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Journal ArticleDOI

The Hurwitz Enumeration Problem of Branched Covers and Hodge Integrals

TL;DR: In this article, the simple Hurwitz numbers for arbitrary source and target Riemann surfaces were computed using algebraic methods, motivated by the Gromov-Witten potentials.
About: This article is published in Journal of Geometry and Physics.The article was published on 2004-04-01 and is currently open access. It has received 33 citations till now. The article focuses on the topics: Riemann–Hurwitz formula & Hurwitz's automorphisms theorem.

Summary (3 min read)

1 Introduction

  • Many classical problems in enumerative geometry have been receiving renewed interests in recent years, the main reason being that they can be translated into the modern language of Gromov-Witten theory and, moreover, that they can be consequently solved.
  • Generalizing this analogy, the authors have determined the generating functions for arbitrary targets in terms of the representation theory of the symmetric group Sn.
  • The character of the irreducible representation γ.

2 Computations of Simple Hurwitz Numbers

  • This section describes their computations of the simple Hurwitz numbers µ̃g,nh,n.
  • The simple covers of an elliptic curve by elliptic curves are actually unramified, and the authors obtain the numbers µ̃1,n1,n by using the standard theory of two-dimensional lattices 2.
  • For other values of g and h, the authors simplify the general formulas of Mednykh [M1] and explicitly compute the numbers for low degrees.

2.2 Low Degree Computations from the Work of Mednykh

  • The most general Hurwitz enumeration problem for an arbitrary branch type has been formally solved by Mednykh in [M1].
  • His answers are based on the original idea of Hurwitz of reformulating the ramified covers in terms of the representation theory of Sn [H].
  • Thus, the Hurwitz enumeration problem reduces to counting the number of orbits in Tn,h,σ under the action of Sn by conjugation.
  • Interestingly, the general formula (2.5) still has some applicability.
  • In [M2], Mednykh considers the special case of branch points whose orders are all equal to the degree of the cover and obtains a simplified formula which is suitable for practical applications.

2.2.3 Even Degree Covers

  • The computations of fixed-degree-n simple Hurwitz numbers are thus reduced to com- puting the two numbers.
  • The authors now compute these numbers for some low degrees and arbitrary genera h and g.
  • The nature of the computations is such that the authors only need to know the characters of the identity and the transposition elements in Sn.
  • The first equality follows from the fact that the order of a finite group is equal to the sum of squares of the dimension of its irreducible representations.
  • The second equality follows by noticing that the expression for Tn,0,(s̃p k )/n! is the n-th coefficient of the formal q-expansion of log( ∑∞ n=0 q n/n!)the authors.

2.3 Cautionary Remarks

  • The master formula obtained by Mednykh uses the Burnside’s formula to account for the fixed points.
  • In the case of simple Hurwitz numbers, this will lead to an apparent discrepancy between their results and those obtained by others for even degree covers, the precise reason being that for even degree covers, say of degree-2n, the action of (2n) ∈ S2n on T2n,h,σ has fixed points which are counted by the second term in (2.17).
  • For odd degree cases, there is no non-trivial fixed points, and their formula needs no adjustment.
  • Using the approach described in the previous subsection, the authors have obtained the closed-form formulas forNn,h,r for n < 8.
  • For h = 1, its ` = 1 parts agree with the known free energies Fg. Although their results are rewarding in that they give explicit answers for all g and h, further computations become somewhat cumbersome beyond degree 8.

3 Generating Functions for Simple Hurwitz Numbers

  • Recently, Göttsche has conjectured an expression for the generating function for the number of nodal curves on a surface S, with a very ample line bundle L, in terms of certain universal power series and basic invariants [G].
  • In a kindred spirit, it would be interesting to see whether such universal structures exist for Hurwitz numbers.
  • For a curve, the analogues of KS and c2(S) would be the genus of the target and L the degree of the branched cover.
  • It turns out that for simple Hurwitz numbers, the authors are able to find their generating functions in closed-forms, but the resulting structure is seen to be more complicated than that for the case of surfaces.

3.1 Summing up the String Coupling Expansions

  • That is, the authors are summing up the string coupling expansions, and this computation is a counterpart of “summing up the world-sheet instantons” which string theorists are accustomed to studying.
  • The coefficient of λ2g−2qn in Φ(1) contains precisely what the authors are looking for, namely µg,n1,n.
  • This recursive method also works for determining the general simple Hurwitz numbers µg,nh,n, upon using the general “partition function” (3.10) in place of Z.

3.2 The Generating Functions for Target Curves of Arbitrary Genus

  • For arbitrary genus targets, there is a natural generalization of the above discussion on the generating functions.
  • Then, the authors have 5Unfortunately, they have previously used the notation tpk to denote the branching matrix.
  • The proof is exactly the same as that of Claim 3.1.
  • For definitions of Hodge integrals, see [FP, FP2].

4.1 Generating Functions for Hodge Integrals

  • The Hurwitz enumeration problem has been so far investigated intensely mainly for branched covers of the Riemann sphere.
  • After long and tedious computations, the authors arrive at the following results: (a) There are 2k “one-hook” diagrams.

4.3 Possible Extensions

  • T 2g. (4.34) 7For k non-positive integers and half-integers, the below expression of G(t,−k) appears to be divergent.
  • For these cases, one might try first expanding G(t,−k) in t and setting k equal to the desired values.
  • Sn and the geometry of the moduli space of marked Riemann surfaces.
  • Of course, Gn(t,−1) can be also explicitly computed from their previous computations of the simple Hurwitz numbers Hg,n.

5 Conclusion, or An Epilogue of Questions Unanswered

  • To recapitulate, the first part of their paper studies the simple branched covers of Riemann surfaces by Riemann surfaces of arbitrary genera.
  • Upon fixing the degree of the irreducible covers, the authors have obtained closed form answers for simple Hurwitz numbers for arbitrary source and target Riemann surfaces, up to degree 7.
  • For higher degrees, the authors have given a general prescription for extending their results.
  • The authors general answer (3.10) for an arbitrary target genus differs from the elliptic curve case only by the prefactor (n!/fγ)2h−2.
  • The recursion relations and the Virasoro constraints seem to lose their efficacy when one considers the Gromov-Witten invariants of an elliptic curve.

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Citations
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TL;DR: In this paper, an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of points, and conjecture this to be true for an arbitrary number of Ramification points.
Abstract: We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification points. In addition, the conjecture is proved for simple coverings of the sphere by the torus. We obtain corresponding expressions for surfaces of higher genera for a small number of ramification points, and conjecture the general form for this number in terms of a symmetric polynomial that appears to be new. The approach involves the analysis of the action of a transposition to derive a system of linear partial differential equations that give the generating series for the desired numbers.

96 citations

Journal ArticleDOI
TL;DR: In this article, the existence of a branched covering e!U between closed surfaces is studied in terms of. eU/,.U/, orientability, the total degree, and the local degree at the branching points.
Abstract: For the existence of a branched covering e!U between closed surfaces there are easy necessary conditions in terms of . eU/, .U/ , orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also sufficient. Thanks to the work of many authors, the problem remains open only when U is the sphere, in which case exceptions to existence are known to occur. In this paper we describe new infinite series of exceptions, in particular previously unknown exceptions with e not the sphere and with more than three branching points. All our series come with systematic explanations, based on several different techniques (including dessins d’enfants and decomposability) that we exploit to attack the problem, besides Hurwitz’s classical technique based on permutations. Using decomposability we also establish an easy existence result. 57M12; 57M30, 57N05

56 citations

Posted Content
TL;DR: In this paper, Liu and Kefeng Liu proved some combinatorial results related to a formula on Hodge integrals conjectured by Mari\~no and Vafa.
Abstract: We prove some combinatorial results related to a formula on Hodge integrals conjectured by Mari\~no and Vafa These results play important roles in the proof and applications of this formula by the author jointly with Chiu-Chu Melissa Liu and Kefeng Liu We also compare with some related results on Hurwitz numbers and obtain some closed expressions for the generating series of Hurwitz numbers and the related Hodge integrals

51 citations

Journal ArticleDOI
TL;DR: For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula as mentioned in this paper.
Abstract: For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary relations have been khown to be also sufficient for the existence of the covering except when the base surface is the sphere (and when it is the projective plane, but this case reduces to the case of the sphere). If the base surface is the sphere, many exceptions are known to occur and the problem is widely open. Generalizing methods of Baranski, we prove in this paper that the necessary relations are actually sufficient in a specific but rather interesting situation. Namely under the assumption that the base surface is the sphere, that there are three branching points, that one of these branching points has only two preimages with one being a double point, and either that the covering surface is the sphere and that the degree is odd, or that the covering surface has genus at least one, with a single specific exception. For the case of the covering surface the sphere we also show that for each even degree there are precisely two exceptions.

50 citations


Cites methods from "The Hurwitz Enumeration Problem of ..."

  • ...A variation of Mednykh’s formulae, where some explicit computation is possible, was recently derived in [16] for special types of branched coverings....

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TL;DR: In this article, the Riemann-Hurwitz formula for a branched cover between two closed orientable surfaces is analyzed from the point of view of the geometry of 2-orbifolds.
Abstract: For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whether a collection of partitions of an integer having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever the candidate base surface is not the 2-sphere, while for the 2-sphere exceptions do occur. A long-standing conjecture however asserts that when the candidate degree is a prime number, a candidate cover is always realizable. In this paper we analyze the question from the point of view of the geometry of 2-orbifolds, and we provide strong supporting evidence for the conjecture. In particular, we exhibit three different sequences of candidate covers, indexed by their degree, such that for each sequence: (1) The degrees giving realizable covers have asymptotically zero density in the naturals; (2) Each prime degree gives a realizable cover.

18 citations

References
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Journal ArticleDOI
TL;DR: The Riemannsche Theorie der algebraischen Funktionen as discussed by the authors is a Theorie von der graphisch uber der komplexen Zahlenebene.
Abstract: Die grundlegende Bedeutung des vorliegenden Themas fur die Riemann’sche Theorie der algebraischen Funktionen brauche ich wohl kaum hervorzuheben. Geht doch diese Theorie von der graphisch uber der komplexen Zahlenebene konstruierten Riemann’schen Flache aus, um erst sodann die Funktionen, welche durch diese Flache bestimmt sind, zu untersuchen.

605 citations

Journal ArticleDOI
TL;DR: In this article, a universal system of differential equations is proposed to determine the generating function of the Chern classes of the Hodge bundle in Gromov-Witten theory for any target X. The genus g, degree d multiple cover contribution of a rational curve is found to be simply proportional to the Euler characteristic of M_g.
Abstract: Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these integrals from the standard descendent potential (for any target X). We use virtual localization and classical degeneracy calculations to find trigonometric closed form solutions for special Hodge integrals over the moduli space of pointed curves. These formulas are applied to two computations in Gromov-Witten theory for Calabi-Yau 3-folds. The genus g, degree d multiple cover contribution of a rational curve is found to be simply proportional to the Euler characteristic of M_g. The genus g, degree 0 Gromov-Witten invariant is calculated (in agreement with recent string theoretic calculations of Gopakumar-Vafa and Marino-Moore). Finally, with Zagier's help, our Hodge integral formulas imply a general genus prediction of the punctual Virasoro constraints applied to the projective line.

603 citations


"The Hurwitz Enumeration Problem of ..." refers methods in this paper

  • ...egrals over the moduli space Mg,1. More precisely, the formula F(t,k) := 1 + X g≥1 t2g Xg i=0 ki Z Mg,1 ψ2g−2+i λ g−i = t/2 sin(t/2) k+1 (4.3) which was first obtained by Faber and Pandharipande in [FP] by using virtual localization techniques has been rederived by Ekedahl et al. in [ELSV] by using the generating function for Hurwitz numbers for branched covers whose only non-simple branch point has...

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TL;DR: In this paper, the authors explore the possibility of finding an equivalent string representation of two-dimensional QCD, and develop the large N expansion of the QCD2 partition function on an arbitrary 2-dimensional euclidean manifold.

498 citations

Journal ArticleDOI
TL;DR: In this article, Teke et al. presented a survey of the state-of-the-art mathematics departments in Sweden, including the Department of Mathematics, University of Stockholm, S-10691, Stockholm, Sweden (e.g.
Abstract: 1 Department of Mathematics, University of Stockholm, S-10691, Stockholm, Sweden (e-mail: teke@matematik.su.se) 2 Higher College of Mathematics, Independent University of Moscow and Institute for System Research RAS, Moscow, Russia (e-mail: lando@mccme.ru) 3 Department of Mathematics, Royal Institute of Technology, S-10044, Stockholm, Sweden (e-mail: mshapiro@math.kth.se) 4 Department of Mathematics and Department of Computer Science, University of Haifa, Haifa 31905, Israel (e-mail: alek@mathcs.haifa.ac.il)

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Book ChapterDOI
01 Jan 1995
TL;DR: In this article, the authors review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are quasimodular forms for the mirror family; they can be computed by a summation over trivalent Feynman graphs.
Abstract: I review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: (1) counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are ‘quasimodular forms’ for the mirror family; (2) they can be computed by a summation over trivalent Feynman graphs.

250 citations

Frequently Asked Questions (16)
Q1. What are the contributions in "The hurwitz enumeration problem of branched covers and hodge integrals" ?

Motivated by the Gromov-Witten potentials, the authors find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. 

In particular, they arise naturally in combinatorics, as they count factorizations of permutations into transpositions, and the original idea of Hurwitz expresses them in terms of the representation theory of the symmetric group. 

by relating the simple Hurwitz numbers to descendant Gromov-Witten invariants, the authors have obtained the explicit generating functions (3.10) for the number of inequivalent reducible covers for arbitrary source and target Riemann surfaces. 

(2.8)Because v determines the range of the first sum in the master formula, the authors need to distinguish when the degree n is odd or even. 

An interesting application of this development is to use the generating function for Hurwitz numbers µg,d0,1(d) to derive a generating function for Hodge integrals over the moduli spaceMg,1. 

More precisely, when the summation variable s in their formula equals (g + 1)/2, for an odd genus g, there is a symmetry factor of 1/2 in labeling the edges because the two disconnected graphs are identical except for the labels. 

Many classical problems in enumerative geometry have been receiving renewed interests in recent years, the main reason being that they can be translated into the modern language of Gromov-Witten theory and, moreover, that they can be consequently solved. 

To each irreducible representation labeled by ρ = (ρ1, . . . , ρj) ` 2k, the authors can associate a Young diagram with j rows, the ith row having length ρi. 

the equivalence relation of covers gets translated into conjugation by a permutation in Sn, i.e. two elements of Tn,h,σ are now equivalent iff they are conjugates. 

for a given lattice L associated with the target elliptic curve, the authors need to find the number of inequivalent sublattices L′ ⊂ L of index [L : L′] = n. 

The generalized “partition function” Z(h) = exp(Φ(h)) for all h is given byZ(h) = 1 + q + ∑ n≥2 ∑ γ∈Rn ( n! fγ )2h−2 cosh [( n 2 ) χγ(2) fγ λ ] qn. (3.10)Proof: 

these recursion relations require as initial data the knowledge of simple Hurwitz numbers, and their work would be useful for applying the relations as well. 

Upon fixing the degree of the irreduciblecovers, the authors have obtained closed form answers for simple Hurwitz numbers for arbitrary source and target Riemann surfaces, up to degree 7. 

In particular, Li et al. have obtained a set of recursion relations for the numbers µg,nh,w(α) by applying the gluing formula to the relevant relative GW invariants [LZZ]. 

The generating function G(t, k) can be evaluated at k = −1 to beG(t,−1) = 1 2 − 1 t2( cos t+ t22 − 1) = 12( sin(t/2)t/2)2 , (4.8)and similarly at k = 0 to beG(t, 0) = 12( tsin t) = 12t/2sin(t/2)1cos(t/2) . (4.9)Proof: 

The asymptotic limit of G(t, k) isG(t k 1 2 , k−1) k→0−→ exp( t2/3 ) 2 t √ πErf [ t 2 ] , (4.15)and thus, the integrals can be evaluated to be∫ Mg,2 1 (1− ψ1)(1− ψ2) = 1 2 g∑ m=01 m! 12m (g −m)! (2g − 2m+ 1)! . (4.16)Proof: