Quasi-parabolic Higgs bundles and null hyperpolygon spaces

TL;DR

This paper introduces a new moduli space of quasi-parabolic Higgs bundles on Riemann surfaces and links its fixed points to null polygons in Minkowski space, revealing geometric structures in complex algebraic geometry.

Contribution

It defines the moduli space of quasi-parabolic Higgs bundles and establishes a novel correspondence with null hyperpolygon spaces in Minkowski 3-space.

Findings

Identification of fixed point loci with null polygon moduli spaces

Establishment of a geometric link between Higgs bundles and Minkowski space polygons

New insights into the structure of Higgs bundle moduli spaces

Abstract

We introduce the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over a compact Riemann surface $\Sigma$ and consider a natural involution, studying its fixed point locus when $\Sigma$ is $\mathbb{C} \mathbb{P}^1$ and establishing an identification with a moduli space of null polygons in Minkowski $3$-space.

Equations

$$(E,\Phi )\to (E,-\Phi )$$

$$(E,\Phi )⟼(E^{\ast },{\Phi }^t)$$

$$K\,:=\,\Big{(}{\rm SU}(2)\times{\rm U}(1)^{n}\Big{)}/(\mathbb{Z}/2\mathbb{Z}),$$

$$v\circ w=-x_1{x}_2-y_1{y}_2+t_1{t}_2,$$

$$I:=\{1,\dots ,n\}.$$

$$E(\mathcal{Q},V):=\underset{i\in I}{⨁}\text{Hom}(V_i,V_0)={\mathbb{C}}^{2n}.$$

$$\{(\lambda \cdot {\text{Id}}_{{\mathbb{C}}^2},\lambda ,\dots ,\lambda ):|\lambda |=1\}\subset \mathrm{U}(2)\times \mathrm{U}(1{)}^n.$$

$$K\,:=\,\Big{(}{\rm U}(2)\times{\rm U}(1)^{n}\Big{)}/{\rm U}(1)\,=\,\Big{(}{\rm SU}(2)\times{\rm U}(1)^{n}\Big{)}/(\mathbb{Z}/2\mathbb{Z})\,,$$

$$p_{i}=\left(\begin{array}[]{ll}a_{i}&b_{i}\end{array}\right)\in(\mathbb{C}^{2})^{*}$$

$$(p,q)\cdot[A;e_{1},\ldots,e_{n}]=\Big{(}(e_{1}^{-1}p_{1}A,\ldots,e_{n}^{-1}p_{n}A),(A^{-1}q_{1}e_{1},\ldots,A^{-1}q_{n}e_{n})\Big{)}.$$

$$\mu_{HK}:=\mu_{\mathbb{R}}\oplus\mu_{\mathbb{C}}:T^{*}\mathbb{C}^{2n}\longrightarrow\big{(}\mathfrak{su}(2)^{*}\oplus\mathbb{R}^{n}\big{)}\oplus\big{(}\mathfrak{sl}(2,\mathbb{C})^{*}\oplus(\mathbb{C}^{n})^{*}\big{)},\,$$

$$\mu_{\mathbb{R}}(p,q)\,=\,\frac{1}{2}\sum_{i=1}^{n}(q_{i}q_{i}^{*}-p_{i}^{*}p_{i})_{0}\oplus\Big{(}-\frac{1}{2}(|q_{1}|^{2}-|p_{1}|^{2}),\ldots,-\frac{1}{2}(|q_{n}|^{2}-|p_{n}|^{2})\Big{)}\,,$$

$${\mu }_{\mathbb{C}}(p,q)=-\sqrt{-1}\sum _{i=1}^n(q_i{p}_i{)}_0\oplus (\sqrt{-1}{p}_1{q}_1,\dots ,\sqrt{-1}{p}_n{q}_n).$$

$${\mathcal{P}}_0^n:=\left\{(p,q)\in {\mu }_{HK}^{-1}\left((0,0),(0,0)\right):|p_i{|}^2+|q_i{|}^2\ne 0,\forall i=1,\dots ,n\right\}.$$

$$p_i{q}_i=0\text{and}\sum _{i=1}^n(q_i{p}_i{)}_0=0,$$

$$a_i{c}_i+b_i{d}_i=0$$

$$\sum _{i=1}^n{a}_i{c}_i=0,\sum _{i=1}^n{a}_i{d}_i=0,\sum _{i=1}^n{b}_i{c}_i=0.$$

$$\lvert q_{i}\rvert=\lvert p_{i}\rvert\quad\text{and}\quad\sum_{i=1}^{n}\big{(}q_{i}q_{i}^{*}-p_{i}^{*}p_{i}\big{)}_{0}\,=\,0\,,$$

$$|c_i{|}^2+|d_i{|}^2=|a_i{|}^2+|b_i{|}^2$$

$$\sum _{i=1}^n|c_i{|}^2=\sum _{i=1}^n|a_i{|}^2,\sum _{i=1}^n|b_i{|}^2=\sum _{i=1}^n|d_i{|}^2,\sum _{i=1}^n{a}_i\overline{b_i}-\overline{c_i}{d}_i=0.$$

$$(p,q)\cdot [A;e_1,\dots ,e_n]=(p,q)$$

$$e_i^{-1}{p}_{i}A=p_i\text{and}{A}^{-1}{q}_i{e}_i=q_i$$

$$A{p}_i^{\ast }=e_i{p}_i^{\ast }\text{and}A{q}_i=e_i{q}_i$$

$$A=\left[\begin{array}[]{cc}\lambda&0\\ 0&\lambda^{-1}\end{array}\right]$$

$$q_{i}=\left(\begin{array}[]{l}c_{i}\\ 0\end{array}\right),\quad p_{i}=\left(\begin{array}[]{ll}a_{i}&0\end{array}\right),\quad\text{for}\quad i\in S,$$

$$q_{i}=\left(\begin{array}[]{l}0\\ d_{i}\end{array}\right),\quad p_{i}=\left(\begin{array}[]{ll}0&b_{i}\end{array}\right),\quad\text{for}\quad i\in S^{c},$$

$$[A;e_1,\dots ,e_n]=[\text{Id};1,\dots ,1].$$

$$X_0^n:={\mathcal{P}}_0^n/K$$

$$K^{\mathbb{C}}:=(\mathrm{SL}(2,\mathbb{C})\times ({\mathbb{C}}^{\ast }{)}^n)/(\mathbb{Z}/2\mathbb{Z}),$$

$$(p,q)\cdot [A;e_1,\dots ,e_n]=(p,q)$$

Full text

1. Introduction

Moduli spaces of Higgs bundles over a compact Riemann surface $\Sigma$ were introduced by Hitchin in [Hi1] and have innumerable applications in many areas of mathematics and mathematical physics. In particular, Hitchin establishes in [Hi1] a correspondence between isomorphism classes of rank-$2$ $SL(2,\mathbb{C})$-Higgs bundles over $\Sigma$ and the moduli space of representations of the fundamental group of $\Sigma$ in $SL(2,\mathbb{C})$. Since then these spaces have been generalized in several different ways.

Simpson in [Si1, Si2] extended these spaces to higher dimensions and, in another direction, to Higgs bundles on punctured Riemann surfaces, introducing the definition of parabolic Higgs bundles. Here he established a correspondence between isomorphism classes of these bundles and representations of the fundamental group of the punctured surface with fixed holonomy around the punctures.

On the other hand, in [Hi2], Hitchin replaced $SL(2,\mathbb{C})$ by any real reductive group $G$, introducing the notion of a $G$-Higgs bundle. In particular, he showed that, when $G^{\mathbb{C}}$ is a complex semisimple Lie group and $G$ is the split real form of $G^{\mathbb{C}}$, the moduli space of representations of the fundamental group of $\Sigma$ in $G$ has at least one connected component homeomorphic to a Euclidean space (often referred to as Hitchin components). The definition of $G$-Higgs bundles was further extended in [BGM] to include parabolic $G$-Higgs bundles.

In [G-P1] García-Prada considers involutions in the moduli space of $SL(n,\mathbb{C})$-Higgs bundles and studies their fixed point sets. For example, some of the connected components of the sets obtained coincide with the moduli space of $SL(n,\mathbb{C})$-bundles and others are formed by $SL(n,\mathbb{R})$-Higgs bundles. The study of involutions and their fixed point sets was further pursued in [BGH].

The fixed point sets of involutions on moduli spaces of $G$-Higgs bundles have attracted much attention recently as they are a source of branes [KW]. Indeed, as explained in [HT] and [DP], the moduli space of $G$-Higgs bundles for a semisimple group $G$ and the moduli space of ${}^{L}G$-Higgs bundles, for the Langlands dual ${}^{L}G$ of $G$, are mirror manifolds in the sense of Strominger, Yau and Zaslow [SYZ] and the fixed point sets of involutions on these spaces are branes, i.e. special subvarieties which can be of one of two types: $A$-branes (which are Lagrangian subvarieties) and $B$-branes (which are complex subvarieties). Since the moduli space of $G$-Higgs bundles is hyperkähler it admits three complex structures111It admits a complex structure $I$ coming from the complex structure on the Riemann surface $\Sigma$, a complex structure $J$ coming from the one on the group $G^{\mathbb{C}}$ and a third one $K=IJ$. The three satisfy the quaternionic equations and are compatible with a common metric., so it is possible to have branes that are simultaneously of type $A$ and $B$ with respect to different complex structures.

The study of involutions on the moduli space of parabolic $SL(2,\mathbb{C})$-Higgs bundles on $\mathbb{C}P^{1}$ was introduced in [BFGM], where the natural involution

[TABLE]

was considered. It is holomorphic with respect to the complex structure $I$ arising from the complex structure on $\mathbb{C}\mathbb{P}^{1}$ and anti-holomorphic with respect to $J$ and $K$. Using the isomorphism constructed in [GM] between the moduli space of hyperpolygons (a hyperkähler quotient of $T^{*}\mathbb{C}^{2n}$) and the moduli space of parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ with trivial underlying vector bundle $E$, the authors study the fixed point set of (1) (formed by $(B,A,A)$-branes). In particular, it is shown that the non-compact components of this manifold correspond to $SL(2,\mathbb{R})$-representations of the fundamental group of the punctured sphere (cf. [BGM]) and can be identified with moduli spaces of polygons in Minkowski $3$-space with edges of fixed Minkowski lengths, with some of the edges lying in future pseudospheres and the others in past pseudospheres. This provided a nice geometrical interpretation of these $(B,A,A)$-branes in terms of moduli spaces of another related problem.

Mirror symmetry is often associated to wall-crossing behavior. As the Langland dual of $SL(2,\mathbb{C})$ is $PGL(2,C)=SL(2,\mathbb{C})/\mathbb{Z}_{2}$, the moduli space $H(\alpha)$ of $SL(2,\mathbb{C})$-parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ with parabolic weight vector $\alpha$ can be identified with its mirror.222In general, the mirror $\hat{H}(\alpha)$ of the moduli space $H(\alpha)$ of parabolic $SL(2,\mathbb{C})$-Higgs bundles over a Riemann surface of genus $g$ is $H(\alpha)/\mathbb{Z}_{2}^{2g}$. In this setting, one can think of the mirror of $\hat{H}(\alpha)$ as the moduli space $\hat{H}(-\alpha)$ of $SL(2,\mathbb{C})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ where the parabolic weights are interchanged and the stability condition is changed accordingly. When passing from one space to its mirror, one should expect a wall crossing with an intermediate step where one considers a new moduli space of $G$-Higgs bundles: the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$. Quasi-parabolic bundles were introduced by Mukai in [M] and by Mehta and Seshadri in [MS]. Here we generalize their definition to quasi-parabolic Higgs bundles with an appropriate definition of stability. One should think of these spaces as limit spaces of parabolic Higgs bundles where the two parabolic weights at each parabolic point are allowed to coincide.

In this space we consider the holomorphic involution

[TABLE]

and give a detailed description of its fixed point set when restricted to the (generic) case when the underlying rank-$2$ vector bundle over $\mathbb{C}\mathbb{P}^{1}$ is trivial. We obtain $2^{n-1}-(n+1)$ connected components (where $n$ is the number of quasi-parabolic points) composed of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles that admit a direct sum decomposition $E=L_{0}\oplus L_{1}$ where $L_{0}$ and $L_{1}$ are trivial bundles over $\mathbb{C}\mathbb{P}^{1}$ and the $1$-dimensional flag components of $E$ coincide with the fibers of $L_{0}$ over some of the quasi-parabolic points and with the fibers of $L_{1}$ over the others.

To obtain this characterization of the involution fixed point set we establish a correspondence between classes of quasi-parabolic stable Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ and regular points of a special singular hyperkähler quotient of $T^{*}\mathbb{C}^{2n}$ by

[TABLE]

here called null hyperpolygons.

We further establish a correspondence between the components of the fixed point set of (2) and moduli spaces of null polygons in Minkowski $3$-space, consisting of $SU(1,1)$-equivalence classes of closed polygons in $\mathbb{R}^{2,1}$ (i.e. $\mathbb{R}^{3}$ equipped with the Minkowski inner product

[TABLE]

for $v=(x_{1},y_{1},t_{1})$ and $w\,=\,(x_{2},y_{2},t_{2})$) with some of the edges in the future light cone and the others in the past. This gives a geometrical interpretation of the $(B,A,A)$-branes obtained in terms of moduli spaces in a related space.

The paper is organized as follows. Null hyperpolygons are defined in Section 2 as the set of regular points of a singular hyperkähler quotient of $T^{*}\mathbb{C}^{2n}$ by the group $K$ and as a GIT quotient by $K^{\mathbb{C}}$. In Section 3 we define quasi-parabolic Higgs bundles and establish an isomorphism between the moduli space of null hyperpolygons and the moduli space $\mathcal{H}^{n}_{0}$ of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at a divisor $D$ consisting of $n$ marked points, where the underlying vector bundle is holomorphically trivial. In Section 4 we consider the involution in (2) and study its fixed-point set, showing that it is formed by quasi-parabolic $SL(2,\mathbb{R})$-Higgs bundles. For that we use the isomorphism of Section 3. We then define the spaces of null polygons in Minkowski $3$-space in Section 5 and, in Section 6, we show that the connected components of the fixed point set of the involution (2) can be identified with null polygon spaces in $\mathbb{R}^{2,1}$. Finally, in Section 7, we study the example of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ with four quasi-parabolic points.

Acknowledgements The authors would like to thank the referee for many valuable suggestions and remarks. Both authors were partially supported by FCT/Portugal through project PTDC/MAT-PUR/29447/2017.

2. Null Hyperpolygon spaces

In this section we extend the definition of hyperpolygon spaces introduced in [K].

As usual, let $n$ be a positive integer and let us consider a star-shaped quiver $\mathcal{Q}$ with vertex and arrow sets parametrized respectively by $I\cup\{0\}$ and $I$, where

[TABLE]

Moreover, assume that, for each $i\,\in\,I$, the corresponding arrow goes from $i$ to [math]. The representations of $\mathcal{Q}$ with $V_{i}\,=\,\mathbb{C}$ for $i\,\in\,I$ and $V_{0}\,=\,\mathbb{C}^{2}$ are parametrized by

[TABLE]

Using the standard diagonal action of ${\rm U}(2)\times{\rm U}(1)^{n}$ on $E(\mathcal{Q},V)$, one obtains a hyper-Hamiltonian action of ${\rm U}(2)\times{\rm U}(1)^{n}$ on the cotangent bundle $T^{*}E(\mathcal{Q},V)\,=\,T^{*}\mathbb{C}^{2n}$ for the natural hyperkähler structure on $T^{*}E(\mathcal{Q},V)$ [K]. Note that $T^{*}E(\mathcal{Q},V)$ can be identified with the space of representations of the doubled quiver (the quiver with the same set of vertices, where we add an arrow in the opposite direction for every arrow of $\mathcal{Q}$).

The hyper-Hamiltonian action of ${\rm U}(2)\times{\rm U}(1)^{n}$ is not effective since every point in $T^{*}E(\mathcal{Q},V)$ is fixed by the diagonal circle

[TABLE]

Therefore one considers the quotient group

[TABLE]

where $\mathbb{Z}/2\mathbb{Z}$ acts by multiplication of each factor by $-1$.

Let us consider coordinates $(p,q)$ on $T^{*}\mathbb{C}^{2n}$, where $p=(p_{1},\ldots,p_{n})$ is the $n$-tuple of row vectors

[TABLE]

and $q=(q_{1},\ldots,q_{n})$ is the $n$-tuple of column vectors q_{i}=\Big{(}\begin{array}[]{c}c_{i}\\ d_{i}\end{array}\Big{)}\in\mathbb{C}^{2}. Then, the action of $K$ on $T^{*}\mathbb{C}^{2n}$ is given by

[TABLE]

It is hyper-Hamiltonian333For the symplectic forms $\omega_{I}$, $\omega_{J}$, $\omega_{K}$ associated to the standard triple of complex structures $I,J,K$ on $T^{*}\mathbb{C}^{2n}$ that satisfy the quaternionic relations. with hyperkähler moment map

[TABLE]

where $\mu_{\mathbb{R}}$, the real moment map, is

[TABLE]

with $(\,)_{0}$ representing the traceless part, and $\mu_{\mathbb{C}}$, the complex moment map, is given by

[TABLE]

Let us consider the set

[TABLE]

Note that an element $(p,q)\,\in\,T^{*}\mathbb{C}^{2n}$ is in $\mu_{\mathbb{C}}^{-1}(0,0)$ if and only if

[TABLE]

that is, if and only if

[TABLE]

and

[TABLE]

Similarly, $(p,q)$ is in $\mu_{\mathbb{R}}^{-1}(0,0)$ if and only if

[TABLE]

i.e., if and only if

[TABLE]

and

[TABLE]

Proposition 2.1**.**

The group $K$ acts freely on $\mathcal{P}_{0}^{n}$.

Proof.

If

[TABLE]

for some $[A;e_{1},\ldots,e_{n}]\in K$ and $(p,q)\in T^{*}\mathbb{C}^{2n}$, then

[TABLE]

for $i=1,\ldots,n$, and so

[TABLE]

for $i=1,\ldots,n$.

Since $q_{i}\neq 0$ and $p_{i}\neq 0$ (as $\lvert p_{i}\rvert=\lvert q_{i}\rvert$ and $\lvert p_{i}\rvert^{2}+\lvert q_{i}\rvert^{2}\neq 0$ on $\mathcal{P}_{0}^{n}$), we have that $q_{i}$ and $p_{i}^{*}$ are eigenvectors of $A$ with eigenvalue $e_{i}$. If $A$ has two different eigenvalues $\lambda$ and $\lambda^{-1}$, we may assume that

[TABLE]

and so, if $S:=\{i\in I:\,e_{i}=\lambda\}$, we have that

[TABLE]

and

[TABLE]

for some $a_{i},b_{i},c_{i},d_{i}\in\mathbb{C}\setminus\{0\}$.

By moment map conditions (7) and (9), we conclude that $p_{i}=q_{i}=0$ for every $i\in I$, which is impossible in $\mathcal{P}_{0}^{n}$.

We conclude that $A\in SU(2)$ has only one eigenvalue and so, since $\det A=1$, we have $A=\pm\text{Id}$. Moreover, since, for every $i\in I$, we have that $e_{i}$ is an eigenvalue of $A$ (as $p_{i},q_{i}\neq 0$ for every $i$), we conclude that

[TABLE]

∎

Remark 2.2**.**

The points in $\mu_{HK}^{-1}\left((0,0),(0,0)\right)$ for which $p_{i}=q_{i}=0$ for $i$ in some subset $S\subsetneq\{1,\ldots,n\}$ are fixed by a subtorus of $K$ of dimension $\lvert S\rvert$. If $(p,q)=(0,0)$ then this point is fixed by $K$.

Since $K$ acts freely on $\mathcal{P}_{0}^{n}$, we have that [math] is a regular value of the restriction of $\mu_{HK}$ to $\mathcal{P}_{0}^{n}$ and so $\mathcal{P}_{0}^{n}$ is a smooth manifold of dimension $5(n-3)$. Moreover, by Proposition 2.1, we obtain the following result.

Proposition 2.3**.**

For $n\geq 3$ the hyperkähler quotient

[TABLE]

is a non-empty smooth hyperkähler manifold of dimension $4(n-3)$.

We call $X_{0}^{n}$ the space of null hyperpolygons.

This space can be described as an algebro-geometric quotient by the complexified group

[TABLE]

of $K$. For that, we first need to give a suitable definition of stability.

Given $(p,q)\in T^{*}\mathbb{C}^{2n}$, a subset $S\,\subset\,\{1,\ldots,n\}$ is called straight at $(p,q)$ if $q_{i}$ is proportional to $q_{j}$ for all $i,j\,\in\,S$.

Definition 2.4**.**

A point $(p,q)\,\in\,T^{*}\mathbb{C}^{2n}$ is called stable if the following two conditions hold:

• (i)

$q_{i},p_{i}\,\neq\,0$* for all $i\in\{1,\ldots,n\}$ and*

• (ii)

the set $\{1,\ldots,n\}$ is not straight at $(p,q)$.

Let $\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}$ denote the set of points in $\mu_{\mathbb{C}}^{-1}(0,0)$ which are stable. Then we have the following result.

Proposition 2.5**.**

The group $K^{\mathbb{C}}$ acts freely on $\mu_{\mathbb{C}}^{-1}(0,0)^{\text{\rm st}}$.

Proof.

If

[TABLE]

for some $[A;e_{1},\ldots,e_{n}]\in K^{\mathbb{C}}$ and $(p,q)\in\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}$, then

[TABLE]

for $i=1,\ldots,n$. Since, by stability condition $(i)$, we have $q_{i},p_{i}\neq 0$ for all $i$, we conclude that $q_{i}$ is an eigenvector of $A$ with eigenvalue $e_{i}$ and $p_{i}^{t}$ is an eigenvector of $A^{T}$ also with eigenvalue $e_{i}$. If $A$ has two different eigenvalues $\lambda$ and $\lambda^{-1}$, we may assume that

[TABLE]

and so, if $S:=\{i\in I:\,e_{i}=\lambda\}$, we have that

[TABLE]

and

[TABLE]

for some $a_{i},b_{i},c_{i},d_{i}\in\mathbb{C}\setminus\{0\}$. By moment map conditions (7) and (9), we conclude that $p_{i}=q_{i}=0$ for every $i\in I$, which is impossible by stability condition $(i)$ in Definition 2.4.

If $A$ is not diagonalizable then the eigenspace $U$ of its unique eigenvalue $\lambda$ has dimension $1$ and $q_{i}\in U$ for all $U$, implying that $\{1,\ldots,n\}$ is straight at $(p,q)$, which is impossible by stability condition $(ii)$. We conclude that $A$ is diagonalizable with only one eigenvalue and then, since $\det A=1$, we have

[TABLE]

∎

Before describing $X_{0}^{n}$ as a GIT quotient we need the following definition.

Definition 2.6**.**

For each $(p,q)\in T^{*}\mathbb{C}^{2n}$, let $d_{(p,q)}:K^{\mathbb{C}}\to\mathbb{R}$ be the function given by

[TABLE]

where $\hat{q}_{i}:=A^{-1}q_{i}\,e_{i}$ and $\hat{p}_{i}=e_{i}^{-1}p_{i}\,A$.

These functions satisfy a useful property.

Lemma 2.7**.**

For each $(p,q)\in T^{*}\mathbb{C}^{2n}$ let $d_{(p,q)}:K^{\mathbb{C}}\to\mathbb{R}$ be the function in (12). Then

$d_{(p,q)}([A;e_{1},\ldots,e_{n}]\cdot[B;\tilde{e}_{1},\ldots,\tilde{e}_{n}])$**

[TABLE]

for any $[A;e_{1},\ldots,e_{n}],[B;\tilde{e}_{1},\ldots,\tilde{e}_{n}]\in K^{\mathbb{C}}$.

Proof.

[TABLE]

where $\hat{q}_{i}:=(AB)^{-1}q_{i}e_{i}\tilde{e}_{i}=B^{-1}(A^{-1}q_{i}e_{i})\tilde{e_{i}}$ and $\hat{p}_{i}=\tilde{e}_{i}^{-1}(e_{i}^{-1}p_{i}A)B$. Then,

[TABLE]

∎

By the polar decomposition (a particular case of the Cartan decomposition), we can write every matrix in $SL(2,\mathbb{C})$ as

[TABLE]

with $R\in SU(2)$ and $S\in\mathfrak{su}(2)$ (see, for example, [S, Theorem 6.1]), and every element $[A;e_{1},\ldots,e_{n}]\in K^{\mathbb{C}}\simeq\text{exp}(\sqrt{-1}\mathfrak{k})\cdot K$ as

[TABLE]

with $R\in SU(2)$, $\hat{e}_{i}\in S^{1}$, $S\in\mathfrak{su}(2)$ and $\hat{\xi}_{i}=\sqrt{-1}\xi_{i}\in\sqrt{-1}\mathbb{R}$, for every $i=1,\ldots,n$. Then, using Lemma 2.7, we have

[TABLE]

since $R\in SU(2)$ and $\hat{e}_{i}\in S^{1}$. Hence, $d_{(p,q)}$ induces a function on $K^{\mathbb{C}}/K$.

Lemma 2.8**.**

For each $(p,q)\in T^{*}\mathbb{C}^{2n}$ let $d_{(p,q)}:K^{\mathbb{C}}\to\mathbb{R}$ be the function in (12).

Then, for $(S,\xi_{1},\ldots,\xi_{n})\in\mathfrak{su}(2)\oplus\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we have

1. (1)

$\frac{d}{dt}\left(d_{(p,q)}([e^{t\sqrt{-1}S};e^{-t\xi_{1}},\ldots,e^{-t\xi_{n}}])\right)=-\sqrt{-1}\langle\mu_{\mathbb{R}}(\hat{p},\hat{q}),(S,\sqrt{-1}\xi_{1},\ldots,\sqrt{-1}\xi_{n})\rangle$**

with $\hat{q}:=(\hat{q}_{1},\ldots,\hat{q}_{n})$ and $\hat{p}:=(\hat{p}_{1},\ldots,\hat{p}_{n})$, where

[TABLE] 2. (2)

$\frac{d^{2}}{dt^{2}}\left(d_{(p,q)}([e^{t\sqrt{-1}S};e^{-t\xi_{1}},\ldots,e^{-t\xi_{n}}])\right)\geq 0$.

Proof.

Since $S\in\mathfrak{su}(2)$ is diagonalizable and we can take an orthonormal basis of eigenvectors of $S$, we have $S=ADA^{-1}$ with $A\in SU(2)$ and

[TABLE]

for some $x>0$ (note that $\text{Tr}(S)=0)$.

Then, writing

[TABLE]

with $a_{i},b_{i},c_{i},d_{i}\in\mathbb{C}$, we obtain, for $[e^{t\sqrt{-1}S};e^{-t\xi_{1}},\ldots,e^{-t\xi_{n}}]\in\text{exp}(\sqrt{-1}\mathfrak{k})$,

[TABLE]

with

[TABLE]

and

[TABLE]

We conclude that

[TABLE]

On the other hand, by (3) we have

[TABLE]

with

[TABLE]

and

[TABLE]

Hence,

[TABLE]

and $(1)$ follows.

Differentiating the RHS of (15) we obtain

[TABLE]

and $(2)$ follows.

∎

Lemma 2.9**.**

For each $(p,q)\in T^{*}\mathbb{C}^{2n}$ let $d_{(p,q)}:K^{\mathbb{C}}\to\mathbb{R}$ be the function in (12). Then

1. (1)

$[A;e_{1},\ldots,e_{n}]\in K^{\mathbb{C}}$* is a critical point of $d_{(p,q)}$ if and only if*

[TABLE] 2. (2)

If $(p,q)\in\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}$ then $d_{(p,q)}$ induces a strictly convex function on $K^{\mathbb{C}}/K$.

Proof.

If

[TABLE]

with $R\in SU(2)$, $\hat{e}_{i}\in S^{1}$, $S\in\mathfrak{su}(2)$ and $\hat{\xi}_{i}=\sqrt{-1}\xi_{i}\in\sqrt{-1}\mathbb{R}$ ($i=1,\ldots,n$) is a critical point of $d_{(p,q)}$, then

[TABLE]

is the zero map. Let

[TABLE]

Then,

[TABLE]

where $c:[1-\varepsilon,1+\varepsilon]\to K^{\mathbb{C}}$ is given by

[TABLE]

(Note that $c(1)=[A;e_{1},\ldots,e_{n}]$ and $\dot{c}(1)=v$.) Then, by Lemma 2.7

[TABLE]

and, by Lemma 2.8, we have that

[TABLE]

Since $[R;\hat{e}_{1},\ldots,\hat{e}_{n}]\in K$ and $\mu_{\mathbb{R}}^{-1}(0,0)$ is $K$-invariant, we conclude that

[TABLE]

Conversely, if $(p,q)\cdot[A;e_{1},\ldots,e_{n}]\in\mu_{\mathbb{R}}^{-1}(0,0)$, then, using the decomposition

[TABLE]

in (13), we have that

[TABLE]

(as $\mu_{\mathbb{R}}^{-1}(0,0)$, is $K$-invariant). Moreover, for any

[TABLE]

we can take the path $c:[-\varepsilon,\varepsilon]\to K^{\mathbb{C}}$ given by

[TABLE]

(note that $c(0)=[A;e_{1},\ldots,e_{n}]$ and $\dot{c}(0)=v$). Then,

[TABLE]

since

[TABLE]

and so $[A;e_{1},\ldots,e_{n}]$ is a critical point of $d_{(p,q)}$.

By Lemma 2.8 and (16), if $(p,q)\in\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}$ then

[TABLE]

since $p_{i}\neq 0$ and $q_{i}\neq 0$ for every $i=1,\ldots,n$. We conclude that, if $(p,q)\in\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}$, we have

[TABLE]

for all $[e^{\sqrt{-1}S};e^{-\xi_{1}},\ldots,e^{-\xi_{n}}]\neq[\text{Id};1,\ldots,1]$, proving $(2)$. ∎

We can now describe $X_{0}^{n}$ as a finite dimensional GIT quotient:

Theorem 2.10**.**

Let $\mathcal{P}_{0}^{n}$ be the space defined in (5). Then,

1. (1)

$\mathcal{P}_{0}^{n}\,\subset\,\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}$* and* 2. (2)

there exists a natural bijection

[TABLE]

Remark 2.11**.**

This result is a consequence of the original Theorem of Kempf and Ness [KN], further explored by Mumford, Fogarty and Kirwan in [MFK, Chapter $8$ and Appendix $2$ C]. Indeed, the function $d_{(p,q)}$ in (12) measures how far ${\bf p}_{v}$ gets from $\lvert v\rvert^{2}$, where $v=(p,q)$, and ${\bf p}_{v}$ is the function used in [KN] to keep track of the changes in length along the orbit of $v$. Even though this theorem applies to our spaces, we include a direct, concrete adaptation of the proof to our context for completeness.

Proof.

$(1)$ Let $(p,q)\in\mathcal{P}_{0}^{n}\subset\mu_{\mathbb{C}}^{-1}(0,0)$. By (9), we have that $p_{i},q_{i}\neq 0$ for all $i\in\{1,\ldots,n\}$ and so the stability condition $(i)$ in Definition 2.4 holds. If $\{1,\ldots,n\}$ is straight at $(p,q)$ then, by (7), we can assume that

[TABLE]

with $b_{i},c_{i}\in\mathbb{C}\setminus\{0\}$. Then, by (10) we have

[TABLE]

implying that $p_{i}=q_{i}=0$ for every $i$, which is impossible. We conclude that condition $(ii)$ in Definition 2.4 is also satisfied.

$(2)$ Let us first see that $\iota$ is injective. If $\iota([p,q]_{\mathbb{R}})=\iota([\hat{p},\hat{q}]_{\mathbb{R}})$ for some $[p,q]_{\mathbb{R}},[\hat{p},\hat{q}]_{\mathbb{R}}\in X_{0}^{n}$ then $[p,q]_{\mathbb{C}}=[\hat{p},\hat{q}]_{\mathbb{C}}$, implying that

[TABLE]

for some $[A;e_{1},\ldots,e_{n}]\in K^{\mathbb{C}}$. Since $(\hat{p},\hat{q})\in\mathcal{P}_{0}^{n}$ we have that $[A;e_{1},\ldots,e_{n}]$ is a critical point of the function $d_{(p,q)}$ defined in (12) (see Lemma 2.9) and then, as $d_{(p,q)}$ induces a strictly convex function in $K^{\mathbb{C}}/K$, we conclude that $[A;e_{1},\ldots,e_{n}]\in K$, implying that $[p,q]_{\mathbb{R}}=[\hat{p},\hat{q}]_{\mathbb{R}}$.

To see that $\iota$ is surjective, take any $(p,q)\in\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}$. In order to find $[\hat{p},\hat{q}]_{\mathbb{R}}$ such that $\iota([\hat{p},\hat{q}]_{\mathbb{R}})=[p,q]_{\mathbb{C}}$, it is enough to show that $d_{(p,q)}$ has a minimum. Indeed, if that is the case and $[A;e_{1},\ldots,e_{n}]\in K^{\mathbb{C}}$ is a minimizer, we know, by Lemma 2.9, that

[TABLE]

and then $\iota([\hat{p},\hat{q}]_{\mathbb{R}})=[p,q]_{\mathbb{C}}$. Note that $(\hat{p},\hat{q})\in\mu^{-1}_{HK}((0,0),(0,0))$ and, since by stability condition $(i)$ we have $\hat{p}_{i},\hat{q}_{i}\neq 0$ for every $i=1,\ldots,n$, we conclude that $(\hat{p},\hat{q})\in\mathcal{P}_{0}^{n}$.

Now to show that $d_{(p,q)}$ has a minimum, we know, by Lemma 2.8, that $d_{(p,n)}$ induces a strictly convex function on $K^{\mathbb{C}}/K$. Hence, using the polar decomposition of $K^{\mathbb{C}}$ in (13), it is enough to show that

[TABLE]

for every nonzero $(S;\xi_{1},\ldots,\xi_{n})\in\mathfrak{su}(2)\oplus\mathbb{R}^{n}$.

If $S=0$ then, since $(0;\xi_{1},\ldots,\xi_{n})\neq 0$, there exists some $i_{0}\in\{1,\ldots,n\}$ such that $\xi_{i_{0}}\neq 0$. Moreover, since

[TABLE]

we have that (17) holds since $\xi_{0},p_{i_{0}},q_{i_{0}}\neq 0$.

If $S\neq 0$ then, writing $S=ADA^{-1}$ with $A\in SU(2)$ and

[TABLE]

for some $x>0$, we have

[TABLE]

If there exists $i\in\{1,\ldots,n\}$ such that

[TABLE]

then (17) holds. Since the case where $x\geq\xi_{i}\geq-x$ and $a_{i}=d_{i}=0$ is impossible, as we would have

[TABLE]

contradicting stability condition $(ii)$ in Definition 2.4, the result follows. (Note that if $\xi_{i}>x$ then (18) holds as one of $a_{i},b_{i}$ is necessarily non zero and similarly, if $\xi_{i}<-x$, as one of $c_{i},d_{i}$ is non zero.)

∎

From Theorem 2.10 it follows that

[TABLE]

We denote the elements in $\mu_{\mathbb{C}}^{-1}(0,0)^{\text{st}}/K^{\mathbb{C}}$ by $[p,q]_{\text{st}}$, and by $[p,q]_{\mathbb{R}}$ the elements in $\mathcal{P}_{0}^{n}/K$, when we need to make an explicit use of one of the two constructions. In all other cases, we will simply write $[p,q]$ for a null hyperpolygon in $X_{0}^{n}$.

3. Spaces of Quasi-Parabolic Higgs bundles

Let $\Sigma$ be a compact Riemann surface and consider a set of $n$ ordered distinguished marked points $D\,=\,\{x_{1},\ldots,x_{n}\}$ in $\Sigma$. A quasi-parabolic vector bundle over $\Sigma$ at $D$ consists of a holomorphic vector bundle $E$ of rank $r$ over $\Sigma$ and a collection of flags

[TABLE]

on the fibers $E_{x}$ of $E$ at each marked point $x$ in $D$, where, if $r>1$, we have $s_{x}>1$.

Definition 3.1**.**

Let $E$ be a rank-$r$ quasi-parabolic bundle over $\Sigma$ at $D$ with quasi-parabolic structure as in (20). The quasi-parabolic degree of $E$ is defined as

[TABLE]

Moreover, its quasi-parabolic slope is

[TABLE]

Remark 3.2**.**

The quasi-parabolic degree coincides with the usual parabolic degree of a parabolic vector bundle when all the parabolic weights at all points of $D$ are equal to $\frac{1}{2}$ (see [M, Definition 12.4.5]).

A quasi-parabolic isomorphism $\varphi:E\to F$ between two quasi-parabolic bundles at $D$ with flags of the same type is a bundle isomorphism such that

[TABLE]

Definition 3.3**.**

Let $E$ be a vector bundle over $\Sigma$ with a quasi-parabolic structure at $D$. A quasi-parabolic bundle $F$ over $\Sigma$ at $D^{\prime}\subset D$ is a quasi-parabolic subbundle of $E$ if

• •

$F$* is a subbundle of $E$ in the usual sense and*

• •

for every $x\in D^{\prime}$, the flag of $F_{x}$ is a nontrivial subflag of $E_{x}$, meaning that for every $i=1,\ldots,s_{x}^{F}$, there exists a $j\in\{1,\ldots,s_{x}^{E}\}$, such that $F_{x,i}=E_{x,j}$.

Remark 3.4**.**

In the above definition any subbundle of the underlying vector bundle of a quasi-parabolic bundle is also a quasi-parabolic subbundle since we may assume that $D^{\prime}=\varnothing$.

We also introduce a notion of stability.

Definition 3.5**.**

A quasi-parabolic bundle $E$ over $\Sigma$ at $D$ is said to be stable if

[TABLE]

for every proper quasi-parabolic subbundle $L$ of $E$.

Let us now restrict to the rank-$2$ case. A quasi-parabolic bundle of rank-$2$ over $\Sigma$ at $D$ is a rank-$2$ vector bundle $E$ over $\Sigma$ together with a collection of complete flags

[TABLE]

on the fibers $E_{x}$ of each marked point $x\in D$, where $\dim_{\mathbb{C}}E_{x,2}=1$, for every $x\in D$. Note that

[TABLE]

The dual $E^{*}$ of a rank-$2$ quasi-parabolic bundle $E$ at $D$ is the dual of the holomorphic bundle with the collection of the dual flags at the points of $D$.

If $L$ is a quasi-parabolic line subbundle of $E$ then it has a (trivial) quasi-parabolic structure at the points in the set

[TABLE]

Moreover,

[TABLE]

Then, by Definition 3.5, $E$ is a stable quasi-parabolic bundle if and only if

[TABLE]

for every quasi-parabolic line subbundle $L$ of $E$.

Remark 3.6**.**

If, in addition, $E$ is a trivial vector bundle of rank $2$ over $\Sigma$ with a quasi-parabolic structure at $D$, then the corresponding quasi-parabolic bundle is stable if and only if

[TABLE]

for every quasi-parabolic line subbundle $L$.

If $\Sigma=\mathbb{C}\mathbb{P}^{1}$ then $\deg(L)\leq 0$ and we conclude that (23) is always satisfied except when $L$ is the trivial bundle and $\lvert S_{L}\rvert=\lvert D\rvert=n$. This happens only when all the vector spaces $E_{x,2}$ in (21) are the same for every $x\in D$.

We conclude that all holomorphically trivial rank-$2$ quasi-parabolic bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$ are stable except those for which all the $1$-dimensional flag elements are equal for all $x\in D$.

Let $K_{\Sigma}$ denote the holomorphic cotangent bundle of $\Sigma$ and let ${\mathcal{O}}_{\Sigma}(D)$ be the line bundle on $\Sigma$ defined by the divisor $D$. A rank-$2$ quasi-parabolic Higgs bundle over $\Sigma$ at $D$ is a pair ${\bf E}\,:=\,(E,\Phi)$, where $E$ is a rank-$2$ quasi-parabolic vector bundle over $\Sigma$ at $D$ and

[TABLE]

is a Higgs field on $E$. Here $SQParEnd(E)$ is the subsheaf of $End(E)$ formed by strongly quasi-parabolic endomorphisms $\varphi\,:\,E\,\longrightarrow\,E$, meaning that

[TABLE]

Note that $\Phi$ is a meromorphic endomorphism-valued one-form with simple poles along $D$ whose residue at each $x\,\in\,D$ is nilpotent with respect to the flag, i.e.,

[TABLE]

for all $i=1,2$ and $x\in D$, where we consider $E_{x,3}=\{0\}$.

The definition of stability extends to quasi-parabolic Higgs bundles in the following way.

Definition 3.7**.**

A rank-$2$ quasi-parabolic Higgs bundle ${\bf E}=(E,\Phi)$ at $D$ is stable if

1. (1)

the residue of the Higgs field $\Phi$ at each $x\in D$ is nonzero and 2. (2)

for all quasi-parabolic line subbundles $L\subset E$ which are preserved by $\Phi$ we have

[TABLE]

Remark 3.8**.**

By Remark 3.6 we know that when $\Sigma=\mathbb{C}\mathbb{P}^{1}$ and the bundle $E$ is trivial, condition (24) is always satisfied except when all the flag elements $E_{x,2}$ in (21) are the same for every point $x$ in $D$ and $L$ is the trivial bundle $L=E_{x,2}\times\mathbb{C}$. Such line bundles are always preserved by the Higgs field $\Psi$. We conclude that all holomorphically trivial rank-$2$ quasi-parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$ with nonzero residues at all points of $D$ are stable except those for which the $1$-dimensional flag elements are all equal.

If $G$ is a complex reductive Lie group, a $G$-Higgs bundle over $\Sigma$ is a pair $(P,\Phi)$, where $P$ is a principal $G$-bundle on $\Sigma$ and $\Phi$ is a holomorphic section of $P(\mathfrak{g})\otimes K$, where $P(\mathfrak{g}):=P\times_{\text{Ad}}{\mathfrak{g}}$ is the adjoint bundle associated with $P$ [Hi1]. For $G=GL(2,\mathbb{C})$ one obtains classical Higgs bundles. Moreover, for matrix groups, this definition can be restated in terms of vector bundles. In particular, to any principal $SL(2,\mathbb{C})$-bundle we can associate a rank-$2$ vector bundle with trivial determinant, using the fundamental $2$-dimensional representation of $SL(2,\mathbb{C})$. Conversely, if $E$ is a rank-$2$ vector bundle, its frame bundle is a $GL(2,\mathbb{C})$-principal bundle and, if in addition its determinant is trivial, the structure group of this principal bundle can be reduced to $SL(2,\mathbb{C})$ [HS]. Consequently, we can think of $SL(2,\mathbb{C})$-Higgs bundles as pairs $(E,\Phi)$, where $E$ is a rank-$2$ holomorphic bundle over $\Sigma$ with trivial determinant, and $\Phi$ is a traceless holomorphic section of $End(E)\otimes K$ [G-P2]. We can further generalize this definition and consider rank-$2$ quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\Sigma$.

Definition 3.9**.**

A rank-$2$ quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundle over $\Sigma$ at $D$ is a rank-$2$ quasi-parabolic Higgs bundle ${\bf E}=(E,\Phi)$ over $\Sigma$ at $D$, where the underlying vector bundle of $E$ has trivial determinant and the Higgs field $\Phi$ is traceless.

Let $\mathcal{H}_{0}^{n}$ be the moduli space of stable rank-$2$ quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\Sigma=\mathbb{C}\mathbb{P}^{1}$ at $D$, for which the underlying holomorphic vector bundle is trivial. In Appendix 8 we define a structure of complex manifold on $\mathcal{H}_{0}^{n}$ which turns it into a moduli space of quasi-parabolic Higgs bundles in the usual sense in algebraic/complex geometry.

We will now see that it is isomorphic to the space of null hyperpolygons $X_{0}^{n}$ in (19). The correspondence between these two spaces is given by the map

[TABLE]

where $E_{(p,q)}$ is the trivial vector bundle $\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}^{2}\,\longrightarrow\,\mathbb{C}\mathbb{P}^{1}$ with the quasi-parabolic structure given by the flags

[TABLE]

over the $n$ marked points in $D=\{x_{1},\ldots,x_{n}\}\,\subset\,\mathbb{C}\mathbb{P}^{1}$ and

[TABLE]

is the Higgs field uniquely determined by the residues:

[TABLE]

for each $x_{i}\,\in\,D$ (see the proof of Theorem 3.10 below for a justification that $\Phi_{(p,q)}$ is uniquely determined by its residues).

Theorem 3.10**.**

The spaces $X_{0}^{n}$ and $\mathcal{H}_{0}^{n}$ are isomorphic.

Proof.

We first show that the map $\mathcal{I}$ is well-defined: the Higgs field $\Phi_{(p,q)}$ is uniquely defined, the QPHB $\mathbf{E}_{(p,q)}$ is stable, and the map $\mathcal{I}$ is independent of the choice of representative in ${[p,q]}_{\textnormal{st}}$.

• •

Let $[p,q]\in X_{0}^{n}$. Since $(p,q)\in\mu_{\mathbb{C}}^{-1}(0,0)$ we have by (6) and (7) that

[TABLE]

Consequently the residues in (26) uniquely determine the meromorphic endomorphism valued $1$-form $\Phi_{(p,q)}$ on $\mathbb{C}\mathbb{P}^{1}$.

• •

By the stability of $(p,q)$ we have that the set $\{1,\ldots,n\}$ is not straight at $(p,q)$ and then, by Remark 3.6, the quasi-parabolic Higgs bundle is stable.

• •

To show that $\mathcal{I}$ is independent of the choice of a representative in ${[p,q]}_{\textnormal{st}}$, let $(\tilde{p},\tilde{q})$ be in the $K^{\mathbb{C}}$-orbit of $(p,q)$ and consider $[E_{(\tilde{p},\tilde{q})},\Phi_{(\tilde{p},\tilde{q})}]$ as before. The Higgs field $\Phi_{(\tilde{p},\tilde{q})}$ is defined by the residues

[TABLE]

for some $A\in SL(2,\mathbb{C})$ and $z_{i}\in\mathbb{C}^{*}$. Moreover, the flags in $E_{(\tilde{p},\tilde{q})}$ are determined by $\tilde{q}_{i}=A^{-1}q_{i}z_{i}$. Since $q_{i}z_{i}\in\langle q_{i}\rangle$, we have that $A^{-1}\langle q_{i}\rangle=\langle\tilde{q}_{i}\rangle$ for every $i\in\{1,\ldots,n\}$ and so $[E_{(p,q)},\Phi_{(p,q)}]=[E_{(\tilde{p},\tilde{q})},\Phi_{(\tilde{p},\tilde{q})}]$.

Let us now see that $\mathcal{I}$ is injective. Let $[p,q]_{\text{st}},[\hat{p},\hat{q}]_{\text{st}}\in X_{0}^{n}$ be such that

[TABLE]

Then there exists $A\in SL(2,\mathbb{C})$ such that $\langle q_{i}\rangle=A\langle\hat{q}_{i}\rangle$ for every $i=1,\ldots,n$, and so

[TABLE]

for some $[A;e_{1},\ldots,e_{n}]\in K^{\mathbb{C}}$. Moreover,

[TABLE]

and so

[TABLE]

implying that $q_{i}p_{i}A=q_{i}\,\hat{p}_{i}\,e_{i}$. Multiplying both sides by $q^{*}_{i}$, we obtain

[TABLE]

and, since $\lvert q_{i}\rvert\neq 0$, we get

[TABLE]

We conclude that

[TABLE]

for some $[A;e_{1},\ldots,e_{n}]\in K^{\mathbb{C}}$ and so $[p,q]_{\text{st}}=[\hat{p},\hat{q}]_{\text{st}}$.

To show that $\mathcal{I}$ is surjective, let $[E,\Phi]\in\mathcal{H}_{0}^{n}$. For each point $x_{i}\in D,$ let $q_{i}=(c_{i},d_{i})^{t}$ be a generator of $E_{x_{i},2}$ and, representing the residue of the Higgs field $\Phi$ at $x_{i}$ by a traceless matrix

[TABLE]

we take $p_{i}$ to be

[TABLE]

Note that $\lvert q_{i}\rvert\neq 0$ for all $i\in\{1,\ldots,n\}$ since the flags are nontrivial.

The pair $(p,q)$ constructed in this way is in $\mu_{\mathbb{C}}^{-1}(0,0)$ and is stable:

• •

Since $N_{i}$ is nilpotent with respect to the flag, we have that

[TABLE]

Moreover, since $N_{i}$ is traceless and $N_{i}q_{i}=0$, we get

[TABLE]

and then, since the sum of the residues $N_{i}$ is [math], we obtain $\sum_{i=1}^{n}(q_{i}p_{i})_{0}=0$. We conclude that $(p,q)\in\mu_{\mathbb{C}}^{-1}(0,0)$;

• •

To show that $(p,q)$ is stable, we need to check conditions $(i)$ and $(ii)$ of Definition 2.4. The first one ($q_{i}\neq 0$ and $p_{i}\neq 0$ for all $i$) is trivially verified since the flags are complete and, by assumption, the residue of the Higgs field never vanishes on $D$ (note that $q_{i}p_{i}=N_{i}$). For $(ii)$, if $S=\{1,\ldots,n\}$ were straight at $(p,q)$, then all the flag elements $E_{x,2}$ for $x\in D$ would be the same complex line $\ell$. We could therefore consider the trivial line bundle $L$ over $\mathbb{C}\mathbb{P}^{1}$ with total space $\mathbb{C}\mathbb{P}^{1}\times\ell$ which is trivially a quasi-parabolic subbundle of $E$ at $D$ (i.e. $S_{L}=D$, where $S_{L}$ is the set defined in (22)). Moreover, it is also preserved by the Higgs field $\Phi$ since

[TABLE]

and then, by Remark 3.6, the quasi-parabolic Higgs bundle $\mathbf{E}$ would not be stable.

It is easy to check that $\mathcal{I}([p,q]_{\text{st}})=[E,\Phi]$ and so $\mathcal{I}$ is surjective.

From the local description of $\mathcal{H}_{0}^{n}$ given in Proposition 8.1 it is easy to check that the map $\mathcal{I}$ is a biholomorphism between the two complex manifolds (where we consider in $X_{0}^{n}$ the complex structure induced by the complex structure $I$).

∎

4. Involutions

The compact real form $\mathfrak{su}(2)$ of $\mathfrak{sl}(2,\mathbb{C})$ is the fixed point set of the anti-linear involution

[TABLE]

and the involution

[TABLE]

gives the Cartan decomposition

[TABLE]

where

[TABLE]

and

[TABLE]

Then, denoting by $\mathfrak{m}$ the Lie algebra $\sqrt{-1}\mathfrak{h}$ of traceless symmetric endomorphisms of $\mathbb{R}^{2}$, the split real form $\mathfrak{k}\oplus\mathfrak{m}$ of $\mathfrak{sl}(2,\mathbb{C})$ corresponding to (29) is $\mathfrak{sl}(2,\mathbb{R})$. It is the fixed point set of the extension of the involution $\iota_{2}$ to $\mathfrak{sl}(2,\mathbb{C})$.

At the group level, the compact real form $SU(2)$ of $SL(2,\mathbb{C})$ is the fixed point set of the anti-holomorphic involution

[TABLE]

while the anti-holomorphic involution

[TABLE]

has fixed point set the split real form $SL(2,\mathbb{R})$ of $SL(2,\mathbb{C})$.

The involution $\theta:=\iota_{2}\circ\iota_{1}$ given by $\theta(X)=-X^{t}$ is the Cartan involution of the split real form $SL(2,\mathbb{R})$ of $SL(2,\mathbb{C})$ (see [OV, Chapter 5]). In the moduli space $\mathcal{H}_{0}^{n}$ of quasi-parabolic Higgs bundles, the map $\theta$ induces the involution

[TABLE]

Recall that a $SL(2,\mathbb{R})$-Higgs bundle over $\Sigma$ is a pair $(P,\Phi)$, where $P$ is a holomorphic principal $SO(2,\mathbb{C})$-bundle over $\Sigma$ (i.e. a line bundle $L$) and $\Phi$ is a holomorphic section of $P(\mathfrak{m}^{\mathbb{C}})\otimes K$, where $P(\mathfrak{m}^{\mathbb{C}}):=P\times_{\text{Ad}}\mathfrak{m}^{\mathbb{C}}=L^{2}\oplus L^{-2}$ is the adjoint bundle associated with $P$ (see for example [Mu]). Indeed, considering the standard complex structure

[TABLE]

we can take the decomposition of $\mathbb{R}^{2}\otimes\mathbb{C}\simeq\mathbb{C}^{2}=V_{+}\oplus V_{-}$, where $V_{+}$ and $V_{-}$ are the eigenspaces $V_{+}:=\langle(1,-\sqrt{-1})\rangle$ and $V_{-}:=\langle(1,\sqrt{-1})\rangle$ of $I$ (corresponding to the eigenvalues $\pm\sqrt{-1}$). Taking the basis $\{v_{+},v_{-}\}$ of $\mathbb{C}^{2}$ with $v_{+}:=(1,-\sqrt{-1})\in V_{+}$ and $v_{-}:=\frac{1}{2}(1,\sqrt{-1})\in V_{-}$, a traceless symmetric endomorphism $\psi$ of $\mathbb{R}^{2}\otimes\mathbb{C}$ for the standard Euclidean structure (denoted by $\cdot$) on $\mathbb{R}^{2}$ (i.e. an element of $\mathfrak{m}^{\mathbb{C}}$) can be written in this basis as a matrix

[TABLE]

with

[TABLE]

since $\psi(v_{+})\cdot v_{-}=v_{+}\cdot\psi(v_{-})$. Hence

[TABLE]

Moreover, the adjoint action of $SO(2,\mathbb{C})\simeq\mathbb{C}^{*}$ on $\mathfrak{m}^{\mathbb{C}}$ is given by

[TABLE]

Consequently, we can think of an $SL(2,\mathbb{R})$-Higgs bundle over $\Sigma$ as a Higgs bundle $(E,\Phi)$ whose underlying vector bundle is $E=L^{2}\oplus L^{-2}$ and the Higgs field $\Phi$ is

[TABLE]

Following [BGM] we can again generalize this definition to the quasi-parabolic case.

Definition 4.1**.**

A rank-$2$ quasi-parabolic $SL(2,\mathbb{R})$-Higgs bundle over $\Sigma$ at $D$ is a rank-$2$ quasi-parabolic Higgs bundle ${\bf E}=(E,\Phi)$ over $\Sigma$ at $D$, such that

• •

the underlying vector bundle is $E=L\oplus L^{-1}$ for some line bundle $L$ over $\Sigma$ with quasi-parabolic structure at $D$ given by

[TABLE]

with $E_{x,2}=L_{x}$ for $x$ in some subset $S_{L}\subset D$ and $E_{x,2}=L^{-1}_{x}$ for $x\in D\setminus S_{L}$;

• •

the Higgs field is of the form

[TABLE]

*with $\beta=0$ on the fibers over $S_{L}$ and $\alpha=0$ on those of $D\setminus S_{L}$. *

We will see that the fixed-point set of the involution $\Theta:\mathcal{H}_{0}^{n}\to\mathcal{H}_{0}^{n}$ defined in (30) is the moduli space of rank-$2$ quasi-parabolic $SL(2,\mathbb{R})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$ for which the underlying vector bundle is holomorphically trivial (i.e. those for which the line bundle $L$ in Definition 4.1 is the trivial line bundle over $\mathbb{C}\mathbb{P}^{1}$).

For that we will use the isomorphism in (25) and study the fixed point set of the corresponding involution on the space $X_{0}^{n}$ of null hyperpolygons, that is the map

[TABLE]

It is holomorphic with respect to the complex structure induced by $I$ and anti-holomorphic with respect to those induced by $J$ and $K$, where $I,J,K$ are the standard hyperkähler complex structures on $T^{*}\mathbb{C}^{2n}$.

Remark 4.2**.**

The dual of a rank-$2$ trivial quasi-parabolic bundle $E$ with quasi-parabolic structure given by a collection of complete flags

[TABLE]

on the fibers $E_{x}$ at the parabolic points $x\in D$ is the rank-$2$ trivial quasi-parabolic bundle $E$ with quasi-parabolic structure given by the collection of dual flags

[TABLE]

where

[TABLE]

Then, if $[p,q]_{st}\in X_{0}^{n}$ is such that

[TABLE]

where $\mathcal{I}:X_{0}^{n}\to\mathcal{H}_{0}^{n}$ is the map defined in (25), we have

[TABLE]

for every $i=1,\ldots,\lvert D\rvert$ and so, by (6), we have

[TABLE]

Moreover, from (28) we have

[TABLE]

and so

[TABLE]

Hence

[TABLE]

and the involution $\Theta:\mathcal{H}_{0}^{n}\to\mathcal{H}_{0}^{n}$ defined in (30) induces the map $\iota:X_{0}^{n}\to X_{0}^{n}$ on (31).

Note that this map is well-defined. Indeed, if $[\widetilde{p},\widetilde{q}\,]=[p,q]\in X_{0}^{n}$, then there exists $[A;e_{1},\ldots,e_{n}]\in K$ such that

[TABLE]

and so

[TABLE]

implying that

[TABLE]

and so $[\widetilde{q}^{\,\,t},\widetilde{p}^{\,\,t}\,]=[q^{\,t},p^{\,t}]$. Note that, since $A\in SU(2)$, we have

[TABLE]

and $(\bar{A})^{-1}=A^{T}$.

Let $\mathcal{S}$ be the collection of subsets $S\subset\{1,\ldots,n\}$ such that $2\leq\lvert S\rvert\leq n-2$ and $1\in S$. For $S\in\mathcal{S}$, consider the sets

[TABLE]

Then we have the following result.

Theorem 4.3**.**

The fixed-point set of the involution in (31) is

[TABLE]

where $Z_{S}$ is defined above. Moreover, each $Z_{S}$ is a non-compact symplectic manifold of dimension $2(n-3)$ and $(X_{0}^{n})^{\,\iota}$ has $2^{\,n-1}-(n+1)$ connected components.

Proof.

Suppose that $[p,q]\,\in\,X_{0}^{n}$ is a fixed point of $\iota$. Then there exists an element

[TABLE]

such that

[TABLE]

In particular,

[TABLE]

and so $p_{i}^{\,t}$ is an eigenvector of $(A\bar{A})^{T}$ associated to the eigenvalue $1$, for all $i=1,\ldots,n$. Since $(A\bar{A})^{T}\in SU(2)$, we conclude that

[TABLE]

Writing

[TABLE]

with $\alpha,\beta\in\mathbb{C}$ such that $\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}=1$, we have

[TABLE]

and so

[TABLE]

for some $\alpha\in\mathbb{C}$ and $k\in\mathbb{R}$ such that $\lvert{\alpha}\rvert^{2}+k^{2}=1$.

Assuming, without loss of generality, that

[TABLE]

with $b,c\in\mathbb{C}\setminus\{0\}$ such that $\lvert b\rvert=\lvert c\rvert$, we have from (33) and (34) that

[TABLE]

and so $\alpha=0$, impliying that

[TABLE]

Then, writing, as usual,

[TABLE]

with $(a_{i},b_{i}),(c_{i},d_{i})\in\mathbb{C}^{2}\setminus\{0\}$, we have, from (33),

[TABLE]

Then, by (7), we conclude that $a_{i}b_{i}=0$ and so there exists an index set $S\subset\{1,\ldots,n\}$ with $1\in S$ such that

[TABLE]

Since $\lvert q_{i}\rvert=\lvert p_{i}\rvert$, we conclude that

[TABLE]

Moreover, since $\sum_{i=1}^{n}(q_{i}q_{i}^{*}-p_{i}^{*}p_{i})_{0}=0$, we obtain by (10) that

[TABLE]

On the other hand, since $\sum_{i=1}^{n}(q_{i}p_{i})_{0}\,=\,0$, we have by (8) that

[TABLE]

Since by (39) we have $b_{i},c_{i}\neq 0$ for all $i\in S$ (as $p_{i},q_{i}\neq 0$) and $1\in S$, it follows from (46) that $S$ has cardinality at least two. On the other hand, since $\{1,\ldots,n\}$ is not straight at $(p,q)$ (by stability condition $(ii)$ in Definition 2.4), we have that $S^{c}\neq\varnothing$. Since by (39) we have $a_{i},d_{i}\neq 0$ for all $i\in S$ (as $p_{i},q_{i}\neq 0$), it follows from (46) that $S^{c}$ has cardinality at least two. Ws conclude that $S\in\mathcal{S}$ and $[p,q]\in Z_{S}$.

Conversely, given any subset $S\in\mathcal{S}$, we have that all the points in $Z_{S}$ are fixed by $\iota$. Indeed, if $[p,q]\in Z_{S}$, then $S$ and $S^{c}$ are straight at $(p,q)$ and so we can assume that

[TABLE]

Then

[TABLE]

with

[TABLE]

and

[TABLE]

Finally, since for every subset $S\subset\{1,\ldots,n\}$, we have that either $1\in S$ or $1\in S^{c}$ , the number of subsets in $\mathcal{S}$ is

[TABLE]

∎

Using the isomorphism in (25) we conclude that the fixed point set of the involution (30) on the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles $\mathcal{H}_{0}^{n}$ is formed by the quasi-parabolic $SL(2,\mathbb{R})$-Higgs bundles in $\mathcal{H}_{0}^{n}$.

Theorem 4.4**.**

The fixed-point set of the involution $\Theta:\mathcal{H}_{0}^{n}\to\mathcal{H}_{0}^{n}$ in (30) on the space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles $\mathcal{H}_{0}^{n}$ is

[TABLE]

where $\mathcal{Z}_{S}\,\subset\,\mathcal{H}_{0}^{n}$ is formed by the quasi-parabolic $SL(2,\mathbb{R})$-Higgs bundles $\mathbf{E}\,=\,(E,\Phi)\in\mathcal{H}_{0}^{n}$ such that

1. (i)

the quasi-parabolic vector bundle $E$ admits a direct sum decomposition

[TABLE]

where $L_{0}$ and $L_{1}$ are trivial line bundles over $\mathbb{C}\mathbb{P}^{1}$ and the quasi-parabolic structure at $D$ is given by

[TABLE]

with $E_{x,2}=(L_{0})_{x}$ for $x\in\{x_{i}\in D:\,i\in S\}\subset D$ and $E_{x,2}=(L_{1})_{x}$ otherwise; 2. (1)

the Higgs field is of the form

[TABLE]

with $\beta=0$ on the fibers over $x_{i}\in D$ such that $i\in S$ and $\alpha=0$ on those over $x_{i}$ with $i\in S^{c}$. (i.e. it is either upper or lower triangular with respect to the above decomposition, according to whether $i$ is in $S$ or in $S^{c}$).

Moreover each $\mathcal{Z}_{S}$ is a non-compact manifold of dimension $2(n-3)$ and $\mathcal{H}_{0,n}^{\mathbb{R}}$ has

[TABLE]

connected components.

5. Null Polygons in Minkowski $3$-space

Let $\mathbb{R}^{2,1}$ be the Minkowski $3$-space, consisting of $\mathbb{R}^{3}$ equipped with the signature $(-,-,+)$-inner product given by

[TABLE]

The Minkowski norm of a vector $v\in\mathbb{R}^{2,1}$ is

[TABLE]

and the vectors $v\in\mathbb{R}^{2,1}$ are classified according to the sign of $v\circ v$. Those $v$ for which $v\circ v=0$ are called light-like or null vectors and form the light cone. If $v\circ v\,>\,0$, then $v$ is a time-like vector and, if $v\circ v\,<\,0$, it is a space-like vector. A null vector is said to be lying in the future (respectively past) light cone if $v_{3}>0$ (respectively $v_{3}<0$).

The Minkowski cross product $\dot{\times}$ on $\mathbb{R}^{2,1}$ is defined as

[TABLE]

for $v=(v_{1},v_{2},v_{3}),w=(w_{1},w_{2},w_{3})$ and $e_{1},e_{2},e_{3}$ the standard unit vectors in $\mathbb{R}^{3}$ and the Lie algebra $(\mathbb{R}^{3},\dot{\times})$ is isomorphic to $\mathfrak{su}(1,1)$ via the map

[TABLE]

Note that $\mathfrak{su}(1,1)^{*}\simeq\sqrt{-1}\cdot\mathfrak{su}(1,1)$ is also identified with $\mathbb{R}^{3}$.

Under this identification, the Minkowski inner product $(\circ)$ corresponds to the pairing between $\mathfrak{su}(1,1)^{*}$ and $\mathfrak{su}(1,1)$ given by

[TABLE]

for $A\in\mathfrak{su}(1,1)^{*}$ and $B\in\mathfrak{su}(1,1)$.

The coadjoint action of (the non-compact group) ${\rm SU}(1,1)$ on $\mathfrak{su}(1,1)^{*}\simeq\mathbb{R}^{2,1}$ is defined by

[TABLE]

for every $v\in\mathfrak{su}(1,1)$, where $A\in SU(1,1)$ and $u\in\mathfrak{su}(1,1)^{*}$. In particular, if

[TABLE]

(with $\alpha,\beta\in\mathbb{C}$ such that $\lvert\alpha\rvert^{2}-\lvert\beta\rvert^{2}=1$), then $A\cdot u$ corresponds to multiplication by the $3\times 3$-matrix

[TABLE]

The ${\rm SU}(1,1)$ coadjoint orbits in $\mathbb{R}^{2,1}$ are subsets of the surfaces

[TABLE]

If $R>0$ one obtains a two-sheeted hyperboloid and each sheet is a coadjoint orbit. If $R<0$ one obtains a one-sheeted hyperboloid which is also a coadjoint orbit. If $R=0$ one obtains three distinct coadjoint orbits: the origin and the two components $C^{\pm}$ of the light cone defined by

[TABLE]

We will call $C^{\pm}$ the future and past light cones.

Each coadjoint orbit of $SU(1,1)$ is a symplectic manifold having an invariant symplectic structure (the Kostant–Kirillov form). Moreover, the action of $SU(1,1)$ is Hamiltonian and the corresponding moment map is the inclusion map to $\mathfrak{su}(1,1)^{*}$.

We will study the geometry of the symplectic quotients of products of several copies of future and past light cones $C^{\pm}$ with respect to the diagonal ${\rm SU}(1,1)$-action. For that, let us fix two positive integers $k_{1},k_{2}$ with $k_{1}+k_{2}\,=\,n$. We will consider null polygons in Minkowski $3$-space that have the first $k_{1}$ edges in the past light cone and the last $k_{2}$ edges in the future light cone. A closed null polygon is one whose sum of the first $k_{1}$ sides in the past light cone is symmetric to the sum of the last $k_{2}$ sides in the future light cone. The space of all such closed null polygons can be identified with the zero level set of the moment map

[TABLE]

for the diagonal ${\rm SU}(1,1)$–action, where $\mathcal{O}_{i}\cong C^{-}$ is the past light cone if $1\leq i\leq k_{1}$, and $\mathcal{O}_{i}\cong C^{+}$ is the future light cone if $k_{1}+1\leq i\leq n$, equipped with the Kostant-Kirillov symplectic form on coadjoint orbits.

Let $\left(\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}\right)^{\text{reg}}$ be the set of regular points of $\mu$ in $\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}$. By equivariance, it is invariant under $SU(1,1)$ and it is easy to check that the action of $SU(1,1)$ on $\left(\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}\right)^{\text{reg}}$ is free (cf. Remark 5.1). If

[TABLE]

is nonempty then it is a smooth submanifold of $\left(\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}\right)^{\text{reg}}$ (by the Implicit Function Theorem). Moreover, if $\omega$ is the symplectic form on $\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}$ and $x\in\mu^{-1}(0)_{\text{reg}}$, then the kernel of $\omega_{x}$ on the tangent space

[TABLE]

is

[TABLE]

Hence, $i^{*}\omega$, where $i:\mu^{-1}(0)_{\text{reg}}\to\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}$ is the inclusion map, is a closed $2$-form on $\mu^{-1}(0)_{\text{reg}}$ and the leaves of its null foliation are the orbits of $SU(1,1)$. Consequently, the space

[TABLE]

is a symplectic manifold.

Remark 5.1**.**

A point $u=(u_{1},\ldots,u_{n})\in\mathcal{O}_{1}\times\cdots\times\mathcal{O}_{n}$ is regular if and only if its $SU(1,1)$-stabilizer is trivial. Indeed, if it exists $A\in SU(1,1)$ such that $A\neq Id$ and $Au_{i}=u_{i}$ for $i=1,\ldots,n$ then $\dim\langle u_{1},\ldots,u_{n}\rangle=1$ (meaning that the null polygon lies along a line) and the stabilizer of $u$ is $1$-dimensional.

To see this, let us consider a matrix $A_{\alpha,\beta}$ as in (47) fixing a vector $u\in C^{\pm}$. Using a rotation around the $t$-axis we can assume, without loss of generality, that $u=(0,1,\pm 1)$. Then, up to conjugation by a rotation around the $t$-axis, we have that $A_{\alpha,\beta}$ is of the form

[TABLE]

with $k\in\mathbb{R}$ and so the stabilizer of $u$ has dimension $1$. Moreover, the space of eigenvectors of $A_{\alpha,k\sqrt{-1}}$ associated with the eigenvalue $1$ is always one-dimensional for every $k\neq 0$.

We conclude that for $n\geq 4$ and $k_{1},k_{2}\geq 2$ with $k_{1}+k_{2}=n$ the space $M^{k_{1},k_{2}}_{0}$ is nonempty and its points represent the null polygons that do not lie along a line. Moreover, the spaces $M^{k_{1},k_{2}}_{0}$ and $M^{k_{2},k_{1}}_{0}$ are symplectomorphic. Note that if $k_{1}=1$ or $k_{2}=1$ the space is empty since the sum of two noncolinear past null vectors is a past timelike vector and the sum of a past timelike vector with a past null vector is still a past timelike vector.

Theorem 5.2**.**

The space $M^{k_{1},k_{2}}_{0}$ is non-compact.

Proof.

Let $[v]=[v_{1},\ldots,v_{n}]$ by a polygon in $M^{k_{1},k_{2}}_{0}$, where $[v]=[v_{1},\ldots,v_{n}]$ with $v_{i}\in C^{-}$ for $i=1,\ldots,k_{1}$ and $v_{i}\in C^{+}$ otherwise. Note that the vectors $v_{1},\ldots,v_{k_{1}}$ are not all aligned since, by definition, $(v_{1},\ldots,v_{n})$ is a regular value of the moment map $\mu$ defined in (48) (cf. Remark 5.1). Therefore, $w:=v_{1}+\cdots+v_{k_{1}}$ is a time-like vector444The sum of two noncolinear past null vectors is a past timelike vector and the sum of a past timelike vector with a past null vector is still a past timelike vector. and we can use a rotation around the $t$-axis followed by a boost

[TABLE]

along the $y$-direction to place the vector $w$ along the $t$-axis. Let $\ell$ be the Minkowski length of $w$. Note that $\ell$ never vanishes and can take any value in $(0,\infty)$. Indeed if we take, for instance,

[TABLE]

and

[TABLE]

with $m\in\mathbb{N}$, then $w=v_{1}+\cdots+v_{k_{1}}=(0,0,-2m)$ and $\lvert\lvert w\rvert\rvert_{2,1}=2m$.

We can therefore consider the (continuous) map $\ell:M^{k_{1},k_{2}}_{0}\to(0,\infty)$ which for each $[v]\in M^{k_{1},k_{2}}_{0}$ gives the Minkowski length of the vector $v_{1}+\cdots+v_{k_{1}}$ and the result follows. Note that $\ell$ is the moment map of the circle action obtained by rotating the section of the polygon formed by the first $k_{1}$ edges around the $t$-axis. This map has no critical values so all level sets are diffeomorphic and we obtain a diffeomorphism between $M^{k_{1},k_{2}}_{0}$ and $P\times(0,\infty)$ where $P$ is a level set of $\ell$.

∎

6. Null hyperpolygons, quasi-parabolic $SL(2,\mathbb{R})$-Higgs bundles and null polygons in $\mathbb{R}^{2,1}$

Let $[p,q]\in Z_{S}$ be a fixed point of the involution (31) in the null hyperpolygon space, where $Z_{S}$ is defined in (32) and consider the vectors $u_{i}\in\mathbb{R}^{3}$ given by

[TABLE]

where we use identifications $\mathfrak{su}(2)^{*}\cong(\mathbb{R}^{3})^{*}\cong\mathfrak{su}(1,1)^{*}$. Assuming, without loss of generality, that

[TABLE]

with $\lvert b_{i}\rvert=\lvert c_{i}\rvert$, for $i\in S$ and

[TABLE]

with $\lvert a_{i}\rvert=\lvert d_{i}\rvert$, for $i\in S^{c}$, we have

[TABLE]

and

[TABLE]

Note that for $i\in S$ we have

[TABLE]

and, for $i\in S^{c}$,

[TABLE]

and so the vectors $u_{i}$ are null vectors in $\mathbb{R}^{2,1}$. By (45) and (46), we have that

[TABLE]

and so the vectors $u_{i}$ form a closed null polygon in Minkowski $3$-space with $\lvert S\rvert$ sides in the past light cone and $n-\lvert S\rvert$ sides in the future light cone.

Moreover, the vectors $u_{i}$ are not all aligned. Indeed, if the vectors $u_{i}$ were colinear for $i\in S$, there would exist $i_{0}\in S$ and $k_{i}\in\mathbb{R}^{+}$ such that

[TABLE]

Then

[TABLE]

contradicting (46). We conclude that the vectors $u_{i}$ form a regular null polygon in $\mathbb{R}^{2,1}$ (cf. Remark 5.1).

Theorem 6.1**.**

For any $S\,\subset\,\{1,\ldots,n\}$, such that $1\in S$ and $2\leq\lvert S\rvert\leq n-2$, the components $\mathcal{Z}_{S}$ and $Z_{S}$, of the fixed-point sets of the involutions (30) and (31) in $\mathcal{H}_{0}^{n}$ and $X_{0}^{n}$ respectively, are diffeomorphic to the moduli space

[TABLE]

of closed null polygons in Minkowski $3$-space.

Proof.

Let $S\,\subset\,\{1,\ldots,n\}$ be such that $1\in S$ and $2\leq\lvert S\rvert\leq n-2$. After a suitable reshuffling, consider the map $\varphi:Z_{S}\longrightarrow M^{\lvert S\rvert,\lvert S^{c}\rvert}_{0}$ defined by (49) and (50). Hence $\varphi([p,q])$ is the element of $M^{\lvert S\rvert,\lvert S^{c}\rvert}_{0}$ represented by the null polygon in $\mathbb{R}^{2,1}$ whose first $\lvert S\rvert$ sides are the vectors $u_{i}$ given by (49), and the last $n-\lvert S\rvert$ sides are the vectors $u_{i}$ given by (50).

Let us first see that $\varphi$ is well defined. For that, consider two representatives $(p,q)$ and $(p^{\prime},q^{\prime})$ of the same class $[p,q]$ in $Z_{S}$. Then there exists $[A;e_{1},\ldots,e_{n}]\,\in\,K$ such that

[TABLE]

We can assume, without loss of generality, that p_{i}=\left(\begin{array}[]{ll}a_{i}&b_{i}\end{array}\right), p_{i}^{\prime}=\left(\begin{array}[]{ll}a_{i}^{\prime}&b_{i}^{\prime}\end{array}\right) with $a_{i}=a_{i}^{\prime}=0$ for $i\in S$ and $b_{i}=b_{i}^{\prime}=0$ for $i\in S^{c}$, while q_{i}=\left(\begin{array}[]{ll}c_{i}&d_{i}\end{array}\right)^{t}, q_{i}^{\prime}=\left(\begin{array}[]{ll}c_{i}^{\prime}&d_{i}^{\prime}\end{array}\right)^{t} with $d_{i}\,=\,d_{i}^{\prime}\,=\,0$ for $i\in S$ and $c_{i}\,=\,c_{i}^{\prime}=0$ for $i\,\in\,S^{c}$. Hence,

[TABLE]

for some $\alpha=e^{\sqrt{-1}\,\theta_{0}}\in S^{1}$. Then we have

[TABLE]

and

[TABLE]

where

[TABLE]

is a rotation around the $t$-axis (an element of ${\rm SU}(1,1)$).

To see that $\varphi$ is injective, let $[p,q],[p^{\prime},q^{\prime}]$ be two points in $Z_{S}$ with

[TABLE]

Then, without loss of generality, we can write

[TABLE]

for $i=1,\ldots,\lvert S\rvert$ with

[TABLE]

(cf. (46)) and

[TABLE]

for $i=\lvert S\rvert+1,\ldots,n$, with

[TABLE]

and there exists $A_{\alpha,\beta}\in SU(1,1)$ as in (47) such that

[TABLE]

and

[TABLE]

Then, we have

[TABLE]

for $i=1,\ldots,\lvert S\rvert$ and so, by (51), we conclude that $\alpha\beta=0$, implying that $\beta=0$ (since $\lvert\alpha\rvert^{2}-\lvert\beta\rvert^{2}=1$).

Consequently,

[TABLE]

is a rotation around the $t$-axis.

Hence

[TABLE]

for some $\theta_{0}\in[0,2\pi)$ and, by (53), we have

[TABLE]

Moreover, since

[TABLE]

and

[TABLE]

we have

[TABLE]

and

[TABLE]

for some $\gamma_{i},\phi_{i}\in[0,2\pi)$.

We conclude that

[TABLE]

with A\,=\left(\begin{array}[]{cc}e^{-\sqrt{-1}\,\theta_{0}/2}&0\\ 0&e^{\sqrt{-1}\,\theta_{0}/2}\end{array}\right), and

[TABLE]

implying that $[p,q]=[p^{\prime},q^{\prime}]$.

To show that $\varphi$ is surjective, let us take $[v]\in M^{\lvert S\rvert,\lvert S^{c}\rvert}_{0}$, where $[v]=[v_{1},\ldots,v_{n}]$ with $v_{i}\in C^{-}$ for $i=1,\ldots,\lvert S\rvert$ and $v_{i}\in C^{+}$ otherwise. Note that the vectors $v_{1},\ldots,v_{\lvert S\rvert}$ are not all aligned since, by definition, $(v_{1},\ldots,v_{n})$ is a regular value of the moment map $\mu$ defined in (48) (cf. Remark 5.1). Therefore, $w:=v_{1}+\cdots+v_{\lvert S\rvert}$ is a time-like vector and we can use a rotation around the $t$-axis followed by a boost along the $y$-direction to place the vector $w$ along the $t$-axis. Hence, we can assume that $[v]$ is represented by a polygon with the first $\lvert S\rvert$ sides past null vectors $(x_{i},y_{i},-t_{i})$ and the last $n-\lvert S\rvert$ sides future null vectors $(x_{i},y_{i},t_{i})$ with $t_{i}=\sqrt{x_{i}^{2}+y_{i}^{2}}$, satisfying

[TABLE]

Then $[v]$ is the image of the hyperpolygon $[p,q]$, with

[TABLE]

and

[TABLE]

with

[TABLE]

Note that $[p,q]\in Z_{S}\subset X_{0}^{n}$, since

[TABLE]

and

[TABLE]

where as usual we write p_{i}=\left(\begin{array}[]{ll}a_{i}&b_{i}\end{array}\right) and q_{i}=\left(\begin{array}[]{ll}c_{i}&d_{i}\end{array}\right)^{t}, for $i=1,\ldots,n$.

Note that clearly $\varphi$ and its inverse are differentiable and the result follows. ∎

7. An Example

As an example, we consider the case where $n=4$.

Let $\mathcal{H}_{0}^{4}$ be the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles $(E,\Phi)$ of rank two over $\mathbb{C}\mathbb{P}^{1}$ at $D$ with $\lvert D\rvert=4$, where the underlying holomorphic vector bundle $E$ is trivial and let us consider the space of null hyperpolygons $X_{0}^{4}$.

A subset $S$ of $\{1,2,3,4\}$ such that $\lvert S\rvert\geq 2$ and $\lvert S^{c}\rvert\geq 2$ must have exactly two elements. If, in addition $1\in S$, there are exactly three possibilities. Let us denote these sets by $S_{1}$, $S_{2}$ and $S_{3}$, where $S_{j}:=\{1,1+j\}$.

Then the fixed point set of the involution in (31) defined in the space of null hyperpolygons $X_{0}^{4}$ has exactly three connected components $Z_{S_{1}},Z_{S_{2}}$ and $Z_{S_{3}}$ (cf. Theorem 4.3).

By Theorem 6.1 we know that each component $Z_{S_{i}}$ is diffeomorphic to $M^{2,2}_{0}$, formed by classes of closed polygons in Minkowski $3$-space with the first two sides $u_{1},u_{2}$ in the past light cone and the last two, namely $u_{3}$ and $u_{4}$, in the future light cone. Let us consider the diagonal vector $w=u_{1}+u_{2}$ connecting the origin to the third vertex of the polygon. Since the vectors $u_{1}$ and $u_{2}$ are not aligned, $w$ is a past time-like vector and we can consider its Minkowski length $\ell$. Using a rotation around the $t$-axis followed by a boost we can assume that $w$ lies along the $t$-axis.

For each value of $\ell\in(0,\infty)$ we have a circle of possible classes of polygons obtained by rotating the last two sides of the polygon around the diagonal, while fixing the other two. The length $\ell$ of $w$ is the moment map for the bending flow obtained by this rotation of the last two sides of the polygon around the diagonal with a constant angular speed while fixing the other two vectors. Hence, $Z_{S_{i}}$ is a non-compact toric manifold of dimension $2$ with moment map $\ell$. Moreover, $\ell$ has no critical values in $(0,\infty)$ so $Z_{S_{i}}$ is diffeomorphic to $\mathbb{C}\setminus\{0\}$.

We conclude that the fixed point sets $(X_{0}^{4})^{\iota}$ and $\mathcal{H}_{0,4}^{\mathbb{R}}$ (the space of quasi-parabolic $SL(2,\mathbb{R})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ with four quasi-parabolic points) of the involutions (31) and (30) have three non-compact components diffeomorphic to $\mathbb{C}\setminus\{0\}$.

8. Appendix: Existence of the moduli space

We will now see that the space $\mathcal{H}^{n}_{0}$ defined in Section 3, of isomorphism classes of stable rank-$2$ quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$ for which the underlying holomorphic vector bundle is trivial, is a moduli space and a complex manifold. For this we use the similar proof given by Furuta and Steer for quasi-parabolic bundles in [FS] and adapt it to quasi-parabolic Higgs bundles.

Let $T$ be a complex manifold. A holomorphic family $(E_{t},\phi_{t})_{t\in T}$ of stable rank-$2$ trivial holomorphic quasi-parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D=\{p_{1},\ldots,p_{n}\}$ parametrized by $T$ is

• •

a holomorphic family of stable rank-$2$ trivial quasi-parabolic vector bundles $\widetilde{E}\to\mathbb{C}\mathbb{P}^{1}\times T$ parametrized by $T$ (see [FS])555i.e a holomorphic rank-$2$ vector bundle $\widetilde{E}\to\mathbb{C}\mathbb{P}^{1}\times T$ such that $\widetilde{E}_{\lvert_{\{p_{i}\}\times T}}\simeq\varphi^{*}\widetilde{E}\to T$ is a rank-$2$ vector bundle over $T$ (where $\varphi_{i}:T\to\mathbb{C}\mathbb{P}^{1}\times T$ is the map $\varphi_{i}(t)=(p_{i},t)$ for $i=1,\ldots,n$) and a filtration

$\varphi_{i}^{*}\widetilde{E}=\widetilde{E}_{i,1}\supsetneq\widetilde{E}_{i,2}\supsetneq\{0\},$

for each $i=1,\ldots,n$, where $\widetilde{E}_{i,2}\to T$ is a line subbundle of $\varphi^{*}\widetilde{E}$.;

• •

a holomorphic section $\Phi$ of $SQParEnd(\widetilde{E})\otimes\rho^{*}K_{\mathbb{C}\mathbb{P}^{1}}(D)$, where $\rho:\mathbb{C}\mathbb{P}^{1}\times T\to\mathbb{C}\mathbb{P}^{1}$ is the projection map, such that

[TABLE]

Note that $\Phi$ is an $End(\widetilde{E})$-valued $1$-form on $\mathbb{C}\mathbb{P}^{1}\setminus D$ as $\rho^{*}K_{\mathbb{C}\mathbb{P}^{1}}(D)\subset K_{\mathbb{C}\mathbb{P}^{1}\times T}(\rho^{*}D)$.

Given such a family we have a set theoretical map

[TABLE]

classifying isomorphism classes of parametrized bundles. We will start by constructing a collection of holomorphic families of stable trivial holomorphic quasi-parabolic Higgs bundles whose images under the map $f$ cover the set ${\mathcal{H}}_{0}^{n}$.

Proposition 8.1**.**

There exists a finite collection of holomorphic families of stable rank-$2$ trivial holomorphic quasi-parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$ parametrized by open sets of the form $\mathbb{C}^{n-3}\times(\mathbb{C}^{*})^{n-3}$ whose images under the map $f$ cover the set $\mathcal{H}^{n}_{0}$.

Proof.

Let $\Sigma$ be $\mathbb{C}\mathbb{P}^{1}$ and consider a set of $n$ marked points $\{x_{1},\ldots,x_{n}\}\neq\varnothing$ in $\Sigma$. Let $E$ be a stable rank-$2$ quasi-parabolic bundle over $\Sigma$ at $D$ for which the underlying holomorphic vector bundle is trivial (then $E\simeq\mathcal{O}(0)\oplus\mathcal{O}(0)$).

For a fixed rank-$2$ holomorphic trivial vector bundle $E$, quasi-parabolic structures as in (21) are parametrized by a product of flag manifolds

[TABLE]

where $Iso(\mathbb{C}^{2},E_{x_{1},1})$ is the set of linear isomorphisms from $\mathbb{C}^{2}$ to $E_{x_{1},1}$, and $B$ is the parabolic subgroup of $SL(2,\mathbb{C})$ of upper triangular matrices (which leave the standard full flag in $\mathbb{C}^{2}$ invariant).

Moreover, the automorphism group of $E$ is $SL(2,\mathbb{C})$ and, by Remark 3.6, a point $(X_{1},\ldots,X_{n})\in\left(\mathbb{C}\mathbb{P}^{1}\right)^{n}$ represents a stable quasi-parabolic bundle if and only if

[TABLE]

Considering disjoint index sets $A,B\subset\{1,\ldots,n\}$ such that $A\cup B=\{1,\ldots,n\}$, we can cover $\left(\mathbb{C}\mathbb{P}^{1}\right)^{n}$ by a finite number of coordinate charts

[TABLE]

Now the possible Higgs fields on a quasi-parabolic bundle represented by a point in $U_{A,B}$ are determined by the collections of their residue matrices. These are traceless nilpotent matrices $N_{1},\ldots,N_{n}$ such that $N_{i}(E_{x_{i},1})=E_{x_{i},2}$. In particular, if the quasi-parabolic Higgs bundle is stable, $E_{x_{i},2}$ is the kernel of $N_{i}$ (since, by stability, we have $N_{i}\neq 0$). Each of these matrices is of the form

[TABLE]

with $\lambda_{i}\in\mathbb{C}^{*}$, when the quasi-parabolic bundle is represented by a point

[TABLE]

Assuming, without loss of generality, that $X_{1}\neq X_{2}$, there is a unique element of $SL(2,\mathbb{C})/\{\pm Id\}$ that takes $X_{1}$ to $[1:0]$, $X_{2}$ to $[0:1]$ and $N_{1}$ to the matrix

[TABLE]

Note that this element of $SL(2,\mathbb{C})/\{\pm Id\}$ takes $N_{2}$ to a matrix of the form

[TABLE]

Consequently, a point

[TABLE]

with $X_{i}=[1:w_{i}]$ for $i\in A$, $X_{i}=[z_{i}:1]$ for $i\in B$ and $\lambda_{i}\in\mathbb{C}^{*}$, represents an isomorphism class of a stable quasi-parabolic Higgs bundle whose underlying quasi-parabolic bundle is represented by a point in $U_{A,B}$ with $1\in A$ and $2\in B$ (and $w_{1}=z_{2}=0$).

Since the sum of the residues must be zero, the $z_{i},w_{i},\lambda_{i}$ must satisfy the following compatibility conditions

[TABLE]

If $A\neq\{1\}$ then, by (55) and (56), we can take one $i_{0}\in A\setminus\{1\}$ and determine

[TABLE]

and

[TABLE]

Moreover, $\lambda_{2}$ can be determined by $\eqref{eq:sumzero3}$.

If $A=\{1\}$, then by (55), there exists $i_{0}\in B\setminus\{2\}$ such that $z_{i_{0}}\neq 0$ and so we can determine $\lambda_{i_{0}}$ and $z_{i_{0}}$ from (55) and (56):

[TABLE]

Again $\lambda_{2}$ can be determined by $\eqref{eq:sumzero3}$.

Consequently, the quotient by $SL(2,\mathbb{C})/\{\pm Id\}$ of the set $H_{A,B}$ of trivial quasi-parabolic Higgs bundles with quasi-parabolic structures given by $(X_{1},\ldots,X_{n})\in U_{A,B}$, such that

[TABLE]

can be identified with

[TABLE]

Remark 8.2**.**

Considering injective functions $\sigma:\{1,2\}\to\{1,\ldots,n\}$ with $\sigma(1)<\sigma(2)$ and the sets

[TABLE]

as well as the corresponding sets of stable quasi-parabolic Higgs bundles $H_{A,B}^{\sigma}$, we conclude that the set $H_{0}$ of stable rank-$2$ quasi-parabolic bundles over $\Sigma$ at $D$ for which the underlying holomorphic vector bundle is trivial is covered by

[TABLE]

As we have seen before, the set of isomorphism classes of elements of each set $H_{A,B}^{\sigma}$ can be identified with $\mathbb{C}^{n-3}\times(\mathbb{C}^{*})^{n-3}$ and so, by Remark 8.2, we conclude that the space $\mathcal{H}^{n}_{0}$ of isomorphism classes of rank-$2$ quasi-parabolic Higgs bundles over $\Sigma$ at $D$ for which the underlying holomorphic vector bundle is trivial, can be covered by a finite number of copies of $\mathbb{C}^{n-3}\times(\mathbb{C}^{*})^{n-3}$. Moreover, the patching maps among these sets are holomorphic. ∎

Remark 8.3**.**

It is easy to check that $\mathcal{H}^{n}_{0}$ with the quotient topology is Hausdorff and so Proposition 8.1 shows that $\mathcal{H}^{n}_{0}$ has the structure of a complex manifold of (real) dimension $4(n-3)$ covered by finitely many open sets of the form $\mathbb{C}^{n-3}\times(\mathbb{C}^{*})^{n-3}$.

Let $T$ be any complex manifold parametrizing a holomorphic family of rank-$2$ trivial holomorphic quasi-parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$ and consider the corresponding map $f:T\to\mathcal{H}^{n}_{0}$ classifying isomorphism classes of parametrized bundles. We will see that $f$ is holomorphic and so $\mathcal{H}^{n}_{0}$ is a moduli space.

Proposition 8.4**.**

Given a complex manifold $T$ parametrizing a holomorphic family $(E_{t},\phi_{t})_{t\in T}$ of stable rank-$2$ trivial holomorphic quasi-parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$, the classifying map $f$ is holomorphic.

Proof.

Since $\widetilde{E}$ is a rank-$2$ vector bundle over $\mathbb{C}\mathbb{P}^{1}\times T$, we know that $\mathcal{O}(\widetilde{E})$ is a rank-$2$ locally free sheaf over $\mathbb{C}\mathbb{P}^{1}\times T$. Let $\pi:\mathbb{C}\mathbb{P}^{1}\times T\to T$ be the projection map and let

[TABLE]

Then

[TABLE]

(as $\mathcal{O}(\widetilde{E})\lvert_{\mathbb{C}\mathbb{P}^{1}\times\{t\}}=\mathcal{O}(\widetilde{E}\lvert_{\mathbb{C}\mathbb{P}^{1}\times\{t\}})$ and $\widetilde{E}_{\lvert_{\mathbb{C}\mathbb{P}^{1}\times\{t\}}}\simeq\mathbb{C}\mathbb{P}^{1}\times\mathbb{C}^{2}$).

Consequently, the sheaf $R^{0}\pi_{*}(\widetilde{E})$ over $T$ defined by

[TABLE]

for open Zariski open subsets $U$ of $T$, is a rank-$2$ locally free sheaf and so there is an open cover $\{U_{i}\}_{i}$ of $T$ such that

[TABLE]

This gives a trivialization of $R^{0}\pi_{*}(\widetilde{E})$ on $U_{i}$

[TABLE]

which, in turn, gives a trivialization of $\widetilde{E}$ on the product $\mathbb{C}\mathbb{P}^{1}\times U_{i}$,

[TABLE]

Indeed,

[TABLE]

and, since $\varphi_{i}$ is a trivialization, we can define

[TABLE]

as well as the isomorphism

[TABLE]

Hence, there exists an open subset $U$ of $T$ such that $\widetilde{E}\lvert_{\mathbb{C}\mathbb{P}^{1}\times U}\simeq(\mathbb{C}\mathbb{P}^{1}\times U)\times\mathbb{C}^{2}$ and this set $U$ parametrizes holomorphically trivial stable quasi-parabolic Higgs bundles over $\mathbb{C}\mathbb{P}^{1}$ at $D$. In particular, we have a map

[TABLE]

with

[TABLE]

Assuming, without loss of generality, that for each $t\in U$ we have $X_{1}(t)\neq X_{2}(t)$ and $(X_{1}(t),\ldots,X_{n}(t))\in U_{A,B}$ for some $A,B$ satisfying (62), (63) and (64), it is enough to show that the quotient map from the set $H_{A,B}$ which sends $(X_{1},\ldots,X_{n})$ to

[TABLE]

and $(N_{1},\ldots,N_{n})$ to $(gN_{3}g^{-1},\ldots,\widehat{gN_{i_{0}}g^{-1}},\ldots,gN_{n}g^{-1})$, with $gX_{1}=[1:0]$, $gX_{2}=[0:1]$,

[TABLE]

with $\widetilde{\lambda}_{2}$ determined by $\eqref{eq:sumzero3}$ (and consequently sends $(\lambda_{1},\ldots,\lambda_{n})$ to

[TABLE]

with $g\cdot\lambda_{1}=1$ and $g\cdot\lambda_{2}=\widetilde{\lambda}_{2}=(1-w_{1}z_{2})^{2}\lambda_{1}\lambda_{2}$) is holomorphic.

Here

[TABLE]

[TABLE]

and $i_{0}$ is the index used in (58), (59), (60) and (61). The corresponding values of $X_{i_{0}}$ and $\lambda_{i_{0}}$ are determined by the equations (55), (56) and (57), a consequence of the fact that the sum of the residues is zero.

The quotient map is the composition $h_{3}\circ h_{2}\circ h_{1}$ of maps

[TABLE]

[TABLE]

and

[TABLE]

where, for $i\in A$, we have $X_{i}=[1:w_{i}]$ and $a_{i}=w_{i}$ and, for $i\in B$, we have $X_{i}=[z_{1}:1]$ and $a_{i}=z_{i}$. Moreover, $f_{\lvert_{U}}=h_{3}\circ h_{2}\circ h_{1}\circ\widetilde{f}$.

Note that the map $h_{1}$ is holomorphic since the map $g$ is holomorphic and depends holomorphicaly on $X_{1}$, $X_{2}$ and $\lambda_{1}$:

[TABLE]

∎