# Principal bundles on metric graphs: the $\operatorname{GL}_{n}$ case

Andreas Gross  Institut für Mathematik, Goethe–Universität Frankfurt, 60325
Frankfurt am Main, Germany<EMAIL_ADDRESS>,  Martin Ulirsch
Institut für Mathematik, Goethe–Universität Frankfurt, 60325 Frankfurt am
Main, Germany<EMAIL_ADDRESS>and  Dmitry Zakharov  Department
of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA
<EMAIL_ADDRESS>
###### Abstract.

Using the notion of a root datum of a reductive group $G$ we propose a
tropical analogue of a principal $G$-bundle on a metric graph. We focus on the
case $G=\operatorname{GL}_{n}$, i.e. the case of vector bundles. Here we give
a characterization of vector bundles in terms of multidivisors and use this
description to prove analogues of the Weil–Riemann–Roch theorem and the
Narasimhan–Seshadri correspondence. We proceed by studying the process of
tropicalization. In particular, we show that the non-Archimedean skeleton of
the moduli space of semistable vector bundles on a Tate curve is isomorphic to
a certain component of the moduli space of semistable tropical vector bundles
on its dual metric graph.

###### Contents

  1. 1 Line bundles on metric graphs
  2. 2 Principal bundles and vector bundles on metric graphs
  3. 3 Vector bundles and multidivisors
  4. 4 Baker–Norine rank and a Weil–Riemann–Roch theorem
  5. 5 Semistable bundles and a Narasimhan–Seshadri correspondence
  6. 6 The process of tropicalization
  7. 7 The case of a Tate curve

## Introduction

Since its introduction to the mathematical world in [MZ08], the theory of
divisors and linear systems on metric graphs has stimulated a multitude of
further work during the last decade. It naturally aligned with the
simultaneously developed theory of divisors, linear systems, and chip-firing
on finite graphs introduced in [BN07], whose first highlight is the
Riemann–Roch theorem in [BN07] (also see [MZ08, GK08] for its generalization
to metric graphs). Using Baker’s method of specialization [Bak08], new
applications to the classical subject of Brill–Noether theory on an algebraic
curve were found (see [BJ16] for an overview). We mention the groundbreaking
paper [CDPR12], which includes a new proof of the Brill–Noether theorem and
point to [Pfl17, JR21, CPJ19, CPJ20, LU21] for (Hurwitz–)Brill–Noether
theorems with gonality conditions (also see [Lar21] for a non-tropical
approach) and [FJP20, FJP21] for applications to the birational geometry of
moduli spaces, which both take us significantly beyond what was known
classically.

What all of these works have in common is that they always focus on the case
of divisors or equivalently of line bundles, i.e. the (very much abelian) rank
one situation. In this article we endeavour towards a (non-abelian) higher
rank generalization and use the notion of a root datum (coming from the
classification of reductive groups) to propose a tropical analogue of
principal $G$-bundles for a reductive group $G$.

Let $\Gamma$ be a metric graph and let $\mathcal{H}_{\Gamma}$ be the sheaf of
real-valued harmonic functions with integer slopes on $\Gamma$. When
$G=\operatorname{GL}_{n}$, our definition amounts to considering
$S_{n}\ltimes\mathcal{H}_{\Gamma}^{n}$-torsors on $\Gamma$, an object that we
refer to as a _vector bundle of rank_ $n$ on $\Gamma$. This definition agrees
with the geometric version of tropical vector bundles as defined in [All12].
In Section 3 we show that any vector bundle of rank $n$ on $\Gamma$ is the
pushforward of a line bundle along a suitable free cover of $\Gamma$ of degree
$n$, and use this observation to describe isomorphism classes vector bundles
on $\Gamma$ as linear equivalence classes of so-called _multidivisors_ on
$\Gamma$, a notion inspired by Weil’s matrix divisors coming from [Wei38].
This description, for example, allows for the classification of vector bundles
on metric graphs of genus zero in analogy with the Birkhoff–Grothendieck
theorem [Gro57] (see Example 3.3 below) and of genus one in analogy with
Atiyah’s classication in [Ati57] (see Section 3.3 below).

### A tropical Weil–Riemann–Roch theorem

In [Wei38], Weil generalized the classical Riemann–Roch theorem to the case of
vector bundles on a compact Riemann surface. From a modern perspective, this
is now an immediate corollary of the Hirzebruch–Riemann–Roch theorem. Given a
vector bundle $E$ on a compact Riemann surface $X$ of genus $g$, the
Hirzebruch–Riemann–Roch formula states that

$h^{0}(X,E)-h^{1}(X,E)=\int_{X}\operatorname{ch}(E)\cdot\operatorname{td}(X)\
.$

The Chern character of $E$ and the Todd class of $X$ are given respectively by

$\operatorname{ch}(E)=c_{0}(E)+c_{1}(E)=\operatorname{rk}(E)+\deg(E)\cdot[X]$

and

$\operatorname{td}(X)=c_{0}(T_{X})+\frac{1}{2}c_{1}(T_{X})=1-(g-1)\cdot[X],$

while $h^{1}(X,E)=h^{0}(X,E^{\ast}\otimes\omega_{X})$ by Serre duality. Hence
we integrate over $X$ to obtain

(1)
$h^{0}(X,E)-h^{0}(X,E^{\ast}\otimes\omega_{X})=\deg(E)-\operatorname{rk}(E)\cdot(g-1)\
.$

In Section 4 we use multidivisors to define a generalization of the _Baker-
Norine rank_ $r_{\Gamma}$ (originally introduced in [BN07]) to vector bundles
on a compact metric graph and prove an analogue of (1).

######  Theorem A (Weil–Riemann–Roch theorem, Theorem 4.3).

Let $E$ be a vector bundle of rank $n$ on a compact and connected metric graph
$\Gamma$ of genus $g$. Writing $\omega_{\Gamma}$ for the line bundle
associated to the canonical divisor $K_{\Gamma}$, we have

$r_{\Gamma}(E)-r_{\Gamma}(E^{\ast}\otimes\omega_{\Gamma})=\deg(E)-n\cdot(g-1)\
.$

The proof of Theorem A uses multidivisors and a special case of the
Riemann–Hurwitz formula to deduce the statement from the rank one Riemann–Roch
theorem of [BN07, MZ08, GK08] for possibly disconnected metric graphs.

### Semistable bundles and a Narasimhan–Seshadri correspondence

The Abel–Jacobi theorem on a compact Riemann surface $X$ provides a natural
correspondence between topology (the Jacobian) and algebraic geometry (the
Picard group $\operatorname{Pic}_{0}(X))$. In order to generalize this
correspondence to the higher rank case, Narasimhan and Seshadri [NS65],
building on ideas of Weil [Wei38], prove that there is a natural one-to-one
correspondence between irreducible unitary representations of the fundamental
group of $X$ (or, alternatively, irreducible unitary local systems on $X$) and
stable vector bundles of degree zero on $X$.

The Abel–Jacobi theorem has a natural analogue on a metric graph $\Gamma$ (see
[MZ08, BF11]), as recalled in Section 1.4 below. In Section 5 we prove an
analogue of the Narasimhan–Seshadri correspondence in our framework.

A section of the sheaf $S_{n}\ltimes\mathcal{H}_{\Gamma}^{n}$ can be viewed as
a $\operatorname{GL}_{n}(\mathbb{T})$-valued function on $\Gamma$, and an
element of $H^{1}(S_{n}\ltimes\mathcal{H}_{\Gamma}^{n})$ defines a rank-$n$
vector bundle on $\Gamma$. We interpret sections of the natural subsheaf
$S_{n}\ltimes\mathbb{R}^{n}$ of locally constant functions as functions valued
in a tropical _unitary group_ , by showing that the corresponding vector
bundles are semistable. Indeed, let $\lambda$ be a local system on $\Gamma$
with fiber $S_{n}\ltimes\mathbb{R}^{n}$. This datum produces a free degree $n$
cover $f:\widetilde{\Gamma}\to\Gamma$ as well as a local system
$\widetilde{\lambda}$ with fiber $\mathbb{R}$ (both unique up to isomorphism)
such that $f_{\ast}\widetilde{\lambda}=\lambda$. At the same time, $\lambda$
also naturally defines a vector bundle $E(\lambda)$ with constant transition
maps.

######  Theorem B (Narasimhan–Seshadri correspondence, Theorem 5.4).

Let $\Gamma$ be a compact and connected metric graph.

  1. (i)

A vector bundle $E$ of rank $n$ on $\Gamma$ is associated to an
$S_{n}\ltimes\mathbb{R}^{n}$-local system $\lambda$ if and only if it is
semistable and of degree zero. The vector bundle $E(\lambda)$ is stable if and
only if the corresponding $S_{n}$-representation of $\pi_{1}(\Gamma)$ is
indecomposable.

  2. (ii)

Two $S_{n}\ltimes\mathbb{R}^{n}$-local systems $\lambda_{i}$ (for $i=1,2$)
give rise to the same vector bundle if and only if they define the same cover
$f\colon\widetilde{\Gamma}\rightarrow\Gamma$ and the induced classes of the
$\widetilde{\lambda}_{i}$ in $\operatorname{Jac}(\widetilde{\Gamma})$ are
equal.

We point out that all representations of $\pi_{1}(\Gamma)$ into
$S_{n}\ltimes\mathbb{R}^{n}$ are finite direct sums of irreducible
representations. So, contrary to the classical situation, we do not need to
assume semisimplicity on the representation side for our tropical result to
hold.

### The process of tropicalization

Our approach to vector bundles on a metric graph is completely independent on
any kind of base field. In general, however, the notion of a vector bundle on
an algebraic curve strongly depends on the chosen base field. So one should
not expect there to be a natural tropicalization of an arbitrary algebraic to
a tropical vector bundle, at least as defined here. We can see this via a
simple dimension count. As we will see in Section 2.3, the tropical linear
group $\operatorname{GL}_{n}(\mathbb{T})$ has real dimension $n$, not $n^{2}$,
and for this reason the moduli space of rank-$n$ vector bundles on a metric
graph $\Gamma$ of genus $g$ has dimension

$\dim_{\mathbb{R}}\operatorname{Bun}_{n}(\Gamma)=n(g-1)+1\ .$

On the other hand, for a projective algebraic curve $X$ the non-stacky
dimension of the moduli space of rank $n$ vector bundles is $n^{2}(g-1)+1$.
Hence for $g\geq 2$ we cannot expect the tropicalization map to be defined for
all vector bundles. Instead, we consider vector bundles that are linearized
with respect to the $n$-dimensional subgroup
$S_{n}\ltimes\mathbb{G}_{m}^{n}\subset\operatorname{GL}_{n}$ of generalized
permutation matrices.

Let $X$ be a Mumford curve over an algebraically closed non-Archimedean field
$K$ and consider a non-Archimedean skeleton $\Gamma_{X}$ of the Berkovich
analytic space $X^{an}$ in the sense of [Ber90]. Denote by
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}(X)$ the moduli stack of
$S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector bundles of rank $n$ on $X$
and $\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)$
for the open substack of _freely $S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized_
vector bundles of rank $n$ on $X$ (see Section 6.1 below). Write
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}$
for the Berkovich analytification of
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)$ and
$\operatorname{Bun}_{n}(\Gamma)$ for the moduli space of vector bundles on
$\Gamma_{X}$, as introduced in Section 3.2. We construct a natural
tropicalization map

${\mathrm{trop}}\colon\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}\longrightarrow\operatorname{Bun}_{n}(\Gamma_{X})$

that, on $K$-valued points, associates to a freely
$S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector bundle on $X$ a vector
bundle on $\Gamma_{X}$ (essentially by taking valuations of the transition
maps).

From a different perspective, the rigidification of the moduli stack
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)$
naturally admits the structure of a disjoint union of a finite quotient of
torsors over abelian varieties. So, using Raynaud’s non-Archimedean
uniformization, we find that there is a natural strong deformation retraction
map

$\rho\colon\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}\longrightarrow\Sigma\big{(}\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)\big{)}$

onto a closed subset of
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}$,
its _non-Archimedean skeleton_ (see [Ber90, Section 6.5] for details). These
two constructions are naturally compatible.

######  Theorem C (Theorem 6.2).

Let $X$ be a Mumford curve over $K$. There is a natural isomorphism

$J\colon\operatorname{Bun}_{n}(\Gamma_{X})\xlongrightarrow{\sim}\Sigma\big{(}\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)\big{)}$

that makes the diagram

${\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}}$${\operatorname{Bun}_{n}(\Gamma_{X})}$${\Sigma\big{(}\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)\big{)}}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\rho}$$\scriptstyle{J}$$\scriptstyle{\sim}$

commute.

In Section 6 we also show that the process of tropicalization is naturally
compatible with forming direct sums, tensor products, duals, and taking
determinants, and that a higher rank version of Baker’s specialization [Bak08]
inequality holds.

### The case of a Tate curve

Let $K$ be an algebraically closed non-Archimedean field of characteristic
zero and $X$ a Tate curve over $K$, a smooth projective curve of genus one,
whose analytification is given by
$X^{\mathrm{an}}=\mathbb{G}_{m}^{\mathrm{an}}/q^{\mathbb{Z}}$ with
$\operatorname{val}(q)>0$. Then the minimal non-Archimedean skeleton
$\Gamma_{X}$ is isomorphic to a circle of length $\operatorname{val}(q)$. In
this case, our tropicalization map can be extended to the whole good moduli
space $M_{n,d}(X)$ (as constructed in [Tu93] using results of [Ati57]), whose
$K$-points are in natural one-to-one correspondence with equivalence classes
of semistable vector bundles of rank $n$ and degree $d$ on $X$. This moduli
space is isomorphic to a suitable symmetric power of $X$, as shown in [Tu93],
and thus admits a polystable model and a natural non-Archimedean skeleton
$\rho\colon M_{n,d}(X)^{\mathrm{an}}\rightarrow\Sigma\big{(}M_{n,d}(X)\big{)}$
in the sense of Berkovich [Ber99] (see [BU21] but also [She16] for details).

The tropical counterpart of $M_{n,d}(X)$ is the moduli space
$M_{n,d}(\Gamma_{X})$ of isomorphism classes of semistable vector bundles on
$\Gamma_{X}$, which is explicitly described in Example 5.3. In Theorem 7.1
below we use ideas of [Oda71], [PZ98], and [BBDG06, Theorem 2.18] to prove a
characterization of indecomposable vector bundles on a Tate curve that is
equivalent to Atiyah’s classification. This result allows us to expand on the
above construction in Section 7.3 in order to construct a natural
tropicalization map

${\mathrm{trop}}\colon M_{n,d}(X)^{\mathrm{an}}\longrightarrow
M_{n,d}(\Gamma_{X})$

that lands in a component $M_{n,d}^{\oplus}(\Gamma_{X})$ of
$M_{n,d}(\Gamma_{X})$, which may be identified with a suitable symmetric power
of $\Gamma_{X}$.

######  Theorem D (Theorem 7.5).

Let $X$ be a Tate curve. There is a natural isomorphism $J\colon
M_{n,d}^{\oplus}(\Gamma_{X})\xrightarrow{\sim}\Sigma\big{(}M_{n,d}(X)\big{)}$
that makes the diagram

${M_{n,d}(X)^{\mathrm{an}}}$${M_{n,d}^{\oplus}(\Gamma_{X})}$${\Sigma\big{(}M_{n,d}(X)\big{)}}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\rho}$$\scriptstyle{J}$$\scriptstyle{\sim}$

commute.

Theorem D, in particular, implies that the tropicalization map is continuous,
surjective onto $M_{n,d}^{\oplus}(\Gamma_{X})$, and proper, since the
retraction $\rho$ has all these properties. In Section 7.4, we prove a
refinement of the higher rank specialization inequality. In Sections 7.5 and
7.6, this allows us to describe the tropicalization of natural Brill–Noether
loci and generalized $\Theta$-divisors in the moduli space $M_{n,0}(X)$ that
were introduced classically in [Tu93].

In [KS06] Kontsevich and Soibelman propose a non-Archimedean incarnation of
the SYZ-fibration coming from mirror symmetry. The basic idea is that for a
Calabi-Yau variety $X$ the retraction of $X^{\mathrm{an}}$ to its _essential
skeleton_ in the sense of [KS06, MN15] should function as a natural fibration
in affinoid tori. The essential skeleton of the Tate curve is its minimal
skeleton and in [BM19, Proposition 6.1.11] the authors show that the formation
of essential skeletons is compatible with taking symmetric powers. So Theorem
D also provides us with another explicit example of a non-Archimedean SYZ-
fibration for the Calabi-Yau variety $M_{n,d}(X)$.

By [FGPN14, Proposition 4.24], the moduli space of semistable Higgs bundles of
degree $d$ on an elliptic curve $X$ is naturally isomorphic to a product of
$M_{n,d}(X)$ with the symmetric power
$\operatorname{Sym}^{n}(\mathbb{A}^{1})$. So Theorem D also provides us with a
non-Archimedean SYZ-fibration for the moduli of semistable Higgs bundles on a
Tate curve $X$. We point out that, since $X$ is its own mirror dual and
$\operatorname{GL}_{n}$ is its own Langlands dual, the moduli space of
semistable Higgs bundles of rank $n$ will be its own mirror as well.

### Further discussion, complements, and remarks

Theorems C and D are two new additions to a long line of results identifying
tropical moduli spaces with non-Archimedean skeletons of the Berkovich
analytic spaces associated to their algebraic counterparts. We refer the
reader to [BR15] for the case of Jacobians and to [ACP15] for the moduli space
of curves, as the first two examples. These results establish a tight
dictionary between tropical and algebraic geometry and lie of the heart of
many applications. They also justify that our synthetic definition of tropical
objects (in our case vector bundles) is not just hot air, but is actually
related to the corresponding classical notion in a non-trivial way.

In this article we focus on the $\operatorname{GL}_{n}$-case. We also expect
analogues of Theorems C and D to hold for other reductive groups $G$. The
technical input for such a result will most likely come from the work of
Laszlo [Las98] and Frăţilă [Fră21], where they describe the moduli space of
semistable principal $G$-bundles as a moduli space of $T$-linearized principal
bundles for a big torus $T$ in $G$ (up to a quotient by the Weyl group). A
careful analysis of this situation from a tropical point of view will appear
in a follow-up article.

A step towards a more comprehensive theory in higher genus appears to be
closely related to the construction of polystable models of the moduli space
of vector bundles over a semistable one-parameter degeneration of a smooth
projective curve. We refer the reader to the work of Nagaraj and Seshadri
[NS97, NS99, Ses00] for the already very intricate story, when the special
fiber has only one node, building on the work of Gieseker [Gie84] in rank two.
An underlying problem, which might have prevented further generalizations so
far, seems to lie in the fact that, by [Teo02, MT15], an analogue of the
Birkhoff–Grothendieck theorem holds on a rational nodal curve only when the
dual graph is a chain or a cycle. As soon as the dual graph has vertices of
valence $>2$ an immediate generalization of the Birkhoff–Grothendieck theorem
is false (see in particular [MT15, Example 4.5] for an illustration of the
central difficulty).

While generalizations of our story to the higher genus situation appears to be
challenging, a generalization of Theorem D to the case of abelian varieties
seems very much within reach. The crucial point here is that Miyanishi [Miy73]
and Mukai [Muk78] provide us with a classification of homogeneous vector
bundles on abelian varieties in the spirit of Atiyah’s work [Ati57]. As a
coarse heuristic to explain our expectation, we note that the fundamental
group of an abelian variety is abelian, whereas the fundamental group of a
curve of genus $g\geq 2$ is not. Tropical vector bundles on abelian varieties
might eventually also provide us with a new perspective on the logarithmic
approach to Donaldson–Thomas theory introduced in [MR20].

We have already mentioned that our approach is the same as the one introduced
by Allermann in [All12]. In a similar vein, what we propose also appears to be
compatible with the theory of tropical vector bundles on a toric tropical
scheme developed in [JMT20]. A precise relationship would, however, require
the theory in [JMT20] to be generalized to a local situation (possibly by
introducing an analogue of rigid-analytic methods to tropical scheme theory).
Once this is achieved, one could then consider $\mathbb{Z}$-invariant vector
bundles on $\mathbb{G}_{m}$ and rediscover vector bundles on the (scheme-
theoretic) tropicalization of the Tate curve. For higher genus curves we
expect that a more intricate theory will be necessary, e.g. expanding on the
matroidal notion of tropical ideals coming from [MR18].

### Acknowledgements

The authors would like to thank Chiara Damiolini, Lorenzo Fantini, Emilio
Franco, Johannes Horn, Inder Kaur, Enrica Mazzon, Mirko Mauri, Dhruv
Ranganathan, Mattia Talpo, Annette Werner, and Jonathan Wise for several
helpful discussions en route to and while writing this article.

A.G. and M.U. have received funding from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) TRR 326 _Geometry and Arithmetic of
Uniformized Structures_ , project number 444845124, by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) Sachbeihilfe _From
Riemann surfaces to tropical curves (and back again)_ , project number
456557832, and from the LOEWE grant _Uniformized Structures in Algebra and
Geometry_.

##  1\. Line bundles on metric graphs

In this section we recall the story of divisors and line bundles on metric
graphs that was originally introduced in [MZ08]. Section 1.3 and 1.4 put a
particular emphasis on the nature of line bundles as torsors over the sheaf of
harmonic functions. This perspective is inspired by the logarithmic Picard
group, as popularized in [MW21], and sets the stage for what is to come in the
later sections.

###  1.1. Metric graphs

A _star-shaped domain_ of valence $n$ and radius $r$ is a metric space of the
form

$S(n,r)=\big{\\{}z=te^{\frac{2k\pi i}{n}}\in\mathbb{C}\ \big{|}\ 0\leq t\leq
r\textrm{ and }k=1,\ldots,n-1\big{\\}}\ ,$

where the distance between two points is the length of the shortest path
connecting them. A _metric graph_ is a metric space $\Gamma$ such that every
point $p\in\Gamma$ has an open neighborhood that is isometric to a star-shaped
domain $S(n,r)$ for some $n\geq 0$ and some $r>0$.

Let $G$ be a finite graph (possibly with loops and multiple edges). We write
$V(G)$ and $E(G)$ for the sets of vertices and edges of $G$, respectively. An
_orientation_ on $G$ determines _source_ and _target_ maps $s,t:E(G)\to V(G)$.
Given a length function $\ell\colon E(G)\rightarrow\mathbb{R}_{>0}$, we may
construct a compact metric graph $\Gamma(G,\ell)$ by gluing line segments of
length $\ell(e)$ for every edge according to the incidences determined by $G$.
Given a compact metric graph $\Gamma$, a pair $(G,\ell)$ such that
$\Gamma=\Gamma(G,\ell)$ is called a _model_ of $\Gamma$; we say that the model
is _simple_ if $G$ has no loops or multiedges. Given a simple model $(G,\ell)$
of $\Gamma$, we define the _star cover_ $\mathcal{U}(G)=\\{U_{v}\\}_{v\in
V(G)}$ of $\Gamma$, where $U_{v}$ is the open set consisting of $v$ together
with the open intervals joining $v$ to all neighboring vertices. Each $U_{v}$
is contractible, the pairwise intersections $U_{v}\cap U_{v^{\prime}}$ are
open intervals or empty, and all triple intersections are empty, making the
star cover convenient for cohomological calculations.

The _Euler characteristic_ of a compact metric graph $\Gamma$ is given by
$\chi(\Gamma)=h^{0}(\Gamma)-h^{1}(\Gamma)$; this quantity is additive in
connected components. Given a model $G$ of $\Gamma$, the Euler characteristic
is given by $\chi(\Gamma)=\\#V(G)-\\#E(G)$. If $\Gamma$ is connected, we call
the number

$g=1-\chi(\Gamma)=h^{1}(\Gamma)=\\#E(G)-\\#V(G)+1$

the _genus_ of $\Gamma$.

A _free cover_ of a metric graph $\Gamma$ is an isometric covering space
$f\colon\widetilde{\Gamma}\rightarrow\Gamma$. For a free cover
$f\colon\widetilde{\Gamma}\rightarrow\Gamma$ of degree $d$, we naturally have

$\chi(\widetilde{\Gamma})=d\cdot\chi(\Gamma)\ .$

This is a special case of the tropical Riemann-Hurwitz formula.

######  Remark 1.1.

In many approaches to tropical covers, e.g. in [ABBR15a, ABBR15b] or [CMR16],
the authors work with more general _harmonic or admissible covers_ of metric
graphs that allow for dilation along vertices and edges. We explicitly refrain
from including this extra level of intricacy, since, for different reasons,
neither Theorem B nor Theorem C appear to generalize to this situation and it
is not needed for Theorem D. For us the natural next step is the case of
logarithmic curves, expanding on [MW21], or, equivalently, of metrized curve
complexes in the sense of [AB15].

###  1.2. Piecewise linear functions, divisors, and the Picard group

Let $\Gamma$ be a metric graph. A _piecewise linear function_ on an open
subset $U\subseteq\Gamma$ is a continuous function $f\colon
U\rightarrow\mathbb{R}$ whose restriction to a line segment is piecewise
linear with integer slopes. We denote by
$\operatorname{\mathcal{PL}}_{\Gamma}(U)$ the abelian group of piecewise
linear functions on $U$, and observe that the association
$U\mapsto\operatorname{\mathcal{PL}}_{\Gamma}(U)$ defines a sheaf
$\operatorname{\mathcal{PL}}_{\Gamma}$ of abelian groups on $\Gamma$. A global
section of $\operatorname{\mathcal{PL}}_{\Gamma}$ is called a _rational
function_ on $\Gamma$, and we write
$\operatorname{Rat}(\Gamma)=\operatorname{\mathcal{PL}}_{\Gamma}(\Gamma)$.

A _divisor_ on an open subset $U\subset\Gamma$ is a formal sum $D=\sum_{p\in
U}D(p)\cdot p$ over the points $p$ of $U$ such that the set $\\{p\in
U|D(p)\neq 0\\}$ is discrete. We denote by $\operatorname{Div}(U)$ the abelian
group of divisors on $U$, this defines a sheaf $\operatorname{Div}$ of abelian
groups on $\Gamma$. When $\Gamma$ is compact, we define the _degree
homomorphism_

(2)
$\begin{split}\deg\colon\operatorname{Div}(\Gamma)&\longrightarrow\mathbb{Z}\\\
D&\longmapsto\sum_{p\in\Gamma}D(p)\ .\end{split}$

There is a natural sheaf homomorphism
$\operatorname{div}\colon\operatorname{\mathcal{PL}}_{\Gamma}\rightarrow\operatorname{Div}$
given by

(3)
$\begin{split}\operatorname{\mathcal{PL}}_{\Gamma}(U)&\longrightarrow\operatorname{Div}(U)\\\
f&\longmapsto\sum_{p\in\Gamma}\operatorname{ord}_{p}(f)\cdot p\ ,\end{split}$

for $U\subseteq\Gamma$ open, where $\operatorname{ord}_{p}(f)$ is the sum of
the outgoing slopes of $f$ at $p$ (note that
$\Delta(f)=-\operatorname{div}(f)$, the sum of the incoming slopes of $f$, is
the _Laplacian_ of $f$). The image of $\operatorname{Rat}(\Gamma)$ is the
subgroup of _principal divisors_
$\operatorname{PDiv}(\Gamma)\subseteq\operatorname{Div}(\Gamma)$. We say that
two divisors $D,D^{\prime}\in\operatorname{Div}(\Gamma)$ on $\Gamma$ are
_linearly equivalent_ , written as $D\sim D^{\prime}$, if $D-D^{\prime}$ is a
principal divisor, i.e. if $D-D^{\prime}\in\operatorname{div}(f)$ for a
rational function $f\in\operatorname{Rat}(\Gamma)$. The _Picard group_ of
$\Gamma$ is the quotient

$\operatorname{Pic}(\Gamma)=\operatorname{Div}(\Gamma)/\operatorname{PDiv}(\Gamma)\
.$

If $\Gamma$ is compact, then $\deg(\operatorname{div}(f))=0$ for all
$f\in\operatorname{Rat}(\Gamma)$ by continuity, and the degree map descends to
$\operatorname{Pic}(\Gamma)$. We write $\operatorname{Pic}_{d}(\Gamma)$ for
the set of linear equivalence classes of divisors of degree $d$ on $\Gamma$;
it is naturally a torsor over the group $\operatorname{Pic}_{0}(\Gamma)$.

Given a free cover $f:\widetilde{\Gamma}\to\Gamma$ of degree $d$ and divisors
$\widetilde{D}\in\operatorname{Div}(\widetilde{\Gamma})$ and
$D\in\operatorname{Div}(\Gamma)$, we define the _pushforward_ and _pullback_
as

$f_{*}(\widetilde{D})=\sum_{\widetilde{p}\in\widetilde{\Gamma}}\widetilde{D}(\widetilde{p})\cdot
f(\widetilde{p})\in\operatorname{Div}(\Gamma)\quad\textrm{ and }\quad
f^{*}(D)=\sum_{p\in\Gamma}D(p)\cdot\sum_{\widetilde{p}\in
f^{-1}(p)}\widetilde{p}\in\operatorname{Div}(\widetilde{\Gamma})$

respectively. The maps $f_{*}$ and $f^{*}$ respect linear equivalence, and
hence descend to maps
$f_{*}:\operatorname{Pic}(\widetilde{\Gamma})\to\operatorname{Pic}(\Gamma)$
and
$f^{*}:\operatorname{Pic}(\Gamma)\to\operatorname{Pic}(\widetilde{\Gamma})$.
For $f^{\ast}$ this is elementary and for $f_{\ast}$ we refer to [LU21,
Section 1.4].

###  1.3. Harmonic functions and line bundles

The kernel of the divisor map (3) is called the _sheaf of harmonic functions_
on $\Gamma$ and is denoted $\mathcal{H}_{\Gamma}$. Any divisor on an open tree
is principal, so
$\operatorname{Div}=\operatorname{\mathcal{PL}}_{\Gamma}/{\mathcal{H}}_{\Gamma}$
and we have a short exact sequence

(4)
$0\longrightarrow\mathcal{H}_{\Gamma}\longrightarrow\operatorname{\mathcal{PL}}_{\Gamma}\xlongrightarrow{\operatorname{div}}\operatorname{Div}\longrightarrow
0\ $

of sheaves of abelian groups on $\Gamma$. The sheaves $\mathcal{H}_{\Gamma}$
and $\operatorname{\mathcal{PL}}_{\Gamma}$ play the role of respectively the
sheaves $\mathcal{O}_{X}^{\ast}$ and $\mathcal{M}_{X}^{\ast}$ of invertible
holomorphic and meromorphic functions on a Riemann surface $X$. We now extend
this analogy by describing the relationship between divisors and line bundles
on $\Gamma$.

A _line bundle_ ${\mathcal{L}}$ on $\Gamma$ is an
${\mathcal{H}}_{\Gamma}$-torsor, that is to say a sheaf of
${\mathcal{H}}_{\Gamma}$-sets such that $\Gamma$ can be covered by open
subsets $U$ for which ${\mathcal{L}}|_{U}$ and
${\mathcal{H}}_{U}={\mathcal{H}}_{\Gamma}|_{U}$ are isomorphic as sheaves of
${\mathcal{H}}_{U}$-sets. The set of isomorphism classes of line bundles is
the sheaf cohomology group $H^{1}(\Gamma,{\mathcal{H}}_{\Gamma})$, and the
group structure can be described as follows. Let ${\mathcal{L}}$ and
${\mathcal{L}}^{\prime}$ be two line bundles on $\Gamma$. For an open subset
$U\subset\Gamma$, consider the diagonal action of ${\mathcal{H}}_{\Gamma}(U)$
on ${\mathcal{L}}(U)\times{\mathcal{L}}^{\prime}(U)$ given by
$f(s,s^{\prime})=(fs,(-f)s^{\prime})$, and equip the quotient
$\big{[}{\mathcal{L}}(U)\times{\mathcal{L}}^{\prime}(U)\big{]}/{{\mathcal{H}}_{\Gamma}(U)}$
with the ${\mathcal{H}}_{\Gamma}(U)$-action given by
$f\overline{(s,s^{\prime})}=\overline{(fs,s^{\prime})}$. The assignment
$U\mapsto\big{[}{\mathcal{L}}(U)\times{\mathcal{L}}^{\prime}(U)\big{]}/{{\mathcal{H}}_{\Gamma}(U)}$
defines an ${\mathcal{H}}_{\Gamma}$-torsor called the _tensor product_
${\mathcal{L}}\otimes{\mathcal{L}}^{\prime}$. Similarly, given a line bundle
${\mathcal{L}}$, we define ${\mathcal{L}}^{-1}$ as the
${\mathcal{H}}_{\Gamma}$-torsor whose underlying sheaf of sets is
${\mathcal{L}}$, but equipped with the opposite action $(f,s)\mapsto(-f)s$. It
is clear that the map $f\mapsto\overline{(fs,s)}$ defines an isomorphism
${\mathcal{H}}_{\Gamma}$-torsors
${\mathcal{H}}_{\Gamma}\simeq{\mathcal{L}}\otimes{\mathcal{L}}^{-1}$. The
identity element and group operation on $H^{1}(\Gamma,{\mathcal{H}}_{\Gamma})$
correspond to the isomorphism class of ${\mathcal{H}}_{\Gamma}$ and the tensor
product, respectively.

We can also describe a line bundle ${\mathcal{L}}$ on $\Gamma$ in terms of a
Čech cocycle as follows. Choose an oriented simple model $(G,\ell)$ of
$\Gamma$, and let $\mathcal{U}(G)=\\{U_{v}\\}_{v\in v(G)}$ be the associated
star cover. Since each $U_{v}$ is contractible, we can find isomorphisms
$g_{v}:{\mathcal{L}}|_{U_{v}}\to{\mathcal{H}}_{U_{v}}$ of sheaves of
${\mathcal{H}}_{U_{v}}$-sets. Each edge $e\in E(G)$ with start and end
vertices $s(e)$ and $t(e)$ corresponds to a nonempty pairwise intersection
$U_{s(e)}\cap U_{t(e)}$, which we also denote by $e$ by abuse of notation, and
all other pairwise intersections are empty. Hence the line bundle
${\mathcal{L}}$ determines a cocycle $\\{g_{e}\\}_{e\in E(\Gamma)}$, where
$g^{e}$ is the harmonic function on $e$ (that is to say, a linear function
with integer slope) defined by the composition $g^{e}=\big{[}g_{t(e)}\circ
g_{s(e)}^{-1}\big{]}\big{|}_{e}:{\mathcal{H}}_{e}\to{\mathcal{H}}_{e}$. Two
cocycles $\\{g_{e}\\}$ and $\\{g^{\prime e}\\}$ determine the same line bundle
if and only if $g^{e}-g^{\prime e}=f_{t(e)}-f_{s(e)}$ for some harmonic
functions $f_{v}\in{\mathcal{H}}_{\Gamma}(U_{v})$, and the cocycle condition
is trivially satisfied because all triple intersections are empty. We note
that changing the orientation of an edge $e$ replaces $g^{e}$ with $-g^{e}$.

Given a line bundle ${\mathcal{L}}$ defined by a Čech cocycle
$\\{g^{e}\\}_{e\in E(\Gamma)}$ with respect to an oriented simple model
$(G,\ell)$, the _degree_ of ${\mathcal{L}}$ is defined as

$\deg{\mathcal{L}}=\sum_{e\in E(G)}\dot{g}^{e},$

where $\dot{g}^{e}$ denotes the slope of $g^{e}$ in the direction of the
oriented edge $e$. Is is easy to verify that $\deg{\mathcal{L}}$ does not
depend on the choice of cocycle $\\{g^{e}\\}$ or model $G$, and that
$\deg{\mathcal{L}}_{1}\otimes{\mathcal{L}}_{2}=\deg{\mathcal{L}}_{1}+\deg{\mathcal{L}}_{2}$
and $\deg{\mathcal{L}}^{-1}=-\deg{\mathcal{L}}$.

######  Proposition 1.2.

Let $\Gamma$ be a metric graph. There is a natural isomorphism

$\operatorname{Pic}(\Gamma)\xlongrightarrow{\sim}H^{1}(\Gamma,\mathcal{H}_{\Gamma})\
.$

######  Remark 1.3.

In [MZ08] the term _line bundle_ is used for a locally trivial fibration with
fiber the tropical affine line over $\Gamma$ in the category integral affine
manifolds. In this language, Proposition 1.2 has appeared as [MZ08,
Proposition 4.6].

###### Proof of Proposition 1.2.

Given a divisor $D\in\operatorname{Div}(\Gamma)$, we define the associated
line bundle $\mathcal{H}_{\Gamma}(D)$ by

$U\longmapsto\mathcal{H}_{\Gamma}(D)(U)=\big{\\{}s\in\operatorname{\mathcal{PL}}_{\Gamma}(U)\
\big{|}\ \operatorname{div}(s)=D|_{U}\big{\\}}$

for $U\subseteq\Gamma$ open. For $s\in\mathcal{H}_{\Gamma}(D)(U)$ and
$f\in\mathcal{H}_{\Gamma}(U)$ we have

$\operatorname{div}(s+f)=\operatorname{div}(s)+\operatorname{div}(f)=\operatorname{div}(s)=D|_{U}\
,$

since $f$ is harmonic, so ${\mathcal{H}}_{\Gamma}(D)$ is a sheaf of
${\mathcal{H}}_{\Gamma}$-sets.

Let $G$ be an oriented simple model of $\Gamma$, and let
$U_{v}\in\mathcal{U}(G)$ be an element of the star cover of $\Gamma$. Since
$U_{v}$ is a tree, we can find a function
$f_{v}\in\operatorname{\mathcal{PL}}_{\Gamma}(U_{v})$ such that
$\operatorname{div}f_{v}=D|_{U_{v}}$. Therefore we have an isomorphism
$\mathcal{H}_{\Gamma}(D)|_{U_{v}}\xrightarrow{\sim}\mathcal{H}_{U_{v}}$ of
$\mathcal{H}_{U_{v}}$-torsors given by

$s\longmapsto s-f_{v}|_{V}$

for $s\in\mathcal{H}_{\Gamma}(D)(V)$ and $V\subseteq U_{v}$ open. Therefore
$\mathcal{H}_{\Gamma}(D)$ is an $\mathcal{H}_{\Gamma}$-torsor on $\Gamma$. It
is easy to verify that
${\mathcal{H}}_{\Gamma}(D_{1})\otimes{\mathcal{H}}_{\Gamma}(D_{1})\simeq{\mathcal{H}}_{\Gamma}(D_{1}+D_{2})$
and ${\mathcal{H}}_{\Gamma}(D)^{-1}\simeq{\mathcal{H}}_{\Gamma}(-D)$, so the
map $\operatorname{Div}(\Gamma)\rightarrow H^{1}(\Gamma,\mathcal{H}_{\Gamma})$
given by $D\mapsto\mathcal{H}_{\Gamma}(D)$ is a homomorphism, and furthermore
$\deg{\mathcal{H}}_{\Gamma}(D)=\deg D$.

Let $D_{1},D_{2}\in\operatorname{Div}(\Gamma)$ be two linearly equivalent
divisors on $\Gamma$, so that there is a rational function
$f\in\operatorname{Rat}(\Gamma)$ such that
$D_{1}=D_{2}+\operatorname{div}(f)$. The bijection

$\begin{split}\mathcal{H}_{\Gamma}(D_{1})(U)&\xlongrightarrow{\sim}\mathcal{H}_{\Gamma}(D_{2})(U)\\\
s&\longmapsto s-f|_{U}\end{split}$

for $U\subseteq\Gamma$ open is $\mathcal{H}_{\Gamma}(U)$-equivariant and
compatible with restriction to open subsets $V\subseteq U$; thus it defines an
isomorphism
$\mathcal{H}_{\Gamma}(D_{1})\xrightarrow{\sim}\mathcal{H}_{\Gamma}(D_{2})$ of
$\mathcal{H}_{\Gamma}$-torsors. Therefore the map
$\operatorname{Div}(\Gamma)\rightarrow H^{1}(\Gamma,\mathcal{H}_{\Gamma})$
descends to a homomorphism $\operatorname{Pic}(\Gamma)\rightarrow
H^{1}(\Gamma,\mathcal{H}_{\Gamma})$

In order to see that this map is an isomorphism, we consider the long exact
sequence associated to the short exact sequence (4)

$0\longrightarrow\mathcal{H}_{\Gamma}(\Gamma)\longrightarrow\operatorname{Rat}(\Gamma)\xlongrightarrow{\operatorname{div}}\operatorname{Div}(\Gamma)\xlongrightarrow{\delta}H^{1}(\Gamma,\mathcal{H}_{\Gamma})\
.$

By definition, $\operatorname{Pic}(\Gamma)$ is the cokernel of
$\operatorname{div}$, and furthermore the boundary homomorphism $\delta$ is
precisely the map $D\mapsto\mathcal{H}(D)$ described above. By [MZ08,
Proposition 4.7], the morphism $\delta$ is surjective, which implies that
there is an induced isomorphism $\operatorname{Pic}(\Gamma)\to
H^{1}(\Gamma,\mathcal{H}_{\Gamma})$. ∎

###  1.4. The Abel-Jacobi theorem

Let $\Gamma$ be a metric graph. We define the sheaf $\Omega_{\Gamma}$ of
_harmonic $1$-forms_ on $\Gamma$ by the short exact sequence

(5)
$0\longrightarrow\mathbb{R}\longrightarrow\mathcal{H}_{\Gamma}\longrightarrow\Omega_{\Gamma}\longrightarrow
0$

of sheaves of abelian groups. Choose an oriented simple model $(G,\ell)$ of
$\Gamma$, and let $s,t:E(G)\to V(G)$ be the associated source and target maps.
A harmonic $1$-form $\alpha\in\Omega_{\Gamma}(\Gamma)$ may be written as a
formal sum

$\alpha=\sum_{e\in E(G)}a_{e}de$

with $a_{e}\in\mathbb{Z}$, such that at every vertex $v\in V(G)$ we have

$\sum_{e\textrm{ with }s(e)=v}a_{e}=\sum_{e\textrm{ with }t(e)=v}a_{e}.$

Hence, for a compact metric graph $\Gamma$, we have a natural identification
between $H^{0}(\Gamma,\Omega_{\Gamma})$ and the simplicial homology group
$H_{1}(G,\mathbb{Z})$, which, in turn, is naturally isomorphic to
$H_{1}(\Gamma,\mathbb{Z})$.

######  Lemma 1.4.

Let $\Gamma$ be a connected metric graph. Then there is a natural degree
isomorphism

$\deg:H^{1}(\Gamma,\Omega_{\Gamma})\xlongrightarrow{\sim}\mathbb{Z}.$

###### Proof.

Choose an oriented simple model $(G,\ell)$ of $\Gamma$ with $n=\\#V(G)$
vertices and $m=\\#E(G)$ edges. As for line bundles, we compute
$H^{1}(\Gamma,\Omega_{\Gamma})$ as the first Čech cohomology group with
respect to the star cover $\mathcal{U}(G)=\\{U_{v}\\}_{v\in v(G)}$. Since all
triple intersections are empty, we have an exact sequence

$0\longrightarrow\Omega_{\Gamma}(\Gamma)\longrightarrow\prod_{v\in
V(G)}\Omega_{\Gamma}(U_{v})\longrightarrow\prod_{e\in
E(G)}\Omega_{\Gamma}(e)\longrightarrow\check{H}^{1}(\Gamma,\Omega_{\Gamma})\longrightarrow
0\ ,$

where all terms are free abelian groups of finite rank. Define the _degree_ of
a $1$-cochain $\gamma=\\{c^{e}\\}_{e\in E(G)}$ by

$\deg\gamma=\sum_{e\in E(G)}c^{e}.$

The degree of a $1$-coboundary is zero, and we claim that the induced map
$\deg:\check{H}^{1}(\Gamma,\Omega_{\Gamma})\to\mathbb{Z}$ is an isomorphism.
It is clearly surjective, and to complete the proof we compute the ranks in
the exact sequence. Since $\Omega_{\Gamma}(\Gamma)\simeq
H_{1}(\Gamma,\mathbb{Z})$ we have

$\operatorname{rk}\Omega_{\Gamma}(\Gamma)=g(\Gamma)=m-n+1\ .$

Let $v\in V(G)$, then an element $\Omega_{\Gamma}(U_{v})$ is determined by
$\operatorname{val}(v)$ integers subject to a single relation, so
$\operatorname{rk}\Omega_{\Gamma}(U_{v})=\operatorname{val}(v)-1$ and
therefore

$\operatorname{rk}\prod_{v\in V(G)}\Omega_{\Gamma}(U_{v})=\sum_{v\in
V(G)}(\operatorname{val}(v)-1)=2m-n\ .$

Finally, a harmonic $1$-form on an edge is simply an integer, so
$\Omega_{\Gamma}(e)\simeq\mathbb{Z}$ and hence

$\operatorname{rk}\prod_{e\in E(G)}\Omega_{\Gamma}(e)=m\ .$

Putting all this together, we see that

$\operatorname{rk}\check{H}^{1}(\Gamma,\Omega_{\Gamma})=(m-n+1)-(2m-n)+m=1\ ,$

which completes the proof. ∎

######  Remark 1.5.

This result appears to be well-known. In the framework of tropical homology
(as introduced by [IKMZ19]), Poincaré duality establishes a natural
isomorphism $H^{1}(\Gamma,\Omega_{\Gamma})\simeq H_{0,0}(\Gamma,\mathbb{Z})$,
which is isomorphic to $\mathbb{Z}$ via the degree map (see [JRS18]).
Alternatively one could also deduce this from [MW21, Section 3.4].

Let us continue to assume that $\Gamma$ is compact and connected. The short
exact sequence (5) provides us with a long exact sequence

(6)
$0\longrightarrow\mathbb{R}\longrightarrow\mathcal{H}_{\Gamma}(\Gamma)\longrightarrow\Omega_{\Gamma}(\Gamma)\longrightarrow
H^{1}(\Gamma,\mathbb{R})\longrightarrow\operatorname{Pic}(\Gamma)\xlongrightarrow{\deg}\mathbb{Z}\rightarrow
0\ ,$

where the second arrow is an isomorphism and we identify the last arrow with
the degree homomorphism using Proposition 1.2 and Lemma 1.4. Hence the middle
arrow establishes an isomorphism between $\operatorname{Pic}_{0}(\Gamma)$ and
the real $g$-dimensional torus
$H^{1}(\Gamma,\mathbb{R})/\Omega_{\Gamma}(\Gamma)$ known as the _Jacobian_
$\operatorname{Jac}(\Gamma)$. We note that according to some authors,
$H^{1}(\Gamma,\mathbb{R})/\Omega_{\Gamma}(\Gamma)$ is the _Albanese variety_
$\operatorname{Alb}(\Gamma)$, while $\operatorname{Jac}(\Gamma)$ is the dual
torus $\Omega_{\Gamma}(\Gamma)^{*}/H_{1}(\Gamma,\mathbb{Z})$.

This isomorphism also admits a description in terms of an Abel–Jacobi map
$\Gamma\rightarrow\operatorname{Jac}(\Gamma)$ inducing a homomorphism
$\operatorname{Div}_{0}(\Gamma)\rightarrow\operatorname{Jac}(\Gamma)$ and
subsequently an isomorphism
$\operatorname{Pic}_{0}(\Gamma)=\operatorname{Div}_{0}(\Gamma)/\operatorname{PDiv}(\Gamma)\xrightarrow{\sim}\operatorname{Jac}(\Gamma)$;
for more details we refer the interested reader to [MZ08] and [BF11].

##  2\. Principal bundles and vector bundles on metric graphs

###  2.1. Root data of reductive groups

It is well known that the category of split algebraic tori is naturally
equivalent to the category of finitely generated free abelian groups in two
(dual) ways: to go from a split algebraic torus to a finitely generated free
abelian group we either take the character lattice or the cocharacter lattice.

Split connected reductive algebraic groups over a fixed field are classified
in terms of root data (see [Spr98, Ch. 7 and 8] for details). A _root datum_
is a quadruple $\Phi=(M,R,M^{\vee},R^{\vee})$ consisting of

  * •

a finitely generated free abelian group $M$ and its dual group
$M^{\vee}=\operatorname{Hom}(M,\mathbb{Z})$ (with duality pairing
$\langle.,.\rangle$) and

  * •

a subset of _roots_ $R$ of $M$ and of _coroots_ $R^{\vee}$ of $M^{\vee}$
together with a bijection $(.)^{\vee}\colon R\rightarrow R^{\vee}$

subject to the following two axioms:

  1. (i)

For all $\alpha\in R$ we have $\langle\alpha,\alpha^{\vee}\rangle=2$;

  2. (ii)

The reflection homomorphisms $s_{\alpha}\colon M\rightarrow M$ and
$s_{\alpha}^{\vee}\colon M^{\vee}\rightarrow M^{\vee}$ given by

$u\longmapsto u-\langle u,\alpha^{\vee}\rangle\alpha\qquad\textrm{and}\qquad
v\longmapsto v-\langle\alpha,v\rangle\alpha$

fulfill

$s_{\alpha}(R)=R\qquad\textrm{and}\qquad s_{\alpha}^{\vee}(R^{\vee})=R^{\vee}$

for all $\alpha\in R$.

Unlike the case of tori, this classification is not a categorical equivalence.
When $G$ is semisimple the associated root datum is nothing but a root system
and this recovers the well-known classification of semisimple Lie algebras.

The _Weyl group_ $W=W_{\Phi}$ of a root datum $\Phi$ is the (finite)
automorphism group of $M$ generated by all reflections $s_{\alpha}$ for
$\alpha\in R$.

Our main example is the group $\operatorname{GL}_{n}$. In this case
$M=\mathbb{Z}^{n}$ is the character lattice of its diagonal torus
$\mathbb{G}_{m}^{n}$, its dual lattice is the cocharacter lattice of
$\mathbb{G}_{m}^{n}$ and the roots (and coroots) are given by

$\big{\\{}e_{i}-e_{j}\ \big{|}\ 1\leq i,j\leq n\quad\textrm{and}\quad i\neq
j\big{\\}}$

where $e_{1},\ldots,e_{n}$ denote the standard basis vectors of
$\mathbb{Z}^{n}$. The Weyl group is isomorphic to the symmetric group $S_{n}$
operating via permutation matrices. The character lattice of the group
$\operatorname{SL}_{n}$ is the sublattice $M_{0}\subset\mathbb{Z}^{n}$
consisting of vectors whose coordinates sum to zero, while the Weyl group
$W=S_{n}$ is the same.

###  2.2. Principal bundles on metric graphs.

Let $\Gamma$ be a metric graph, let $\mathcal{H}_{\Gamma}$ be the sheaf of
harmonic functions on $\Gamma$, and let $\Phi=(M,R,N,R^{\vee})$ be a root
datum with Weyl group $W$. For an open subset $U\subset\Gamma$, the $W$-action
on $M$ extends to a $W$-action on
$\mathcal{H}_{\Gamma}(U)\otimes_{\mathbb{Z}}M$, where $W$ acts trivially on
the first summand. Taking the semidirect product, we obtain a sheaf
$W\ltimes\mathcal{H}_{\Gamma}\otimes_{\mathbb{Z}}M$ of (generally speaking)
non-abelian groups on $\Gamma$.

######  Definition 2.1.

Let $\Gamma$ be a metric graph and let $\Phi=(M,R,M^{\vee},R^{\vee})$ be a
root datum with Weyl group $W$. A _principal $\Phi$-bundle_ on $\Gamma$ is a
$W\ltimes\mathcal{H}_{\Gamma}\otimes_{\mathbb{Z}}M$-torsor on $\Gamma$.

When $G$ is a connected split reductive group and $\Phi$ is the associated
root datum, we also sometimes refer to a principal $\Phi$-bundle as a
_principal $G$-bundle_ on $\Gamma$. In the case
$G=\operatorname{GL}_{1}=\mathbb{G}_{m}$, a principal $\mathbb{G}_{m}$-bundle
is nothing but a $\mathcal{H}_{\Gamma}$-torsor on $\Gamma$. So, in this case,
we recover the theory of line bundles on metric graphs outlined in Section 1.3
above.

######  Remark 2.2.

Let $G$ be a connected split reductive group over a field $k$. Our definition
of a tropical analogue of principal $G$-bundle on a metric graph $\Gamma$ is
completely independent of $k$. So one may think of this as being defined over
the ”field $\mathbb{F}_{1}$ with one element“. This goes back to a suggestion
by Tits [Tit57]. For the state of the art concerning analogues of reductive
groups over $\mathbb{F}_{1}$ we refer the reader to [Lor18].

The set of isomorphism classes of principal $\Phi$-bundles over a metric graph
$\Gamma$ is given by the non-abelian Čech cohomology
$\check{H}^{1}(\Gamma,W\ltimes\mathcal{H}_{\Gamma}\otimes M)$. This set is
convenient to describe in terms of an oriented simple model $(G,\ell)$ of
$\Gamma$. Indeed, let $E$ be a principal $G$-bundle on $\Gamma$, then we can
find trivializations $g_{v}:E|_{U_{v}}\to
W\ltimes\mathcal{H}_{\Gamma}(U_{v})\otimes_{\mathbb{Z}}M$, and $E$ is
described in terms of the transition functions

$g^{e}=\big{[}g_{t(e)}\circ
g^{-1}_{s(e)}\big{]}:W\ltimes\mathcal{H}_{\Gamma}(e)\otimes_{\mathbb{Z}}M\longrightarrow
W\ltimes\mathcal{H}_{\Gamma}(e)\otimes_{\mathbb{Z}}M.$

Choosing a basis $M\simeq\mathbb{Z}^{n}$, we can write the transition
functions as $g^{e}=(w^{e},g_{1}^{e},\ldots,g_{n}^{e})$, where $w^{e}\in W$
and the $g_{i}^{e}\in{\mathcal{H}}_{\Gamma}(e)$ are linear functions with
integer slope. The cocycle condition is trivially verified since triple
intersections are empty.

###  2.3. Vector bundles on metric graphs.

From now on we will focus on principal $\operatorname{GL}_{n}$-bundles on a
metric graph $\Gamma$ or, in other words,
$S_{n}\ltimes\mathcal{H}_{\Gamma}^{n}$-torsors on $\Gamma$. To avoid clunky
terminology, we will refer to such an object simply as a _vector bundle_ of
rank $n$ on $\Gamma$.

It is convenient to represent transition functions of vector bundles on
$\Gamma$ in terms of the tropical semifield
$\mathbb{T}=\mathbb{R}\cup\\{\infty\\}$ as follows. By [All12, Lemma 1.4], the
invertible $n\times n$ matrices over $\mathbb{T}$ are precisely the
compositions of invertible diagonal matrices (having finite entries on the
diagonal and $\infty$ elsewhere) and permutation matrices. In other words,
$\operatorname{GL}_{n}(\mathbb{T})=S_{n}\ltimes\mathbb{R}^{n}$ is the group of
matrices having a unique finite entry in each row and each column, while
$\operatorname{SL}_{n}(\mathbb{T})\subset\operatorname{GL}_{n}(\mathbb{T})$ is
the subgroup of matrices whose _tropical determinant_ , defined as the sum of
the finite entries, is zero.

Given a vector bundle $E$ of rank $n$ on a metric graph $\Gamma$, we can
represent its transition functions as
$\operatorname{GL}_{n}(\mathbb{T})$-valued harmonic functions as follows.
Choose an oriented simple model, and let
$g^{e}=(\sigma^{e},g^{e}_{1},\ldots,g^{e}_{n})$ for $e\in E(G)$ be the
transition functions of $E$, where $\sigma^{e}\in S_{n}$ and
$g^{e}_{i}\in{\mathcal{H}}_{\Gamma}(e)$. We can view the $g^{e}$ as
$\operatorname{GL}_{n}(\mathbb{T})$-matrices in the following way:

$g^{e}_{ij}=\left\\{\begin{array}[]{cc}g^{e}_{i},&\textrm{if
}j=\sigma^{e}(i),\\\ \infty,&\textrm{if
}j\neq\sigma^{e}(i).\end{array}\right.$

Note also that our notion of a tropical vector bundle coincides with the one
proposed in [All12].

We note that we may construct the _total space_ of a vector bundle $E$ on a
metric graph $\Gamma$ as an integral affine manifold by gluing trivializations
$U_{v}\times\mathbb{T}^{n}$ along the tropical linear maps $g^{e}_{ij}$. A
principal $\operatorname{PGL}_{n}(\mathbb{T})$-bundle, where
$\operatorname{PGL}_{n}(\mathbb{T})=\operatorname{GL}_{n}(\mathbb{T})/\mathbb{T}$,
then corresponds to a fibration by tropical projective spaces
$\mathbb{T}\mathbb{P}^{n-1}=\big{(}\mathbb{T}^{n}\backslash\\{\infty,\ldots,\infty\\}\big{)}/\mathbb{T}$.
We leave the details of this construction to the avid reader.

###  2.4. Basic operations

We now describe a number of standard constructions for vector bundles in our
tropical situation. Let $E$ and $F$ be two vector bundles of rank $m$ and $n$
on a metric graph $\Gamma$. Choose an oriented simple model $(G,\ell)$ of
$\Gamma$, and let $\\{g^{e}_{ij}\\}_{e\in E(G)}$ and $\\{h^{e}_{kl}\\}_{e\in
E(G)}$ for $e\in E(G)$ be the transition functions of $E$ and $F$,
respectively.

  1. (i)

The _direct sum_ $E\oplus F$ is the vector bundle of rank $m+n$ whose
transition functions on an edge $e\in E(G)$ are given by the block diagonal
matrices

$\begin{bmatrix}g^{e}_{ij}&\infty\\\ \infty&h^{e}_{kl}\end{bmatrix}\ .$

  2. (ii)

The _tensor product_ $E\otimes F$ is the vector bundle of rank $mn$ whose
transition functions on an edge $e\in E(G)$ are given by the Kronecker product
of the transition functions of $E$ and $F$, in other words by $mn\times mn$
matrices whose $((i,k),(j,l))$-th entry is

$g^{e}_{ij}+h^{e}_{kl}\ .$

  3. (iii)

The _dual vector bundle_ $E^{\ast}$ is the vector bundle of rank $n$ whose
transition functions on an edge $e\in E(G)$ are

$\widetilde{g}^{e}_{ij}=\begin{cases}-g^{e}_{ij},&\textrm{if
}g^{e}_{ij}\neq\infty\ ,\\\ \infty,&\textrm{if }g^{e}_{ij}=\infty\
.\end{cases}$

  4. (iv)

The _determinant_ $\det(E)$ is the line bundle on $\Gamma$ whose transition
functions are given by the tropical determinants of $g^{e}_{ij}$, in other
words the sums of all finite entries.

  5. (v)

The _degree_ of $E$ is the sum of the slopes of the finite entries of the
$g^{e}_{ij}$ over all edges $e\in E(G)$.

We leave it to the avid reader to verify that these constructions do not
depend on the choice of model $G$ and transition functions. We furthermore
note that a rank-$n$ vector bundle $E$ admits the structure of an
$\operatorname{SL}_{n}$-bundle if and only if $\det(E)$ is the trivial line
bundle. Finally, it is clear that the equations

$\begin{split}\deg E\oplus F&=\deg E+\deg F\\\ \deg E\otimes F&=\deg
E\cdot\deg F\\\ \deg E^{\ast}&=-\deg E\\\ \deg\det E&=\deg E\end{split}$

hold for all vector bundles $E$ and $F$.

######  Remark 2.3.

There are several different options on how one could define the notion of a
_homomorphism_ between two vector bundles $E$ and $F$: as a section of
$E^{\ast}\otimes F$, in terms of its total space, or by mimicking morphisms of
principal bundles. In fact, there does not even seem to be a generally agreed-
upon definition of a tropical linear map (which would immediately provide us
with a notion of a homomorphism of trivial bundles). This is why we refrain
from opening Pandora’s box and prefer not to propose a definite definition at
this point.

We point out, however, that two vector bundles are isomorphic (as torsors) if
and only if their transition functions define the same class in
$H^{1}(\Gamma,S_{n}\ltimes\mathcal{H}_{\Gamma}^{n})$. Thus, it is
straightforward to define subbundles:

######  Definition 2.4.

Let $E$ and $F$ be vector bundles of rank $n$ and $m\leq n$ on $\Gamma$. We
say that $F$ is a _subbundle_ of $E$, if we may choose transition functions
$h_{ij}^{e}$ of $F$ and $g_{ij}^{e}$ of $E$ in such a way that

$\big{[}g_{ij}^{e}\big{]}=\begin{bmatrix}h_{ij}^{e}&\ast\\\
\infty&\ast\end{bmatrix}$

for all $e\in E(G)$.

Since the matrix $h^{e}_{ij}$ is invertible, the unique finite entry in each
of the first $m$ rows occurs in one of the first $m$ columns. In other words,
an invertible block-triangular matrix is block-diagonal, and we have the
following result:

######  Proposition 2.5.

Every subbundle $F$ of a vector bundle $E$ on $\Gamma$ is _split_ , i.e. there
exists another subbundle $H$ of $E$ such that $E\simeq F\oplus H$.

We also define the pushforward and pullback of vector bundles along a free
cover. Let $f:\widetilde{\Gamma}\to\Gamma$ be a free cover of degree $d$.
Choose models $G$ and $\widetilde{G}$ for $\Gamma$ and $\widetilde{\Gamma}$
such that $f$ corresponds to a graph morphism $\widetilde{G}\to G$. Choose a
labeling $f^{-1}(v)=\\{\widetilde{v}_{1},\ldots,\widetilde{v}_{d}\\}$ of the
preimages of each vertex $v\in V(G)$. Similarly, for each edge $e\in E(G)$ we
label the preimages
$f^{-1}(e)=\\{\widetilde{e}_{1},\ldots,\widetilde{e}_{d}\\}$, where we assume
that $s(\widetilde{e}_{k})=\widetilde{s(e)}_{k}$. The cover is then determined
by an $S_{d}$-valued cocycle $\\{\sigma^{e}\\}_{e\in E(G)}$, where for each
$e\in E(G)$, the permutation $\sigma^{e}\in S_{d}$ is determined by the
formula $t(\widetilde{e}_{k})=\widetilde{t(e)}_{\sigma^{e}(k)}$.

######  Definition 2.6.

Let $f:\widetilde{\Gamma}\to\Gamma$ be a free cover of metric graphs of degree
$d$ with models $\widetilde{G}$ and $G$.

  1. (i)

Let $E$ be a vector bundle of rank $n$ on $\Gamma$, given by transition
functions $\\{g^{e}_{ij}\\}_{e\in E(G)}$ with respect to the model $G$. We
define the _pullback_ $f^{*}E$ as the rank-$n$ vector bundle on
$\widetilde{\Gamma}$ with transition functions

$\widetilde{g}^{\widetilde{e}}_{ij}=g^{f(\widetilde{e})}_{ij}$

for $\widetilde{e}\in E(\widetilde{G})$.

  2. (ii)

Let $\widetilde{E}$ be a vector bundle of rank $n$ on $\widetilde{\Gamma}$
defined by the cocycle
$\\{\widetilde{g}^{\widetilde{e}}_{ij}\\}_{\widetilde{e}\in
E(\widetilde{G})}$. The _pushforward_ $f_{*}E$ is the rank-$n\cdot d$ vector
bundle on $\Gamma$ whose transition function over $e\in E(G)$ is the $n\cdot
d$ block matrix whose $(k,l)$-th $n\times n$ block is
$\widetilde{g}^{\widetilde{e}_{k}}_{ij}$ if $l=\sigma^{e}(k)$ and $\infty$
otherwise.

##  3\. Vector bundles and multidivisors

###  3.1. Multidivisors

Line bundles on a metric graph $\Gamma$ may be described in terms of linear
equivalence classes of divisors on $\Gamma$. In the higher-rank situation, we
have a similar description in terms of so-called _multidivisors_. In this
section we establish a dictionary between the two.

######  Definition 3.1.

Let $\Gamma$ be a compact metric graph.

  1. (i)

A _multidivisor_ $(f\colon\widetilde{\Gamma}\to\Gamma,D)$ on $\Gamma$ of rank
$n$ consists of a (possibly disconnected) topological cover
$f\colon\widetilde{\Gamma}\rightarrow\Gamma$ of degree $n$ and a divisor $D$
on $\widetilde{\Gamma}$.

  2. (ii)

Two multidivisors $(f\colon\widetilde{\Gamma}\to\Gamma,D)$ and
$(f^{\prime}\colon\widetilde{\Gamma}^{\prime}\to\Gamma,D^{\prime})$ are said
to be _linearly equivalent_ if there is an isometry
$\phi\colon\widetilde{\Gamma}\xrightarrow{\sim}\widetilde{\Gamma}^{\prime}$ of
metric graphs such that $f^{\prime}\circ\phi=f$ and such that the difference
$D-\phi^{\ast}D^{\prime}$ is a principal divisor on $\widetilde{\Gamma}$.

One may think of think of the notion of a multidivisor as a tropical
incarnation of the Weil’s notion of a matrix divisor from [Wei38]. Given a
multidivisor $(f,D)$ on a compact metric graph $\Gamma$, the pushforward

$\mathcal{H}(f,D):=\mathcal{H}_{\Gamma}(f,D):=f_{\ast}\mathcal{H}_{\widetilde{\Gamma}}(D)$

is a vector bundle of rank $n$ on $\Gamma$. We now show that these two notions
are essentially equivalent.

######  Proposition 3.2.

Let $\Gamma$ be a compact metric graph. The association

$(f,D)\longmapsto\mathcal{H}(f,D)$

induces a natural bijection between linear equivalence classes of
multidivisors of rank $n$ and isomorphism classes of vector bundles of rank
$n$ on $\Gamma$.

Under this correspondence, the basic operations correspond to the following:

  1. (i)

Given two multidivisors $(f_{1},D_{1})$ and $(f_{2},D_{2})$, we write
$f_{1}\oplus
g_{1}\colon\widetilde{\Gamma}_{1}\sqcup\widetilde{\Gamma}_{2}\rightarrow\Gamma$
for the disjoint union of the covers and $D_{1}\oplus D_{2}$ for the induced
divisor on $\widetilde{\Gamma}_{1}\sqcup\widetilde{\Gamma}_{2}$. Then

$\mathcal{H}\big{(}f_{1}\oplus f_{2},D_{1}\oplus
D_{2}\big{)}\simeq\mathcal{H}(f_{1},D_{1})\oplus\mathcal{H}(f_{2},D_{2})\ .$

  2. (ii)

Given two multidivisors $(f_{1},D_{1})$ and $(f_{2},D_{2})$, we write
$f_{1}\times_{\Gamma}f_{2}\colon\widetilde{\Gamma}_{1}\times_{\Gamma}\widetilde{\Gamma}_{2}\rightarrow\Gamma$
for the fibered product of topological covers and

$D_{1}\boxtimes
D_{2}=\sum_{(p,q)\in\widetilde{\Gamma}_{1}\times_{\Gamma}\widetilde{\Gamma}_{2}}(D_{1}(p)+D_{2}(q))(p,q)$

for the induced divisor on
$\widetilde{\Gamma}_{1}\times_{\Gamma}\widetilde{\Gamma}_{2}$. Then

$\mathcal{H}\big{(}f_{1}\times_{\Gamma}f_{2},D_{1}\boxtimes
D_{2}\big{)}\simeq\mathcal{H}(f_{1},D_{1})\otimes\mathcal{H}(f_{2},D_{2})\ .$

  3. (iii)

Given a multidivisor $(f,D)$ on $\Gamma$, we have

    1. (a)

$\mathcal{H}(f,-D)\simeq\mathcal{H}(f,D)^{\ast}$

as well as

    2. (b)

$\det\mathcal{H}(f,D)\simeq\mathcal{H}(f_{\ast}D)$

and

    3. (c)

$\deg\mathcal{H}(f,D)=\deg(D).$

Furthermore, under this equivalence, indecomposable vector bundles on $\Gamma$
correspond to those multidivisors $(f,D)$ for which $f$ is a connected
topological cover.

###### Proof.

Choose a simple model $G$ for $\Gamma$. Let $E$ be a rank-$n$ vector bundle on
$\Gamma$ with transition functions
$\\{\sigma^{e},g^{e}_{1},\ldots,g^{n}_{e}\\}_{e\in E(G)}$, where
$\sigma^{e}\in S_{n}$ and $g^{i}_{e}\in{\mathcal{H}}(e)$. The $S_{n}$-cocycle
$\\{\sigma^{e}\\}_{e\in E(G)}$ determines a free cover
$f:\widetilde{\Gamma}\to\Gamma$ of degree $n$, which we describe using the
notation from the paragraph preceding Definition 2.6. Let $\mathcal{L}_{E}$ be
the line bundle on $\widetilde{\Gamma}$ whose transition function on the edge
$\widetilde{e}_{i}$ over $e\in E(G)$ is equal to $g^{i}_{e}$. It follows from
Definition 2.6 that $f_{*}\mathcal{L}_{E}\simeq E$. Choosing a divisor
$D\in\operatorname{Div}(\widetilde{\Gamma})$ such that
$\mathcal{L}_{E}\simeq{\mathcal{H}}_{\widetilde{\Gamma}}(D)$, it is elementary
to verify that the isomorphism class of the pair $(f,D)$ does not depend on
the choices made. The verification of (i)-(iii) follows immediately by
calculating cocycles with respect to the cover $G$ (see the definitions of the
basic operation in Section 2.4) and is left to the avid reader. ∎

######  Example 3.3 (Vector bundles on metric trees).

Let $\Gamma$ be a compact and connected metric tree. Two divisors on $\Gamma$
are linearly equivalent if and only if they have the same degree. So, up to
isomorphism, there is exactly one line bundle $\mathcal{H}_{\Gamma}(d)$ of
degree $d$ on $\Gamma$.

Since the fundamental group of $\Gamma$ is trivial, the only connected
covering space of $\Gamma$ is $\Gamma$ itself. Hence for each $n$ there is a
unique degree $n$ free cover $\widetilde{\Gamma}\rightarrow\Gamma$ consisting
of $n$ disjoint copies of $\Gamma$. By Proposition 3.2, this means that every
vector bundle $E$ of rank $n$ on $\Gamma$ splits as a sum of line bundles

$E\simeq\mathcal{H}_{\Gamma}(a_{1})\oplus\cdots\oplus\mathcal{H}_{\Gamma}(a_{n})$

for a unique multiset of $n$ integers $a_{1},\ldots,a_{n}$. This is a tropical
analogue of the Birkhoff–Grothendieck theorem which states that any vector
bundle on $\mathbb{P}^{1}$ is a direct sum of line bundles (see [Gro57]).

###  3.2. The moduli space $\operatorname{Bun}_{n}(\Gamma)$

Proposition 3.2 may be used to describe the moduli space
$\operatorname{Bun}_{n}(\Gamma)$ of all vector bundles on a compact and
connected metric graph $\Gamma$. One may think of this in analogy with [NS64],
where a moduli space of (semi)-stable vector bundles is constructed using the
Narasimhan-Seshadri correspondence.

######  Corollary 3.4.

Let $\Gamma$ be a compact and connected metric graph. The set
$\operatorname{Bun}_{n}(\Gamma)$ of isomorphism classes of rank $n$ vector
bundles on $\Gamma$ is naturally given by

$\operatorname{Bun}_{n}(\Gamma)=\bigsqcup_{f\colon\widetilde{\Gamma}\rightarrow\Gamma}\operatorname{Pic}(\widetilde{\Gamma})/\operatorname{Aut}(f)\
,$

where the union is taken over isomorphism classes of topological covers
$f\colon\widetilde{\Gamma}\rightarrow\Gamma$ of degree $n$, and
$\operatorname{Aut}(f)$ is the group of deck transformations of $f$ acting on
$\operatorname{Pic}(\Gamma)$ via pullback.

###### Proof.

This is an immediate consequence of Proposition 3.2. ∎

For a fixed free cover $f\colon\widetilde{\Gamma}\rightarrow\Gamma$, we write

$\widetilde{\Gamma}=\widetilde{\Gamma}_{1}\sqcup\cdots\sqcup\widetilde{\Gamma}_{k}$

for the decomposition into connected components. We then have

$\operatorname{Pic}(\widetilde{\Gamma})=\operatorname{Pic}(\widetilde{\Gamma}_{1})\times\cdots\times\operatorname{Pic}(\widetilde{\Gamma}_{k})$

and every $\operatorname{Pic}_{d}(\widetilde{\Gamma}_{i})$ is naturally a
torsor over the real torus
$\operatorname{Pic}_{0}(\widetilde{\Gamma}_{i})\simeq\operatorname{Jac}(\widetilde{\Gamma}_{i})$.
So $\operatorname{Bun}_{n}(\Gamma)$ is naturally a disjoint union of finite
group quotients of torsors over real tori.

We point out that if $\Gamma$ has genus $g$, then $\widetilde{\Gamma}$ has
genus $\widetilde{g}=n(g-1)+1$. Hence we obtain that

$\dim_{\mathbb{R}}\operatorname{Bun}_{n}(\Gamma)=n(g-1)+1.$

Classically, the dimension of the moduli space of vector bundles on an
algebraic curve of genus $g$ is equal to $d(g-1)+1$, where $d$ is the
dimension of the structure group, and in particular the dimension is
$n^{2}(g-1)+1$ when the structure group is $\operatorname{GL}_{n}$. This
discrepancy is explained by noting that
$\dim_{\mathbb{R}}\operatorname{GL}_{n}(\mathbb{T})=n$, not $n^{2}$, and is
further clarified by Theorem C, which states that tropical vector bundles
(according to our definition) arise as tropicalizations of geometrically
$S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector bundles; so the algebraic
structure group actually has dimension $n$, not $n^{2}$. In the case $g=1$,
the two dimensions agree and we have a much stronger tropicalization result in
Theorem D.

###  3.3. Vector bundles on tropical elliptic curves

Let $\ell>0$ be a real number, and let $\Gamma=\mathbb{R}/\ell\mathbb{Z}$ be a
circle of length $\ell$ (for example, $\Gamma$ could be the minimal skeleton
of a Tate curve $X^{\mathrm{an}}=\mathbb{G}_{m}^{\mathrm{an}}/q^{\mathbb{Z}}$,
where $\ell$ is the valuation of $q$). We determine the structure of the set
$M_{n,d}^{\mathrm{ind}}(\Gamma)$ of isomorphism classes of indecomposable
vector bundles of rank $n$ and degree $d$ on $\Gamma$.

Since $\pi_{1}(\Gamma)=\mathbb{Z}$, for each $n\geq 1$ there is a unique
connected free cover $f:\widetilde{\Gamma}\to\Gamma$ of degree $n$, where
$\widetilde{\Gamma}=\mathbb{R}/n\ell\mathbb{Z}$ is a circle of length $n\ell$
and the covering map $f\colon\widetilde{\Gamma}\to\Gamma$ is the quotient map
$\mathbb{R}/n\ell\mathbb{Z}\to\mathbb{R}/\ell\mathbb{Z}$. For a real number
$x\in\mathbb{R}$, we denote the corresponding points on $\Gamma$ and
$\widetilde{\Gamma}$ by $p_{x}=x+\ell\mathbb{Z}$ and
$\widetilde{p}_{x}=x+n\ell\mathbb{Z}$, respectively. The group
$\operatorname{Aut}(f)$ is the cyclic group of order $n$ generated by the deck
transformation $g(\widetilde{p}_{x})=\widetilde{p}_{x+\ell}$.

We can identify $\Gamma=\operatorname{Pic}^{0}(\Gamma)$ via the map
$p_{x}\mapsto[p_{x}-p_{0}]$ (which is in fact an isomorphism of groups), and
similarly $\widetilde{\Gamma}$ is identified with
$\operatorname{Pic}^{0}(\widetilde{\Gamma})$. In terms of these
identifications, the pullback morphism

$f^{*}\colon\Gamma=\operatorname{Pic}^{0}(\Gamma)\longrightarrow\operatorname{Pic}^{0}(\widetilde{\Gamma})=\widetilde{\Gamma}$

is given by multiplication by $n$, in other words it is the bijection
$p_{x}-p_{0}\mapsto\widetilde{p}_{nx}-\widetilde{p}_{0}$. The set
$\operatorname{Pic}^{d}(\widetilde{\Gamma})$ is a torsor over
$\operatorname{Pic}^{0}(\widetilde{\Gamma})$, and we translate by
$d\widetilde{p}_{0}$ to obtain the bijection

$\begin{split}\Gamma&\longrightarrow\operatorname{Pic}^{d}(\widetilde{\Gamma})\\\
p_{x}&\longmapsto
d\widetilde{p}_{0}+f^{*}(p_{x})=(d-1)\widetilde{p}_{0}+\widetilde{p}_{nx}.\end{split}$

By Corollary 3.4 we can identify $M_{n,d}^{\mathrm{ind}}(\Gamma)$ with the
quotient $\operatorname{Pic}^{d}(\widetilde{\Gamma})/\operatorname{Aut}(f)$.
The generator $g$ of $\operatorname{Aut}(f)$ acts on
$\operatorname{Pic}^{d}(\widetilde{\Gamma})$ (identified with
$\widetilde{\Gamma}$) by translation by $d\ell$, hence $p_{x}$ and
$p_{x^{\prime}}$ define the same vector bundle if and only if $x-x^{\prime}$
is a multiple of $\frac{d\ell}{n}$. The multiples of $\frac{d\ell}{n}$
correspond to the $n^{\prime}=\frac{n}{\gcd(d,n)}$-torsion points of $\Gamma$
and constitute the kernel of the multiplication by $n^{\prime}$ map

$\begin{split}\Gamma&\longrightarrow\Gamma\\\ p_{x}&\longmapsto
p_{n^{\prime}x}\ .\end{split}$

We now summarize our results for future use and rephrase them in the notation
of Proposition 3.2. Consider the vector bundle

$E^{\mathrm{trop}}(n,d)={\mathcal{H}}(f,d\cdot\widetilde{p}_{0})$

of degree $d$ and rank $n$ on $\Gamma$. Then the formula

$\Psi(p_{x})={\mathcal{H}}\big{(}f,d\widetilde{p}_{0}+f^{*}(p_{x}-p_{0})\big{)}=E^{\mathrm{trop}}(n,d)\otimes{\mathcal{H}}_{\Gamma}(p_{x}-p_{0})\
.$

defines a surjective map $\Psi:\Gamma\to M_{n,d}^{\mathrm{ind}}(\Gamma)$.
Hence the map $\Psi$ fits into the diagram

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where $\Gamma\xrightarrow{\cdot n^{\prime}}\Gamma$ is the multiplication by
$n^{\prime}$ map, and we have an identification of
$M_{n,d}^{\mathrm{ind}}(\Gamma)$ with $\Gamma$.

We may view this identification as a tropical analogue of Atiyah’s
classification of indecomposable vector bundles on an elliptic curve from
[Ati57].

##  4\. Baker–Norine rank and a Weil–Riemann–Roch theorem

###  4.1. Linear systems and Baker–Norine rank

Let $\Gamma$ be a metric graph. A divisor $D\in\operatorname{Div}(\Gamma)$ is
said to be _effective_ , written as $D\geq 0$, if $D(p)\geq 0$ for all
$p\in\Gamma$. We write $D\geq D^{\prime}$ for divisors
$D,D^{\prime}\in\operatorname{Div}(\Gamma)$ if $D-D^{\prime}\geq 0$. The
_(complete) linear system_ $|D|$ associated to
$D\in\operatorname{Div}(\Gamma)$ is the set

$|D|=\big{\\{}D^{\prime}\geq 0\mid D\sim D^{\prime}\big{\\}}\ .$

Let $\Gamma$ be a compact metric graph. The _Baker–Norine rank_
$r_{\Gamma}(D)$ of a divisor $D$ on $\Gamma$ was introduced in [BN07] and is
defined as follows. Suppose first that $\Gamma$ is connected. If
$|D|=\emptyset$, we set $r(D)=-1$. Otherwise we define $r(D)$ to the maximal
$r\geq 0$ such that $|D-E|\neq\emptyset$ for all effective divisors $E$ of
degree $r$. Now let $\Gamma$ be a disconnected curve, and let
$\Gamma=\Gamma_{1}\sqcup\cdots\sqcup\Gamma_{k}$ be the decomposition into
connected components. We recall that for a divisor $D$ on an algebraic curve
$X$ we have $r(D)=h^{0}(\mathcal{O}_{X}(D))-1$, and that the quantity $h^{0}$
is additive in connected components. Motivated by this, we define
$r_{\Gamma}(D)$ by the identity

$r_{\Gamma}(D)=\sum_{i=1}^{k}(r_{\Gamma_{i}}(D|_{\Gamma_{i}})+1)-1=\sum_{i=1}^{k}r_{\Gamma_{i}}(D|_{\Gamma_{i}})+k-1\
.$

The _canonical divisor_ $K_{\Gamma}$ on a metric graph $\Gamma$ is defined to
be

$K_{\Gamma}=\sum_{p\in\Gamma}(\operatorname{val}(p)-2)\cdot p\ ,$

where $\operatorname{val}(p)$ denotes the valence of $\Gamma$ at $p$, i.e. the
number of outgoing half-edges from $p$.

The Riemann–Roch formula from [BN07, MZ08, GK08] states that, for every
divisor $D$ of degree $d$ on a compact and connected metric graph $\Gamma$ of
genus $g$, we have

$r_{\Gamma}(D)-r_{\Gamma}(K_{\Gamma}-D)=d-g+1\ .$

As its classical counterpart, this combinatorial Riemann–Roch formula has
already found numerous applications; we refer the reader to [BJ16, Section
4.3] for a concise survey of these applications as well as an outline of its
history and proof.

We first observe that the following extension of the Riemann–Roch formula
holds for the Baker–Norine rank on arbitrary (possibly disconnected) compact
metric graphs.

######  Proposition 4.1.

Let $\Gamma$ be a compact metric graph, and let $D$ be a divisor of degree $d$
on $\Gamma$. Then

$r_{\Gamma}(D)-r_{\Gamma}(K_{\Gamma}-D)=d+\chi(\Gamma)\ .$

###### Proof.

Let $\Gamma=\Gamma_{1}\sqcup\cdots\sqcup\Gamma_{k}$ be the decomposition into
connected components and write $D_{i}=D|_{\Gamma_{i}}$ for the restriction of
$D$ to $\Gamma_{i}$ as well as $g_{i}$ for the genus of each $\Gamma_{i}$.
Then $K_{\Gamma_{i}}=K_{\Gamma}|_{\Gamma_{i}}$, and by the definition of
$r_{\Gamma}$ we have

$\begin{split}r_{\Gamma}(D)-r_{\Gamma}(K_{\Gamma}-D)&=\sum_{i=1}^{k}r_{\Gamma_{i}}\big{(}D|_{\Gamma_{i}}\big{)}+k-1-\left[\sum_{i=1}^{k}r_{\Gamma_{i}}\big{(}K_{\Gamma}|_{\Gamma_{i}}-D|_{\Gamma_{i}}\big{)}+k-1\right]\\\
&=\sum_{i=1}^{k}\big{(}r_{\Gamma_{i}}(D_{i})-r_{\Gamma_{i}}(K_{\Gamma_{i}}-D_{i})\big{)}\\\
&=\sum_{i=1}^{k}\big{(}\deg(D_{i})-g_{i}+1\big{)}\\\ &=d+\chi(\Gamma)\
,\end{split}$

since

$\chi(\Gamma)=\sum_{i=1}^{k}\chi(\Gamma_{i})=\sum_{i=1}^{k}(1-g_{i})\ .$

∎

###  4.2. The higher rank case

Multidivisors allow us to generalize the definition of the Baker–Norine rank
of a complete linear system on $\Gamma$ to tropical vector bundles.

######  Definition 4.2.

Let $\Gamma$ be a compact metric graph. We define the _Baker–Norine rank_
$r_{\Gamma}(f,D)$ of a multidivisor $(f\colon\widetilde{\Gamma}\to\Gamma,D)$
simply to be the Baker–Norine rank $r_{\widetilde{\Gamma}}(D)$ of the divisor
$D$ on $\widetilde{\Gamma}$. Given a vector bundle $E$ on $\Gamma$, we define
its _Baker–Norine rank_ by

$r_{\Gamma}(E)=r_{\Gamma}(f,D)$

for one (and automatically all) multidivisors $(f,D)$ such that
$f_{\ast}\mathcal{H}(D)\simeq E$.

######  Theorem 4.3 (Weil–Riemann–Roch).

Let $\Gamma$ be a compact metric graph. For a vector bundle $E$ of degree $d$
on $\Gamma$ the formula

$r_{\Gamma}(E)-r_{\Gamma}\big{(}\mathcal{H}(K_{\Gamma})\otimes
E^{\ast}\big{)}=d+n\chi(\Gamma)$

holds.

###### Proof.

Let $(f,D)$ be a multidivisor such that $f_{\ast}\mathcal{H}(D)\simeq E$. By
Proposition 3.2, the multidivisor $(f,K_{\widetilde{\Gamma}}-D)$ represents
the vector bundle $\mathcal{H}(K_{\Gamma})\otimes E^{\ast}$ on $\Gamma$. Then
the Riemann–Roch formula for divisors from [BN07, MZ08, GK08] (generalized to
disconnected metric graphs as in Proposition 4.1) can be applied to $D$ on
$\widetilde{\Gamma}$ and tells us that

$r_{\widetilde{\Gamma}}(D)-r_{\widetilde{\Gamma}}(K_{\widetilde{\Gamma}}-D)=d+\chi(\widetilde{\Gamma})\
.$

Since $f\colon\widetilde{\Gamma}\rightarrow\Gamma$ is a covering space of
degree $n$, we have

$\chi(\widetilde{\Gamma})=n\chi(\Gamma)\ .$

Thus we obtain the desired formula

$r_{\Gamma}(E)-r_{\Gamma}\big{(}\mathcal{H}(K_{\Gamma})\otimes
E^{\ast}\big{)}=d+n\chi(\Gamma)\ .$

∎

##  5\. Semistable bundles and a Narasimhan–Seshadri correspondence

In this section we prove an analogue of the Narasimhan–Seshadri correspondence
on a metric graph.

######  Definition 5.1.

Let $\Gamma$ be a compact metric graph.

  1. (i)

The _slope_ of a vector bundle $E$ on $\Gamma$ is defined to be the quotient

$\mu(E)=\frac{\deg E}{\operatorname{rk}E}$

  2. (ii)

A vector bundle $E$ on $\Gamma$ is said to be _semistable_ if for all
subbundles $E^{\prime}\subseteq E$, we have

$\mu(E^{\prime})\leq\mu(E)\ .$

If this inequality is strict for all proper subbundles, we say that $E$ is
_stable_.

######  Proposition 5.2.

Let $\Gamma$ be a compact and connected metric graph.

  1. (i)

A vector bundle on $\Gamma$ is stable if and only if it is indecomposable.

  2. (ii)

A vector bundle $\mathcal{E}$ that decomposes as $E=E_{1}\oplus\cdots\oplus
E_{r}$ with all $E_{i}$ indecomposable is semistable if and only if

$\mu(E_{1})=\cdots=\mu(E_{r})\ .$

###### Proof.

Let $E=E_{1}\oplus E_{2}$ for two vector bundles $E_{1}$ and $E_{2}$ on
$\Gamma$ of ranks $n_{1}$ and $n_{2}$. Assume that $\mu(E_{1})\leq\mu(E_{2})$.
Then we have

$\mu(E)=\frac{\deg(E_{1})+\deg(E_{2})}{n_{1}+n_{2}}=\frac{n_{1}}{n_{1}+n_{2}}\mu(E_{1})+\frac{n_{2}}{n_{1}+n_{2}}\mu(E_{2})\
.$

This implies that

$\mu(E_{1})\leq\mu(E)\leq\mu(E_{2})\ ,$

and both inequalities are strict unless $\mu(E_{1})=\mu(E_{2})$. Since every
sub-bundle splits by Proposition 2.5 above this immediately implies Part (i).
Part (ii) follows by induction over the number of indecomposable summands. ∎

######  Example 5.3.

Let $M_{n,d}(\Gamma)$ denote the locus of semistable bundles in
$\operatorname{Bun}_{n}(\Gamma)$ of degree $d$. By Proposition 3.2, a
component of $M_{n,d}(\Gamma)$ parametrizes vector bundles associated to a
disjoint union of connected multidivisors $(f_{i},D_{i})$ such that
$\mu\big{(}\mathcal{H}(f_{i},D_{i})\big{)}=\mu\big{(}\mathcal{H}(f_{j},D_{j})\big{)}$
for all $i$ and $j$. Unlike its algebraic counterpart, the tropical moduli
space $M_{n,d}(\Gamma)$ is, in general, not connected.

Now suppose that $\Gamma$ is a circle, and denote $h=(n,d)$. There is a main
component $M_{n,d}^{\oplus}(\Gamma)$ that parametrizes those semistable
bundles of rank $n$ and degree $d$ that are direct sums of $h$ stable bundles
(all of which must automatically be of rank $n/h$ and degree $d/h$ by
Proposition 5.2 above). Combining Sections 3.2 and 3.3, we find that this
component may be identified with the symmetric product
$\operatorname{Sym}^{h}\Gamma$.

Let $\Gamma$ be a metric graph. The sheaf
$S_{n}\ltimes{\mathcal{H}}_{\Gamma}^{n}$ has a natural subsheaf
$S_{n}\ltimes\mathbb{R}^{n}$, where $\mathbb{R}\subset{\mathcal{H}}_{\Gamma}$
is the sheaf of constant functions. We view sections of
$S_{n}\ltimes{\mathcal{H}}_{\Gamma}^{n}$ as
$\operatorname{GL}_{n}(\mathbb{T})$-valued functions on $\Gamma$. We now show
that sections of $S_{n}\ltimes\mathbb{R}^{n}$ can be viewed as functions
valued in a tropical _unitary_ group, by proving an analogue of the
Narasimhan–Seshadri correspondence. Indeed, consider a local system $\lambda$
on $\Gamma$ with fiber $S_{n}\ltimes\mathbb{R}^{n}$. The $S_{n}$-factor of
$\lambda$ corresponds to a free degree $n$ cover
$f:\widetilde{\Gamma}\to\Gamma$, and there exists a local system
$\widetilde{\lambda}$ on $\widetilde{\Gamma}$ with fiber $\mathbb{R}$ such
that $f_{\ast}\widetilde{\lambda}=\lambda$. At the same time, $\lambda$ also
naturally defines a rank-$n$ vector bundle $E(\lambda)$ on $\Gamma$ with
constant transition functions.

######  Theorem 5.4 (Narasimhan–Seshadri correspondence).

Let $\Gamma$ be a compact and connected metric graph.

  1. (i)

A vector bundle $E$ of rank $n$ on $\Gamma$ is associated to an
$S_{n}\ltimes\mathbb{R}^{n}$-local system $\lambda$ if and only if it is
semistable and of degree zero. The vector bundle $E(\lambda)$ is stable if and
only if the corresponding $S_{n}$-representation of $\pi_{1}(\Gamma)$ is
indecomposable.

  2. (ii)

Two $S_{n}\ltimes\mathbb{R}^{n}$-local systems $\lambda_{i}$ (for $i=1,2$)
give rise to the same vector bundle if and only if they define the same cover
$f\colon\widetilde{\Gamma}\rightarrow\Gamma$ and the induced classes of the
$\widetilde{\lambda}_{i}$ in $\operatorname{Jac}(\widetilde{\Gamma})$ are
equal.

###### Proof.

We first consider the case $n=1$, where we need to show that any line bundle
of degree zero (which is trivially semistable) is associated to an
$\mathbb{R}$-local system, corresponding to an element
$H^{1}(\Gamma,\mathbb{R})$. Consider the long exact sequence (6), written in
Čech cohomology with respect to an appropriate cover. The set of isomorphism
classes of line bundles is
$\operatorname{Pic}(\Gamma)=\check{H}^{1}(\Gamma,{\mathcal{H}}_{\Gamma})$. The
subset of degree zero line bundles is the image of
$\check{H}^{1}(\Gamma,\mathbb{R})$. Hence any line bundle of degree zero can
be represented by an $\mathbb{R}$-valued cocycle. Two $\mathbb{R}$-local
systems $\lambda_{i}$ (for $i=1,2$) give rise to the same line bundle if and
only if their classes in the quotient
$\operatorname{Jac}(\Gamma)=\check{H}^{1}(\Gamma,\mathbb{R})/\Omega_{\Gamma}(\Gamma)$
agree.

We now consider the general case. By Proposition 5.2, a vector bundle $E$ of
degree (and hence slope) zero is semistable if and only if it is a direct sum
of stable vector bundles of degree zero. By Proposition 3.2, this is
equivalent to saying that $E$ is the pushforward $f_{*}L$, along a degree $n$
free cover $f:\widetilde{\Gamma}\to\Gamma$, of a line bundle $L$ having degree
zero on each connected component of $\widetilde{\Gamma}$. Hence if $E$ is
semistable, then we can choose the transition functions of $L$ on each
connected component to be constant, and then the transition functions of $E$
are also constant by the definition of pushforward. Conversely, if $E$ is
associated to a $S_{n}\ltimes\mathbb{R}^{n}$-local system, then the transition
functions of $L$ can be chosen to be constant, so $L$ has degree zero on each
connected component and therefore $E$ is semistable.

By Proposition 5.2 the vector bundle $E(\lambda)$ is stable if and only if
$\widetilde{\Gamma}$ is connected, which is the case if and only if the
representation $\pi_{1}(\Gamma)\rightarrow S_{n}$ giving rise to
$f\colon\widetilde{\Gamma}\rightarrow\Gamma$ is indecomposable. ∎

######  Remark 5.5.

We point out that every representation $\pi_{1}(\Gamma)\rightarrow
S_{n}\ltimes\mathbb{R}^{n}$ is already a direct sum of irreducible
representations. This is why we do not have to require the representation in
Theorem 5.4 to be “semisimple” in any suitable sense.

##  6\. The process of tropicalization

In this section and the next, we work over an algebraically closed field $K$
that is complete with respect to a non-trivial non-Archimedean absolute value
$|.|$. Write $\operatorname{val}$ for the associated valuation on $K$, as well
as $R$ for the valuation ring of $K$ and $k$ for its (automatically
algebraically closed) residue field.

###  6.1. Freely $S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector bundles

Let $X$ be a _Mumford curve_ , i.e. a (connected) smooth and projective curve
over $K$ that admits a prestable reduction $\mathcal{X}$ whose special fiber
is a nodal curve, all of whose components normalize to $\mathbb{P}^{1}$. A
vector bundle $E$ on $X$ is said to be
$S_{n}\ltimes\mathbb{G}_{m}^{n}$-_linearized_ if the structure group of $E$ is
$S_{n}\ltimes\mathbb{G}_{m}^{n}$; this means that we may trivialize $E$ on an
open cover $U_{i}$ of $X$ such that the transition maps, which a priori live
in $\operatorname{GL}_{n}\big{(}\mathcal{O}(U_{i}\cap U_{j})\big{)}$, may be
chosen to land in
$S_{n}\ltimes\mathbb{G}_{m}^{n}\subseteq\operatorname{GL}_{n}$.

A vector bundle is $S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized if and only if
there is an étale degree $n$ cover $f\colon\widetilde{X}\rightarrow X$ (with
$\widetilde{X}$ possibly disconnected) and a line bundle $L$ on
$\widetilde{X}$ such that $f_{\ast}L\simeq E$. We say that $E$ is _freely
$S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized_ if the cover
$f\colon\widetilde{X}\rightarrow X$ is induced from an étale cover
$\widetilde{\mathcal{X}}\rightarrow\mathcal{X}$ of the prestable model
$\mathcal{X}$ of $X$ (and not from an admissible cover in the sense of
[HM82]). Note that this already implies that $\widetilde{X}$ is also a Mumford
curve. Being freely linearized is equivalent to the condition that the induced
map on the dual metric graphs (respectively on the skeletons) is a free cover
(in general, this map is a harmonic morphism with nontrivial dilation).
Equivalently, we may require the induced map
$f^{\mathrm{an}}\colon\widetilde{X}^{\mathrm{an}}\rightarrow X^{\mathrm{an}}$
of Berkovich analytic spaces to be a topological covering space. Hence this
condition is also independent of the choice of the particular prestable model
$\mathcal{X}$ of $X$.

Write $\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}(X)$ for the moduli
stack of vector bundles of rank $n$, which are
$S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized. Arguing as in Corollary 3.4, we
find:

######  Lemma 6.1.

The moduli stack $\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}(X)$ is
given by the disjoint union

$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}(X)=\bigsqcup_{f\colon\widetilde{X}\rightarrow
X}\big{[}\operatorname{Bun}_{\mathbb{G}_{m}}(\widetilde{X})\big{/}\operatorname{Aut}(f)\big{]}$

taken over isomorphism classes of étale covers
$f\colon\widetilde{X}\rightarrow X$ of degree $n$, where
$\operatorname{Aut}(f)$ is the group of deck transformations of $f$ acting on
$\operatorname{Bun}_{\mathbb{G}_{m}}(\widetilde{X})$ via pullback.

For fixed $f\colon\widetilde{X}\rightarrow X$ we write

$\widetilde{X}=\widetilde{X}_{1}\sqcup\cdots\sqcup\widetilde{X}_{k}$

for the decomposition into connected components. Then we have

$\operatorname{Bun}_{\mathbb{G}_{m}}(X)=\operatorname{Bun}_{\mathbb{G}_{m}}(\widetilde{X}_{1})\times\cdots\times\operatorname{Bun}_{\mathbb{G}_{m}}(\widetilde{X}_{k})\
.$

Rigidifying by $\mathbb{G}_{m}$-automorphism groups, we find

$\big{[}\operatorname{Bun}_{\mathbb{G}_{m}}(\widetilde{X}_{i})\fatslash\
\mathbf{B}\mathbb{G}_{m}\big{]}\simeq\operatorname{Pic}(\widetilde{X}_{i})$

and every $\operatorname{Pic}_{d}(\widetilde{X}_{i})$ is naturally a torsor
over the Jacobian
$\operatorname{Pic}_{0}(\widetilde{X}_{i})\simeq\operatorname{Jac}(\widetilde{X}_{i})$.

The stack $\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}(X)$ is a smooth
and proper Artin stack that arises as a disjoint union of finite group
quotients of the $\operatorname{Bun}_{\mathbb{G}_{m}}(\widetilde{X}_{i})$. The
subspace
$\operatorname{Bun}^{{\mathrm{free}}}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}(X)$ of
freely $S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector bundles consists
precisely of those components, for which the cover
$f\colon\widetilde{X}\rightarrow X$ is induces a topological covering space
$f^{\mathrm{an}}\colon\widetilde{X}^{\mathrm{an}}\rightarrow X^{\mathrm{an}}$.

###  6.2. Tropicalization

Let $X$ be a Mumford curve and write $\Gamma_{X}$ for a _non-Archimedean
skeleton_ of the Berkovich analytic space $X^{\mathrm{an}}$ that is associated
to a prestable model $\mathcal{X}$ of $X$ over $R$ (e.g. the minimal
semistable model when $g(X)\geq 2$). Write
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}$
for the Berkovich analytification of
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)$
and, implicitly, also for its underlying topological space (see [Uli17,
Section 3] for details).

Then there is a natural _tropicalization map_

${\mathrm{trop}}\colon\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}\longrightarrow\operatorname{Bun}_{n}(\Gamma_{X})$

given by the following: A point in
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}$
may be represented by an $K^{\prime}$-valued point of
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)$ for
a non-Archimedean extension $K^{\prime}$ of $K$. This datum corresponds to a
freely $S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector bundle $E$ on the
base change $X_{K^{\prime}}$ of $X$ to $K^{\prime}$. Let
$f\colon\widetilde{X}\rightarrow X_{K^{\prime}}$ be an étale cover of degree
$n$ and $L$ a line bundle on $\widetilde{X}$ such that $f_{\ast}L=E$. The
cover $f$ is the generic fiber of a degree $n$ étale cover
$\widetilde{\mathcal{X}}\rightarrow\mathcal{X}$ of prestable models over $S$,
where $S$ is the valuation ring of $K^{\prime}$; thus the induced map
$f^{{\mathrm{trop}}}\colon\Gamma_{\widetilde{X}}\rightarrow\Gamma_{X}$ is a
free cover of degree $n$. Then we set

${\mathrm{trop}}(E)=\big{(}f^{{\mathrm{trop}}},{\mathrm{trop}}(L)\big{)}$

where ${\mathrm{trop}}(L)$ is the usual tropicalization of a line bundle $L$
on $\widetilde{X}_{L}$ thought of as a $K^{\prime}$-valued point of
$\operatorname{Pic}(X)^{\mathrm{an}}$.

To describe ${\mathrm{trop}}(L)$ we expand on [BJ16, Section 6.3]: We may
assume that $K^{\prime}$ is algebraically closed. Write
$L=\mathcal{O}_{X_{K^{\prime}}}(D)$ for a split divisor

$D=\sum_{p\in X(K^{\prime})}D(p)\cdot p$

on $\widetilde{X}_{K^{\prime}}$. Then ${\mathrm{trop}}(L)$ is defined to be
the linear equivalence class of the pointwise pushforward of $D$ to the
skeleton $\Gamma_{\widetilde{X}}=\Gamma_{\widetilde{X}_{K^{\prime}}}$ of
$\widetilde{X}^{\mathrm{an}}$. This is well-defined thanks to the slope
formula from [BPR16] (also see [BPR13]).

There is a second way to associate to
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)$ a
disjoint union of real torus torsors. For this note first that the underlying
topological space of
$\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}$
is naturally homeomorphic to the underlying topological space of

(8) $\bigsqcup_{f\colon\widetilde{X}\rightarrow
X}\big{[}\operatorname{Pic}(\widetilde{X})\big{/}\operatorname{Aut}(f)\big{]}^{\mathrm{an}}\
,$

since rigidifying by $\mathbf{B}\mathbb{G}_{m}^{\mathrm{an}}$ does not change
the points of a non-Archimedean stack. Every component in (8) is a finite
quotient of torsors over an abelian variety and, using Raynaud’s
uniformization, one can show that the Berkovich analytification of each
component admits a strong deformation retraction onto a non-Archimedean
skeleton that is a torsor over a real torus (see [Ber90, Section 6.5] and
[BR15, Section 4]). We write

$\rho\colon\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}\longrightarrow\Sigma\big{(}\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)\big{)}$

for the disjoint union of these retractions.

The tropicalization map agrees with $\rho$ in the following sense:

######  Theorem 6.2.

There is a natural isomorphism
$J\colon\operatorname{Bun}_{n}(\Gamma_{X})\xrightarrow{\sim}\Sigma\big{(}\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)\big{)}$
that makes the diagram

${\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)^{\mathrm{an}}}$${\operatorname{Bun}_{n}(\Gamma_{X})}$${\Sigma\big{(}\operatorname{Bun}_{S_{n}\ltimes\mathbb{G}_{m}^{n}}^{{\mathrm{free}}}(X)\big{)}}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\rho}$$\scriptstyle{J}$$\scriptstyle{\sim}$

commute.

Theorem 6.2, in particular, tells us that the tropicalization map is well-
defined, continuous, proper, and surjective, since these are well-known
properties of the retraction map to the skeleton.

###### Proof of Theorem 6.2.

The case $n=1$ is an immediate consequence of the main result of [BR15]. We
have
$\operatorname{Pic}(X)=\bigsqcup_{d\in\mathbb{Z}}\operatorname{Pic}_{d}(X)$ as
well as
$\operatorname{Pic}(\Gamma_{X})=\bigsqcup_{d\in\mathbb{Z}}\operatorname{Pic}_{d}(\Gamma_{X})$
and, since tropicalization preserves degree, we may consider each component
separately.

Choose a base point $p\in X(K)$ and write $\alpha\colon
X\rightarrow\operatorname{Pic}_{0}(X)\simeq\operatorname{Jac}(X)$ as well as
$\alpha^{{\mathrm{trop}}}\colon\Gamma_{X}\rightarrow\operatorname{Jac}(\Gamma_{X})$
for the associated Abel–Jacobi maps, both given by $q\mapsto[q-p]$. By [BR15]
we have a natural isomorphism
$\operatorname{Jac}(\Gamma_{X})\xrightarrow{\sim}\Sigma(\operatorname{Jac}(X))$
between the Jacobian $\operatorname{Jac}(\Gamma_{X})$ and the non-Archimedean
skeleton of $\operatorname{Jac}(X)^{\mathrm{an}}$ that makes the diagram

${X^{\mathrm{an}}}$${\operatorname{Jac}(X)^{\mathrm{an}}}$${\Gamma_{X}}$${\operatorname{Jac}(\Gamma_{X})}$${\Sigma(\operatorname{Jac}(X))}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\alpha^{\mathrm{an}}}$$\scriptstyle{\rho}$$\scriptstyle{\alpha^{{\mathrm{trop}}}}$$\scriptstyle{\sim}$

commute. This immediately implies the commutativity of

${\operatorname{Pic}_{0}(X)^{\mathrm{an}}}$${\operatorname{Jac}(X)^{\mathrm{an}}}$${\operatorname{Pic}_{0}(\Gamma_{X})}$${\operatorname{Jac}(\Gamma_{X})}$${\Sigma(\operatorname{Jac}(X))}$$\scriptstyle{\sim}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\rho}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

and thus the claim for the component $\operatorname{Pic}_{0}(X)$.

Using the base point $p\in X(K)$ and its image $q=\rho(p)\in\Gamma_{X}$, we
consider the isomorphisms

$\begin{split}\operatorname{Pic}_{0}(X)&\xlongrightarrow{\sim}\operatorname{Pic}_{d}(X)\\\
[D]&\longmapsto[D+dp]\end{split}$

and

$\begin{split}\operatorname{Pic}_{0}(\Gamma_{X})&\xlongrightarrow{\sim}\operatorname{Pic}_{d}(\Gamma_{X})\\\
[D]&\longmapsto[D+dq]\ .\end{split}$

We obtain an isomorphism
$J_{d}\colon\operatorname{Pic}_{d}(\Gamma)\xrightarrow{\sim}\Sigma(\operatorname{Pic}_{d}(X))$
that makes the diagram

${\operatorname{Pic}_{d}(X)^{\mathrm{an}}}$${\operatorname{Jac}(X)^{\mathrm{an}}}$${\operatorname{Pic}_{d}(\Gamma_{X})}$${\Sigma\big{(}\operatorname{Pic}_{d}(X)\big{)}}$$\scriptstyle{\sim}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\rho}$$\scriptstyle{\sim}$

commute. This vertical arrows here do not depend on the choice of the base
point $p$. Since

$\big{[}\operatorname{Bun}_{\mathbb{G}_{m}}(X)\fatslash\
\mathbf{B}\mathbb{G}_{m}\big{]}\simeq\operatorname{Pic}(X)$

this is our claim for $n=1$.

The general case $n\geq 1$ follows from the fact that there is a natural one-
to-one correspondence between étale covers of $\mathcal{X}$ and free covers of
$\Gamma_{X}$ and that pullback along automorphisms commutes with the formation
of skeleton and tropicalization. ∎

######  Remark 6.3.

Any free cover $\widetilde{\Gamma}\rightarrow\Gamma_{X}$ of finite degree
naturally arises as the skeleton of an algebraic cover that is induced by an
étale cover on the level of prestable models, since the local Hurwitz numbers
are all equal to one. See [ABBR15a, ABBR15b, CMR16] for background on this.

######  Proposition 6.4.

Tropicalization is naturally compatible with direct sums, tensor products,
dualization, determinants, and degrees of vector bundles.

###### Proof.

This follows from the compatibility of tropicalization with the natural
constructions for covers from Proposition 3.2.

To give an explicit example, we consider the case of the tensor product. Given
two freely $S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector bundles $E_{1}$
and $E_{2}$, we write $E_{i}=(f_{i})_{\ast}L_{i}$ (for $i=1,2$), where
$f_{i}\colon\widetilde{X}_{i}\rightarrow X$ is an étale cover (induced from an
étale cover $\widetilde{\mathcal{X}}_{i}\rightarrow\mathcal{X}$) and $L_{i}$
is a line bundle on $\widetilde{X}_{i}$. Write
$L_{i}=\mathcal{O}_{\widetilde{X}_{i}}(D_{i})$ for a divisor $D_{i}$ on
$\widetilde{X}_{i}$. Consider the fibered product
$f_{1}\times_{X}f_{2}\colon\widetilde{X}_{1}\otimes_{X}\widetilde{X}_{2}\rightarrow
X$ of the two covers as well as the divisor $D_{1}\boxtimes D_{2}$ on
$\widetilde{X}_{1}\times_{X}\widetilde{X}_{2}$ defined by

$(D_{1}\boxtimes D_{2})(p,q)=D_{1}(p)+D_{2}(q)\ .$

Then we have
$(f_{1}\times_{X}f_{2})_{\ast}\mathcal{O}_{\widetilde{X}_{1}\times_{X}\widetilde{X}_{2}}(D_{1}\boxtimes
D_{2})\simeq E_{1}\otimes E_{2}$ and $f_{1}\times_{X}f_{2}$ is the generic
fiber of the étale cover
$\widetilde{\mathcal{X}}_{1}\times_{\mathcal{X}}\widetilde{\mathcal{X}}_{2}\rightarrow\mathcal{X}$
on the level of prestable models, which induces the free cover
$f_{1}^{{\mathrm{trop}}}\times_{\Gamma_{X}}f_{2}^{{\mathrm{trop}}}\colon\Gamma_{\widetilde{X}_{1}}\times_{\Gamma_{X}}\Gamma_{\widetilde{X}_{2}}\rightarrow\Gamma_{X}$
on the level of dual graphs. The other cases follow with an analogous
argument. ∎

The following is a higher rank generalization of Baker’s specialization
inequality from [Bak08].

######  Proposition 6.5.

Let $X$ be a Mumford curve over $K$ and $\Gamma_{X}$ a skeleton of $X^{an}$.
Then for every freely $S_{n}\ltimes\mathbb{G}_{m}^{n}$-linearized vector
bundle $E$ on $X$ we have

$h_{X}^{0}(E)\leq r_{\Gamma}\big{(}{\mathrm{trop}}_{X}(E)\big{)}+1\ .$

###### Proof.

Let $f\colon\widetilde{X}\rightarrow X$ be an étale cover induced from an
étale cover of prestable models and $L$ a line bundle on $\widetilde{X}$ such
that $f_{\ast}L=E$. Write
$\widetilde{X}=\widetilde{X}_{1}\sqcup\cdots\sqcup\widetilde{X}_{k}$ for its
decomposition into connected components and
$\widetilde{\Gamma}=\widetilde{\Gamma}_{1}\sqcup\cdots\sqcup\widetilde{\Gamma}_{k}$
for the corresponding decomposition of dual metric graphs. Then we have

$h^{0}(E)=h^{0}(L)=\sum_{i=1}^{k}h^{0}\big{(}L|_{\widetilde{X}_{i}}\big{)}\leq\sum_{i=1}^{k}\big{(}r_{\widetilde{\Gamma}_{i}}({\mathrm{trop}}(L|_{\widetilde{X}_{i}}))+1\big{)}=r_{\Gamma}\big{(}{\mathrm{trop}}(E)\big{)}+1$

by Baker’s specialization inequality [Bak08, Lemma 2.8 and Cor. 2.10] applied
to each line bundle $L|_{\widetilde{X}_{i}}$. ∎

##  7\. The case of a Tate curve

From now on let $K$ be also of characteristic zero. In this section we
consider the special case of a _Tate curve_ , i.e. of a (connected) smooth and
projective curve $X$ over $K$ of genus one, for which
$\operatorname{val}(j(X))<0$. In this case, we may write $X^{\mathrm{an}}$ as
a quotient

$X^{\mathrm{an}}=\mathbb{G}_{m}^{\mathrm{an}}/q^{\mathbb{Z}}$

for a unique $q\in K^{\ast}$ with
$\operatorname{val}(q)=-\operatorname{val}(j)$. In particular, the Tate curve
$X$ admits a semistable model $\mathcal{X}$ over $R$, whose special fiber is a
circle of projective lines. The (minimal) non-Archimedean skeleton
$\Gamma_{X}$ of $X^{\mathrm{an}}$ is a circle of length
$\operatorname{val}(q)$, as considered in Section 3.3 above.

###  7.1. Vector bundles on elliptic curves

In [Ati57] Atiyah provides us with a classification of all indecomposable
vector bundles of rank $n$ and degree $d$ on an elliptic curve $X$. For this
we assume from now on that $\operatorname{char}(K)=0$.

Atiyah first constructs for each $h\geq 1$ a unique indecomposable vector
bundle $F_{h}$ of rank $h$ and degree $0$ such that $h^{0}(X,E)=1$ (by the
Riemann–Roch theorem, $h^{0}(X,E)=0$ or $h^{0}(X,E)=1$ are the only
possibilities). The line bundle $F_{1}$ is the trivial line bundle on $E$ and,
inductively, for $h\geq 2$ the vector bundle $F_{h}$ is the unique nontrivial
extension

$0\longrightarrow\mathcal{O}_{X}\longrightarrow F_{h}\longrightarrow
F_{h-1}\longrightarrow 0\ .$

Now fix a base point $p$ in $X$ and let $h=(n,d)$ as well as
$n^{\prime}=\frac{n}{h}$ and $d^{\prime}=\frac{d}{h}$. Atiyah constructs a
canonical indecomposable vector bundle $E(n,d)$ of rank $n$ and degree $d$ for
every $n\geq 1$ and $d\in\mathbb{Z}$ (depending only on the chosen base point)
such that every indecomposable vector bundle $E$ of rank $n$ and degree $d$ on
$X$ is of the form

$E=E(n,d)\otimes L$

for a line bundle $L\in\operatorname{Pic}_{0}(X)$. The line bundle $L$ is
unique up to multiplication by $n^{\prime}$-torsion elements. In degree zero
we have $E(h,0)=F_{h}$ and the compatibilities

$E(n,d)\simeq E(n^{\prime},d^{\prime})\otimes F_{h}$

as well as

$E(n,d+n)\simeq E(n,d)\otimes\mathcal{O}_{X}(p)$

hold, where $\mathcal{O}_{X}(p)$ denotes the line bundle associated to the
chosen base point $p\in X$.

We now rephrase Atiyah’s classification in the case of the Tate curve using
ideas of Oda [Oda71] (see in particular [BBDG06, Theorem 2.18] for a complex-
analytic version of this).

For this we need cyclic covers of $X$ that are themselves Tate curves. There
are two ways to understand such covers, one using analytic uniformization and
the other tropicalizations of torsion line bundles. For the analytic point of
view, we pick an isomorphism
$X^{\mathrm{an}}=\mathbb{G}_{m}^{\mathrm{an}}/q^{\mathbb{Z}}$, where $q\in
K^{*}$ is uniquely determined. Then the quotient morphism
$\mathbb{G}_{m}^{\mathrm{an}}/q^{n\mathbb{Z}}\to\mathbb{G}_{m}^{\mathrm{an}}/q^{\mathbb{Z}}$
is the unique free connected degree $n$ cover of $X^{\mathrm{an}}$, which in
turn induces a cover of Tate curves up to unique isomorphism. We denote this
cover by $f\colon X_{n}\to X$.

For the alternative construction using tropicalizations of line bundles,
consider the tropicalization map on $n$-torsion points of the Jacobian

$\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\simeq\operatorname{Jac}_{n}(X)(K)\to\operatorname{Jac}_{n}(\Gamma)\simeq\mathbb{Z}/n\mathbb{Z}\
,$

where $\Gamma$ is the tropicalization of $X$. By [JL18, Theorem 3.1] this
morphism is surjective and therefore its kernel is cyclic of order $n$. Let
$L$ be a generator. After picking a suitable semistable model ${\mathcal{X}}$
of $X$, we can extend $L$ to a line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$
of multidegree $0$. Then ${\mathcal{L}}^{\otimes n}$ is a line bundle of
multidegree $0$ that extends the trivial bundle on $X$. Therefore,
${\mathcal{L}}^{\otimes n}$ is trivial. If
$s\in\Gamma({\mathcal{X}},{\mathcal{L}}^{\otimes n})$ is a nowhere vanishing
section, then

$\mathcal{X}_{n}=\operatorname{\mathcal{S}pec}\Big{(}\bigoplus_{k\geq
0}\mathcal{L}^{\otimes k}\big{/}_{1\sim s}\Big{)}$

is a cyclic $n$-cover of ${\mathcal{X}}$. In particular, the generic fiber of
${\mathcal{X}}_{n}$ is a cyclic $n$-cover of $X$ that is itself a Tate curve
and therefore it is isomorphic to the cover $X_{n}$ from above. Note that
different choices of $L$ produce the same underlying étale cover, but the
$\mathbb{Z}/n\mathbb{Z}$-action differs by the action of
$(\mathbb{Z}/n\mathbb{Z})^{*}$. We will not actually need the
$\mathbb{Z}/n\mathbb{Z}$-action and therefore the cover $f\colon X_{n}\to X$
is unique for our purposes.

######  Theorem 7.1.

Let $E$ be an indecomposable vector bundle of rank $n$ and degree $d$ on a
Tate curve $X$. Write $h=(n,d)$ as well as $n=n^{\prime}\cdot h$ and
$d=d^{\prime}\cdot h$, and let $f:X_{n^{\prime}}\to X$ be the unique connected
free cover of degree $n^{\prime}$. Then there exists a unique line bundle $L$
on $X_{n^{\prime}}$ such that

$E\simeq f_{\ast}L\otimes F_{h}\ .$

###### Proof.

Let $\widetilde{L}$ be a line bundle on $X_{n^{\prime}}$ of degree
$d^{\prime}$. The proof of [BBDG06, Theorem 2.18] word-for-word shows that
$f_{\ast}\widetilde{L}$ is an indecomposable vector bundle on $X$ of rank
$n^{\prime}$ and degree $d^{\prime}$. The central geometric point in this
proof is that $X_{n^{\prime}}$ is connected.

Now Atiyah’s classification allows us to find a line bundle
$M\in\operatorname{Pic}_{0}(X)$ such that

$E\simeq f_{\ast}\widetilde{L}\otimes M\otimes F_{h}\ ,$

and the projection formula tells us that

$f_{\ast}\widetilde{L}\otimes M\simeq f_{\ast}\big{(}\widetilde{L}\otimes
f^{\ast}M\big{)}\ .$

The line bundle $M\in\operatorname{Pic}_{0}(X)$ is unique up to multiplication
by an $n^{\prime}$-torsion element, but passing to the pullback $f^{\ast}M$
avoids this ambiguity. Therefore $L:=\widetilde{L}\otimes f^{\ast}M$ is the
unique line bundle such that $f_{\ast}L\otimes F_{h}\simeq E$. ∎

######  Proposition 7.2.

Let $X$ be a Tate curve and let $n,d$ be integers with $(n,d)=1$. Furthermore,
let $f\colon X_{n}\to X$ be the natural étale cover of degree $n$, and let
$p_{0}^{(n)}$ be the origin of $X_{n}$. Then we have

$E(n,d)\simeq f_{*}\big{(}{\mathcal{O}}(dp_{0}^{(n)})\otimes M\big{)}\ ,$

where $M$ is the unique order-$(n,2)$ point of $\operatorname{Jac}(X_{n})$
with trivial tropicalization.

###### Proof.

By [Ati57, Corollary to Theorem 7], two indecomposable bundles of rank $n$ and
degree $d$ are isomorphic if and only if their determinants are, so it
suffices to compute the determinants. By [Ati57, Theorem 6] we have $\det
E(n,d)={\mathcal{O}}(dp_{0})$, where $p_{0}=f\big{(}p_{0}^{(n)}\big{)}$. On
the other hand, we have

$\det\big{(}f_{*}{\mathcal{O}}(dp_{0}^{(n)})\big{)}={\mathcal{O}}(dp_{0})\otimes\det(f_{*}{\mathcal{O}}_{X_{n}})\
.$

By the discussion above, we have

$f_{*}{\mathcal{O}}_{X_{n}}=\bigoplus_{k=0}^{n-1}L^{\otimes k}$

for some order-$n$ point of $\operatorname{Jac}(X)$ with trivial
tropicalization and therefore
$\det(f_{*}{\mathcal{O}}_{X_{n}})={\mathcal{L}}^{\frac{n(n+1)}{2}}$, which has
trivial tropicalization and order $(n,2)$. In particular, if $n$ is odd we are
done. If $n$ is even, let $P$ be an element of order $2n$ of
$\operatorname{Jac}(X)$ with trivial tropicalization. Then $P^{\otimes n}$ has
order $2$ and trivial tropicalization, hence $P^{\otimes
n}=\det(f_{*}{\mathcal{O}}_{X_{n}})$. Let $M=f^{*}P$. As $P^{\otimes 2}$ is a
power of $L$ and $\ker(f^{*})=\langle L\rangle$, the line bundle $M$ has order
$2$ and trivial tropicalization. Furthermore, we have

$\det f_{*}\big{(}{\mathcal{O}}(dp_{0}^{(n)})\otimes
M\big{)}=\det\big{(}f_{*}{\mathcal{O}}(dp_{0}^{(n)})\otimes
P\big{)}=\det\big{(}f_{*}{\mathcal{O}}(dp_{0}^{(n)})\big{)}\otimes P^{\otimes
n}={\mathcal{O}}(dp_{0})\ ,$

finishing the proof. ∎

###  7.2. Moduli spaces of semistable bundles

Given a vector bundle $E$ of rank $n$ and degree $d$ on a smooth projective
algebraic curve $X$, we write $\mu(E)=\frac{d}{n}$ for its _slope_. Recall
that the vector bundle $E$ is said to be _semistable_ if the inequality
$\mu(E^{\prime})\leq\mu(E)$ holds for all subbundles $E^{\prime}$ of $E$; it
is called _stable_ if the inequality is strict for all proper non-zero
subbundles. For a vector bundle of slope $\mu$ there always is a natural
_Jordan–Hölder filtration_

$0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V_{s}=E$

by subbundles such that the quotients $V_{i}/V_{i-1}$, the so-called
_Jordan–Hölder factors_ , are stable vector bundles of slope $\mu$ for all
$i=1,\ldots,s$. We associate to $E$ its _graded vector bundle_

$\operatorname{gr}(E)=\bigoplus_{i=1}^{s}V_{i}/V_{i-1}$

and say that two semistable vector bundles $E$ and $E^{\prime}$ are
_equivalent_ if $\operatorname{gr}(E)\simeq\operatorname{gr}(E^{\prime})$. We
write $M_{n,d}(X)$ for the moduli space of semistable vector bundles of rank
$n$ and degree $d$ on $X$, whose $K$-points are in one-to-one correspondence
with equivalence classes of semistable vector bundles.

Let now $X$ be of genus one. It is well-known that Atiyah’s classification
implies that an indecomposable vector bundle of rank $n$ and degree $d$ on $X$
is always semistable, and that it is stable if and only $n$ and $d$ are
coprime. We refer the reader to [Tu93, Appendix A] for a proof of these facts.

Write again $h=(n,d)$ as well as $n^{\prime}=\frac{n}{h}$ and
$d^{\prime}=\frac{d}{h}$. In [Tu93, Theorem 1] Tu proves that there is a
natural isomorphism

$M_{n,d}(X)\simeq\operatorname{Sym}^{h}X$

with the $h$-th symmetric power $\operatorname{Sym}^{h}X=X^{h}/S_{h}$ of $X$.
The isomorphism is given as follows. A semistable vector bundle $E\in
M_{n,d}(X)$ is equivalent to its graded vector bundle $\operatorname{gr}(E)$.
Thus we may assume that $E$ is a direct sum

$E=\bigoplus_{i=1}^{s}E_{i}$

of stable bundles $E_{i}$ on $X$. By an argument analogous to the one in
Proposition 5.2 (ii), we must have $\mu(E_{i})=\mu(E_{j})$ for all
$i=1,\ldots,s$. Moreover, since stable bundles are precisely the
indecomposable bundles of coprime rank and degree, we must have $s=h$ and
$\mu(E_{i})=\frac{d^{\prime}}{n^{\prime}}$ for all $i=1,\ldots,s$. By Atiyah’s
classification each $E_{i}$ is of the form

$E_{i}=E(n^{\prime},d^{\prime})\otimes M_{i}$

for a line bundle $M_{i}\in\operatorname{Pic}_{0}(X)$ that is unique up to
multiplication by an $n^{\prime}$-torsion element. The isomorphism
$M_{n,d}(X)\xrightarrow{\sim}\operatorname{Sym}^{h}\operatorname{Pic}_{1}(X)\simeq\operatorname{Sym}^{h}X$
is then given by the association

$E=\bigoplus_{i=1}^{h}E(n^{\prime},d^{\prime})\otimes
M_{i}\longmapsto\sum_{i=1}^{h}M_{i}^{n^{\prime}}\otimes\mathcal{O}_{X}(p)\ ,$

where we canonically identify $X$ with $\operatorname{Pic}_{1}(X)$.

###  7.3. Tropicalization

Let $X$ be a Tate curve with
$X^{\mathrm{an}}=\mathbb{G}_{m}^{\mathrm{an}}/q^{\mathbb{Z}}$ and $\Gamma_{X}$
its (minimal) non-Archimedean skeleton, which is a circle of length
$\operatorname{val}(q)$. Let $n\geq 1$ and $d\in\mathbb{Z}$. Write again
$h=(n,d)$ as well as $n^{\prime}=\frac{n}{h}$ and $d^{\prime}=\frac{d}{h}$.
The moduli space of stable bundles on $\Gamma_{X}$ is described in Section 3.3
above.

There is a natural tropicalization map

${\mathrm{trop}}\colon
M_{n,d}(X)^{\mathrm{an}}\longrightarrow\operatorname{Bun}_{n}(\Gamma_{X})$

defined as follows:

  * •

A point in $M_{n,d}(X)^{\mathrm{an}}$ is represented by a semistable vector
bundle $E$ of rank $n$ and degree $d$ on $X_{K^{\prime}}$ for a non-
Archimedean extension $K^{\prime}$ of $K$; to avoid unnecessary notation we
may assume that $K^{\prime}$ is algebraically closed. Then, as above, the
vector bundle $E$ is equivalent to a direct sum

$E=\bigoplus_{i=1}^{h}E_{i}$

where each $E_{i}$ is a stable vector bundle of rank $n^{\prime}$ and degree
$d^{\prime}$ on $X_{K^{\prime}}$.

  * •

By Theorem 7.1, we find unique line bundles $L_{i}$ on
$(X_{K^{\prime}})_{n^{\prime}}$ such that $E_{i}=f_{\ast}L_{i}$. We can define
each ${\mathrm{trop}}(E_{i})$ as the pushforward
$f_{\ast}^{{\mathrm{trop}}}{\mathrm{trop}}(L_{i})$, as in Section 6 above. The
tropicalization ${\mathrm{trop}}(E)$ of $E$ is given by

${\mathrm{trop}}(E)=\bigoplus_{i=1}^{h}f_{\ast}^{{\mathrm{trop}}}{\mathrm{trop}}(L_{i})\
.$

We begin by stating the following immediate consequence of Proposition 7.2
above that essentially tells us that Atiyah’s classification is compatible
with tropicalization when $(n,d)=1$.

######  Proposition 7.3.

Let $n,d$ be integers with $(n,d)=1$. Then we have

${\mathrm{trop}}\big{(}E(n,d)\big{)}=E^{\mathrm{trop}}(n,d)\ ,$

where $E^{\mathrm{trop}}(n,d)$ is the vector bundle on $\Gamma_{X}$ defined in
Section 3.3.

###### Proof.

By Proposition 7.2, we have

$E(n,d)=f_{*}\big{(}{\mathcal{O}}(dp_{0}^{(n)})\otimes M\big{)}\ ,$

where $f\colon X_{n}\to X$ is the covering map from above, $p_{0}^{(n)}$ is
the identity element on $X_{n}$, and $M$ is a line bundle with trivial
tropicalization. By the definitions of the tropicalization and of
$E^{\mathrm{trop}}(n,d)$, it follows that

${\mathrm{trop}}\big{(}E(n,d)\big{)}=f^{\mathrm{trop}}_{*}{\mathrm{trop}}\big{(}{\mathcal{O}}(dp_{0}^{(n)})\otimes
M\big{)}=f^{\mathrm{trop}}_{*}{\mathrm{trop}}\big{(}{\mathcal{O}}(dp_{0}^{(n)})\big{)}=E^{\mathrm{trop}}(n,d)\
.$

∎

######  Remark 7.4.

Note that the statement of Proposition 7.3 is not true if $n$ and $d$ are not
coprime. In fact, if $(n,d)=h$ and $n^{\prime}$ and $d^{\prime}$ are defined
by $n^{\prime}h=n$ and $d^{\prime}h=d$, then
${\mathrm{trop}}\big{(}E(n,d)\big{)}=E^{\mathrm{trop}}(n^{\prime},d^{\prime})^{h}$.
The reason for this is that $E^{\mathrm{trop}}(n,d)$ is stable tropically,
while $E(n,d)$ is not stable algebraically.

Recall now that $M_{n,d}^{\oplus}(\Gamma_{X})$ denotes the component of
$M_{n,d}(\Gamma_{X})$ that parametrizes semistable bundles of rank $n$ and
degree $d$ that arise as the direct sums of $h$ stable bundles (all of which
automatically must be of rank $n^{\prime}$ and degree $d^{\prime}$). As we saw
in Example 5.3, this component may be identified with the symmetric product
$\operatorname{Sym}^{h}\Gamma_{X}$. We note that the tropicalization map
naturally lands in $M_{n,d}^{\oplus}(\Gamma_{X})$.

######  Theorem 7.5.

Let $X$ be a Tate curve and $n\geq 1$ as well as $d\in\mathbb{Z}$. Set
$h=(n,d)$. Then the diagram

${M_{n,d}(X)^{\mathrm{an}}}$${\operatorname{Sym}^{h}X^{\mathrm{an}}}$${M_{n,d}^{\oplus}(\Gamma_{X})}$${\operatorname{Sym}^{h}\Gamma_{X}\
,}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\sim}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\sim}$

commutes.

###### Proof.

By definition, the decompositions into direct sums of stable bundles in
$M_{n,d}(X)$ and $M_{n,d}^{\oplus}(\Gamma_{X})$ are compatible with
tropicalization. The deformation retraction to the skeleton is naturally
compatible with the formation of symmetric powers (see [BU21, BM19, She16] for
details). Thus we may reduce our statement to the case $h=1$ and we only need
to prove the commutativity of the diagram

${M_{n,d}(X)^{\mathrm{an}}}$${X^{\mathrm{an}}}$${M_{n,d}^{\oplus}(\Gamma_{X})}$${\Gamma_{X}\
,}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\sim}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\sim}$

when $(n,d)=1$. We note that the two horizontal identifications factor as

${M_{n,d}(X)^{\mathrm{an}}}$${\operatorname{Pic}_{d}(X)^{\mathrm{an}}}$${X^{\mathrm{an}}}$${M_{n,d}^{\oplus}(\Gamma_{X})}$${\operatorname{Pic}_{d}(\Gamma_{X})}$${\Gamma_{X}\
,}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\sim}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\sim}$$\scriptstyle{{\mathrm{trop}}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the two horizontal arrows on the left are given by $E\mapsto\det(E)$ (by
[Ati57, Cor. to Thm. 7] and Section 3.3) and the horizontal arrows on the
right are induced by the Abel–Jacobi theorems and tensoring with the base
point. The diagram on the left commutes by Proposition 6.4 and the one on the
right by [BR15, Theorem 1.3]. ∎

Theorem 7.5 together with the compatibility of skeletons and tropicalizations
with symmetric powers that is treated in [BU21, Theorem A] (also see [She16,
Section 6]) implies Theorem D from the introduction. This also shows that the
tropicalization map

${\mathrm{trop}}\colon M_{n,d}(X)^{\mathrm{an}}\longrightarrow
M_{n,d}^{\oplus}(\Gamma_{X})$

is continuous, surjective onto $M^{\oplus}_{n,d}(\Gamma_{X})$, and proper.

For a closed subscheme $Y\subseteq M_{n,d}(X)$ we set

$\operatorname{Trop}(Y):={\mathrm{trop}}(Y^{\mathrm{an}})\subseteq
M_{n,d}^{\oplus}(\Gamma)\ .$

By a version of the classical Bieri–Groves theorem [BU21, Theorem C] the set
$\operatorname{Trop}(Y)$ is the support of a $\operatorname{val}(K)$-rational
polyhedral complex in
$M_{n,d}^{\oplus}(\Gamma_{X})\simeq\operatorname{Sym}^{h}(\Gamma_{X})$.

###  7.4. A refined specialization inequality

For semistable vector bundles on an elliptic curve we can make the
specialization inequality from Proposition 6.5 more precise.

######  Proposition 7.6.

Let $E$ be a semistable vector bundle of rank $n$ and degree $d$ on a Tate
curve $X$. Then we have the inequality

$h^{0}(X,E)\leq r_{\Gamma_{X}}\big{(}{\mathrm{trop}}(E)\big{)}+1\ .$

Moreover, this is an equality if and only if $\deg(E)\neq 0$, or $\deg(E)=0$
and $E$ is a direct sum of a trivial bundle and a semistable bundle whose
Jordan–Hölder summands are all line bundles with non-trivial tropicalization.

###### Proof.

All components of the covering defining ${\mathrm{trop}}(E)$ are tropical
elliptic curves, so a divisor $D$ with non-zero degree on any of them has rank
$\max\\{-1,\deg(D)-1\\}$. As each component of ${\mathrm{trop}}(E)$ has the
same slope as $E$, it follows that in the case where $\deg(E)<0$ we have

$r_{\Gamma_{X}}\big{(}{\mathrm{trop}}(E)\big{)}+1=k(-1+1)=0=h^{0}(X,E)\ ,$

where $k$ is the number of Jordan–Hölder factors of $E$ (it is shown in [Tu93]
that $k=(n,d)$). If $\deg(E)>0$, then we have

$r_{\Gamma_{X}}\big{(}{\mathrm{trop}}(E)\big{)}+1=\sum_{i=1}^{k}(d_{i}-1+1)=d=h^{0}(X,E)\
,$

where the $d_{i}$ are the degrees of the components of ${\mathrm{trop}}(E)$
and the last equality follows from [Tu93, Lemma 17]. Now assume that
$\deg(E)=0$. Let $E=E_{1}\oplus\cdots\oplus E_{k}$ be a decomposition of $E$
into indecomposables. We have $h^{0}(X,E)=\sum h^{0}(X,E_{i})$ as well as
$r_{\Gamma_{X}}\big{(}{\mathrm{trop}}(E)\big{)}+1=\sum(r_{\Gamma_{X}}\big{(}{\mathrm{trop}}(E_{i})\big{)}+1)$,
and the multiset of Jordan–Hölder factors of $E$ is the union of the multisets
of Jordan–Hölder factors of the $E_{i}$. Therefore, we may assume that
$E=E_{1}$ is indecomposable. The Jordan–Hölder factors of $E$ are all line
bundles of degree $0$, so $r_{\Gamma_{X}}\big{(}{\mathrm{trop}}(E)\big{)}+1$
is the number of Jordan–Hölder factors of $E$ with trivial tropicalization. In
particular, if $h^{0}(X,E)=0$ (in which case the inequality is trivially true)
we have equality if and only if all Jordan–Hölder factors of $E$ have non-
trivial tropicalization. On the other hand, if $h^{0}(X,E)\neq 0$, then
$E\cong F_{n}$. The Jordan–Hölder factors of $F_{n}$ are all trivial, hence
$r_{\Gamma_{X}}\big{(}{\mathrm{trop}}(E)\big{)}+1=n$. But $h^{0}(X,F_{n})=1$
by [Ati57, Lemma 15], implying that the inequality holds, with equality if and
only if $n=1$ and $E=F_{1}={\mathcal{O}}_{X}$ is trivial. ∎

###  7.5. Brill–Noether loci

On an elliptic curve $X$ all semistable vector bundle of degree $d>0$ are non-
special. In fact, we have that
$h^{0}(E^{\ast}\otimes\omega_{X})=h^{0}(E^{\ast})=0$, since $\deg
E^{\ast}=-d<0$ and thus by the Weil–Riemann–Roch formula $h^{0}(X,E)=d$. Thus,
the only interesting Brill–Noether loci are in degree $d=0$. Following [Tu93],
we therefore consider for $r\geq-1$ the Brill–Noether locus

$W_{n,0}^{r}(X):=\big{\\{}[E]\in M_{n,0}(X)\ \big{|}\ h^{0}(X,E^{\prime})\geq
r+1\textrm{ for some }E^{\prime}\sim E\big{\\}}$

in $M_{n,0}(X)$. By [Tu93, Lemma 19] the condition $h^{0}(X,E^{\prime})\geq
r+1$ for some $E^{\prime}\sim E$ simplifies to
$h^{0}\big{(}X,\operatorname{gr}(E)\big{)}\geq r+1$, and this is the case if
and only at least $r+1$ summands are trivial. So we find [Tu93, Theorem 4]
stating that

$W_{n,0}^{r}(X)\simeq\operatorname{Sym}^{n-r-1}(X)\ .$

For $r\geq-1$ we consider a tropical analogue of $W_{n,0}^{r}(X)$ by setting

$W_{n,0}^{r}(\Gamma_{X})=\big{\\{}E\in M_{n,0}^{\oplus}(\Gamma_{X})\ \big{|}\
r_{\Gamma_{X}}(E)\geq r\big{\\}}\ .$

An element $E$ in $M_{n,0}^{\oplus}(\Gamma_{X})$ is a direct sum of line
bundles of degree zero, and, in order for $r_{\Gamma_{X}}(E)\geq r$ to hold,
at least $r+1$ of them need to be trivial. Therefore we also have

$W_{n,0}^{r}(\Gamma_{X})\simeq\operatorname{Sym}^{n-r-1}(\Gamma_{X})\ .$

######  Proposition 7.7.

Let $X$ be a Tate curve and write $\Gamma_{X}$ for its minimal skeleton. Then
we have

$\operatorname{Trop}\big{(}W_{n,0}^{r}(X)\big{)}=W_{n,0}^{r}(\Gamma_{X})\ .$

###### Proof.

The inclusion $\subseteq$ is an immediate consequence of the specialization
inequality coming from Proposition 7.6. For the reverse inclusion $\supseteq$
we consider $E\in W_{n,0}^{r}(\Gamma_{X})$. Then $E$ is a direct sum of $n$
line bundles of degree $0$, at least $r+1$ of which are trivial. So we may
lift $E$ to the corresponding direct sum $F$ of line bundles on
$X_{K^{\prime}}$ (over a suitable extension $K^{\prime}$ of $K$), where we
lift each trivial line bundle to a trivial line bundle on $X_{K^{\prime}}$ and
each non-trivial line bundle to an automatically non-trivial line bundle. Then
we have $h^{0}(X,F)-1=r_{\Gamma_{X}}(E)\geq r$ and thus $F\in
W^{r}_{n,0}(X)(K^{\prime})$ with ${\mathrm{trop}}(F)=E$. ∎

###  7.6. Generalized $\Theta$-divisors

Fix a stable vector bundle $F$ of rank $n^{\prime}=\frac{n}{(n,d)}$ and degree
$-d^{\prime}=-\frac{d}{(n,d)}$. Then a natural generalized $\Theta$-divisor in
$M_{n,d}(X)$ is defined by

$\Theta_{F}:=\big{\\{}E\in M_{n,d}(X)\ \big{|}\ h^{0}(E\otimes F)\neq
0\big{\\}}\ ,$

which is well-defined by [Tu93, Lemma 18 (i) and 20]. Then, by [Tu93, Theorem
6], under a suitable isomorphism $M_{n,d}(X)\simeq\operatorname{Sym}^{h}X$, we
have

$\Theta_{F}=\operatorname{Sym}^{h-1}X\subseteq\operatorname{Sym}^{h}X\ ,$

where one summand is fixed.

Denote by $F^{{\mathrm{trop}}}$ the tropicalization of $F$. A natural tropical
analogue of the generalized $\Theta$-divisor is given by the locus

$\Theta_{F^{{\mathrm{trop}}}}=\big{\\{}E\in M_{n,d}^{\oplus}(\Gamma_{X})\
\big{|}\ r_{\Gamma_{X}}(E\otimes F^{{\mathrm{trop}}})\neq-1\big{\\}}\ .$

######  Proposition 7.8.

Let $E\in M_{n,d}^{\oplus}(\Gamma_{X})$. Then we have $r_{\Gamma_{X}}(E\otimes
F^{\mathrm{trop}})\geq 0$ if and only if $(F^{\mathrm{trop}})^{*}$ is a direct
summand of $E$. In particular, we have

$\Theta_{F^{{\mathrm{trop}}}}\simeq\operatorname{Sym}^{h-1}(\Gamma_{X})\ .$

###### Proof.

It follows from Proposition 3.2 that tensor products distribute over sums.
Since the quantity $r_{\Gamma_{X}}(\cdot)+1$ is additive with respect to
direct sums, we may assume that $E$ is stable of rank $n^{\prime}$ and degree
$d^{\prime}$. In this case, the bundles $E$ and $F^{\mathrm{trop}}$ are
represented divisors $D$ and $D^{\prime}$ of degree $d^{\prime}$ and
$-d^{\prime}$, respectively, on the domain of the unique degree $n^{\prime}$
cover $\widetilde{\Gamma}\to\Gamma_{X}$. By Proposition 3.2, the bundle
$E\otimes F$ is represented by the divisor $D\boxtimes D^{\prime}$ on
$\widetilde{\Gamma}\times_{\Gamma_{X}}\widetilde{\Gamma}$. If
$\sigma_{1},\ldots,\sigma_{n^{\prime}}$ denote the automorphisms of the cover,
we obtain a homeomorphism

$\bigsqcup_{i=1}^{n^{\prime}}\widetilde{\Gamma}\to\widetilde{\Gamma}\times_{\Gamma_{X}}\widetilde{\Gamma}$

given by $\sigma_{i}\times\operatorname{id}$ on the $i$-th copy of
$\widetilde{\Gamma}$. With this identification, $D\boxtimes D^{\prime}$ is
given by $D+\sigma_{i}^{*}(D^{\prime})$ on the $i$-th copy of
$\widetilde{\Gamma}$. We have $r_{\Gamma_{X}}(E\otimes F^{\mathrm{trop}})\geq
0$ if and only if
$r_{\widetilde{\Gamma}\times_{\Gamma_{X}}\widetilde{\Gamma}}(D+\sigma_{i}^{*}(D^{\prime}))\geq
0$ for some $i$. Because each $D+\sigma_{i}^{*}(D^{\prime})$ has degree zero,
this is the case if and only if $D\sim\sigma_{i}^{*}(-D^{\prime})$ for some
$i$. This in turn happens if and only if $D$ and $-D^{\prime}$ define the same
bundle. As $-D^{\prime}$ defines $(F^{\mathrm{trop}})^{*}$ by Proposition 3.2,
this happens if and only if $E=(F^{\mathrm{trop}})^{*}$.

To show that
$\Theta_{F^{{\mathrm{trop}}}}\simeq\operatorname{Sym}^{h-1}(\Gamma_{X})$, we
observe that $M_{n,d}^{\oplus}(\Gamma_{X})$ is identified with
$\operatorname{Sym}^{h}(\Gamma_{X})$ by decomposing a semistable bundle into
its summands and then identifying a stable bundle with a point on
$\Gamma_{X}$. By what we have just proved, $\Theta_{F^{\mathrm{trop}}}$ is
precisely the locus in $M_{n,d}^{\oplus}(\Gamma_{X})$ where one summand is
equal to $(F^{\mathrm{trop}})^{*}$. This fixes one of the $h$ points of
$\Gamma_{X}$ corresponding to a bundle in $\Theta_{F^{\mathrm{trop}}}$,
allowing us to identify $\Theta_{F^{\mathrm{trop}}}$ with
$\operatorname{Sym}^{h-1}(\Gamma_{X})$ ∎

######  Remark 7.9.

A caveat: The vector bundle $E\otimes F^{\mathrm{trop}}$ in the proof of
Proposition 7.8 above is not contained in $M^{\oplus}_{n\cdot
n^{\prime},0}(\Gamma_{X})$. In particular, for $E\in M_{n,d}(X)$ we have
${\mathrm{trop}}(E\otimes F)\neq{\mathrm{trop}}(E)\otimes F^{\mathrm{trop}}$.
Thus, tensor products do not commute with the tropicalization on the moduli
space of semistable vector bundles on a Tate curve. This is in stark contrast
to Proposition 6.4.

######  Proposition 7.10.

Let $X$ be a Tate curve and write $\Gamma_{X}$ for its minimal skeleton. Then
we have

$\operatorname{Trop}(\Theta_{F})=\Theta_{F^{{\mathrm{trop}}}}\ .$

###### Proof.

As shown in the proof of [Tu93, Theorem 6], the theta divisor $\Theta_{F}$
consists precisely of those semistable bundles of rank $n$ and degree $d$ that
have a Jordan–Hölder factor equal to $F^{*}$. This shows that
$\operatorname{Trop}(\Theta_{F})\subseteq\Theta_{F^{\mathrm{trop}}}$. For the
converse, let $E\in\Theta_{F^{{\mathrm{trop}}}}$. By Proposition 7.8, one
summand of $E$ is equal to $(F^{\mathrm{trop}})^{*}$; we denote by
$E_{2},\ldots,E_{h}$ the remaining summands. By Theorem 7.5, for each
$i=2,\ldots,h$ there exists a stable vector bundle $G_{i}$ on $X$ of rank
$n^{\prime}$ and degree $d^{\prime}$ with ${\mathrm{trop}}(G_{i})=E_{i}$. Let
$G=F^{*}\oplus\bigoplus_{i=2}^{h}G_{i}$. By construction, we have
$G\in\Theta_{F}$ and ${\mathrm{trop}}(G)=E$. ∎

## References

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