t is thus a topological parameter . ORMULAE FOR CALCULATING HURWITZ NUMBERS 24 The Cut-and-Join Equation. Hurwitz numbers satisfy combinatorial conditions of partialdifferential equations (PDEs) called the cut-and-join equation. These PDEs are only useful forvery specific branched covering with a given branch profile. In particular, single hurwitz numberssatisfy a cut-and-join equation of Goulden-Jackson in [13]. Namely, H • = exp(H) (7.3)where the exponential generating function for single Hurwitz numbers is defined to be exp (cid:0) H( t, p) (cid:1) = 1 + H( t, p) + H( t, p) 2! + H( t, p) 3! + · · · and counts disconnected single branched coverings and the power of H( t, p) is the number ofconnected components. Then the cut and join recursion takes the following form: Lemma 7.2. ∂ H • ∂t = X i,j ≥ (cid:18) p i + j · ( i · j ) · ∂∂p i · ∂∂p j | {z } τ m joins + p i · p j · ( i + j ) · ∂∂p i + j | {z } τ m cuts (cid:19) H • (7.4)We immediately deduce the cut-and-join equation of Goulden-Jackson for the generating function H( t, p) of the number of connected single Hurwitz numbers. Theorem 7.3 (Cut and Join equation, [13]) . 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