TWISTED HURWITZ NUMBERS TROPICAL AND POLYNOMIAL STRUCTURES MARVIN ANAS HAHN AND HANNAH MARKWIG Abstract. Hurwitz numbers count covers of curves satisfying fixed ramification data. Via monodromy

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TWISTED HURWITZ NUMBERS: TROPICAL AND POLYNOMIAL STRUCTURES
MARVIN ANAS HAHN AND HANNAH MARKWIG
Abstract.
Hurwitz numbers count covers of curves satisfying xed ramication data. Via monodromy
representation, this counting problem can be transformed to a problem of counting factorizations in the
symmetric group. This and other beautiful connections make Hurwitz numbers a longstanding active
research topic. In recent work [CD22], a new enumerative invariant called
𝑏
-Hurwitz number was
introduced, which enumerates non-orientable branched coverings. For
𝑏=1
, we obtain twisted Hurwitz
numbers which were linked to surgery theory in [BF21] and admit a representation as factorisations in
the symmetric group. In this paper, we derive a tropical interperetation of twisted Hurwitz numbers in
terms of tropical covers and study their polynomial structure.
1. Introduction
Hurwitz numbers are enumerations of branched morphisms between Riemann surfaces with
xed numerical data. They go back to work by Adolf Hurwitz in the 1890s [Hur92] and are now
important invariants in enumerative geometry. They admit various equivalent descriptions in the
language of dierent areas of mathematics, e.g., as shown by Hurwitz in his above-mentioned work,
they can be computed by an enumeration of transitive factorisations in the symmetric group. This
equivalence gives rise to a deep connection between Hurwitz theory and the representation theory
of the symmetric group; it will also play a key role in the present work. Moreover, Hurwitz numbers
are closely related to the algebraic topology underlying Riemann surfaces, since they turn out to be
topological invariants. While the theory of Hurwitz numbers has been dormant for most of the 20th
century, the close relationship between Hurwitz and Gromov-Witten theory discovered in the 1990s
has rekindled interest in these enumerative invariants and led to several exciting developments.
1.1. Hurwitz numbers, Gromov–Witten theory and variants. When studying relations between
Hurwitz numbers and Gromov–Witten theory, certain classes of Hurwitz numbers with particularly
well–behaved structures take center stage. Among these classes are so-called double Hurwitz numbers,
which are dened as follows.
Denition 1 (Double Hurwitz numbers).Let
𝑔≥0
be a non-negative integer,
𝑛>0
a positive integer
and
𝜇, 𝜈
partitions of
𝑛
. Moreover, we x
𝑝1, . . . , 𝑝𝑏∈P1
, where
𝑏=2𝑔−2+ℓ(𝜇) + ℓ(𝜈)
. Then, we
dene a cover of type (𝑔, 𝜇, 𝜈)to be a map 𝑓:𝑆→P1, such that
•𝑆is a connected Riemann surface of genus 𝑔;
•the ramication prole of 0is 𝜇;
•the ramication prole of ∞is 𝜈;
•the ramication prole of 𝑝1, . . . , 𝑝𝑏is (2,1. . . , 1).
Two covers
𝑓:𝑆→P1
and
𝑓′:𝑆′→P1
are called equivalent if there exists a homeomorphism
𝑔:𝑆→𝑆′, such that 𝑓=𝑓′◦𝑔.
Key words and phrases. Tropical geometry, Hurwitz numbers.
2010 Mathematics Subject Classication: 14T15, 14N10, 57M12, 05C30.
1
arXiv:2210.00595v3 [math.CO] 6 Dec 2023
2 M. A. HAHN AND H. MARKWIG
Then, we dene double Hurwitz numbers as
ℎ𝑔(𝜇, 𝜈)=
[𝑓]
1
|Aut(𝑓)| ,
where the sum runs over all equivalence classes of covers of type (𝑔, 𝜇, 𝜈).
When 𝜈=(1, . . . , 1), we call ℎ𝑔(𝜇, 𝜈)asingle Hurwitz number and denote it by ℎ𝑔(𝜇).
At the core of the relationship between double Hurwitz numbers and Gromov–Witten theory is
a polynomial structure in the prescribed ramication data of these enumerative invariants. First
discovered in the seminal work of Goulden, Jackson and Vakil in [GJV05], double Hurwitz numbers
exhibit a piecewise polynomial behaviour. More precisely, we consider the space
H𝑚.𝑛 B{(𝜇, 𝜈) ∈ N𝑚×N𝑛|𝜇𝑖=𝜈𝑗}
of partitions (𝜇, 𝜈)of xed lengths 𝑚, 𝑛 and of the same size. For xed 𝑔, we consider the map
ℎ𝑔:H𝑚,𝑛 →Q
(𝜇, 𝜈) ↦→ ℎ𝑔(𝜇, 𝜈)
which parametrises double Hurwitz numbers.
The authors of [GJV05] showed that there exists a hyperplane arrangement
R𝑚,𝑛
in
H𝑚,𝑛
(called
the resonance arrangement), such that the map
ℎ𝑔
restricted to each connected component (called
chamber) of
H𝑚,𝑛\R𝑚,𝑛
may be represented as a polynomial in the entries of
𝜇
and
𝜈
. In [SSV08;
CJM11; Joh15], the natural question of how the polynomials dier from chamber to chamber was
studied. It was observed that there is a recursive structure in the sense that this dierence can be
expressed by double Hurwitz numbers with smaller input data. This is called a wall-crossing formula.
We want to highlight the work in [CJM11], in which a graph theoretic approach towards the polynomi-
ality of double Hurwitz numbers was established. The key technique in this paper revolves around the
eld of tropical geometry. Tropical geometry is a relatively new eld of mathematics, which may be
described as a combinatorial shadow of algebraic geometry. The tropical geometry perspective allows
to degenerate algebraic curves to certain metric graphs that are called tropical curves. In this manner,
branched morphisms between Riemann surfaces are tropicalised to maps between tropical curves that
are called tropical covers. Motivated by this point of view, a combinatorial interpretation of double
Hurwitz numbers in terms of tropical covers was derived in [CJM10], which laid the groundwork
for the analysis of the polynomial behaviour of double Hurwitz numbers undertaken in [CJM11].
In particular, by proceeding along an intricate combinatorial analysis of tropical covers in dierent
chambers, the authors of [CJM11] were able to derive the desired wall-crossing structure.
In the past years, several variants of Hurwitz numbers have appeared in the literature in a plethora of
dierent contexts. Among the most prominent ones are so-called pruned Hurwitz numbers [DN18;
Hah20], monotone Hurwitz numbers [GGN14], strictly monotone Hurwitz numbers [KZ15], completed
cycles Hurwitz numbers [OP06] and many more. For all of these variants the piecewise polynomiality
of the double Hurwitz numbers analogue was established and for the majority a wall-crossing struc-
ture as well (see e.g. [Hah17; HKL18; HL20; SSZ12]).
While classical Hurwitz theory deals with the enumeration of branched morphisms between orientable
surfaces, it is very natural to ask for an analogous theory for non-orientable surfaces. Such a new and
exciting theory for so-called 𝑏-Hurwitz numbers was introduced in [CD22] .
1.2. Twisted Hurwitz numbers. The construction of
𝑏
-Hurwitz numbers is based on the following
idea: Let
H
be the compactied complex upper halfplane of
P1
and
J
the corresponding natural
involution on
P1
. A generalised branched covering is a covering
𝑓:𝑆→H
where
𝑆
is a not-necessarily
connected compact orientable surface with orientation double cover
ˆ
𝑆
, such that
𝑓
may be "lifted"
TROPICAL TWISTED HURWITZ NUMBERS 3
to a branched covering
ˆ
𝑆→P1
. We give a precise formulation in Section 2. Via these generalised
coverings the authors of [CD22] introduce a new one-parameter deformation of classical Hurwitz
numbers called
𝑏
-Hurwitz numbers in reference to the
𝑏
-conjecture by Goulden and Jackson in the
context of Jack polynomials [GJ96]. In order to obtain
𝑏
-Hurwitz numbers, the authors of [CD22]
associate a non-negative integer
𝜈𝑝(𝑓)
to any generalised branched covering
𝑓
which "measures" the
non-orientability of the the surface
˜
𝑆
. This non-negative integer is zero if and only if
˜
𝑆
is orientable.
Based on this idea
𝑏
-Hurwitz numbers are dened – depending on a measure of non-orientability
𝑝
–
as a sum over generalised branched coverings weighted by
𝑏𝜈𝑝(𝑓)
. Thus, one obtains a Hurwitz-type
enumerations for any value of
𝑏
. For example, under the convention that
00=1
, one recovers classical
Hurwitz numbers for
𝑏=0
. For
𝑏=1
one obtains enumerations of generalised branched coverings
which are called twisted Hurwitz numbers. This is the case we study in the present paper.
The term twisted Hurwitz numbers was coined in [BF21] in the context of surgery theory. Surgery
theory studies the construction of new manifolds from given ones via cutting and glueing, such that
key properties are preserved. In [BF21], the enumeration of decompositions of a given surface with
boundary and marked points is studied. The term twisted is motivated by the fact in [BF21] gluings
are performed with respect to a twist of the natural boundary orientations. It was proved in [BF21]
that the enumeration of certain decompositions with respect to such a twist may be computed in
terms of factorisations in the symmetric group, reminiscent of Hurwitz’ result in his original work
[Hur92]. More precisely, we x the involution
𝜏=(1𝑛+1)(2𝑛+2). . . (𝑛2𝑛) ∈ S2𝑛
and use the notation
𝐵𝑛=𝐶(𝜏)={𝜎∈S2𝑛|𝜎𝜏𝜎−1=𝜏}, 𝐶∼(𝜏)={𝜎∈S2𝑛|𝜏𝜎𝜏 −1=𝜏𝜎𝜏 =𝜎−1}
where
𝐵𝑛
is the hyperoctahedral group. We further dene the subset
𝐵∼
𝑛⊂𝐶∼(𝜏)
consisting of
those permutations that have no self-symmetric cycles (see [BF21, Lemma 2.1]). We then set, for a
partition
𝜆
of
𝑛
,
𝐵∼
𝜆⊂𝐵∼
𝑛
to consist of those permutations that have
2ℓ(𝜆)
cycles, two of length
𝜆𝑖
for
each i, that pair up under conjugation with 𝜏. We are now ready to dene twisted single Hurwitz
numbers in terms of the symmetric group.
Denition 2 (Twisted single Hurwitz numbers, [BF21]).Fix a partition
𝜆
of
𝑛
and a number
𝑏
(the
number of transpositions). Then dene
˜
ℎ𝑏(𝜆)=1
𝑛!♯n(𝜎1, . . . , 𝜎𝑏) | 𝜎𝑠=(𝑖𝑠𝑗𝑠), 𝑗𝑠≠𝜏(𝑖𝑠), 𝜎1. . . 𝜎𝑏(𝜏𝜎𝑏𝜏). . . (𝜏𝜎1𝜏) ∈ 𝐵∼
𝜆o.
Maybe surprisingly, it was then proved in [BF21, Theorem 3.2] that these numbers coincide with
𝑏
-Hurwitz numbers for
𝑏=1
by showing that the generating series’ of both invariants satisfy the
same PDE with equal initial data.
1.3. Tropical geometry of twisted Hurwitz numbers. The present paper develops a tropical
theory of twisted Hurwitz numbers and demonstrates some rst applications. In Section 2, we dene a
generalisation Denition 2 to twisted double Hurwitz numbers
˜
ℎ𝑔(𝜇, 𝜈)
which by the same arguments
as in [BF21, Theorem 3.2] coincides with
𝑏
-Hurwitz numbers for
𝑏=1
. This generalisation arises
naturally from the symmetric group expression for twisted single Hurwitz numbers. Moreover, we
dene in Section 3 a tropical analogue of twisted double Hurwitz numbers in terms of tropical covers.
We prove in Section 4 that twisted double Hurwitz numbers coincide with their tropical counterpart,
thus giving a tropical correspondence theorem for these enumerative invariants. This allows us to
derive a purely graph-theoretic interpretation of twisted double Hurwitz numbers in Section 5 by
reinterpreting the tropical covers as directed graphs. Finally, we employ this expression of twisted
double Hurwitz numbers as a weighted enumeration of directed graphs to study the polynomiality of
4 M. A. HAHN AND H. MARKWIG
twisted Hurwitz numbers in Section 6 improving [CD22, Theorem 6.6] in the case
𝑏=1
. Finally, we
discuss the wall-crossing behaviour of twisted Hurwitz numbers.
Acknowledgements. We would like to thank an anonymous referee for their thorough work and for
numerous helpful suggestions on how to improve the rst versions of this paper. We thank Raphaël
Fesler and Veronika Körber for useful discussions and comments. The second author acknowledges
support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID
286237555, TRR 195. Computations have been made using the Computer Algebra System gap and
OSCAR [15; 22].
2. Twisted double Hurwitz numbers
In this section, we dene twisted double Hurwitz numbers as a factorization problem in the
symmetric group, generalizing the case of single twisted Hurwitz numbers discussed in Subsection
1.2. We use the notation of Subsection 1.2. To begin with, we recall that
𝐶∼(𝜏)={𝜎∈S2𝑛|𝜏𝜎𝜏 −1=𝜏𝜎𝜏 =𝜎−1}.
It was proved in [BF21, Lemma 2.1] for
𝜎∈𝐶∼(𝜏)
with the decomposition in cycles
𝜎=𝑐1· · · 𝑐𝑚
,
we have for any 𝑖that either
•there exists 𝑗≠𝑖with 𝜏𝑐𝑖𝜏=𝑐−1
𝑗or
•we have 𝜏𝑐𝑖𝜏=𝑐−1
𝑖and 𝑐𝑖has even length.
In the rst case,
𝑐𝑖
and
𝑐𝑗
are called
𝜏
-symmetric, while in the second case
𝑐𝑖
is called self-symmetric.
As mentioned above, we denote by
𝐵∼
𝑛⊂𝐶∼(𝜏)
the set of permutations without self-symmetric cycles
and by
𝐵∼
𝜆⊂𝐵∼
𝑛
the set of permutations in
𝐵∼
𝑛
with
2ℓ(𝜆)
cycles and for each
𝑖
two cycles of length
𝜆𝑖
that are 𝜏-symmetric.
Denition 3 (Cycle type).Let
𝜎∈𝐵∼
𝑛
. We denote its cycle type by
𝐶(𝜎)
, which is a partition of
2𝑛
recording the lengths of the cycles of 𝜎.
For a partition
𝜇
of
𝑛
we denote by
2𝜇
the partition of
2𝑛
with twice as many parts, where each
part is repeated once.
We may now dene twisted double Hurwitz numbers generalising Denition 2.
Denition 4 (Twisted double Hurwitz numbers).Let
𝑔≥0, 𝑛 >0
and
𝜇, 𝜈
partitions of
𝑛
. We dene
𝐶𝑔(𝜇, 𝜈)as the set of tuples (𝜎1, 𝜂1, . . . , 𝜂𝑏, 𝜎2), such that we have:
(1) 𝑏=2𝑔−2+2ℓ(𝜇)+2ℓ(𝜈)
2>0,
(2) 𝐶(𝜎1)=2𝜇,𝐶(𝜎2)=2𝜈,𝜂𝑖are transpositions satisfying 𝜂𝑖≠𝜏𝜂𝑖𝜏,
(3) 𝜎1∈𝐵∼
𝜇,
(4) 𝜂𝑏· · · 𝜂1𝜎1(𝜏𝜂1𝜏) · · · (𝜏𝜂𝑏𝜏)=𝜎2,
(5) the subgroup
⟨𝜎1, 𝜂1, . . . , 𝜂𝑏, 𝜏𝜂1𝜏, . . . , 𝜏𝜂𝑏𝜏, 𝜎2⟩
acts transitively on the set {1, . . . , 2𝑑}
Then, we dene the associated twisted double Hurwitz number as
˜
ℎ𝑔(𝜇, 𝜈)=1
(2𝑛)!! |𝐶𝑔(𝜇, 𝜈)|.
When we drop the transitivity condition, we obtain possibly disconnected twisted double Hurwitz
numbers which we denote ˜
ℎ•
𝑔(𝜇, 𝜈).
Remark 5 (Conventional dierences).Compared with the denition of twisted single Hurwitz
numbers in Denition 2, there are two conventional dierences:
TROPICAL TWISTED HURWITZ NUMBERS 5
(1)
Rather than the number of transpositions, we use the genus (of the source of a twisted tropical
cover, see Denition 11) as subscript in the notation. In the usual case of Hurwitz numbers
(without twisting) counting covers, this corresponds to the genus of the source curves. By
the Riemann-Hurwitz formula, the genus
𝑔
and the number
𝑏
of transpositions (i.e. simple
branch points) are related via
𝑏=2𝑔−2+2ℓ(𝜇) + 2ℓ(𝜈)
2.
(2)
We choose to normalize with the factor
1
(2𝑛)!!
rather than
1
𝑛!
. This leads to nicer formulae and
structural results.
Remark 6 (Connectedness and transitivity).As noted above, one can also drop the transitivity
condition (5) in Denition 4 to obtain
ℎ•
𝑔(𝜇, 𝜈)
. On the tropical side, this amounts to allowing
disconnected tropical curves as source of a twisted tropical cover, see Denition 11. In this setting,
it can happen that a disconnected twisted tropical cover contains two twisted components which
both just correspond to a single edge without any interior vertex. This also happens in the connected
case if we allow
𝑏=0
. For this case, one has to adapt the tropical multiplicity we set in Denition
15: a twisted tropical cover which consists of a pair of twisted single edges of weight
𝜇
each has
multiplicity 1
𝜇. With this adaption, one can easily generalize our results to the disconnected case.
As mentioned in the introduction, twisted Hurwitz numbers rst appeared in the Hurwitz theory
of non-orientable surfaces. More precisely, we denote by
J:P1→P1
the complex conjugation,
by
HB{𝑧∈C|Im(𝑧) ≥ 0}
the complex upper halfplane and let
HBH∪ {∞}
the compactied
complex upper halfplane. Moreover, we denote by 𝜋:P1→Hthe quotient map.
We call continuous maps
𝑓:𝑆→H
generalised branched coverings, if
𝑆
is a not-necessarily orientable
surface and there exists a further map ˆ
𝑓:ˆ
𝑆→P1with
(1) 𝑝:ˆ
𝑆→𝑆is the orientation double cover,
(2) 𝜋◦ˆ
𝑓=𝑓◦𝑝,
(3) all the branch points of ˆ
𝑓are real.
Let
T:ˆ
𝑆→ˆ
𝑆
be an orientation reversing involution without x points, such that
𝑝◦ T =𝑝
. Then,
the second condition of
ˆ
𝑓
may be reformulated as
ˆ
𝑓◦ T =J ◦ ˆ
𝑓
. As
T
has no xed points, for any
branch points
𝑐∈P1
R⊂P1
of
ˆ
𝑓
, the points in its pre-image come in pairs
(𝑎, T (𝑎))
with the same
ramication index. Thus, the degree of
ˆ
𝑓
is even and the ramication prole of
𝑐
repeats any entry
twice, e.g.
(𝜆1, 𝜆1, . . . , 𝜆𝑠, 𝜆𝑠)
. We then say that
𝑓
has ramication prole
(𝜆1, . . . , 𝜆𝑠)
at
𝜋(𝑠) ∈ 𝜕H
.
We further call two generalised branched coverings
𝑓1
and
𝑓2
equivalent if their lifts
ˆ
𝑓1
and
ˆ
𝑓2
are, and
denote the equivalence class of 𝑓by [𝑓]. We are now ready to dene
Denition 7. Let
𝑔≥0
,
𝑛>0
,
𝜇, 𝜈
partitions of
𝑛
. Let
𝑏=2𝑔−2+2ℓ(𝜇)+2ℓ(𝜈)
2
and x
𝑝1, . . . , 𝑝𝑏
pairwise
distinct real points on
H
. We dene
𝐺𝑔(𝜇, 𝜈)
as the set of equivalence classes
[𝑓]
of generalised
branched coverings 𝑓:𝑆→H, such that
•𝑓is of degree 𝑛,
•𝑓has ramication prole 𝜇over 0and 𝜈over ∞,
•𝑓has ramication prole (2,1, . . . , 1)over 𝑝𝑖.
Then, we dene 1-Hurwitz numbers as
ℎ1
𝑔(𝜇, 𝜈)=
[𝑓] ∈𝐺𝑔(𝜇,𝜈)
1
|Aut(𝑓)| .
摘要：

TWISTEDHURWITZNUMBERS:TROPICALANDPOLYNOMIALSTRUCTURESMARVINANASHAHNANDHANNAHMARKWIGAbstract.Hurwitznumberscountcoversofcurvessatisfyingfixedramificationdata.Viamonodromyrepresentation,thiscountingproblemcanbetransformedtoaproblemofcountingfactorizationsinthesymmetricgroup.Thisandotherbeautifulconnec...

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