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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях) Hitchin systems: some recent advances O. K. Sheinman, B. Wang Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Англоязычная версия:
Russian Mathematical Surveys, 2024, Volume 79, Issue 4, Pages DOI: https://doi.org/10.4213/rm10163e 
Реферативные базы данных:
    1. IntroductionThe integrable systems, this survey is devoted to, were invented by Hitchin [26] and bear his name. They are attractive in their geometric simplicity, and have been studied mainly from the geometric point of view. However, methods of heir solution them are much less developed. In this survey we touch on both issues; however, we do not set ourselves an impossible task of a comprehensive review (several mathematical schools and centres work on the subject). We focus on two fields of our own research, namely, on parabolic Hitchin systems and on exact methods of the solution of Hitchin systems. In the introductory section (Section 2) we give the necessary definitions for classical Hitchin systems, and describe briefly their main applications. Parabolic Hitchin systems were invented in the series of works of Seshadri with coauthors. While the original Hitchin systems are related to compact Riemann surfaces, parabolic ones correspond to Riemann surfaces with punctures, and a certain additional structure referred to as a parabolic structure. Parabolic Hitchin systems are the subject of Section 3 of the present paper. We begin with a general definition of a parabolic structure as a set of filtrations and weights associated with punctures. As a motivation, we present the original approach by Mehta and Seshadri, where the parabolic structure is related to a unitary representation of a parabolic subgroup (of the fundamental group of the Riemann surface) at a cusp. Then we define parabolic vector bundles as vector bundles endowed with the parabolic structure, and parabolic Higgs bundles as parabolic bundles endowed with parabolic Higgs fields. These fields are required to satisfy certain nilpotency conditions with respect to the filtrations at punctures. Then the theory of parabolic Hitchin systems is developed by following the same scheme as for original Hitchin systems, with modifications caused by the presence of a parabolic structure, particularly by the nilpotency conditions. It is a fundamental problem of the theory of integrable systems to investigate the fibres of the corresponding Lagrangian foliations. For the classic integrable systems these are Liouville tori. In the context of Hitchin systems fibres are level varieties of the Hitchin map given by the evaluation of a full set of independent Hamiltonians on the Higgs bundle. Typical fibres of a Hitchin map are proved to be the Jacobians of the corresponding spectral curves in the case of structure group $\operatorname{GL}(n)$, or to be the Prym varieties (Prymians) of spectral curves in the case of simple classical structure groups. We discuss here the following two results on fibres of Hitchin maps: the parabolic BNR correspondence and mirror symmetry for parabolic Hitchin systems. The BNR (Beauville–Narasimhan–Ramanan) correspondence [5] is the correspondence between the Higgs bundles in the fibres of Hitchin maps and the line bundles on the corresponding spectral curves. Following [5], we present the BNR correspondence for parabolic Hitchin systems, which requires a certain accuracy in the approach to spectral curves. In the context of the Inverse Spectral Method (Subsection 4.2) the above line bundles on spectral curves emerge as the eigenspace bundles of Lax operators. The Baker–Akhieser vector function defined in Subsection 4.2.3 is nothing but a section of such a line bundle. Turning to mirror symmetry for Hitchin systems, first notice that geometrically it is expressed by a theorem of Donagi and Pantev [11] stating that the fibres of two Hitchin systems with Langlands dual structure groups are torsors over dual Abelian varieties (see Theorem 3.16 below). This can be proved to be an (Abelian) implication of homological mirror symmetry in the sense of Kontsevich. Here we stick to the concept of topological mirror symmetry interpreted as the equality of the stringy Hodge numbers or, equivalently, stringy $E$-polynomials (Subsection 3.4.2) corresponding to the parabolic Hitchin systems with the same base curve and Langlands dual structure groups (Theorems 3.19, 3.22). It is another goal of the present survey to discuss the exact methods of solution of Hitchin systems. It is the subject of Section 4. Systems with structure group1 $\operatorname{GL}(n)$ can explicitly be solved in terms of theta-functions both by means of the inverse spectral method (Krichever [34]), and by means the method of separation of variables (Gorsky, Nekrasov, Rubtsov [16], and Sheinman [47], [7]). In the framework of separation of variables we present an explicit theta-function formula for solutions, which has not been published elsewhere (Subsection 4.3.7). It is pretty clear that the separation of variables technique is applicable also in the case of parabolic Hitchin systems with structure group $\operatorname{GL}(n)$. We hope to revisit this subject in the future. As we can see, to show mirror symmetry, it is crucial to interpret parabolic Higgs bundles in the fibres of Hitchin maps as line bundles on the normalizations of the corresponding spectral curves. It would also be interesting to obtain such an interpretation and comprehend the mirror symmetry for Hitchin systems in terms of separation of variables2. In the case of systems with simple structure groups, no matter parabolic or not, obstructions emerge for both methods. It turns out that the Baker–Akhieser function of the system has a dynamical pole divisor in this case, which makes the inverse spectral method inapplicable, at least in the currently known form. As for separation of variables, the method can be developed much further, to finding action-angle coordinates in many cases. However, no analogue of the theta-function formula in Subsection 4.3.7 can be obtained in general because of the peculiarity of the inversion problem on Prymians. We hope that identifying this problem will serve as an incentive for further research in this direction. 2. Basic results on Hitchin systems2.1. Hitchin systems: the definition (Hitchin [26])Assume that $\Sigma$ is a compact genus $g$ Riemann surface with a conformal structure, $G$ is a complex semisimple or reductive Lie group, $\mathfrak{g}=\operatorname{Lie}(G)$, and $P_0$ is a smooth principal $G$-bundle on $\Sigma$. By a holomorphic structure on $P_0$ we mean a $(0,1)$-connection, that is, a differential operator on the sheaf of sections of the bundle $P_0$ which is locally given as $\bar{\partial}+\omega$, where $\omega \in \Omega^{0,1}(\Sigma,\mathfrak{g})$, and, under the action of a gluing function $g$, $\omega$ transforms as follows:
$$
\begin{equation*}
\omega \mapsto g\omega g^{-1}-(\bar{\partial}g)g^{-1}.
\end{equation*}
\notag
$$
Suppose $\mathcal{A}$ is a space of semistable [26] holomorphic structures on $P_0$ and $\mathcal{G}$ is a group of smooth global gauge transformations. The quotient $\mathcal{N}=\mathcal{A}/\mathcal{G}$ is referred to as the moduli space of semistable holomorphic structures on $P_0$. We define a Hitchin system as a dynamical system with $\mathcal{N}$ as a configuration space and $T^{*}(\mathcal{N})$ as a phase space. A point in $\mathcal{N}$ is a principal holomorphic $G$-bundle on $\Sigma$. In the case when $G$ is semisimple we have $\dim\mathcal{N}=\dim\mathfrak{g}\cdot (g-1)$. We will consider $G=\operatorname{GL}(n)$ as a typical representative of the class of reductive groups. For $G=\operatorname{GL}(n)$ we have
$$
\begin{equation*}
\dim\mathcal{N} =n^2\cdot (g-1)+1=\dim\mathfrak{g}\cdot (g-1)+1.
\end{equation*}
\notag
$$
As defined above, the phase space of a Hitchin system is $T^{*}(\mathcal{N})$. According to the Kodaira–Spencer theory, $T_P(\mathcal{N}) \simeq H^1(\Sigma,\operatorname{Ad} P)$. By Serre duality
$$
\begin{equation*}
T^*_P(\mathcal{N}) \simeq H^0(\Sigma,\operatorname{Ad} P \otimes \mathcal{K})
\end{equation*}
\notag
$$
where $\mathcal{K}$ is the canonical class of $\Sigma$ and $\operatorname{Ad} P\otimes \mathcal{K}$ is a holomorphic vector bundle with fibre $\mathfrak{g}\otimes\mathbb{C}$. We denote the points of $T^*(\mathcal{N})$ by $(P,\Phi)$, where $P \in \mathcal{N}$ and $\Phi\in H^0(\Sigma,\operatorname{Ad} P\otimes \mathcal{K})$. Sections of the sheaf $T^*(\mathcal{N})$ are called Higgs fields. Assume that $\chi_i$ is a homogeneous invariant polynomial on $\mathfrak{g}$ of degree $d_i$. It defines a map
$$
\begin{equation*}
\chi_i(P)\colon H^0(\Sigma,\operatorname{Ad} P\otimes \mathcal{K}) \to H^0(\Sigma,\mathcal{K}^{\otimes d_i})
\end{equation*}
\notag
$$
for each $P\in\mathcal{N}$. Let $\Phi$ denote a Higgs field; then we can define Thus, each point $(P,\Phi)$ of the phase space gets assigned to an element of $H^0(\Sigma, \mathcal{K}^{\otimes d_i})$. Suppose $\{\Omega^i_j\}$ is a basis in $H^0(\Sigma,\mathcal{K}^{\otimes d_i})$; then where the $H_{i,j}(P,\Phi)$ are scalar functions on $T^*(\mathcal{N})$. For any $i$ and $j$ the function $H_{i,j}(P,\Phi)$ is called a Hitchin Hamiltonian. Theorem 2.1 [26]. Whatever be the choice of the basis $\{\Omega^i_j\}$, Hitchin Hamiltonians form a complete set of Hamiltonians in involution on $T^*(\mathcal{N})$ with respect to the natural symplectic structure. The set $d_1,\dots,d_n$ is a characteristic invariant of the Lie algebra $\mathfrak{g}$, hence the space $\mathcal{H}=\bigoplus\limits_{i=1}^nH^0(\Sigma,\mathcal{K}^{\otimes d_i})$ ($n=\operatorname{rank}\mathfrak{g}$) is well defined. It is referred to as the Hitchin base space. Let $\chi=\{\chi_1,\dots,\chi_n\}$ be a base in the space of $\operatorname{Ad} G$-invariant polynomials on $\mathfrak{g}$. Then $\chi$ defines a map $h_\chi \colon T^*(\mathcal{N})\to \mathcal{H}$ called the Hitchin map. Though this map depends on the choice of the base $\chi=\{\chi_1,\dots,\chi_n\}$, its level varieties are also well defined. They are referred to as Hitchin fibres. It is one of our main goals in this survey to describe generalizations of the Hitchin map, and give a description of the corresponding Hitchin fibers. It is also a goal to provide effective alternatives to the above definition of Hitchin systems, along with methods for obtaining exact solutions for them. 2.2. Classical applicationsThe first and best known application of Hitchin systems was given by Hitchin himself in [27]. It is the application to the $2$-dimensional Conformal Field Theory (2dCFT below). By one of the equivalent definitions [23] a 2dCFT is a projectively flat connection in a holomorphic vector bundle on the moduli space of curves (with punctures in general). Then the problem of constructing this type of connection arises. Recall that a connection $\nabla$ is said to be projectively flat if there is a $2$-cocycle $\lambda$ on the Lie algebra of smooth vector fields such that $[\nabla_X,\nabla_Y]=\lambda(X,Y)\cdot \mathbf{1}$ for any pair of smooth vector fields $X$ and $Y$, where $\mathbf{1}$ is the identity operator. Let $M$ be a real compact symplectic manifold with symplectic form $\omega$. By a Kähler polarization of $M$ we mean a complex structure $I\colon H^0(TM)\to H^0(TM)$, $I^2=-1$ on the tangent vector bundle which satisfies the following conditions of compatibility with the symplectic structure and positivity:
$$
\begin{equation}
\omega(X,IY)=\omega(Y,IX), \quad \omega(X,IX)\ \text{is positive definite} \quad (X,Y\in H^0(TM)).
\end{equation}
\tag{2.1}
$$
If the de Rham cohomology class $(2\pi)^{-1}[\omega]$ is integral, then $[\omega]$ is the curvature form of a connection on a principal $\operatorname{U}(1)$-bundle over $M$ whose first Chern class is $(2\pi)^{-1}[\omega]$. When $M$ is simply connected this connection is unique up to gauge equivalence. Instead of considering the principal $\operatorname{U}(1)$-bundle, we consider an associated complex line bundle $L$ and an associated connection $\nabla^L$, both depending on a natural number called the level and coming from the character a $\operatorname{U}(1)$. Hence with every Kähler polarization of $M$ we can canonically associate the finite-dimensional space of globally holomorphic sections of $L$. Then we obtain a vector bundle over the variety of Kähler polarizations. In [27] Hitchin looked for a projectively flat connection in this vector bundle. First we show that a deformation of the Kähler polarization is given by a symmetric $2$-tensor on $M$. Let $I=I_t$ be a path in the variety of Kähler polarizations. The compatibility conditions (2.1) imply that $\omega(X,IY)$ is a Riemannian metric on $M$, hence it is representable in the form $\omega(X,IY)=(X,\widetilde{G}Y)$ where $\displaystyle (X,Y)=\sum_{i}X_iY_i$, $\widetilde{G}$ is a positive definite (in particular, symmetric) matrix. We also have $\omega(X,Y)=(X,\widetilde{G}I^{-1}Y)$. By differentiation with respect to $t$ of the first compatibility relation we obtain that is, $\omega(X,\dot{I}Y)$ is a symmetric bilinear form too and, as such, is representable in the form $\omega(X,\dot{I}Y)=(X,GY)$ where $G$ is also a symmetric matrix. On the other hand It follows that $\widetilde{G}I^{-1}\dot{I}=G$, hence $\dot{I}=(\widetilde{G}I^{-1})^{-1}G$. Since $\widetilde{G}I^{-1}$ is the matrix of $\omega$, which is assumed to be fixed, $\dot{I}$ is determined by the symmetric tensor $G$.Vice versa, assume that we have a multiparameter family of Kähler polarizations and the corresponding family of Kähler manifolds $M_t$, $t=(t_1,\dots,t_A)$. For every $t$ we construct the line bundle $L=L_t$ as above. Consider the vector bundle of the spaces $H^0(L_t)$ over this family. Given a symmetric tensor $G^{ij}$, we can construct a connection in this vector bundle as follows. First, we define locally the operator $\Delta=\nabla^L_i(G^{ij}\nabla^L_j)$ in the space of holomorphic sections of $L$. As proved in [27], for every $a=1,\dots,A$ there exists a locally defined operator $P_a\colon H^0(L)\to H^0(L)$ with highest symbol $\Delta$ such that is a globally defined operator $H^0(L)\to H^0(L)$, for all $a=1,\dots,A$, where $\partial_a=\partial/\partial t_a$.Throughout what follows $M=\mathcal{N}$. We assume additionally the rank and degree of the bundles in $\mathcal{N}$ are fixed and, moreover, coprime. Then $\mathcal{N}$ is compact [41]. These assumptions are not restrictive because $\mathcal{N}$ is just an auxiliary object, while our main goal here is to construct a connection on the moduli space of curves. We observe that $\mathcal{N}$ is canonically symplectic. Indeed, by the Narasimhan–Seshadri theorem [41]
$$
\begin{equation*}
\mathcal{N} \simeq \operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(n))/\operatorname{SU}(n),
\end{equation*}
\notag
$$
and by [2] the latter is just the space of flat connections of the form $\partial+\varphi$, where $\varphi$ is a skew-Hermitian matrix-valued $1$-form on $\Sigma$. Given two such matrix-valued $1$-forms $\alpha$ and $\beta$, now representing tangent vectors to $\mathcal{N}$, we define
$$
\begin{equation*}
\omega(\alpha,\beta) =\int_\Sigma \operatorname{Tr}(\alpha\wedge\beta).
\end{equation*}
\notag
$$
Next, we specify a family of Kähler polarizations and a tensor $G^{ij}$ so that
$$
\begin{equation*}
[\nabla_a,\nabla_b]=\lambda_{ab}\cdot \mathbf{1}, \qquad \lambda_{ab}\in\mathbb{C}.
\end{equation*}
\notag
$$
First, we choose a Hodge star operator (a complex structure) $*$ on $\Omega^1(\Sigma)$:
$$
\begin{equation*}
\begin{aligned} \, & *\colon \Omega^1(\Sigma)\to\Omega^1(\Sigma), \\ & *^2=-1. \end{aligned}
\end{equation*}
\notag
$$
Then the corresponding operator on Higgs fields $I\colon\Phi\to *\Phi$ (obtained by acting entry by entry on the matrix $\Phi$) is a Kähler polarization on $M$. A deformation of the complex structure on $\Sigma$ is given by a Beltrami differential $\beta_a=\phi_a\partial_z\otimes d\overline{z}$. We define the corresponding symmetric $2$-tensor on $T^*\mathcal{N}$ by
$$
\begin{equation}
G_a(\Phi,\Phi) =\int_\Sigma\operatorname{Tr}(\Phi^2)\beta_a.
\end{equation}
\tag{2.3}
$$
Now we possess all we need to define the connection by means of (2.2). Theorem 2.2 [27]. The connection $\nabla$ defined by means of relation (2.2) with symmetric tensor (2.3) is projectively flat. Sketch of proof. Indeed, Now consider the exact sequence of sheaves of differential operators
$$
\begin{equation*}
0 \to \mathcal{D}^1(L) \to \mathcal{D}^2(L) \to S^2T\to 0
\end{equation*}
\notag
$$
(we say that the second-order differential operator is the sum of its principal symbol and a first-order operator), and the exact sequence of cohomology groups
$$
\begin{equation*}
\begin{aligned} \, 0\to H^0(M,\mathcal{D}^1(L)) & \to H^0(M,\mathcal{D}^2(L)) \to H^0(M,S^2T) \\ & \xrightarrow{\;\delta\; }H^1(M,\mathcal{D}^1(L)) \to\cdots\,. \end{aligned}
\end{equation*}
\notag
$$
Without going into details, we state, following [27], that $\delta$ is an injection, and there are no holomorphic vector fields on $M$. The first part of the last assertion implies that $H^0(M,\mathcal{D}^1(L))\to H^0(M,\mathcal{D}^2(L))$ is an epimorphism, that is, any holomorphic second-order operator is actually of the first order. The second part implies that $H^0(M,T)=0$, hence the last operator is of order $0$, that is, a holomorphic function. Since $M$ is compact, such functions are just constants, hence $[\nabla_a,\nabla_b]=\mathrm{const}\cdot \mathbf{1}$. 2.3. Specific approach to rank $2$ genus $2$ Hitchin systemsIn [40] Narasimhan and Ramanan showed that the space of semistable rank $2$ holomorphic vector bundles on a Riemann surface is canonically isomorphic to $\mathbb{P}\Theta_2$. Here $\Theta_2$ denotes the space of level $2$ theta-functions, that is,
$$
\begin{equation*}
\exp(2(\pi i n\cdot\tau n+2\pi i n\cdot u))\theta(u+m+\tau n) =\theta(u)
\end{equation*}
\notag
$$
(where $m,n\in\mathbb{Z}^2$ and $\tau$ is the period matrix), and $\mathbb{P}$ stays for the projectivization. Based on this result, van Geemen and Previato [15] and van Geemen and de Jong [14] gave a more specific description of the rank $2$, genus $2$ Hitchin systems. For genus $2$ curves, $\Theta_2$ is $4$-dimensional, hence $\mathbb{P}\Theta_2\cong \mathbb{P}^3$; thus in [15] they were seeking an integrable system on $T^*\mathbb{P}^3$. The construction of Hamiltonians in [15] is as follows. Let
$$
\begin{equation*}
q\in\mathbb{P}^3, \quad q=(x:y:z:t), \quad\text{and}\quad p\in(\mathbb{P}^3)^*.
\end{equation*}
\notag
$$
We define $\epsilon_i(q)\in(\mathbb{P}^3)^*$ by
$$
\begin{equation*}
\begin{alignedat}{3} \epsilon_1&=(y : -x : t : -z),&\qquad \epsilon_3&=(z : t : -x : -y),&\qquad \epsilon_5&=(t : z : -y: -x), \\ \epsilon_2&=(y : -x : -t : z),&\quad \epsilon_4&=(z : -t : -x : y),&\quad \epsilon_6&=(t : -z : y : -x). \end{alignedat}
\end{equation*}
\notag
$$
Let $x_{ij}$ be the Klein coordinates (see [15] for the definition) of the line in $(\mathbb{P}^3)^*$ incident to the points $\epsilon_i(q)$ and $p$. Then
$$
\begin{equation}
H_i(p,q) =\sum_{1\leqslant j\leqslant 6,\,j\ne i}\frac{x_{ij}^2}{z_i-z_j},
\end{equation}
\tag{2.4}
$$
where $\displaystyle w^2=\prod_{j=1}^{6}(z-z_j)$ is the equation of the base curve. Then the commutativity of the Hamiltonians (2.4) with respect to the symplectic structure on $T^*\mathbb{P}^3$, as well as explicit expressions for them, were established in [15] by using Mathematica software. The identification of (2.4) as Hitchin Hamiltonians was done subsequently in [14], in the course of examining the Hitchin connection (see Subsection 2.2) in the genus $2$ rank $2$ case. Analytically, the commutativity of Hamiltonians was later proved in [13] (see Subsection 4.1).
3. Parabolic and parahoric Hitchin systemsIn the previous section we presented constructions of Hitchin systems on compact Riemann surfaces and of the corresponding flat connections in moduli spaces of curves without punctures. In what follows we define analogues of these objects for Riemann surfaces with punctures (at least for the purposes of Conformal Field Theory). The corresponding integrable systems are referred to as parabolic Hitchin systems, for the reason that the corresponding Higgs bundles have a special structure, called a parabolic structure, at punctures of the underlying Riemann surface. A further generalization of parabolic Hitchin systems is referred to as parahoric Hitchin systems. Since our way to treat parabolic Hitchin systems has an algebraic nature, we start working with Hitchin systems over an algebraically closed field $k$. Actually, some results mentioned in what follows also work over an arbitrary field (with mild assumptions in characteristic $2$), and such kinds of results are important for the proof of ‘topological mirror symmetry’. 3.1. Parabolic vector bundlesIn this subsection we introduce first the concept of parabolic vector bundle. Let $X$ be a smooth projective curve of genus $g$ over a field $k$, which we assume to be algebraically closed. We fix a finite subset $D\subset X$, which we also regard as a reduced effective divisor on $X$. Then we assume that $2g-2+\deg D>0$ ($g=0$ is also allowed, then $\deg D\geqslant 3$). We also fix a positive integer $r$ which will be the rank of vector bundles on $X$. To define a parabolic structure on a vector bundle we need to specify a quasi-parabolic structure and weights for each $x\in D$. Quasi-parabolic structure consists of a finite sequence
$$
\begin{equation*}
m^{\boldsymbol{\cdot}} (x) =(m^1(x),m^2(x), \dots, m^{\sigma_x}(x))
\end{equation*}
\notag
$$
satisfying $\displaystyle \sum_{i=1}^{\sigma_x}m^{i}(x)=r$ for each $x\in D$. We denote this quasi-parabolic structure simply by $P$. The weights are given by a choice of a set of rational numbers3
$$
\begin{equation*}
0\leqslant \alpha_1(x)<\cdots <\alpha_{\sigma_x}(x)<1,
\end{equation*}
\notag
$$
denoted by $\alpha(x)$, for each $x\in D$. We also denote the set of weights $\{\alpha(x)\}_{x\in D}$ by $\alpha$. The set $(X,D,P,\alpha)$ will be referred to as the parabolic type. Definition 3.1. A parabolic vector bundle of type $(X,D,P,\alpha)$ is a rank $r$ vector bundle $\mathcal{E}$ on $X$ which for every $x\in D$ is endowed with a filtration such that $\dim F^{j-1}(x)/ F^{j}(x)=m^j(x)$ and with the weight $\alpha(x)$. Remark. The definitions of (quasi-)parabolic vector bundles over a Riemann surface were first introduced by Mehta and Seshadri [38]. Then they were generalized to varieties parametrized by very general schemes by Yokogawa [55]. Now we define the parabolic degree (or $\alpha$-degree) of $\mathcal{E}$ by
$$
\begin{equation*}
\operatorname{par-deg}(\mathcal{E}) :=\deg \mathcal{E} +\sum_{x\in D}\,\sum_{j=1}^{\sigma_x}\alpha_{j}(x)m^j(x).
\end{equation*}
\notag
$$
Definition 3.2. A parabolic vector bundle $\mathcal{E}$ is said to be stable (respectively, semistable) if for every subbundle $ \mathcal{F} \subsetneq \mathcal{E} $ we have where the parabolic structure on $\mathcal{F}$ is inherited from $\mathcal{E}$. Definition 3.3. Given a semistable parabolic vector bundle $\mathcal{E}$, there is a filtration, called the Harder–Narasimhan filtration: such that for each $1\leqslant i\leqslant\ell$, $\mathcal{E}_{i}/\mathcal{E}_{i-1}$ is stable. We define Though Harder–Narasimhan filtration may not be unique, the associated $\operatorname{gr}\mathcal{E}$ only depends on $\mathcal{E}$. Two semistable parabolic vector bundles $\mathcal{E}$, $\mathcal{E}'$ are said to be equivalent if $\operatorname{gr}\mathcal{E}\cong\operatorname{gr}\mathcal{E}'$. With the help of geometric invariant theory one can construct the coarse moduli space of semistable parabolic vector bundles of parabolic degree4 $0$, which we denote by $\mathcal{N}_{P,\alpha}$, (see [55] where much more general cases were considered). In general, $\mathcal{N}_{P,\alpha}$ is a normal projective variety which parametrizes equivalence classes of semistable parabolic vector bundles of parabolic type $(P,\alpha)$ on $X$. For a generic choice of weights $\alpha$, semistable parabolic bundles are parabolically stable. In this case the moduli space $\mathcal{N}_{P,\alpha}$ is a smooth projective variety. In the rest of the paper we always assume $\alpha$ to be generic, and we drop $\alpha$ from the index for simplicity. Relations with unitary representationsIn what follows we explain the motivation of introducing parabolic vector bundles, following Mehta and Seshadri [38]. Let $\mathbb{H}$ be the upper half-plane and $\Gamma$ be a discrete subgroup of $\operatorname{Aut}(\mathbb{H})=\operatorname{PGL}_{2}(\mathbb{R})$, acting freely on $\mathbb{H}$ and such that $\mathbb{H}/\Gamma$ has a finite volume. We denote by $\mathbb{H}^{+}$ the union of $\mathbb{H}$ and the parabolic cusps5. We set $X=\mathbb{H}^+/\Gamma$, which is a compact Riemann surface, that is, it is a compactification of $X^{\circ}=\mathbb{H}/\Gamma$. We denote the natural quotient map by $p\colon\mathbb{H}\to X^{\circ}$. Let $E$ be a vector space of dimension $r$ endowed with a Hermitian bilinear form and $\rho\colon\Sigma\to \operatorname{SU}(E)$ be a unitary representation. Then $\mathbb{H}\times E$ admits a natural $\Gamma$-action:
$$
\begin{equation*}
\gamma\colon (z,e) \mapsto (\gamma z,\sigma(\gamma)e).
\end{equation*}
\notag
$$
The quotient $\mathbb{H}\times_{\Gamma} E$ can be treated as a vector bundle on $X^{\circ}=\mathbb{H}/\Gamma$, which we denote by $\mathcal{E}^{\circ}$. For a cusp $x\in X\setminus X^{\circ}$, we denote its stabilizer by $\Gamma_{x}$; this is a cyclic subgroup of $\Gamma$ consisting of unipotent elements. If $x$ is the image of $\infty\in\mathbb{H}^{+}$, then for $\delta>0$ we can choose $\Gamma_x$-invariant neighbourhoods $U_{\delta}(\infty):=\{r+is \mid s\geqslant \delta>0 \}$ of $\infty$. Then $\{x\}\cup \{U_{\delta}/\Gamma_{x}\}$ form a basis of neighbourhoods of $x$ on $X$. Since the $\Gamma$-action need not be transitive on the cusps $\mathbb{H}^+\setminus\mathbb{H}$, not all cusps $x\in X\setminus X^{\circ}$ are images of $\infty$. However, for any cusp $y\in \mathbb{H}^+\setminus\mathbb{H}$ we can find $h\in\operatorname{PGL}_{2}(\mathbb{R})$ such that $h\infty=y$, which can also be used to identify their neighbourhoods. Hence we can still define such neighbourhoods $U_{\delta}(y)$ for all $y\in\mathbb{H}^+\setminus\mathbb{H}$. We denote $U_{\delta}/\Gamma_{x}$ by $U_{\delta}(x)$ and $\{x\}\cup \{U_{\delta}/\Gamma_{x}\}$ by $D_{\delta}(x)$. Definition 3.4. A holomorphic section $F\colon\mathbb{H}\to\mathbb{H}\times E$ given by $F(z)=(z,f(z))$ is said to be bounded and $\Gamma$-equivariant if: Remark. In [38; Definition 1.1], Mehta and Seshadri called such sections $\Gamma$-invariant sections in $\mathbb{H}^{+}$. It is quite straightforward to see that there is a local version of boundedness near each cusp. Now let us look at $\Gamma$-equivariant $f=(f_1(z),\dots,f_r(z))$ near a cusp $x$. Without loss of generality we may assume that $x$ is the image of $\infty$ and for simplicity, we assume $\Gamma_{x}$ is generated by $\gamma_{x}\colon z\mapsto z+1$, and choose a basis $\{e_i\}_{i=1}^{r}$ of $E$ such that
$$
\begin{equation*}
\rho(\gamma_{x}) =\begin{pmatrix} \exp^{2\pi\sqrt{-1}\,\alpha_1} &&& \\ & \exp^{2\pi\sqrt{-1}\,\alpha_2} && \\ && \dots && \\ &&& \exp^{2\pi\sqrt{-1}\,\alpha_r} \end{pmatrix}
\end{equation*}
\notag
$$
where $0\leqslant \alpha_1\leqslant\alpha_2\leqslant\dots \leqslant\alpha_r< 1$. Then $\Gamma$-equivariance implies that In particular, we can write here $\tau=\exp^{2\pi\sqrt{-1}\,z}$ can be treated as a local coordinate at the cusp! Hence the boundedness of $F$ amounts to saying that $g$ is holomorphic at $x\in X$. Moreover, we can see that bounded sections near $\infty$,
$$
\begin{equation*}
z\mapsto \exp^{2\pi\sqrt{-1}\alpha_{j}z}e_j, \quad 1\leqslant j\leqslant r,
\end{equation*}
\notag
$$
provide a trivialization of $H\times_{\Gamma}E$ in the neighbourhood $U_{\delta}(x)$. Then we can extend the vector bundle $\mathbb{H}\times_{\Gamma} E$ on $X^{\circ}$ to a vector bundle on $X$ denoted by $\mathcal{E}$. Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be two vector bundles on $X$ coming from two unitary representations $\rho_i\colon\Gamma\to \operatorname{GL}_{r_i}(E_i)$, $i=1,2$. Definition 3.5. We say that a holomorphic morphism is bounded and $\Gamma$-equivariant if: Now for $E_1$ and $E_2$, we choose bases $\{e_{\ell}\}_{\ell\leqslant r_1}$ and $\{d_{m}\}_{m\leqslant r_2}$ such that
$$
\begin{equation*}
\begin{aligned} \, \rho_1(\gamma_{x}) & =\begin{pmatrix} \exp^{2\pi\sqrt{-1}\,a_1} && \\ & \dots &\\ && \exp^{2\pi\sqrt{-1}\,a_{r_1}} \end{pmatrix}, \\ \rho_2(\gamma_{x}) & =\begin{pmatrix} \exp^{2\pi\sqrt{-1}\,b_1} && \\ & \dots &\\ && \exp^{2\pi\sqrt{-1}\,b_{r_2}} \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
We put $\displaystyle \phi(z)e_{\ell}=\sum_{m}\phi_{\ell m}(z)d_m$. Then $\Gamma$-equivariance implies that
$$
\begin{equation*}
\phi_{\ell m}(z+1) =\exp^{2\pi\sqrt{-1}\,(b_{m}-a_{\ell})}\phi_{\ell m}(z).
\end{equation*}
\notag
$$
In other words,
$$
\begin{equation*}
\phi_{\ell m}(z) =\exp^{2\pi\sqrt{-1}(b_{m}-a_{\ell})z}g_{\ell m}(\tau).
\end{equation*}
\notag
$$
Then the boundedness requires that We now look at the special case when $\phi$ is an automorphism of $E$. Then we rewrite $\{a_{\ell}\}_{\ell\leqslant r}$ as
$$
\begin{equation}
\begin{aligned} \, & _{1}=a_{2}=\cdots=a_{m^{1}(x)},\\ & a_{m^1(x)+1}=\cdots=a_{m^{1}(x)+m^{2}(x)},\\ & \dots \dots \dots \dots \dots \dots \dots\\ & a_{\sum_{j=1}^{\sigma_x-1}m^{j}}=\cdots=a_{r}. \end{aligned}
\end{equation}
\tag{3.1}
$$
Now consider a filtration $F^{\boldsymbol{\cdot}}$ on $\mathcal{E}_{x}$ defined by
$$
\begin{equation*}
\begin{aligned} \, F^{0}\mathcal{E}_x & =\mathcal{E}_p; \\ F^{1}\mathcal{E}_x & =\text{subspaces generated by}\ e_{m^1(x)+1},e_{m^1(x)+2},\dots, e_{n}; \\ F^{2}\mathcal{E}_x & =\text{subspaces generated by}\ e_{m^1(x)+m^2(x)+1},e_{m^1(x)+m^2(x)+2},\dots,e_{r}; \\ & \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ F^{\sigma_x-1}\mathcal{E}_x & =\text{subspaces generated by}\ e_{\sum_{j=1}^{\sigma_x-1}m^{j}(x)+1}, e_{\sum_{j=1}^{\sigma_x-1}m^{j}(x)+2}, \dots, e_{r}. \end{aligned}
\end{equation*}
\notag
$$
Then for any bounded $\Gamma$-equivariant morphism $\phi$ we have where $g=(g_{\ell m})$. This motivates the definition of filtrations at cusps (marked points) and also the definition of parabolic endomorphisms. We choose the weights at $x$ to be $0\leqslant \alpha_{1}(x)<\alpha_{2}(x)<\cdots<\alpha_{\sigma_x}(x)<1$ where following (3.1), and so on. To summarize we formulate the following theorem. Theorem 3.6 [38; Corollary 1.10, Theorem 4.1]. Let $X=\mathbb{H}^{+}/\Gamma$ as before. For each $x\in X\setminus X^{\circ}$ let $\{m^{i}(x)\}$ and the weights $0\leqslant \alpha_{1}(x)<\alpha_{2}(x)<\cdots<\alpha_{\sigma_x}(x)<1$ be chosen as above. Then 3.2. Moduli of parabolic Higgs bundlesIn this subsection we talk first about Yokogawa’s work on the construction of coarse moduli spaces of (semi-)stable parabolic pairs on smooth projective varieties. Definition 3.7. A parabolic Higgs bundle is a pair $(\mathcal{E},\theta)$ where $\mathcal{E}$ is a parabolic vector bundle as above and $\theta$, called a parabolic Higgs field, is an $\mathcal{O}_{X}$-homomorphism which takes each $F^j(x)$ to $F^{j+1}(x)\otimes_{\mathcal{O}_{X}}\omega_{X}(D)|_{x}$. By $\omega_X(D)$ we mean the space of meromorphic differentials without poles outside $D$. In particular, a parabolic Higgs field is nilpotent at marked points. An endomorphism of the parabolic bundle $\mathcal{E}$ is a vector bundle endomorphism of $E$ which preserves the filtrations $F^{\boldsymbol\cdot}(x)$. We call it a strongly parabolic endomorphism if it takes $F^i(x)$ to $F^{i+1}(x)$ for all $x\in D$ and $i$. We denote the subspaces of $\operatorname{End}_{\mathcal{O}_X}(E)$ specified by these properties by
$$
\begin{equation*}
\textit{ParEnd}(\mathcal{E}) \quad\text{and}\quad \textit{SParEnd}(\mathcal{E}),
\end{equation*}
\notag
$$
respectively. We can define similarly the sheaf of parabolic endomorphisms and that of strongly parabolic endomorphisms, denoted by
$$
\begin{equation*}
\mathcal{P}\!\textit{ar}\mathcal{E}\!\textit{nd}(\mathcal{E}) \quad\text{and}\quad \mathcal{S}\mathcal{P}\!\textit{ar}\mathcal{E}\!\textit{nd}(\mathcal{E}),
\end{equation*}
\notag
$$
respectively. Hence a parabolic Higgs field $\theta$ lies in $\operatorname{\textit{SParEnd}}(\mathcal{E})\otimes\omega_{X}(D)$. Remark. Unfortunately, conventions regarding parabolic Higgs bundles vary in the literature. For example, a weakly parabolic Higgs bundle in [50] is referred to as a “parabolic Higgs bundle” in [35], and a strongly parabolic Higgs bundle in [50] is referred to as a “parabolic Higgs bundle” in [6] and [43]. What is worse, the two types of parabolic Higgs bundles have very different moduli spaces and Hitchin systems; cf. [55] and [35]. The moduli spaces of strong ones are symplectic while the moduli of weak ones are Poisson. In this survey, for simplicity, a parabolic Higgs bundle is always assumed to be strongly parabolic. By analogy with the non-parabolic case, we can define the (semi-) stability of parabolic Higgs bundles. Definition 3.8. A parabolic Higgs bundle $(\mathcal{E},\theta)$ is said to be stable (respectively, semistable) if for every $\theta$-invariant subbundle $\mathcal{F}\subsetneq \mathcal{E}$ we have where the parabolic structure on $\mathcal{F}$ is inherited from $\mathcal{E}$. Similarly to the construction of coarse moduli spaces of semistable parabolic vector bundles, one can construct coarse moduli space of semistable parabolic Higgs bundles, which we denote by $\mathcal{M}_{P,\alpha}$ (see [55] for example). When the weights $\alpha$ are chosen generically, semistable parabolic Higgs bundles are parabolically stable and in this case the moduli space $\mathcal{M}_{P,\alpha}$ is a smooth quasi-projective variety. In the rest of this paper we always assume $\alpha$ to be generic, and we drop $\alpha$ from the index for simplicity. 3.2.1. The non-Abelian Hodge theoremIn Subsection 3.1, following Mehta–Seshadri we explained the motivation for introducing (stable) parabolic vector bundles and state the correspondence between the stable parabolic vector bundles and the unitary representations of the fundamental group of a punctured Riemann surface (see Theorem 3.6). All of these results can be seen as a parabolic analogue of Narasimhan–Seshadri’s results on unitary local systems and stable bundles on Riemann surfaces. Now we talk about Simpson’s non-Abelian Hodge theorem for parabolic Higgs bundles [49], which can be treated as an analogue of Mehta–Seshadri’s results in the setting of Higgs bundles. We follow the nice and concise description due to Mellit [39; § 7.6]. Let $X$ be a Riemann surface and $D$ be a reduced divisor on $X$. We put $X^{\circ}=X\setminus D$. We start from an irreducible representation
$$
\begin{equation*}
\rho\colon \pi_{1}(X^{\circ})\to\operatorname{GL}_r(\mathbb{C}).
\end{equation*}
\notag
$$
(Notice that the representation here is no longer unitary. Hence the monodromy need not be semisimple.) For each $x\in D$ we denote eigenvalues of the monodromy $\operatorname{C}_x$ around $x$ by
$$
\begin{equation*}
\{\exp^{-2\pi\sqrt{-1}\,a_i(x)+4\pi b_i(x)}\}_{1\leqslant i\leqslant r}.
\end{equation*}
\notag
$$
where the multiplicity of $\exp^{-2\pi\sqrt{-1}\,a_i(x)+4\pi b_i(x)}$ is $m^i(x)$. Note that $\operatorname{C}_x$ need not be diagonal: here we only provide notation for its eigenvalues. Similarly to Subsection 3.1, for each $x\in D$, we put the weights (which are used to define the stability of weakly parabolic Higgs bundles) to be and so on. Notice that, in the following theorem, we only require that $\theta\colon F^{j}(x)\mapsto F^{j}(x)\otimes\omega_{X}(D)|_{x}$, that is, $(\mathcal{E},\theta)$ is a weakly parabolic Higgs bundle. Theorem 3.9. There is a bijection between and Now let us return to (strongly) parabolic Higgs bundles (see Definition 3.7). Corollary 3.10. A stable (strongly) parabolic Higgs bundle $(\mathcal{E},\theta)$ defines uniquely an irreducible representation whose monodromy $\operatorname{C}_x$ is conjugate to a diagonal matrix where each $\exp^{-2\pi\sqrt{-1}\,\alpha_{j}(x)}$ has multiplicity $m^{j}(x)$ for $1\leqslant j\leqslant \sigma_x$ and $x\in D$. Remark. In the previous theorem, we actually consider weakly parabolic Higgs bundles, and in this corollary, we focus again on our parabolic Higgs bundles, which, in fact, are strongly parabolic. In particular, $\theta_x$ acts as $0$ on $F^{i-1}(x)/F^{i}(x)$. This amounts to saying that the Jordan normal form of $\operatorname{C}_x$ is actually the diagonal matrix $\exp^{-2\pi\sqrt{-1}\,\alpha_{i}(x)}\cdot \operatorname{Id}$ of size $m^{i}(x)$. 3.3. Parabolic Hitchin maps and fibersIn this subsection we define parabolic Hitchin maps and focus on their generic fibres. These are parabolic analogues of classical Hitchin maps. The main difference with non-parabolic Hitchin maps is that generic spectral curves are singular and the singularities of generic spectral curves play an important role in the geometry of parabolic Hitchin systems. Yokogawa [55; p. 495] (see also [4] and [50; Section 3]) defined parabolic Hitchin maps as restrictions of the characteristic polynomial map, which we now explain. First, we can define a projective morphism (we call it the characteristic polynomial map) from the moduli space (and even moduli stack) of parabolic Higgs bundles to the affine space $\displaystyle \prod_{i=1}^r \mathbf{H}^0(X,(\omega_X(D))^{\otimes i})$ and given pointwise by the characteristic polynomial of the parabolic Higgs field $\theta$:
$$
\begin{equation*}
\operatorname{char}_P\colon\mathcal{M}_P\to \mathbf{H}_D :=\prod_{i=1}^r \mathbf{H}^0(X,(\omega_X(D))^{\otimes i}), \quad (\mathcal{E},\theta)\mapsto (a_1(\theta),\dots,a_r(\theta)),
\end{equation*}
\notag
$$
where $a_i=\operatorname{Tr}(\wedge^r\theta)$; in particular, these are the coefficients of the characteristic polynomial. By the calculation in [4] the image of $\operatorname{char}_P$ lies in the subspace $\mathcal{H}_P$ defined as follows (see also [50; Subsection 3.1]. For each $x\in D$ we reorder $(m^1(x), \dots, m^{\sigma_x}(x))$ to be
$$
\begin{equation*}
n_1(x)\geqslant n_2(x)\geqslant \cdots \geqslant n_{\sigma_x}.
\end{equation*}
\notag
$$
So $\{n_i(x)\}$ is a partition of $r$ and we use $\mu_j(x)$ to denote the dual partition, which means that
$$
\begin{equation*}
\mu_{j}(x) =\#\{\ell\colon n_{\ell}(x)\geqslant j, 1\leqslant\ell\leqslant \sigma_x\}.
\end{equation*}
\notag
$$
Now we consider the level function $j\mapsto \gamma_{j}(x)$, $1\leqslant j\leqslant r$, such that $\gamma_{j}=l$ if and only if
$$
\begin{equation*}
\sum_{t\leqslant l-1}\mu_{t}(x) <j \leqslant\sum_{t\leqslant l}\mu_{t}(x).
\end{equation*}
\notag
$$
Proposition 3.11 [50; Theorem 4]. The image of $\operatorname{char}_P\colon\mathcal{M}_P\to \mathbf{H}_D$ lies in the subspace which we call the $\operatorname{GL}_r$-parabolic Hitchin base, and we call $h_P\colon \mathcal{M}_P\to \mathcal{H}_P$ the $\operatorname{GL}_r$-parabolic Hitchin map. We denote the natural projection by $\pi\colon\operatorname{Tot}(\omega_{X}(D))\to X$, where $\operatorname{Tot}(\omega_{X}(D))$ is the total space of the line bundle $\omega_{X}(D)$. We denote the tautological section of $\pi^*\omega_{X}(D)$ on $\operatorname{Tot}(\omega_X(D))$ by $\lambda$. Definition 3.12. Given a point $a=(a_1,a_2,\dots,a_r)\in \mathcal{H}_P$, the polynomial is a section of $\pi^*(\omega_{X}(D))^{r}$. The spectral curve $X_a$ (associated with $a$) is defined as the zero divisor of the section. Notice that the spectral curve $X_a$ is a finite branched covering of the base curve $X$, and we denote the projection by $\pi_a\colon X_a\to X$. Defined in this way, spectral curves (cf. [50; § 3.2 and § 4.2]) are never smooth unless all parabolic structures are given by full flags. Example 1. Now consider a simple example. Take the rank $r=4$, and choose a local coordinate $t$ around a marked point $x\in D$. Then the local section $\dfrac{dt}{t}$ can be treated as a generator6 of the line bundle $\omega_{X}(D)$ around $x$. We consider the filtration where $\dim F^1(x)=2$. Then we can write where $\Theta$ is a matrix each entry of which is a Laurent power series: Remark. Since spectral curves are contained in the surface $\operatorname{Tot}(\omega_{X}(D))$, they have planar singularities. To analyze their singularities, we trivialize the line bundle $\omega_{X}(D)$ to obtain local equations explicitly. In the case when generic spectral curves are singular, there is no obvious way to endow rank $1$ torsion-free sheaves on the (singular) spectral curve with a prescribed parabolic structure which fits well with the Higgs fields. In [50; Theorem 6], using some non-obvious commutative algebra arguments, a BNR type correspondence was established in the parabolic case. Theorem 3.13 (parabolic BNR correspondence [50; Theorem 6]). Assume that the base field $k$ is algebraically closed. For generic $a\in \mathcal{H}_P$, if $X_a$ is irreducible and reduced, then there is a one-to-one correspondence between and This theorem was first proved over algebraically closed fields (under some mild assumptions in characteristic $2$) in [50] and then was generalized to arbitrary fields in [51; Theorem 2.3.1]. First, we can define the following local data. Definition 3.14. A local parabolic Higgs bundle is a triple $(V,F^\bullet,\theta)$ satisfying the following conditions: The next step is the resolution of the singularities of the spectral curve $X_a$. By means of successive blow-ups, for generic spectral curves the ramification of the normalized spectral curve $\overline{X}_a$ over a marked point $x\in D$ is described by the conjugate partition of $\{n_i(x)\}$. In particular, for generic characteristic polynomials of parabolic Higgs bundles the local defining equations of spectral curves provide indeed the decomposition required in part (c) of Definition 3.14. Hence, locally around each marked point, generic parabolic Higgs bundles $(\mathcal{E},\theta)$ are distinguished local parabolic Higgs bundle (see Definition 3.14). Remark. The last statement in part (c) of Definition 3.14 has geometric meaning. It says that in the successive blow ups the intersection of the strict transform with the exceptional divisor is as simple as possible. This corresponds to the non-degeneracy condition in toric resolution: see [42; Subsection 3.6]. We write $A:=\mathcal{O}[\lambda]/(f)$, and $A_i:=k[[t]][\lambda]/(f_i(t,\lambda))$. Since each $f_{i}$ is an Eisenstein polynomial, each $A_{i}$ is a DVR (discrete valuation ring), and we put $\displaystyle \widetilde{A}:=\prod_{i=1}^{n_{1}}A_i$. Then we have a natural injection $A\hookrightarrow \widetilde{A}$ and $\widetilde{A}$ can be treated as the normalization of $A$. Theorem 3.15 (local parabolic BNR correspondence). A distinguished local strongly parabolic Higgs bundle $(V,F^\bullet,\theta)$ induces a principal $\widetilde{A}$-module structure on $V$. This theorem implies that parabolic Higgs bundles with generic characteristics polynomials are actually line bundles over normalized spectral curves. Moreover, given a line bundle $\mathcal{L}$ on a normalized spectral curve $\widetilde{X}_a$, there is a natural filtration on $\widetilde{\pi}_{a,*}\mathcal{L}$ which coincides with the parabolic structure we expect. Example 2. Now let us continue the analysis of Example 1. The filtration corresponds to the Young diagram Obviously, the conjugate partition produces the same Young diagram. Now we factorize the characteristic polynomial: where There are two points $x_1$ and $x_2$ on the normalization $\widetilde{X}_a$ over the marked point $x\in X$. Now consider a line bundle $\mathcal{L}$ over $\widetilde{X}_a$; it is endowed with a filtration: Here $\mathfrak{m}$ denotes maximal ideals. Hence on the pushforward $\widetilde{\pi}_{a*}\mathcal{L}$ we have a filtration satisfying $\dim F^1(x)=2$. Remark. As can be seen from the example, the ramification of normalized spectral curves fits perfectly with the filtration at marked points. This condition also holds for generic parabolic symplectic Higgs bundles. But for parabolic $\operatorname{SO}_{2r+1}$-Higgs bundles this does not hold any more. 3.4. Mirror symmetry for Hitchin systemsIn this subsection we talk about mirror symmetry centering around various Hitchin systems. 3.4.1. The geometric Langlands program and mirror symmetryThe geometric Langlands program is a long-lasting and important subject of mathematics, which underwent re-examination from a physical perspective during the last 20 years, for example, by Gukov, Kapustin, and Witten [29], [20], [21]. A physical perspective of this type implies that we can treat the geometric Langlands correspondence as a mirror symmetry for two moduli spaces of stable Higgs bundles $\mathcal{M}_{G}(X)$ and $\mathcal{M}_{{}^LG}(X)$, where $X$ is a Riemann surface. Here the reductive complex algebraic groups $G$ and ${}^LG$ are related by Langlands duality. As explained in [10], in a certain semi-classical limit Kontsevich’s [30] homological mirror symmetry conjecture [29] should take the following form:
$$
\begin{equation*}
\mathcal{D}_{\mathrm{Coh}}^b( \mathcal{M}_{G}(X)) \sim\mathcal{D}_{\mathrm{Coh}}^b( \mathcal{M}_{{}^LG}(X)),
\end{equation*}
\notag
$$
that is, there must be an equivalence of the derived categories of coherent sheaves on moduli spaces of Higgs bundles for Langlands dual groups. In fact, we consider $\mathcal{M}_{G}$ (respectively, $\mathcal{M}_{^{L}G}$) as the cotangent bundles of $\mathcal{N}_{G}$ (respectively, of $\mathcal{N}_{^{L}G}$) which are the moduli of $G$- (respectively, $^{L}G$-)bundles. The derived category of coherent sheaves on $\mathcal{M}_{G}$ and $\mathcal{M}_{^{L}G}$ can be treated as a limit of the derived category of $D$-modules on $\mathcal{N}_{G}$ and $\mathcal{N}_{^{L}G}$. Roughly speaking, we can deform the ring of differential operators on a manifold $X$ to the ring $\operatorname{Sym^{\bullet}}T_{X}$, which can also be treated as the pushforward of $\mathcal{O}_{T^*X}$ to $X$. Donagi and Pantev in [11] provided an “Abelian” version of the above equivalence over an open subset of the Hitchin base. First they showed the following. Theorem 3.16. The two Hitchin systems $\mathcal{M}_{G}(X)$ and $\mathcal{M}_{{}^LG}(X)$ enjoy the Strominger–Yau–Zaslow (SYZ) mirror symmetry, that is, generic fibres are torsors over dual Abelian varieties. Remark. There is a more delicate condition on the torsor structure, which involves certain unitary gerbes. See [25; § 1 and § 3] or [18; § 6] for an explanation in the setting of Hitchin systems. Then for an open subvariety of the Hitchin moduli spaces they proved a Fourier–Mukai type equivalence of derived categories. Theorem 3.17 [11; § 5.3]. Over an open subset of the Hitchin base, over which fibres are torsors over Abelian varieties, there is an equivalence of derived categories via the Fourier–Mukai transform: Remark. Here we omit for simplicity the discussions of the non-neutral components of Hitchin moduli spaces and of unitary gerbes. Both discussions are necessary for the verification of SYZ mirror symmetry in general cases. 3.4.2. Topological mirror symmetryNow we introduce another interpretation of the mirror symmetry from the topological standpoint. According to physicists, from two versions of string theory, type IIA and type IIB, one obtains two Calabi–Yau manifolds $X_A$ and $Y_B$.
One way to describe an abstract and ambiguous “mirror pair” is to compare their topological invariants, for example, Hodge numbers. To be more precise, given a smooth “mirror pair” of compact Calabi–Yau manifolds $(M, M^\vee)$ of same dimension, we expect that If moreover, $M$ (also $M^{\vee}$) is a compact hyper-Kähler manifold, then owing to the symmetry in the Hodge diamond, we expect that $h^{p,q}(M)=h^{p, q}(M^\vee)$.However, in many cases the mirror counterpart may not be a manifold, but rather an orbifold. And we need to use the so-called stringy Hodge numbers, which we define in what follows8. Intermezzo: Stringy Hodge numbersLet $\Gamma$ be a finite group acting generically and fixed-point freely on a smooth quasi-projective complex variety $V$ of dimension $n$. For each element $\gamma\in\Gamma$ we put $C(\gamma)$ to be its centralizer in $\Gamma$. We denote by $[V/\Gamma]$ the quotient stack (orbifold) over $\mathbb{C}$. In what follows we fix a system of primitive roots of unity $\{\xi_{m}\}_{m\in\mathbb{Z}_{>0}}$. Here for each positive integer $m$, $\xi_m$ is an $m$th primitive root of unity, and for all $m,n>0$ we have See [18; Section 1: Conventions].Definition 3.18. The stringy $E$-polynomial of $[V/\Gamma]$ is defined by where $\Gamma/{\operatorname{conj}}$ is the set of conjugacy classes of $\Gamma$ and the second summation is over the connected components of $[V^{\gamma}/C(\gamma)]$. Here $F(\gamma,Z)$ is the Fermionic shift defined as follows: let $x\in V^{\gamma}$ have an image in $Z$; then $\gamma$ acts on $T_xV$ with eigenvalues $\{\xi^{c_i}\}_{i=1,\dots,n}$, where $0\leqslant c_i<\#\langle\gamma\rangle$ for $i=1,\dots,n$ and $\#\langle\gamma\rangle$ is the order of $\gamma$. Then It is not difficult to see that $F(\gamma,x)$ is locally constant for $x\in V^{\gamma}$, hence we use the notation $F(\gamma,Z)$. We write $Z=[W/C(\gamma)]$, where $W$ is the preimage of $Z$ in $V^{\gamma}$; then Remark. In order to fit into SYZ mirror symmetry, the topological mirror symmetry is an equality between the twisted stringy Hodge numbers (or, equivalently, twisted stringy $E$-polynomials). Here, as in the remark after Theorem 3.16, we omit the discussion of how the gerbes have to be used to give the twisted stringy $E$-polynomials. For a more complete definition, see [18; § 2.4]. Now we present the results relevant for Hitchin systems. Let $G=\operatorname{SL}_n$; then ${}^L G=\operatorname{PGL}_n$ and one expects the equality of the (stringy) Hodge numbers. Theorem 3.19 [25; the conjecture], [18; the theorem], [36]. Assume that $d=\deg L$ and $d'=\deg L'$ are coprime to $n$; then Here $\mathcal{M}_{\operatorname{SL}_n, L}$ denotes the moduli space of stable $\operatorname{SL}_n$-Higgs bundles $(E,\theta)$ over $\Sigma$, where $E$ is a vector bundle of rank $n$, together with an isomorphism $\det(E) \cong L$, and $\theta \in \operatorname{H}^0(C,\operatorname{End}(E)\otimes K_\Sigma)$ is trace free. The assumption that $d$ is coprime to $n$ implies that $\mathcal{M}_{\operatorname{SL}_n, L}$ is smooth. It is known that
$$
\begin{equation*}
\mathcal{M}_{\operatorname{PGL}_n, L'} =\mathcal{M}_{\operatorname{SL}_n, L'}/\Gamma,
\end{equation*}
\notag
$$
where $\Gamma=\operatorname{Jac}(\Sigma)[n]$, the subgroup of $n$th torsion points. There is a natural $\mu_n$-gerbe9 whose local sections are liftings of universal $\operatorname{PGL}_n$ bundles to an $\operatorname{SL}_n$ bundle on $\mathcal{M}_{\operatorname{SL}_n, L'}$. By the natural embedding $\mu_n\hookrightarrow \operatorname{U}(1)$ it induces a $\operatorname{U}(1)$-gerbe on $\mathcal{M}_{\operatorname{SL}_n, L'}$, which is $\Gamma$-equivariant and defines a $\operatorname{U}(1)$-gerbe $\alpha$ on $\mathcal{M}_{\operatorname{PGL}_n,L'}$. Now, $E_{\mathrm{st}}(\mathcal{M}_{\operatorname{PGL}_n,L'},\alpha)$ is the twisted stringy $E$-polynomial. The above theorem is also called the topological mirror symmetry; it was originally proposed by Hausel and Thaddeus [25] and proved by them for $n=2$. It was proved for arbitrary $n$ by Groechenig, Wyss and Ziegler [18], and independently by Maulik and Shen [36]. Groechenig, Wyss, and Ziegler [18] first showed that an equality between twisted stringy $E$-polynomials amounts to an equality between certain $p$-adic integrations. They introduced the notion of a dual pair of abstract Hitchin systems [18; Definition 6.9]. In particular, they revealed an important relation between the $\operatorname{U}(1)$-gerbes (which actually can be induced from $\mu_n$-gerbes) and the Tate duality for Abelian varieties over $p$-adic fields. Maulik and Shen [36] used a sheaf-theoretic approach, which relates the cohomology of the orbifold locus (that is, $Z$’s in (3.2)) to the cohomology of moduli of lower-rank Higgs bundles. They showed isomorphisms between graded pieces of perverse filtrations, which was actually a refinement of the equality between twisted stringy Hodge numbers. Such kind of philosophy finally leads to the celebrated proof of the $P=W$ conjecture [8], [9], [37]. (Hausel, Mellit, Minets, and Schiffmann [24] give a different proof of the $P=W$ conjecture.) Topological mirror symmetry for parabolic Hitchin systemsNow we want to close this subsection with the topological mirror symmetry between parabolic $\operatorname{SL}\!/\!\operatorname{PGL}$ Hitchin systems. Now we work over an arbitrary field $k$ instead of an algebraically closed field. First we define a numerical invariant based on the parabolic type $(D,P)$, which will play an important role in the formulation of topological mirror symmetry for parabolic Hitchin systems. Analyzing the resolution of singular spectral curves over arbitrary fields, we can show the existence of rational points. Proposition 3.21. There is a $k$-rational point on $\operatorname{Pic}^{\Delta_P}(\widetilde{X}_a)$, where $\operatorname{Pic}^{\Delta_{P}}$ is the degree $\Delta_{P}$ connected component of the Picard variety. Remark. As mentioned before, Theorem 3.13 holds almost over an arbitrary field, and in particular over the function field of the Hitchin base. Hence this provides some information on the rational points on generic Hitchin fibres which is important for the calculation of $p$-adic integrations. Now we can state the topological mirror symmetry. Theorem 3.22 [51; Theorem 4.3.1]. The following equality10, often referred to as topological mirror symmetry, holds: where $d=\deg\mathcal{L}$ and $e=\deg\mathcal{L}'$. Here we assume that there exists a positive integer $\lambda$ such that $e\equiv \lambda d\ (\operatorname{mod}\Delta_P)$. Here $\hat{\alpha}$ and $\check{\alpha}$ are particular choices of $\mu_n$-gerbes on moduli spaces which satisfy the SYZ mirror symmetry. Remark. Notice that the codimension $2$ condition in the definition of a dual pair of abstract Hitchin systems [18; Definition 6.9] does not hold here because Theorem 3.13 holds over an open subset, whose complement is of codimension $1$. However, this can be compensated by the existence of symplectic forms on $\mathcal{M}_{\operatorname{SL}_n,\mathcal{L}}$. It is even possible to replace the canonical line bundle by other sufficiently ample line bundles over the curve (see [48]). In particular, if at one marked point, the parabolic subgroup $P$ is in fact a Borel subgroup, then $\Delta_{P}=1$, hence we can choose $d$ and $e$ arbitrarily. We can present a generalization of [17; Theorem 3.13], which proved topological mirror symmetry in the cases of rank $2$ and $3$, provided that all parabolic subgroups are Borel. Theorem 3.23. If $\Delta_P=1$, then for all integers $d$ and $e$, In particular, $\Delta_P=1$ holds if there exists a marked point $x$ such that $P_x$ is Borel, that is, a full flag filtration at $x$. In fact, Proposition 3.21 shows that there always are $k$-rational points for all generic fibres over an arbitrary field, hence also over $p$-adic fields. Then by $p$-adic integration the twists given by gerbes are actually integrations of the constant function $1$, hence there is no change of $E$-polynomials after removing the gerbes. 4. Methods of exact solutionAs a non-linear completely integrable system, a Hitchin system admits two methods of exact solution, the inverse spectral method and the classical method of separation of variables. The inverse spectral method for Hitchin systems with structure group $\operatorname{GL}_n$ is due to Krichever [34]. We present it in Subsection 4.2 below. In the course of further attempts by one of the authors some obstructions to the generalization of Krichever’s approach to Hitchin systems with simple classical structure groups were found (Subsection 4.4.1). For Hitchin systems, the method of separation of variables goes back to Hurtubise [28]. In [16] the geometry of separation of variables for Hitchin systems was discussed in more detail. Finally, in [47] we have shown that focusing on a certain class of base curves (hyperelliptic curves in our case) enables us to give a constructive definition of Hitchin systems (including the case of simple structure groups) and find Darboux and action-angle coordinates in terms of separated variables. Here we also present a method for finding theta-function solutions of $\operatorname{GL}_n$-systems and demonstrate obstructions to this in the case of simple structure groups. We begin this section with the Lax representation, due to Gawedzki and Tran-Ngoc-Bich [13], for the $\operatorname{SL}_2$-Hitchin system as it is described in [15] (see Subsection 2.3 of the present paper). It is the historically first Lax representation for a hierarchy of Hitchin flows. 4.1. Lax matrix approach for $\operatorname{SL}_2$-Hitchin systems on genus $2$ curvesIn [13], Gawedzki and Tran-Ngoc-Bich proved that the matrices $(x_{ij})$ (2.4) (which is skew-symmetric by construction) satisfy the following $\operatorname{\mathfrak{so}}_6$-commutation relations:
$$
\begin{equation*}
\begin{alignedat}{2} & \{x_{nm},x_{mp}\}=-x_{np}&\quad &\text{for different}\ n,\ m,\ p, \\ & \{x_{nm},x_{pq}\}=0&\quad &\text{for different}\ n,\ m,\ p,\ q, \end{alignedat}
\end{equation*}
\notag
$$
which imply directly the commutativity of the Hamiltonians (2.4). They invented the following Lax matrix with spectral parameter $\zeta$: and they proved that the quantities $\operatorname{Tr} L(\zeta)^l$ are generating functions for the system of Hamiltonians equivalent to (2.4). Let $\partial_l(\zeta)$ be the generating function for the corresponding time derivatives, that is,
$$
\begin{equation*}
\partial_l(\zeta)(\cdot) =\{ \operatorname{Tr} L(\zeta)^l, \cdot \}.
\end{equation*}
\notag
$$
Proposition 4.1 [13]. The hierarchy of commuting flows given by the Hamiltonians (2.4) is equivalent to the hierarchy given by Lax’s equations where The main result of [13] is in finding the action-angle coordinates for the $\operatorname{SL}_2$-Hitchin system on a genus $2$ curve. The technique developed in [13] certainly enables one to produce a theta-function formula for Hitchin trajectories. By means of another Lax representation this was done by Krichever for $\operatorname{GL}_n$- and $\operatorname{SL}_n$-Hitchin systems of arbitrary rank and genus, as it is described in Subsection 4.2. As for the action–angle variables, they have been found for all simple classical structure groups and hyperelliptic curves of arbitrary genus, as it is described in Subsection 4.3. 4.2. Inverse spectral method4.2.1. The Lax operatorIn the case $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n)$ we define the Hitchin system by its Lax matrix $L$ which is an $n\times n$ matrix-valued meromorphic function on $\Sigma$, depending on parameters called times. It is assumed that $L$ has a stationary (independent of times) pole divisor $\displaystyle D_{\mathrm{st}}=\sum_{P\in\Sigma}m_PP$ ($m_P\geqslant 0$; $m_P=0$ except on a finite subset of $\Sigma$), and $ng$ simple poles $\{\gamma_s\mid s=1,\dots,ng\}$ depending on times. Below $D_{\mathrm{st}}$ is assumed to be the divisor of a holomorphic differential $\xi$ on $\Sigma$ (this property distinguishes Hitchin systems in the class of all integrable systems with spectral parameter on a Riemann surface). In a neighbourhood of a point $\gamma_s$, for every $s$, $L$ is assumed to have an expansion of the form
$$
\begin{equation}
L(z) =\frac{\beta_s\alpha_s^{T} }{z-z_s}+L_{0,s}+O(z-z_s),
\end{equation}
\tag{4.1}
$$
where $\alpha_s$ and $\beta_s$ are $n\times 1$ matrices, $\alpha_s^{T} \alpha_s=0$, $\beta_s^{T} \alpha_s=0$, there exist $\varkappa_s\in\mathbb{C}$ such that $\alpha^{T}L_{0,s}=\alpha^{T}\varkappa_s$, the symbol $^{T} $ denotes transposition, $z$ is a local coordinate in a neighbourhood of $\gamma_s$, and $z_s=z(\gamma_s)$. The elements $\{\gamma_s\mid s=1,\dots,ng\}$ and $\{\alpha_s\mid s=1,\dots,ng\}$ are referred to as the Tyurin parameters. They arise in the following way from the parametrization of semistable holomorphic vector bundles due to Tyurin [53], [54]. By the Riemann–Roch theorem a semistable holomorphic rank $n$ degree $ng$ vector bundle has an $n$-dimensional space of holomorphic sections. Choose a basis in this space. The corresponding sections generically have $ng$ points of linear dependence (these are $\{\gamma_s\mid s=1,\dots,ng\}$), each of which is associated with a set of coefficients of linear dependence, which we organize in the matrices $\alpha_s$. Let $\Phi$ be a Higgs field. Then $\Phi/\xi$ is an endomorphism of the space of meromorphic sections of the bundle. We claim that $L$ is the matrix of the endomorphism $\Phi/\xi$ in the above base [34]. 4.2.2. The hierarchy of Lax’s equationsOur next step is to describe the Hitchin flows by Lax’s equations of the form $\dot L=[M,L]$. Here $M$ is a meromorphic matrix-valued function on $\Sigma$ assumed to have poles at the same points as $L$, with expansions of the form
$$
\begin{equation*}
M(z) =\frac{\mu_s\alpha_s^{T} }{z-z_s}+M_{0,s}+O(z-z_s)
\end{equation*}
\notag
$$
at the points $\gamma_s$, and with no additional relations between $\alpha_s$, $\mu_s$, and $M_{0,s}$. Plugging the expansions for $L$ and $M$ into Lax’s equation we obtain the following important evolution equations of Tyurin parameters:
$$
\begin{equation}
\dot z_s =-\mu_s^{T} \alpha_s, \qquad \dot\alpha_s=-M_{0,s}\alpha_s+\kappa_s\alpha_s
\end{equation}
\tag{4.2}
$$
where $\kappa_s\in\mathbb{C}$ (observe that signs on the right-hand side of the last relation are invariant with respect to the simultaneous change of order of $\alpha_s$ and $\beta_s$ in (4.1), and of $L$ and $M$ in Lax’s equation). According to [34], every $M$-operator is assigned to a certain time, and defines the corresponding flow. The times are enumerated by triples
$$
\begin{equation*}
a=(P,m,k),\qquad P\in\operatorname{supp}(D_{\mathrm{st}}),\quad m>-m_P,\quad k\in\mathbb{Z}_+.
\end{equation*}
\notag
$$
The following theorem states the existence and commutativity of those flows. Theorem 4.2 [34]. $1^\circ$. For every triple $a=(P,m,k)$ of the above form there exists a unique $M$-operator $M_a$ such that it has only one pole $P$ in $\operatorname{supp}(D_{\mathrm{st}})$, $M_a(w)=w^{-m}L^k(w)+O(1)$ ($w$ is a local coordinate in a neighbourhood of $P$), and $M_a(P_0)=0$ ($P_0$ being some point in $\Sigma$, the same for all $a$). $2^\circ$. The equations $\partial_aL=[M_a,L]$ produce a family (called a hierarchy) of commuting flows. A symplectic theory of the systems in question was developed in [34], which implies, in particular, that the flows in Theorem 4.2 correspond exactly to the Hitchin Hamiltonians. 4.2.3. Spectral transformConsider the eigenvalue problem
$$
\begin{equation}
(L(q,t)-\lambda)\psi(Q,t)=0,\qquad q\in\Sigma,\quad Q=(q,\lambda).
\end{equation}
\tag{4.3}
$$
We define the spectral curve of $L$ by the equation and denote it by $\widehat{\Sigma}$. Let $\widehat{g}$ be its genus and $\pi\colon \widehat{\Sigma}\to\Sigma$ be the natural projection. We normalize $\psi$ by the condition Then $\psi(Q)$ becomes a meromorphic function on $\widehat\Sigma$. Its pole divisor plays a special role and is referred to as the dynamical divisor of the problem. It depends on time. We denote it by $D(t)$. It follows from the equation $\partial_aL=[M_a,L]$ that, if $\psi$ is an eigenvector of $L$, then $(\partial_a -M_a)\psi$ is also an eigenvector. Therefore, where $f_a(Q,t)$ is a scalar meromorphic function on $\Sigma$. The vector function
$$
\begin{equation}
\widehat{\psi}(Q,t)=\phi(Q,t)\psi(Q,t), \qquad \phi(Q,t)=\exp \biggl(-\int_{0}^{t_a}f_a(Q,\tau)\,d\tau\biggr),
\end{equation}
\tag{4.6}
$$
satisfies the equations
$$
\begin{equation*}
L(q,t)\widehat{\psi}(Q,t)=\lambda\widehat{\psi}(Q,t), \qquad (\partial_a-M_a(q, t))\widehat{\psi}(Q,t)=0.
\end{equation*}
\notag
$$
It turns out that under the gauge transform (4.6) the pole divisor $D(t)$ is transformed into the time-independent divisor $D=D(0)$ of poles of $\widehat{\psi}$ (see the explanation in the end of Subsection 4.4.1). All the time dependence of $\widehat\psi$ is encoded in the form of its essential singularities, which it acquires at the constant poles of $f_a$. Theorem 4.3 [34]. The function $\widehat{\psi}$ possesses the following properties. $1^\circ$. It is meromorphic except at the preimage of the divisor $(\xi)$. The degree of its divisor of poles is equal to $\widehat{g}+n-1$. $2^\circ$. Let $P\in\operatorname{supp}(D_{\mathrm{st}})$. Then $\widehat{\psi}$ has the following expansion in a neighbourhood of $P^l$:
$$
\begin{equation}
\widehat{\psi}_l(w,t) =\xi_l(w,t) \exp\biggl(\,\sum_{n=1}^\infty\sum_{m\geqslant -m_P} t_{P,m,n}w^{-m}\lambda_l(w)^n\biggr)
\end{equation}
\tag{4.7}
$$
where $w$ is a local parameter on $\widehat{\Sigma}$ in a neighbourhood of $P^l$ (generically, it is independent of $l$), $\xi_l(w,t)$ is a (vector-valued) Taylor series in $w$, $\lambda_l(w)$ is the characteristic root of $L(P(w),t)$ corresponding to the $l$th covering sheet. $3^\circ$. In a neighbourhood of $P_0^l$ the vector function $\psi$ has an expansion of the form where $w$ is a local parameter in a neighbourhood of $P_0^l$ and $\xi_{0,l}(w,t)$ is a (vector-valued) Taylor series in $w$ such that $\xi_{0,l}^i\big|_{w=0}=\delta^i_l$.The function $\widehat{\psi}$ possessing the above properties is called a Baker–Akhieser function associated with the divisor $D$. 4.2.4. A theta-function solution by the inverse spectral methodIn the rest of this section we follow the standard prescriptions of the algebraic-geometric version, due to Krichever [31], [32], of the inverse spectral method, as specified for Hitchin systems in [34]. For technical reasons we enumerate the sheets of the covering $\pi$ in an arbitrary way; what follows does not depend on this enumeration. For a function $\widehat{\psi}$ on $\widehat{\Sigma}$ we denote its restriction to the $l$th sheet by $\widehat{\psi}_l$. Similarly, for $P\in\Sigma$ let $P^l$ denote its preimage on the $l$th sheet. Theorem 4.4 [34]. Given a Baker–Akhieser function $\widehat{\psi}$ satisfying the conditions of Theorem 4.3, for every $a$ there exist unique matrices $L$ and $M_a$ such that where $\partial_a=\partial/\partial t_a$, $L$ satisfies the definition of the Lax matrix, $M_a$ is an $M$-operator, and the equation $\partial_aL=[L,M_a]$ is fulfilled. Theorem 4.4 enables us to express $L$ and $M_a$ explicitly in terms of $\widehat\psi$. Take an open set $U\subset\Sigma$ outside the branch divisor. Then $\pi^{-1}U=\bigcup\limits_{l=1}^n U_l$, where $U_l$ belongs to the $l$th sheet of the covering, and $z$ can serve as a local parameter in $U_l$ for all $l=1,\dots,n$. Let $\widehat\psi_l=\widehat\psi\big|_{U_l}$, and let $\Psi$ be the matrix formed by the vectors $\widehat\psi_l$ ($l=1,\dots,n$) as columns. This $\Psi$ depends on the order of the sheets of the covering; however, the final result does not depend on this ambiguity. The first relation in (4.9) can be written down as $L\Psi=\Psi\Lambda$ where $\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$ is the spectrum of $L$. Similarly, $(\partial_{P,m,n}-M_{P,m,n})\Psi=0$. Finally, we have
$$
\begin{equation}
L=\Psi\Lambda\Psi^{-1}, \quad M_{P,m,n}=-\partial_{P,m,n}\Psi\cdot\Psi^{-1}.
\end{equation}
\tag{4.10}
$$
It is already conventional that the Baker–Akhieser functions can be expressed in terms of theta functions [31]–[33]. In this way we can explicitly express $L$ and $M$ in terms of the spectral curve and the divisor $D$ on it, by means of theta-functions. 4.3. Separation of variables for Hitchin systems with classical structure groups4.3.1. Spectral curveTo define a spectral curve, first we choose and fix a holomorphic $1$-form $\xi$ on $\Sigma$. Given a Higgs field $\Phi$, we define its spectral curve by the equation
$$
\begin{equation*}
\det\biggl(\lambda-\frac{\Phi}{\xi}\biggr) =\lambda^d+\sum_{j=1}^n \lambda^{d-d_j} r_j =0
\end{equation*}
\notag
$$
where $d$ is the rank of the bundle and $n$ is the rank of the Lie algebra $\mathfrak{g}$. Let $\rho_j$, $j=1,\dots,n$, denote the basis invariant polynomials of the Lie algebra $\mathfrak{g}$, and let $\delta_j=\deg \rho_j$. For $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n)$ or $\mathfrak{g}$ a classical simple Lie algebra, we have $r_j=\rho_j(\Phi/\xi)$ and $d_j=\delta_j$, except for $\mathfrak{g}={\mathfrak so}(2n)$ and $j=n$, when $r_{2n}=\rho_n(\Phi/\xi)^2$ and $d_n=2\delta_n$. Indeed, in that case $r_n=\det(\Phi/\xi)$ and $\rho_n(\cdot)=\operatorname{Pf}(\cdot)$. For any divisor $D$ let $\mathcal{O}(D)$ denote the space of meromorphic functions $f$ on the spectral curve such that $(f)+D\geqslant 0$. Below fix $D$ to be the divisor of the differential $\xi$, that is, $D=(\xi)$. Proposition 4.5. The invariant polynomials satisfy $\rho_j\!\in \mathcal{O}(\delta_jD)$ ($j=1,\dots,n$). We also introduce the following shorthand notation for the characteristic polynomial of $\Phi/\xi$:
$$
\begin{equation}
R(\lambda,P) =\lambda^d+\sum_{j=1}^n \lambda^{d-d_j} r_j(P), \qquad P\in\Sigma.
\end{equation}
\tag{4.11}
$$
Proposition 4.6. $[\Phi ]\to R$ is a one-to-one correspondence between the classes of gauge equivalence of Higgs fields and their characteristic polynomials. Proof. This correspondence is obviously a continuous injection. We prove that the dimension of its image is equal to the dimension of the class of characteristic polynomials. Indeed, in case $\mathfrak{g}$ is simple, $\dim\{[\Phi]\}=\dim\mathfrak{g}\cdot(g-1)$ [26]. On the other hand $\displaystyle \dim\{R\}=\sum_{j=1}^n\dim \mathcal{O}(\delta_jD)$. By the Riemann–Roch theorem $\dim\mathcal{O}(\delta_jD)=(2\delta_j-1)(g-1)$, and by the well-known Kostant identity, $\displaystyle \sum_{j=1}^n(2\delta_j-1)=\dim\mathfrak{g}$. For $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n)$, similarly, Hence our claim is true. The proposition is proved. Given a specific class of base curves, Propositions 4.5 and 4.6 provide a description of the family of spectral curves corresponding to all possible Higgs fields $\Phi$. In [47], [7] this was done for hyperelliptic base curves as follows. Let $\Sigma\colon y^2=P_{2g+1}(x)$, and let $\displaystyle \xi=\frac{dx}{y}$ (then $D=2(g-1)\cdot\infty$). Proposition 4.7 [47]. A basis in $\mathcal{O}(\delta_jD)$ is formed by the functions Proposition 4.8 [47]. (The affine part of) the spectral curve for a Hitchin system of types $A_n$, $B_n$, $C_n$ is a complete intersection of two surfaces in $\mathbb{C}^3$, $y^2=P_{2g+1}(x)$ and $R(\lambda,x,y,H)=0$, where and $H_{jk}^{(0)},H_{js}^{(1)}\in\mathbb{C}$. For $\mathfrak{g}=\operatorname{\mathfrak{so}}(2n)$ the bracket in the last summand ($j=n$) must be squared. 4.3.2. HamiltoniansHere we define the Hamiltonians of a Hitchin system. The above coefficients $H_{jk}^{(0)}$ and $H_{js}^{(1)}$ are obviously functions of $\Phi$. These functions are referred to as Hamiltonians of the hyperelliptic Hitchin system. In the case of an arbitrary base curve we choose a basis in $\mathcal{O}(\delta_jD)$ and expand $\rho_j(\Phi/\xi)$ with respect to this basis. We call coefficients of this expansion, as functions of $\Phi$, Hitchin Hamiltonians. The original procedure due to Hitchin prescribes to choose a basis in $H^0(\Sigma,\mathcal{K}^{\delta_j})$, rather than in $\mathcal{O}(\delta_jD)$. However, these two procedures are equivalent because $H^0(\Sigma,\mathcal{K}^{\delta_j})\simeq \mathcal{O}(\delta_jD)\xi^{\otimes \delta_j}$ for all $j$. 4.3.3. Separation of variablesEvery curve can be given by the set of points incident to it. We will show that these points serve as separating coordinates of Hitchin systems. The idea goes back to [16], [3]. However, it was realized in full only in [46], [47] by focusing on a certain class of base curves. Let $h=\dim\{ [\Phi] \}$; $h$ is the same as the number of degrees of freedom of the Hitchin system. The number of coefficients in the equation of the spectral curve is also equal to $h$, as proved in the course of the proof of 4.6. Hence it is sufficient to take $h$ points on the spectral curve to express the coefficients in terms of them. We denote these points by $\gamma_i=(x_i, y_i, \lambda_i)$, $i=1,\dots,h$, where $y_i^2=P_{2g+1}(x_i)$ for all $i$. Let $H$ be the full set of coefficients in (4.12). Then $H$ can be expressed in terms of $(x_i, y_i, \lambda_i)$, $i=1,\dots, h$, from the equations where every equation contains only one triple $(x_i, y_i, \lambda_i)$. A system of equations satisfying the last requirement is referred to as a system of separating equations, and $(x_i, y_i, \lambda_i)$, $i=1,\dots, h$, are separating variables.We stress that for $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n), \operatorname{\mathfrak{sl}}(n), \operatorname{\mathfrak{sp}}(2n), \operatorname{\mathfrak{so}}(2n+1)$ the system of equations (4.13) is linear in $H$ (cf. (4.12)), while in the case $\mathfrak{g}=\operatorname{\mathfrak{so}}(2n)$ it is quadratic. Nevertheless, for $n=2$ it can effectively be solved in the last case (Borisova [7]). 4.3.4. The symplectic form and Poisson structureHere we define the symplectic form of a Hitchin system, and express it in terms of separating coordinates. In [16] the action-angle variables $(I,\alpha)$ were defined by
$$
\begin{equation*}
\lambda dz=\sum_{a=1}^{h} I^a\Omega_a, \qquad \alpha_a=\sum_{i=1}^h \int^{\gamma_i}\Omega_a,
\end{equation*}
\notag
$$
where $z=\displaystyle\int^{\gamma}\widehat{\xi}$ is a (quasi-global) coordinate on $\widehat{\Sigma}$ ($\widehat{\xi}$ is the pullback of $\xi$ to $\widehat{\Sigma}$), the $\Omega_a$ are the normalized Abel/Prym differentials, $h=\dim\mathfrak{g}\cdot(g-1)$ for $\mathfrak{g}=\operatorname{\mathfrak{so}}(n),\operatorname{\mathfrak{sp}}(2n)$, and $h=n^2(g-1)+1$ for $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n)$. For the symplectic form it follows (by definition and by variation of the above integrals in $\gamma_i$) that
$$
\begin{equation*}
\omega=\sum_{a=1}^h \delta I^a\wedge\delta\alpha_a=\sum_{a=1}^h \delta I^a\wedge \sum_{i=1}^h\Omega_a(\gamma_i)
\end{equation*}
\notag
$$
($\delta$ means variation in $\{\gamma_i\}$). Also,
$$
\begin{equation*}
\begin{aligned} \, \lambda dz=\sum_{a=1}^{h} I^a\Omega_a\ & \Longrightarrow\ \lambda_i\delta z_i=\sum_{a=1}^{h} I^a\Omega_a(\gamma_i) \\ & \Longrightarrow\ \delta\lambda_i\wedge\delta z_i=\sum_{a=1}^{h} \delta I^a\wedge \Omega_a(\gamma_i)\quad \forall\,i \end{aligned}
\end{equation*}
\notag
$$
where $z_i=\displaystyle\int^{\gamma_i}\widehat{\xi}$. For hyperelliptic curves we have $dz=\widehat{\xi}$ and $\xi=\dfrac{dx}{y}$, and it follows that
$$
\begin{equation}
\omega =\sum_{i=1}^h \delta\lambda_i\wedge \delta z_i =\sum_{i=1}^h \delta\lambda_i\wedge \frac{\delta x_i}{y_i}.
\end{equation}
\tag{4.14}
$$
Hence the Poisson structure is given by setting and all other brackets vanish. 4.3.5. IntegrabilityThe Hamiltonians defined by the system of equations (4.13), and the Poisson bracket (4.15) produce a Hitchin system in full in terms of the set of points $\{(x_i, y_i, \lambda_i) \mid i=1,\dots, h \}$. It is our next goal to prove the integrability of Hitchin systems in these terms. More generally, consider a system of equations
$$
\begin{equation}
F_i(H_1,\dots,H_h,\lambda_i,x_i)=0,\qquad i=1,\dots,h,
\end{equation}
\tag{4.16}
$$
where the $F_i$, $i=1,\dots,h$, are smooth functions, and the $H_j$, $j=1,\dots,h$, are defined by relations (4.16) as functions of $(\lambda_1,\dots,\lambda_h,x_1,\dots,x_h)$. It is important that for each $i=1,\dots,h$ the function $F_i$ depends explicitly on only one pair of variables $\lambda_i$, $x_i$ (however, it can depend on the rest of the variables via $H_1,\dots,H_h$). Consider a Poisson bracket on $\mathbb{C}^{2n}$ of the form
$$
\begin{equation*}
\{f,g\} =\sum\limits_{j=1}^np_j \biggl( \frac{\partial f}{\partial \lambda_j}\,\frac{\partial g}{\partial x_j} -\frac{\partial g}{\partial \lambda_j}\,\frac{\partial f}{\partial x_j} \biggr)
\end{equation*}
\notag
$$
where the $p_j=p_j(\lambda_j,x_j)$ are smooth functions in only one pair of variables, and
$$
\begin{equation*}
p_k=0 \quad \text{if}\ \ \operatorname{rank}\frac{\partial(F_1,\dots,F_{k-1},F_{k+1},\dots,F_h)} {\partial(H_1,\dots,H_{k-1},H_k,H_{k+1},\dots,H_h)}<n-1.
\end{equation*}
\notag
$$
Proposition 4.9 [46]. The functions $H_1,\dots,H_h$ commute with respect to each Poisson bracket of the above form. The integrability of a Hitchin system is directly implied by 4.9 because the Hitchin Hamiltonians and the Poisson structure (4.15) are precisely of the required form (where $F_j=R$ and $p_j=y_j$ for all $j$). 4.3.6. The method of generating functions and Darboux coordinatesOur next goal is to find the coordinates $\phi_j$, $j=1,\dots,h$, conjugate to the Hamiltonians. These are coordinates in a generic fiber of the Hitchin fibration. They can be found in a standard way by means of the generating functions technique [1]. In the framework of separation of variables a generating function is a function of the form $\displaystyle S=\sum\limits_{i=1}^h S_i$ where $S_i=S_i(z_i,H)$, $z_i=\displaystyle\int^{\gamma_i}\widehat{\xi}$, as in Subsection 4.3.4. Observe that the $(\lambda_i,z_i)$ are the canonical coordinates from (4.14). Each $S_i$ is defined by the relation
$$
\begin{equation*}
R(x_i,y_i,s_i,H)=0, \quad\text{where}\ \ s_i(z_i,H)=\frac{\partial S_i}{\partial z_i};
\end{equation*}
\notag
$$
then $\displaystyle \frac{\partial R}{\partial\lambda} \frac{\partial s_i}{\partial H_j}+\frac{\partial R}{\partial H_j}=0, $ which implies that
$$
\begin{equation*}
\frac{ \partial s_i}{\partial H_j}=-\frac{{\partial R}/\partial H_j}{{\partial R}/\partial\lambda }.
\end{equation*}
\notag
$$
By the above notation $S_i=\displaystyle\int^{\gamma_i}s_i(z_,H)\,dz$. We can write this in the form $S_i=\displaystyle\int^{\gamma_i}s_i\,\frac{dx}{y}$, thus by $\dfrac{dx}{y}$ we actually mean here the pull-back of this differential to $\widehat{\Sigma}$, that is, $\widehat{\xi}$. By a general statement of the generating function technique, the symplectically dual coordinates to $H_j$ are that is, $\{ H_j,\,\phi_j\mid j=1,\dots,h\}$ are Darboux coordinates for the Hitchin system. In the framework of separation of variables
$$
\begin{equation*}
\frac{\partial S}{\partial H_j} =\sum_i \frac{\partial S_i}{\partial H_j} =\sum_i \int^{\gamma_i}\frac{\partial s_i}{\partial H_j}\,\frac{dx}{y},
\end{equation*}
\notag
$$
and, finally,
$$
\begin{equation}
\phi_j =-\sum_{i=1}^{h} \int^{\gamma_i} \frac{{\partial R}/\partial H_j}{{\partial R}/\partial\lambda} \,\frac{dx}{y}.
\end{equation}
\tag{4.17}
$$
Some details of the calculation go back to [28], for the other details we refer to [47]; see also [34; formula (4.61)]. In the case when the equation of a spectral curve is linear in the $H$-coordinates, the Darboux property follows also from the results of [3], [52]. For the root system $D_n$ these results do not work. Proposition 4.10 [7]. In the case of Hitchin systems with the structure group $\operatorname{GL}(n)$ the differentials $\displaystyle \frac{{\partial R}/\partial H_j}{{\partial R}/\partial\lambda}\,\frac{dx}{y}$ ($j=1,\dots,h$) form a basis of holomorphic differentials on the spectral curve. For systems with structure groups $\operatorname{SL}(2)$, $\operatorname{SO}(2n+ 1)$, and $\operatorname{Sp}(2n)$ they form a basis of holomorphic Prym differentials on the spectral curve with respect to the involution $\lambda\mapsto-\lambda$, while for the systems with structure group $\operatorname{SO}(2n)$ they form a basis of holomorphic differentials on the normalization of the spectral curve. In the case of the structure group $\operatorname{SL}(n)$ the basis of holomorphic differentials on the spectral curve consists of the differentials $\displaystyle \frac{{\partial R}/\partial H_j}{{\partial R}/\partial\lambda }\,\frac{dx}{y}$, and of $g$ basis holomorphic differentials pulled back from the base curve. 4.3.7. The theta-function solutionIn the coordinates $(H,\phi)$ the Hamiltonian equations read $\dot{H}=0$, $\dot\phi=H$, hence the trajectories of the system are straight lines, that is, straight windings on fibres of the Hitchin map. We address here the problem of finding trajectories in the coordinates $\{(x_i,y_i,\lambda_i)\}$ representing $\operatorname{Sym^h}\Sigma$. In the case when $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n)$ the transformation $\gamma\mapsto\phi$ given by (4.17) coincides with the Abel transform up to a constant linear transformation coming from the transition from the integrands in (4.17) to normalized holomorphic differentials (observe that $h=n^2(g-1)+1=\operatorname{genus} (\widehat{\Sigma})$ in this case). This difference does not matter in this section. We ignore it and regard (4.17) as the Abel transform. Then it is a natural idea to transfer the above straight windings to $\operatorname{Sym^h}\Sigma$ by means of the Jacobi inverse map. Looking ahead (Subsection 4.3.2), we notice that for other classical Lie algebras there exists a certain obstruction to such approach. The setting of the Jacobi inversion problem is as follows: given a point $\phi$ of the Jacobian of a genus $h$ curve, find points $\gamma_1,\dots,\gamma_h$ on the curve such that $\mathcal{A}(\gamma_1+\cdots+\gamma_h)=\phi$, where $\mathcal{A}$ is the Abel transform. We refer to $\mathcal{A}^{-1}$ as the Jacobi inverse map. In principle, the solution of this problem is given by Riemann’s theorem: $\gamma_1,\dots,\gamma_h$ are the zeros of the function $\theta(\mathcal{A}(\gamma)-\phi-K)$, where $K$ is the vector of Riemann constants, provided that this function does not vanish identically. In [12] Dubrovin stated the problem of finding the points $\gamma_1,\dots,\gamma_h$ analytically. For this purpose he proposed, first, to consider a meromorphic function $f$ on the curve. Then $\displaystyle \sigma_f=\sum_{j=1}^{h}f(\gamma_j)$ is a symmetric function on the curve, hence a meromorphic function of $\phi$, that is, an Abelian function. As such, it can be expressed in terms of the Riemann theta function. Assume that the curve is represented as a covering of $\mathbb{C}P^1$. Then for the functions $f_k(P)=z^k$ ($z=\pi(P)$, where $\pi$ is the covering map, $k=1,\dots,h$) the required expressions can be written out explicitly. Note that $\sigma_{f_k}$ (or just $\sigma_k$ for short) is a Newton polynomial in $z_1,\dots,z_h$, hence the values of $\sigma_1,\dots,\sigma_h$ determine the set $z_1,\dots,z_h$ up to a permutation. Here we realize this program for the spectral curve and the covering map $\pi\colon P\mapsto x$, $P=(x,y,\lambda)$. As a result, we will obtain the set $x_1,\dots, x_h$ up to a permutation, which gives a point of $\operatorname{Sym^h}\widehat{\Sigma}$ locally, that is, we will locally find the trajectories. We note that $\pi$, as a function on the spectral curve, has poles over infinity only, and we start from the following relation due to Dubrovin [12]:
$$
\begin{equation}
\sigma_k(\phi) =\mathrm{const} -\sum_{Q\in\pi^{-1}(\infty)} \operatorname{res}_Q x^k\,d\ln\theta(\mathcal{A}(P)-\phi-K).
\end{equation}
\tag{4.18}
$$
Since $(d\mathcal{A})_i=\omega_i$, we have
$$
\begin{equation*}
d\ln\theta(\mathcal{A}(P)-\phi-K)=\sum_{i=1}^h (\partial_i\ln\theta(\mathcal{A}(P)-\phi-K))\omega_i
\end{equation*}
\notag
$$
where $\{\omega_i\}$ is the normalized basis $\displaystyle\biggl\{\frac{{\partial R}/\partial H_j}{{\partial R}/\partial\lambda }\,\frac{dx}{y}\biggr\}$ and $\partial_i$ means a derivative with respect to the $i$th argument. Choose an arbitrary point ${Q_0\in\pi^{-1}(\infty)}$ as a base point of the Abel transform. In a neighbourhood of $Q_0$ we can regard $\mathcal{A}(P)$ as a small quantity and expand $(\ln\theta(\mathcal{A}(P)-\phi-K))_i$ in a Taylor series. Then we only have to select the terms of order $z^{2k-1}$ in the expansion thus obtained, where $z$ is a local parameter in a neighbourhood of $Q_0$. Indeed, multiplied by $x^k=z^{-2k}$, it gives us the residue at $Q=Q_0$ in (4.18):
$$
\begin{equation*}
\operatorname{res}_{Q_0} x^k\,d\ln\theta(\mathcal{A}(P)-\phi-K) =\sum_{i=1}^h\sum_{|j|\leqslant 2k-1} \varkappa_i^jD^j\partial_i\ln\theta(-\phi-K)
\end{equation*}
\notag
$$
where $j=(j_1,\dots,j_h)$, $|j|=j_1+\cdots+j_h$,
$$
\begin{equation*}
\begin{aligned} \, D^j &=\frac{1}{j_1!\cdots j_h!} \, \frac{\partial^{|j|}}{\partial\phi_1^{j_1}\cdots\partial\phi_h^{j_h}}, \\ \varkappa_i^j &=\sum_{l_i+\sum_{s=1}^h\sum_{p=1}^{j_s} l_{sp}=2k-1} \varphi_i^{(l_i)}\prod_{s=1}^h\ \prod_{p=1}^{j_s}\frac{\varphi_s^{(l_{sp}-1)}}{l_{sp}}, \end{aligned}
\end{equation*}
\notag
$$
and $l_s$ and $\varphi_s^{(l_s)}$ are defined by the relation
$$
\begin{equation*}
\mathcal{A}_s(P)=\sum_{l_s\geqslant 1}\frac{\varphi_s^{(l_s)}}{l_s}\,z^{l_s} \qquad (P=P(z)).
\end{equation*}
\notag
$$
The computation of the contribution of an arbitrary point $Q\in\pi^{-1}(\infty)$ is similar, except that we take the Taylor expansions in the small parameter $\mathcal{A}(P)-\mathcal{A}(Q)$, which results in the shift by $\mathcal{A}(Q)$ of the argument of the theta function. Finally, we obtain
$$
\begin{equation}
\sigma_k(\phi) =c-\sum_{Q\in\pi^{-1}(\infty)} \sum_{i=1}^h \sum_{1\leqslant |j|\leqslant 2k-1} \varkappa_i^jD^j\partial_i\ln\theta(\mathcal{A}(Q)-\phi-\Delta).
\end{equation}
\tag{4.19}
$$
The functions $\sigma_k(\phi)$, $k=1,\dots,h$, provide a full set of symmetric functions of the $x$-coordinates of points in $\mathcal{A}^{-1}(\phi)$. They determine $x_1,\dots,x_h$ up to a permutation. By plugging $\phi=Ht+\phi_0$ we obtain a full set of symmetric functions of $x_1,\dots,x_h$ along trajectories. 4.4. Obstructions in the case of simple groups4.4.1. An obstruction to the inverse spectral methodThe analogue of Theorem 4.2 in the case of classical simple Lie algebra $\mathfrak{g}$, except for $\mathfrak{g}=\operatorname{\mathfrak{sp}}(2n)$, was proved in [44] (see also [45]). As soon as the existence of a commuting hierarchy is established, the problem arises of solving this hierarchy by means of the inverse spectral method, similarly to Subsection 4.2. However, there is an obstruction on this way, which we demonstrate in the case when $\mathfrak{g}$ is the Lie algebra of an orthogonal group. In this case,
$$
\begin{equation*}
L_s(z) =\frac{\beta_s\alpha_s^{T} -\alpha_s\beta_s^{T} }{z-z_s}+L_{0s}+\cdots\,.
\end{equation*}
\notag
$$
To obtain the (global) function $\psi$ on $\widehat\Sigma$ whose values are eigenvectors of $L$ at every point, first we seek eigenvectors either pointwise, or locally, and then apply normalization (for instance, (4.5)). By [45; Lemma 5.12], in a neighbourhood of $\gamma_s$ there is an eigenvector-valued function $\psi_s$ of the form
$$
\begin{equation*}
\psi_s(z,\lambda)=\frac{a_{s,\lambda}}{z-z_{s,\lambda}}+O(1) \quad\text{where}\ \ a_{s,\lambda}=\nu_{s,\lambda}\alpha_s,\ \ \nu_{s,\lambda}\in\mathbb{C}.
\end{equation*}
\notag
$$
Away from the branch points, $z-z_s$ can serve as a local parameter on all sheets of the covering over a neighbourhood of $\gamma_s$, hence we can suppress $\lambda$ in the notation of the Laurent expansions in $z-z_{s,\lambda}$. For brevity we also suppress $\lambda$ in the notation for $\psi(z,\lambda)$ and $a_{s,\lambda}$. Using this notation, the above expression reads A further normalization can change the order of the function at $\gamma_s$, but the coefficient $a_s$ of the leading term is preserved up to scaling. Note that after a normalization $\psi$ does not need the index $s$ either, so that in a neighbourhood of $\gamma_s$ we obtain for some $m\in\mathbb{Z}$. It is an important property proved in [34] (see [45] for a proof in the simple case) that an eigenvalue of $L$ is holomorphic as a function of $z$ at the points $\gamma_s$. For this reason the equation $L\psi=\lambda\psi$ implies that (otherwise the left- and right-hand sides of the equation have different orders), hence
$$
\begin{equation*}
\beta_s(\alpha_s^{T} a_s)-\alpha_s(\beta_s^{T} a_s)=0.
\end{equation*}
\notag
$$
In a generic position $\alpha_s$ and $\beta_s$ are linearly independent. Hence Next, we write down the identity $(\partial_a-M_a)\psi=f_a\psi$ in the form
$$
\begin{equation}
\biggl( \partial_a -\frac{\mu_s\alpha_s^{T} -\alpha_s\mu_s^{T} }{z-z_s}+M_{0s}+\cdots \biggr) (a_s(z-z_s)^m+\cdots)=f_a(a_s(z-z_s)^m+\cdots)
\end{equation}
\tag{4.20}
$$
where dots denote higher-order terms. On the left-hand side the leading term is equal to
$$
\begin{equation*}
a_s\partial_a(z-z_s)^m+\frac{\alpha_s\mu_s^{T} a_s }{z-z_s}(z-z_s)^m =(a_sm(-\partial_az_s)+ \alpha_s\mu_s^{T} a_s)(z-z_s)^{m-1}.
\end{equation*}
\notag
$$
By (4.2), $-\partial_az_s=\mu_s^{T} \alpha_s$. Since $a_s=\nu_s\alpha_s$, we have $\alpha_s\mu_s^{T} a_s=a_s\mu_s^{T} \alpha_s$, hence the leading term is equal to Then (4.20) implies that $f_a=\dfrac{f_{-1}}{z-z_s}+O(1)$, and $f_{-1}=-(m+1)(\partial_az_s)$. Hence
$$
\begin{equation*}
f_a(z) =\frac{f_{-1}}{z-z_s}+O(1) =(m+1)\,\frac{-{\partial_a z_s}}{z-z_s}+O(1) =(m+1)\partial_a\ln(z-z_s)+O(1).
\end{equation*}
\notag
$$
We obtain $\exp\biggl(-\displaystyle\int_0^{t_a} f_a\,dt\biggr)=(z-z_s)^{-(m+1)}\cdot O(1)$, hence
$$
\begin{equation*}
\widehat{\psi} =\exp\biggl(-\int_0^{t_a} f_a\,dt\biggr)\psi =\frac{a_s}{z-z_s}+O(1).
\end{equation*}
\notag
$$
We conclude that $\widehat{\psi}$ has poles at the moving points $\gamma_s$, that is, one of main postulates of the inverse spectral method, namely, the immovability of the pole divisor of the Baker–Akhieser function, cannot be fulfilled. Observe that this obstruction is absent in the case $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n)$ considered in Subsection 4.2, because there is no term $\alpha_s\mu_s^{T} $ in the nominator of the analogue of (4.20). The same calculation as above shows that
$$
\begin{equation*}
\exp\biggl(-\int_0^{t_a} f_a\,dt\biggr) =(z-z_s)^{-m}\cdot O(1)
\end{equation*}
\notag
$$
in that case, hence $\widehat\psi$ is holomorphic at $\gamma_s$ in accordance with what we said in Subsection 4.2. 4.4.2. An obstruction to getting theta-function solutions by means of separation of variables. Integration by quadratures and numerical integrationHere we present an obstruction to obtaining a theta-function solution of Hitchin’s systems of types $B_n$, $C_n$, and $D_n$ using the technique from Subsection 4.3.7. Relation (4.18) is based in the end on Riemann’s theorem on zeros of the Riemann theta function and, in particular, on the fact that the number of zeros is equal to the genus of the spectral curve. In the case $\mathfrak{g}=\operatorname{\mathfrak{gl}}(n)$ the number $h$ of the points $\gamma_1,\dots,\gamma_h$ giving the spectral curve is the same (it can also be characterized as the number of degrees of freedom of the system, the number of independent Hamiltonians, the half dimension of the phase space, and so on). The set of zeros gives the corresponding point of the phase space in that case. However, in the case of a simple structure group the number $h$ of the points specifying the spectral curve is equal to the dimension of its Prym variety, while the number of zeros of the Prym theta-function is twice that dimension [22; Corollary 5.6]. Observe that the above obstruction does not interfere with obtaining a solution by quadratures. Let $\mathcal{A}$ denote the Abel–Prym map. Its Jacobian matrix is equal to $J=\Bigl(\dfrac{\partial\phi_i}{\partial x_k}\Bigr)$. By (4.17),
$$
\begin{equation*}
\frac{\partial\phi_i}{\partial x_k} =-\frac{{\partial R}/\partial H_i}{{\partial R}/\partial\lambda} \,\frac{1}{y}\bigg|_{x=x_k,\,y=y_k,\,\lambda=\lambda_k}.
\end{equation*}
\notag
$$
The last expression is explicitly given in the coordinates $(x_k, y_k, \lambda_k)$, hence we can regard $J^{-1}$ as a matrix explicitly given in these coordinates. Let $X(t)$ denote a point on a Hitchin trajectory in the same coordinates. Then $X(t)=\mathcal{A}^{-1}(\phi(t))$, where $\phi(t)=\phi_0+Ht$ is the image of this trajectory on the fibre. Then $X'(t)=J^{-1}(\phi(t))H$, which implies that
$$
\begin{equation*}
X(t) =X_0 +\biggl(\int_0^t J^{-1}(\phi(t))\,dt\biggr) H.
\end{equation*}
\notag
$$
The last expression is just the solution by quadratures of Hitchin’s system corresponding to the Hamiltonian $H$, and the initial condition $X_0$. The equation $X'(t)=J^{-1}(\phi(t))H$ admits also a numerical solution. The one with mesh width $\Delta t$ can be constructed as follows:
$$
\begin{equation*}
X(0)=X_0,\qquad X(\Delta t)=J^{-1}(X_0)\Delta t,\qquad X(2\Delta t)=J^{-1}(X(\Delta t))\Delta t,
\end{equation*}
\notag
$$
and so on. Observe that we only use the expression for $J$ in the coordinates $(x,y,\lambda)$ in the process.
Цитирование в формате AMSBIB
\RBibitem{SheWan24} |
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