Seiberg–Witten differentials on the Hitchin base

Abstract

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss–Manin) derivative of the Seiberg–Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration.

1 Introduction

The base of the Hitchin integrable system [14] supports a family of cameral curves, and, as a consequence, carries various Hodge-theoretic and differential-geometric structures [6, 7, 20]. In particular, the Zariski open subset of the base, corresponding to smooth cameral curves with generic ramification carries a weight-one variation of Hodge structures (VHS) with a Seiberg–Witten differential. Our goal in this note is to describe the covariant (Gauss–Manin) derivative of the Seiberg–Witten differential explicitly in terms of Lie theory and cameral data.

We recall now the main ingredients and constructions, starting with the Hodge-theoretic ones.

Let ${\mathcal {B}}$ be a complex manifold. Recall that a polarised ${\mathbb {R}}$-VHS of weight $w\in {\mathbb {Z}}$ on ${\mathcal {B}}$ consists of data $({\mathcal {V}},\nabla ,{\mathcal {V}}_{\mathbb {R}},{\mathcal {F}}^\bullet ,S),$ where:

• ${\mathcal {V}}$ is a holomorphic vector bundle on ${\mathcal {B}}$

• $\nabla :{\mathcal {V}}\rightarrow {\mathcal {V}}\otimes \Omega ^1_{\mathcal {B}}$ is a flat (holomorphic) connection, called the Gauss–Manin connection

• ${\mathcal {V}}_{\mathbb {R}}\subseteq {\mathcal {V}}$ is a real, $\nabla$-flat subbundle, satisfying ${\mathcal {V}}={\mathcal {V}}_{\mathbb {R}}\otimes {\mathcal {O}}_{\mathcal {B}},$ called real structure

• ${\mathcal {F}}^\bullet$ is a decreasing filtration of ${\mathcal {V}}={\mathcal {F}}^0,$ the Hodge filtration

• $S:{\mathcal {V}}\otimes {\mathcal {V}}\rightarrow {\mathscr {C}}^\infty _{\mathcal {B}}$ is a non-degenerate, $(-1)^w$-symmetric, $\nabla$-flat pairing, ${\mathbb {R}}$-valued on ${\mathcal {V}}_{\mathbb {R}},$ called polarisation

such that

• (1) $\nabla ({\mathcal {F}}^p)\subseteq {\mathcal {F}}^{p-1}\otimes \Omega _{{\mathcal {B}}}^1$ Griffiths transversality

• (2) ${\mathcal {V}}= {\mathcal {F}}^p\oplus \overline{{\mathcal {F}}^{w+1-p}}$ Hodge structure,

or, in terms of the Hodge bundles ${\mathcal {H}}^{p,w-p}:= {\mathcal {F}}^p\cap \overline{{\mathcal {F}}^{w-p}},$

• (2’) ${\mathcal {V}}=\bigoplus _{p}{\mathcal {H}}^{p,w-p}$

• (3) $S({\mathcal {F}}^p,{\mathcal {F}}^{w+1-p})=0$

• (4) $i^{2p-w}S(v,{\overline{v}})>0$ for $v\in \Gamma ({\mathcal {H}}^{p,w-p}),$ $v\ne 0.$

The notions of polarised ${\mathbb {Z}}$-VHS or ${\mathbb {Q}}$-VHS are introduced analogously, by replacing ${\mathcal {V}}_{\mathbb {R}}$ with appropriate locally constant sheaves ${\mathcal {V}}_{\mathbb {Z}}$ or ${\mathcal {V}}_{\mathbb {Q}}$ of ${\mathbb {Z}}$- or ${\mathbb {Q}}$-modules, respectively.

The prototypical example is that of a geometric VHS, i.e., one arising from a family of compact Kähler (e.g., projective) manifolds.

By Griffiths Transversality, $\nabla$ induces an ${\mathcal {O}}_B$-module homomorphism

$$\begin{aligned} {\mathcal {F}}^p/{\mathcal {F}}^{p+1}\longrightarrow {\mathcal {F}}^{p-1}/{\mathcal {F}}^{p}\otimes \Omega ^1_{\mathcal {B}}\end{aligned}$$

and hence, taking a direct sum over the different p,  an ${\mathcal {O}}_B$-module homomorphism

$$\begin{aligned} \theta =[\nabla ]: \bigoplus _p {\mathcal {F}}^p/{\mathcal {F}}^{p+1} \longrightarrow \left( \bigoplus _p {\mathcal {F}}^p/{\mathcal {F}}^{p+1} \right) \otimes \Omega ^1_{\mathcal {B}}, \end{aligned}$$

which satisfies $\theta \wedge \theta =0.$

The pair $\left( E= \bigoplus _p {\mathcal {F}}^p/{\mathcal {F}}^{p+1},\theta \right)$ is an example of a Higgs bundle on ${\mathcal {B}}.$ This example played an important rôle in Carlos Simpson’s study of Higgs bundles on higher-dimensional varieties [22, 23].

Consider a polarised ${\mathbb {Z}}$-VHS $({\mathcal {V}},\nabla , {\mathcal {V}}_{\mathbb {Z}}, {\mathcal {V}}^\bullet , S,\ldots )$ of weight $w=1.$ An abstract Seiberg–Witten differential on it is a section $\varvec{\lambda }_{SW}\in H^0({\mathcal {B}},{\mathcal {V}}^1),$ for which the ${\mathcal {O}}_{\mathcal {B}}$-module homomorphism

$$\begin{aligned} T_{\mathcal {B}}\longrightarrow {\mathcal {V}}^0,\quad v\longmapsto \nabla _v \varvec{\lambda }_{SW} \end{aligned}$$

factors through an isomorphism

$$\begin{aligned} T_{\mathcal {B}}\simeq {\mathcal {V}}^1. \end{aligned}$$

(1)

Given such data, we obtain a refinement of the weight-1 filtration

$$\begin{aligned} {\mathcal {V}}^1 \subseteq {\mathcal {V}}^0 \end{aligned}$$

to a weight-3 filtration

$$\begin{aligned} \underbrace{{\mathcal {F}}^3}_{=\varvec{\lambda }_{SW}{\mathcal {O}}_{\mathcal {B}}} \subseteq \underbrace{\ {\mathcal {F}}^2 \ }_{{\mathcal {V}}^1 }\subseteq \underbrace{{\mathcal {F}}^1}_{=({\mathcal {F}}^3)^\perp }\subseteq \underbrace{{\mathcal {F}}^0}_{{\mathcal {V}}^0}. \end{aligned}$$

For links to projective special Kähler geometry (“$N=2$ supergravity”) and weight-3 VHS, satisfying the Calabi–Yau condition, one can check [13, §4, §8.3].

Furthermore, given such data, there is an associated fibration of complex tori ${\mathcal {J}}:= {\mathcal {V}}/({\mathcal {V}}^1+{\mathcal {V}}_{\mathbb {Z}})\rightarrow {\mathcal {B}},$ whose vertical bundle is $\text {Vert}={\mathcal {V}}/{\mathcal {V}}^1.$ The polarisation S gives rise to an isomorphism $\text {Vert}\simeq ({\mathcal {V}}^1)^\vee ,$ and hence $\varvec{\lambda }_{SW}$ induces, by composition with the dual of its defining isomorphism $T_{\mathcal {B}}\simeq {\mathcal {V}}^1,$ an isomorphism $i_{\varvec{\lambda }}: \text {Vert}\rightarrow T^\vee _{\mathcal {B}}.$ Such an isomorphism is also induced by a choice of symplectic form on ${\mathcal {J}}.$ There is unique symplectic form $\omega _{\varvec{\lambda }}$ on ${\mathcal {J}},$ which induces $i_{\varvec{\lambda }}$ and such that the 0-section is Lagrangian.

We next recall the construction of the family of cameral covers over the Hitchin base, and introduce a weight-1 VHS with a Seiberg–Witten differential on it.

First, we fix the following data:

• A simple complex Lie group G of rank l,  together with a choice of Borel and Cartan subgroups $T\subset B\subset G.$ We denote by ${\mathfrak {t}}\subset {\mathfrak {b}}\subset {\mathfrak {g}}$ the respective Lie algebras and by W the corresponding Weyl group.

• A compact (connected) Riemann surface X of genus $g\ge 2$ (or equivalently, a non-singular proper algebraic curve over ${\mathbb {C}}$). We do not need to fix a particular projective embedding of X.

Additionally, we choose:

• Homogeneous generators $I_1,\ldots , I_l$ of the ring ${\mathbb {C}}[{\mathfrak {t}}]^W\subset {\mathbb {C}}[{\mathfrak {t}}].$ We write $d_k=\deg I_k.$

• Simple (positive) roots $\{\alpha _1,\ldots , \alpha _l\}.$

These additional choices are not necessary for the entire discussion, but are needed for the explicit calculation in Theorem A.

Two explicit examples of invariant polynomials—for $SL_3({\mathbb {C}})$ and $G_2$—are given in Eqs. (30) and (32), respectively.

Notice that while ${\mathfrak {t}}/W$ is a priori just a cone, the choice of generators $\{I_k\}$ allows us to identify it with ${\mathbb {C}}^l.$ Notice also that we may interpret $\{I_k\}$ as elements of ${\mathbb {C}}[{\mathfrak {g}}]^G,$ via Chevalley’s theorem.

The chosen simple roots determine an isomorphism ${\mathfrak {t}}\simeq {\mathbb {C}}^l,$ $v\mapsto (\alpha _1(v),\ldots , \alpha _l(v)),$ using which we further identify $\chi :{\mathfrak {t}}\rightarrow {\mathfrak {t}}/W$ with a finite map $\textbf{I}: {\mathbb {C}}^l\rightarrow {\mathbb {C}}^l.$ We may abuse the notation for these maps, e.g., write $\chi =(I_1,\ldots , I_l)$ instead of $\textbf{I},$ etc.

We proceed by constructing from these data two rank-l vector bundles on X. The first one is ${\mathfrak {t}}\otimes _{\mathbb {C}}K_X \simeq K_X^{\oplus l},$ whose total space will be denoted by M:

$$\begin{aligned} M=\text {tot }{\mathfrak {t}}\otimes _{\mathbb {C}}K_X. \end{aligned}$$

The group W acts (fibrewise, via its action on ${\mathfrak {t}}$) on M. The resulting quotient ${\varvec{U}}$ is a priori just a cone bundle, but the choice of $\{I_k\}$ allows us to give it the structure of a vector bundle of rank l:

$$\begin{aligned} {\varvec{U}}= {\mathfrak {t}}\otimes _{\mathbb {C}}K_X/W \simeq \bigoplus _{k=1}^l K_X^{d_k}. \end{aligned}$$

(2)

We can also think of ${\varvec{U}}\backslash \{0\}$ as the ${\mathbb {C}}^\times$-bundle with fibre ${\mathfrak {t}}/W,$ associated to the ${\mathbb {C}}^\times$-bundle $K_X\backslash \{0\}.$

The morphism $\chi :{\mathfrak {t}}\rightarrow {\mathfrak {t}}/W$ induces a morphism $\chi :M\rightarrow \text {tot }{\varvec{U}}$ of X-varieties (not of vector bundles!):

(3)

We write ${\mathcal {B}}$ for the Hitchin base—the space of global sections of ${\varvec{U}}$:

$$\begin{aligned} {\mathcal {B}}:= H^0(X,{\varvec{U}})\simeq H^0\left( X, \bigoplus _{k=1}^l K_X^{d_k}\right) \simeq {\mathbb {C}}^{\dim G(g-1)}. \end{aligned}$$

Any $b\in {\mathcal {B}}$ determines a W-cover $p_b:{\widetilde{X}}_b\rightarrow X$ as the pullback of $\chi :M\rightarrow \text {tot }{\varvec{U}}$ via (the evaluation map of) the section b:

This W-cover is called the cameral cover of X (corresponding to b). We may occasionally write $p:{\widetilde{X}}\rightarrow X$ if the point $b\in {\mathcal {B}}$ is fixed or understood.

By construction ${\widetilde{X}}_b$ is a closed subscheme of M that can be singular or non-reduced. The cameral cover ${\widetilde{X}}_b\subset M$ inherits from M a W-action (and thus has lots of automorphisms). For a generic choice of b it is a non-singular ramified Galois W-cover with simple ramification. We write ${\mathscr {B}}\subseteq {\mathcal {B}}$ for the open set of generic cameral covers.

The vector bundle $p_b^*{\varvec{U}}$ is in fact isomorphic to $N_{{\widetilde{X}}_b/M},$ the normal bundle of ${\widetilde{X}}_b\subseteq M,$ see Sect. 2.1.

Example 1.1

Let $G=SL_2({\mathbb {C}}).$ Then $W={\mathbb {Z}}/2{\mathbb {Z}},$ ${\varvec{U}}\simeq K_X^2,$ ${\mathcal {B}}= H^0(X,K^2_X)\simeq {\mathbb {C}}^{3g-3}$ and $p_b: {\widetilde{X}}_b\rightarrow X$ is a 2 : 1 cover. The open set ${\mathscr {B}}\subseteq {\mathcal {B}}$ consists of quadratic differentials with simple roots. For $b\in {\mathscr {B}},$ the cover has genus $g({\widetilde{X}}_b)=4g-3.$

Example 1.2

Let $G=G_2.$ Then $W=D_6$ (dihedral group of order 12) and ${\varvec{U}}\simeq K_X^2\oplus K_X^6.$ Consequently ${\mathcal {B}}= H^0(X,K_X^2)\oplus H^0(X,K_X^6)\simeq {\mathbb {C}}^{14(g-1)}.$ The cameral covers $p_b:{\widetilde{X}}_b\rightarrow X$ are 12 : 1 covers, with $g({\widetilde{X}}_b)= 84(g-1)+1.$

There is a weight-1 ${\mathbb {Z}}$-variation of Hodge structures ${\mathcal {V}}^1\subseteq {\mathcal {V}}^0$ over ${\mathscr {B}}\subseteq {\mathcal {B}},$ whose fibres are respectively ${\mathcal {V}}^1_b=H^0({\widetilde{X}}_b,{\mathfrak {t}}\otimes _{\mathbb {C}}K_{{\widetilde{X}}_b})^W$ and ${\mathcal {V}}_b^0= H^1({\widetilde{X}}_b,{\mathfrak {t}})^W.$ Intrinsically, it is defined as follows. Let $\Lambda \subseteq {\mathfrak {t}}$ be the cocharacter lattice and $p:{\mathcal {X}}\rightarrow {\mathscr {B}}$ the universal cameral curve. Let also $p_*^W$ be the W-invariant pushforward functor. Then we set ${\mathcal {V}}_{\mathbb {Z}}= R^1p_*^W(\Lambda )$ and ${\mathcal {V}}:={\mathcal {V}}_{\mathbb {Z}}\otimes _{\mathbb {Z}}{\mathcal {O}}_{\mathscr {B}}\simeq R^1p_*^W({\mathfrak {t}}\otimes _{\mathbb {C}}\Omega ^\bullet _{{\mathcal {X}}/{\mathscr {B}}}).$ The bundle ${\mathcal {V}}^1=R^0p_*^W\left( {\mathfrak {t}}\otimes _{\mathbb {C}}\Omega ^1_{{\mathcal {X}}/{\mathscr {B}}}\right) ,$ and the Hodge filtration is induced by the naive filtration $\Omega _{{\mathcal {X}}/{\mathscr {B}}}^{\bullet \ge 1}[-1]\subseteq \Omega ^\bullet _{{\mathcal {X}}/{\mathscr {B}}}.$ The Gauss–Manin connection can be identified with the $d_1$ differential of the spectral sequence, induced by the Koszul–Leray filtration on $\Omega ^\bullet _{{\mathcal {X}}}.$ The polarisation pairing S is given by $S_b(\alpha ,\beta )=\left\langle \alpha \cup \beta , \left[ {\widetilde{X}}_b\right] \right\rangle .$ For more details, see Sect. 3.2 and the references therein, as well as [13, 8.1] and [9].

On M there is a canonical ${\mathfrak {t}}$-valued Liouville form $\lambda ,$ see Sect. 2.4. The Liouville form $\lambda$ determines a Seiberg–Witten differential, $\varvec{\lambda }_{SW}\in \Gamma ({\mathscr {B}},{\mathcal {V}}^1),$ via $\varvec{\lambda }_{SW}(b)=\left. \lambda \right| _{{\widetilde{X}}_b},$ and, as in (1), we have that the map

$$\begin{aligned} {\mathcal {B}}= T_{b}{\mathscr {B}}\ni \textbf{g}\longmapsto \left( \nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}\right) _b \in {\mathcal {V}}^0_b \end{aligned}$$

(4)

factors through an isomorphism $T_{b}{\mathscr {B}}\simeq {\mathcal {V}}^1_b,$ i.e.,

(5)

In [16, Proposition 2.11], an isomorphism with the same domain and codomain as in (5) is described as the composition of pullback on global sections (by $\pi$), contraction with $\omega$ and restriction to ${\widetilde{X}}_b,$ see also Proposition 3.1. In [13, Proposition 8.2] it is shown, using a hypercohomology calculation, that the isomorphism described by Hurtubise and Markman coincides with the isomorphism (4). Some of the above relations for $G=SL_2$ are discussed in [5, Proposition 1], see also [21, Eq.(3)].

The above isomorphism can also be considered from an integrable systems viewpoint. Indeed, consider the universal family of generic cameral curves $p:{\mathcal {X}}\rightarrow {\mathscr {B}}\subseteq {\mathcal {B}}.$ The relative Prym fibration $\textbf{Prym}_{{\mathcal {X}}/{\mathscr {B}}}\rightarrow {\mathscr {B}}$ is in fact an algebraic completely integrable system. The fibre $\textbf{Prym}_{{\widetilde{X}}_b}$ over $b\in {\mathscr {B}}$ is an abelian variety, whose tangent space is Serre dual to $H^0({\widetilde{X}}_b,{\mathfrak {t}}\otimes _{\mathbb {C}}K_{{\widetilde{X}}_b})^W,$ the right hand side of (5). The isomorphism (5) actually amounts to lifting a tangent vector in $T_{{\mathcal {B}},b}$ to a vector field along the fibre $\textbf{Prym}_{{\widetilde{X}}_b}$ and then pairing it with the symplectic form on the Prym fibration. This is the viewpoint, taken, e.g., by Hurtubise and Markman.

Our goal in this note is to provide an explicit and global (on X and ${\widetilde{X}}_b$) description of (5) in terms of Lie theory and the covering $p_b:{\widetilde{X}}_b\rightarrow X.$

The simplest case, that of $G=SL_2,$ is given in Example 5.1, where we show that Eq. (5) specialises to

$$\begin{aligned} H^0(X,K_X^2)\ni \textbf{g}\longmapsto \left( \nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}\right) _b= \left. \frac{p^*\textbf{g}}{2\varvec{\alpha }^2}\varvec{\lambda }_{SW}\right| _{{\widetilde{X}}_b}\in H^0({\widetilde{X}}_b, K_{{\widetilde{X}}_b})^{{\mathbb {Z}}/2}_{-}, \end{aligned}$$

where $\varvec{\alpha }^2\in H^0(M,\pi ^*K_X^2)$ is the tautological section and $\varvec{\lambda }_{SW}$ is the Liouville (Seiberg–Witten) form. The expression on the right hand side can in fact also be rewritten as $-\frac{p^*g}{2\lambda },$ and in this form it coincides (up to scaling factors) with [21, (4)], who reference Douady–Hubbard [5, §2].

Our main result is a general formula for $\nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}$ for the case of an arbitrary (complex, simple) group G.

Let $D\textbf{I}$ be the Jacobi matrix of the adjoint quotient $\textbf{I}=(I_1,\ldots , I_l):{\mathbb {C}}^l\rightarrow {\mathbb {C}}^l$ and $\iota$ the natural algebra homomorphism from $\text {Sym}({\mathfrak {t}}^\vee )$ into $H^0\left( M, \bigoplus _{n\ge 0}\pi ^*K_X^n\right) ,$ introduced in Eq. (22). Finally, $\varvec{\alpha }_i=\iota (\alpha _i)$ and $\lambda _i=e_i\otimes \varvec{\alpha }_i,$ where $\{e_i\}$ is the basis of ${\mathfrak {t}},$ dual to $\{\alpha _i\}.$ In this notation, the Liouville form is $\varvec{\lambda }_{SW}=\sum _i\lambda _i.$

Theorem A

Once the main and additional data are chosen, the isomorphism (5) $\nabla ^{GM}\varvec{\lambda }_{SW}$ maps $\textbf{g}\in T_b{\mathscr {B}}=H^0(X,\bigoplus _i K_X^{d_i})$ to the section

$$\begin{aligned} \left( \nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}\right) _b = - \left. \sum _{i=1}^l \frac{\left( \iota (D\textbf{I})^{-1} \cdot \pi ^{*}\textbf{g}\right) _i}{\varvec{\alpha }_i}\lambda _i \right| _{{\widetilde{X}}_b} = - \left. \iota (D\textbf{I})^{-1}\cdot \pi ^*\textbf{g}\right| _{{\widetilde{X}}_b}. \end{aligned}$$

(6)

In particular, for $l=2$ we have that

$$\begin{aligned} \left( \nabla ^{GM}\varvec{\lambda }_{SW}\right) _b:\ H^0(X, K_X^{d_1}\oplus K_X^{d_2}) \longrightarrow H^0({\widetilde{X}}_b, K_{{\widetilde{X}}_b}^{\oplus 2})^W \end{aligned}$$

sends $\textbf{g}= \begin{bmatrix} g_1\\ g_2\\ \end{bmatrix}$ to

$$\begin{aligned} \left( \nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}\right) _b= - \left. \left( \frac{ \begin{vmatrix} \pi ^*g_1&\iota \partial _2 I_1 \\ \pi ^*g_2&\iota \partial _2 I_2 \\ \end{vmatrix}}{\varvec{\alpha }_1 \det \iota D\textbf{I}}\lambda _1 + \frac{ \begin{vmatrix} \iota \partial _1 I_1&\pi ^*g_1 \\ \iota \partial _1 I_2&\pi ^*g_2 \\ \end{vmatrix}}{\varvec{\alpha }_2 \det \iota D\textbf{I}}\lambda _2 \right) \right| _{{\widetilde{X}}_b} =- \frac{1}{\det \iota D\textbf{I}} \left. \begin{bmatrix} \begin{vmatrix} \pi ^*g_1&{} \iota \partial _2 I_1 \\ \pi ^*g_2&{} \iota \partial _2 I_2 \\ \end{vmatrix} \\ \\ \begin{vmatrix} \iota \partial _1 I_1&{} \pi ^*g_1 \\ \iota \partial _1 I_2&{} \pi ^*g_2 \\ \end{vmatrix} \end{bmatrix} \right| _{{\widetilde{X}}_b}.\nonumber \\ \end{aligned}$$

(7)

Knowledge of $\varvec{\lambda }_{SW}$ and $\nabla ^{GM}\varvec{\lambda }_{SW}$ is essential for describing various geometric structures on ${\mathscr {B}}.$ We mention only two examples as an illustration.

First, for the Hitchin integrable system, the Donagi–Markman cubic [6], which is essentially the infinitesimal period map for the family of Hitchin Pryms, is given by the Balduzzi–Pantev formula [2, Theorem 1]. If we consider the cubic as a global section c of $\text {Sym}^3 T^\vee _{{\mathscr {B}}}=\text {Sym}^3{\mathcal {B}}^\vee \otimes {\mathcal {O}}_{{\mathscr {B}}},$ then the Balduzzi–Pantev formula states that the value of c at $b\in {\mathscr {B}}$ is

$$\begin{aligned} c_b(\textbf{g}_1,\textbf{g}_2,\textbf{g}_3)=\frac{1}{2}\sum _{m\in \text {Ram }p_b}\text {Res}^2_m \left( p_b^*\left. \frac{{\mathcal {L}}_{\textbf{g}_1} {\mathfrak {D}}}{{\mathfrak {D}}}\right| _{\{b\}\times X}\left( \nabla ^{GM}_{\textbf{g}_2}\varvec{\lambda }_{SW}\right) _b\cup \left( \nabla ^{GM}_{\textbf{g}_3}\varvec{\lambda }_{SW}\right) _b \right) .\nonumber \\ \end{aligned}$$

(8)

Here ${\mathfrak {D}}$ is the discriminant (see also Sect. 5) and ${\mathcal {L}}$ denotes Lie derivative. In our previous work [3, Theorem A] we have shown that the Balduzzi–Pantev formula holds along the (good) symplectic leaves of the generalised Hitchin system.

The second example which is worth mentioning is the special Kähler metric $g_{SK}$ on ${\mathscr {B}}.$ It is known that for the case of $G=SL_2({\mathbb {C}}),$ the special Kähler metric is given by

$$\begin{aligned} g_{SK}(\textbf{g},\textbf{g})_b=2\int _{{\widetilde{X}}_b}|\nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}|^2, \end{aligned}$$

(9)

see [11, 2.40], [21, §2.3], [8].

We shall discuss additional applications of Theorem A to various aspects of the geometry of ${\mathscr {B}}$ in a forthcoming work.

2 Preliminaries

2.1 The embedding of the cameral curve

We are now going to work at a fixed point $b\in {\mathcal {B}}$ (generic), and hence will write mostly $p:{\widetilde{X}}\rightarrow X$ for the cameral cover. To understand (5) we need to understand $K_{{\widetilde{X}}}$ and for that we need to know more about the normal bundle N of the closed embedding ${\widetilde{X}}\subseteq M.$ This is not difficult, since ${\widetilde{X}}$ is in fact the zero locus of a section of a vector bundle on M.

First, notice that the morphism $\chi :M\rightarrow {\varvec{U}}$ (see (3)) induces a tautological section $\sigma \in H^0(M, \pi ^*{\varvec{U}})$ in a standard way, via

(10)

which on closed points is simply $\sigma (m)= (m, \chi (m))\in M\times _X \text {tot }{\varvec{U}}=\text {tot }\pi ^*{\varvec{U}}.$

Next, the adjunction morphism ${\varvec{U}}\rightarrow \pi _*\pi ^*{\varvec{U}}$ induces on global sections the pullback map ${\mathcal {B}}=H^0(X,{\varvec{U}})\rightarrow H^0(M, \pi ^*{\varvec{U}}),$ which we write as $b\mapsto \pi ^*b.$

Thus the cameral curve $\widetilde{X_b}$ is the zero locus

$$\begin{aligned} {\widetilde{X}}_b=\text {zeros}(s_b),\ s_b=\sigma - \pi ^*(b)\in H^0(M, \pi ^*{\varvec{U}}), \end{aligned}$$

(11)

i.e., is cut out by the equation(s)

$$\begin{aligned} \chi (m)=b(\pi (m)) \end{aligned}$$

(12)

in $M=\text {tot }{\mathfrak {t}}\otimes _{\mathbb {C}}K_X.$ Having fixed basic invariant polynomials $\{I_k\},$ and hence an isomorphism ${\varvec{U}}\simeq \bigoplus _{k=1}^l K_X^{d_k},$ we can express this as the system of equations

$$\begin{aligned} \left| \begin{array}{l} I_1(m)= b_1(\pi (m))\\ \vdots \\ I_l(m)= b_l(\pi (m))\\ \end{array} \right. , \end{aligned}$$

(13)

for $m\in M,$ with $b=(b_1,\ldots ,b_l)\in {\mathcal {B}}$ fixed. These are “global” equations and no choice of local trivialisation is used here: the k-th equation takes values in (the total space of) $K_X^{d_k}.$ Another global description is given in Eq. (23).

From Eq. (11) follows

Proposition 2.1

The normal bundle of ${\widetilde{X}}_b\subseteq M$ is

$$\begin{aligned} N_{{\widetilde{X}}_b/M}\simeq p_b^*{\varvec{U}}= {\mathfrak {t}}\otimes _{\mathbb {C}}p_b^*K_X/W \simeq \bigoplus _{k=1}^l p_b^*K_X^{d_k}. \end{aligned}$$

(14)

Proof

While in general one uses the Koszul complex to compute the normal bundle, here we have that both ${\widetilde{X}}_b$ and M are smooth, and moreover, ${\widetilde{X}}_b$ is a complete intersection. This case is handled by a standard geometric argument, given in, e.g. [10, Proposition 6.15].

The isomorphism $N_{{\widetilde{X}}_b/M}\simeq p_b^*{\varvec{U}}$ is induced by the (vertical component of the) differential $ds_b: T_M\rightarrow s_b^*T_{\pi ^*{\varvec{U}}}$ of the section $s_b:M \rightarrow \text {tot }\pi ^*{\varvec{U}}.$ $\square$

Similarly to the above argument, since M is the total space of a vector bundle (namely ${\mathfrak {t}}\otimes _{\mathbb {C}}K_X$) on X,  its tangent bundle $T_M$ is an extension of $\pi ^*T_X$ by ${\mathfrak {t}}\otimes _{\mathbb {C}}\pi ^*K_X.$ Restricting to ${\widetilde{X}}$ and combining with the previous result, one gets the diagram

(15)

Now, consider $\textbf{g}=(g_1,\ldots ,g_l)\in T_{{\mathcal {B}},b}={\mathcal {B}},$ with $g_i\in H^0(X,K_X^{d_i}).$ It determines a 1-parameter family of deformations of ${\widetilde{X}}_b,$ given by the equation

$$\begin{aligned} \chi (m)= b(\pi (m))+\epsilon \textbf{g}(\pi (m)), \end{aligned}$$

(16)

that is, $\left\{ {\widetilde{X}}_{b+\epsilon g}\right\} _\epsilon .$ For $\epsilon$ in a sufficiently small disk $\Delta _\rho \subseteq {\mathbb {C}}$ the section $b+\epsilon \textbf{g}\in {\mathcal {B}}$ remains generic—which we assume to be the case from now on. The total space of the 1-parameter family is cut out in $M\times \Delta _\rho$ by the Eq. (16).

The section $\textbf{g}$ determines a section of $N_{\widetilde{X_b}/M}= {\mathfrak {t}}\otimes _{\mathbb {C}}p^*_b K_X/W,$ namely, $p_b^*\textbf{g}.$

2.2 Local description

It is not hard to describe the objects from the previous section in local coordinates. A choice of a local (analytic) chart $\psi$ on X,  identifying an open $U\subseteq X$ with a disk $\Delta \subseteq {\mathbb {C}},$ determines a local trivialisation of $K_X$ and a compatible bundle chart $\phi$ on M,  identifying $M_U= \pi ^{-1}(U)\rightarrow U$ with $\text {pr}_1:\Delta \times {\mathfrak {t}}\rightarrow \Delta ,$ as usual:

(17)

Such a local chart determines a trivialisation of $K_X^{d_i}$ over U and hence a section $b_i\in H^0(X,K_X^{d_i})$ is represented locally as $(\psi ^{-1})^*b_i= \beta _i(z) dz^{\otimes d_i}$ on U,  where $\beta _i:\Delta \rightarrow {\mathbb {C}}$ is a holomorphic function.

Using the simple roots as a basis for ${\mathfrak {t}}\simeq {\mathbb {C}}^l,$ we identify ${\widetilde{X}}_U=p_b^{-1}(U)$ (via $\phi$) with the set of solutions of $\textbf{I}(\alpha _1,\ldots ,\alpha _l)=\varvec{\beta }(z)$ for $(z,{\underline{\alpha }})\in \Delta \times {\mathbb {C}}^l,$ giving a local version of Eq. (13).

Next, the trivialisations of $K_X^{d_i}$ $(i=1\ldots l)$ and the choice of roots provide an induced trivialisation $\left. T_{M_U}\right| _{{\widetilde{X}}_U}$ and

$$\begin{aligned} \left. (\psi ^{-1})^*T_{M_U}\right| _{{\widetilde{X}}_U}= \left( {\mathbb {C}}\oplus {\mathfrak {t}}\right) \otimes _{\mathbb {C}}{\mathcal {O}}_{\phi ({\widetilde{X}}_U)}\simeq {\mathcal {O}}_{\phi ({\widetilde{X}}_U)}\left\langle \frac{\partial }{\partial z}, \frac{\partial }{\partial \alpha _1}, \ldots , \frac{\partial }{\partial \alpha _l}\right\rangle \end{aligned}$$

(18)

and, consequently, a local description of the diagram (15):

(19)

Here the bottom vertical map is, in more detail,

$$\begin{aligned} \begin{pmatrix} -\varvec{\beta }'&D\textbf{I}\end{pmatrix}= \left( \begin{array}{cccc} -\beta _1' \qquad &{}\quad \partial _1 I_1\qquad &{} \quad \ldots \qquad &{} \quad \partial _l I_1\\ \vdots &{}\quad \vdots &{} \quad \vdots &{} \quad \vdots \\ -\beta _l'&{} \quad \partial _1 I_l&{} \quad \ldots &{} \quad \partial _l I_l\\ \end{array} \right) \in \text {Mat}_{l\times (l+1)}\left( \Gamma \left( {\mathcal {O}}_{\phi ({\widetilde{X}}_U)}\right) \right) , \end{aligned}$$

(20)

having rank l everywhere on ${\widetilde{X}}_U,$ under the assumption that $b=(b_1,\ldots ,b_l)\in {\mathcal {B}}$ is generic. This is the matrix of the map $\text {pr}_2\circ ds$ from Proposition 2.1. We write $D\textbf{I}$ or $D\chi$ for the Jacobi matrix of $\textbf{I}=(I_1,\ldots , I_l):{\mathbb {C}}^l\rightarrow {\mathbb {C}}^l.$

Finally, given a tangent vector $\textbf{g}=(g_1,\ldots ,g_l)\in T_{{\mathcal {B}},b}={\mathcal {B}},$ with $(\psi ^{-1})^* g_i= \gamma _i(z)dz^{\otimes d_i}$ on U,  the corresponding 1-parameter (analytic) family of deformations of ${\widetilde{X}}_b$ is cut out locally (in $\Delta \times {\mathbb {C}}^l\times \Delta _\rho$) by $\textbf{I}({\underline{\alpha }})=\varvec{\beta }(z) + \epsilon \varvec{\gamma }(z),$ where $\Delta _\rho \subseteq {\mathbb {C}}$ is as before.

We may occasionally suppress the pullbacks by $\phi$ and $\psi ,$ except for the cases when there is a risk of confusion, as when discussing (co)roots and some associated objects.

2.3 Objects, associated with roots

Any linear map $\alpha \in {\mathfrak {t}}^\vee =\text {Hom}({\mathfrak {t}},{\mathbb {C}})$ determines, by extension of scalars, a vector bundle homomorphism ${\mathfrak {t}}\otimes _{\mathbb {C}}K_X\rightarrow K_X,$ denoted by the same letter. Hence, just as $\chi$ in Eq. (10), such an $\alpha$ determines a tautological section $\varvec{\alpha }\in H^0(M, \pi ^*K_X),$ which on (closed) points maps $m\in M$ to $\varvec{\alpha }(m)=(m, \alpha (m))\in M\times _X \text {tot }K_X.$ Furthermore, restricting $\varvec{\alpha }$ to ${\widetilde{X}}\subset M$ gives a section $\varvec{\alpha }_{{\widetilde{X}}}\in H^0({\widetilde{X}},p^*K_X).$ Occasionally, we suppress the subscript ${\widetilde{X}},$ i.e., the restriction.

The section $\varvec{\alpha }$ vanishes along a “hyperplane divisor” $\text {tot }\left( \ker \alpha \otimes _{\mathbb {C}}K_X\right) \subseteq M,$ a rank-$(l-1)$ subbundle of ${\mathfrak {t}}\otimes _{\mathbb {C}}K_X.$ The respective restrictions $\varvec{\alpha }_{i {\widetilde{X}}}$ (of sections arising from roots) vanish along divisors $D_{\alpha _i}$ in ${\widetilde{X}},$ which are the ramification divisors of $p:{\widetilde{X}}\rightarrow X.$

If we choose a local chart $(U,\psi )$ and $\phi :M_U\simeq \Delta \times {\mathfrak {t}},$ as in (17), $\varvec{\alpha }$ is represented by $(z,u)\mapsto \alpha (u)dz,$ where $\alpha (u)=\langle \alpha ,u \rangle$ is the natural pairing between ${\mathfrak {t}}$ and ${\mathfrak {t}}^\vee .$ If we further identify the preimage of $\pi ^{-1}(U)$ in $\text {tot }\pi ^*K_X\rightarrow M$ with $\Delta \times {\mathfrak {t}}\times {\mathbb {C}},$ via $\phi$ and a trivialisation of $K_X,$ then the evaluation map of $\varvec{\alpha }$ is represented by

$$\begin{aligned} \Delta \times {\mathfrak {t}}\ni (z,u) \longmapsto (z,u,\alpha (u)) \in \Delta \times {\mathfrak {t}}\times {\mathbb {C}}. \end{aligned}$$

The linear functional $\alpha \in {\mathfrak {t}}^\vee$ determines a function on $\Delta \times {\mathfrak {t}},$ that we may denote $\text {pr}_2^*\alpha$ if the distinction from $\alpha$ is important. Furthermore, given the choice of $\phi ,$ we may consider $\alpha$ (or rather, $\text {pr}_2^*\alpha$) a function $\phi ^*\alpha \in {\mathcal {O}}_{M_U}(M_U)$ on $M_U.$ Consequently, upon restriction to ${\widetilde{X}}_U,$ we get a local function $\phi ^*\alpha \in {\mathcal {O}}_{{\widetilde{X}}_U}({\widetilde{X}}_U)$ on the cameral curve. Of course, one should really write $\left. \phi ^*\text {pr}_2^*\alpha \right| _{{\widetilde{X}}_U}$ here.

The distinction between the various objects associated to a root $\alpha _i$ becomes important when one considers their differentials. Since $\pi ^*K_X\subseteq \Omega ^1_M,$ $d\varvec{\alpha }_i\in \Omega ^2_M(M).$ At the same time, $d\phi ^*\alpha _i\in \Omega ^1_{M_U}(M_U)$ and $d(\text {pr}_2^*\alpha _i)\in \Omega ^1(\Delta \times {\mathfrak {t}}).$ Naturally, we are going to write $d\alpha _i$ for the penultimate expression, so the distinction between $d\alpha _i$ and $d\varvec{\alpha }_i$ is essential. Finally, we keep in mind that $d\alpha _i=\alpha _i\in \text {Hom}({\mathfrak {t}},{\mathbb {C}}),$ as with any linear map.

The assignment $\alpha _i\mapsto \varvec{\alpha }_i$ determines an (injective) ${\mathbb {C}}$-algebra homomorphism

$$\begin{aligned} \iota : \text {Sym}(\textbf{t}^\vee )\hookrightarrow H^0\left( M, \bigoplus _{n\ge 0}\pi ^*K_X^n\right) \end{aligned}$$

(21)

and, consequently, a homomorphism

$$\begin{aligned} \text {End}({\mathbb {C}}^l)\otimes \text {Sym}({\mathfrak {t}}^\vee )\hookrightarrow \text {End}({\mathbb {C}}^l) \otimes H^0\left( M, \bigoplus _{n\ge 0}\pi ^*K_X^n\right) , \end{aligned}$$

(22)

denoted by $\iota$ as well. Given a $\text {Sym}({\mathfrak {t}}^\vee )$-valued endomorphism A with non-zero determinant $\det A\in \text {Sym}({\mathfrak {t}}^\vee ),$ we write $\iota (A)^{-1}$ for the inverse of $\iota (A)$ in the ring of $l\times l$ matrices with coefficients in the field of fractions $\text {Frac }H^0\left( M, \bigoplus _{n\ge 0}\pi ^*K_X^n\right) ,$ and in fact, in $\text {End}({\mathbb {C}}^l)\otimes H^0\left( M, \bigoplus _{n\ge 0}\pi ^*K_X^n\right) \left[ \frac{1}{\det \iota (A)}\right] .$

We can, more generally, rewrite the global equations for ${\widetilde{X}}_b$ as

$$\begin{aligned} \left| \begin{array}{l} I_1(\varvec{\alpha }_1,\ldots ,\varvec{\alpha }_l)= \pi ^*b_1 \\ \vdots \\ I_l(\varvec{\alpha }_1,\ldots ,\varvec{\alpha }_l)= \pi ^*b_l\\ \end{array} \right. \end{aligned}$$

(23)

that is, the linear system $\iota (I_k)=\pi ^*b_k,$ $k=1\ldots l.$

2.4 Liouville form

On $M=\text {tot }{\mathfrak {t}}\otimes _{\mathbb {C}}K_X$ there is a ${\mathfrak {t}}$-valued 2-form $\omega \in H^0(M, {\mathfrak {t}}\otimes _{\mathbb {C}}\Omega ^2_M).$ Probably the simplest way to introduce it is by setting

$$\begin{aligned} \omega = - d\lambda , \end{aligned}$$

where $\lambda \in H^0(M,{\mathfrak {t}}\otimes _{\mathbb {C}}\pi ^*K_X)\subseteq H^0(M,{\mathfrak {t}}\otimes _{\mathbb {C}}\Omega ^1_M)$ is a tautological section, the “${\mathfrak {t}}$-valued Liouville form”.

We recall some explicit expressions for $\lambda$—although, as usual in symplectic geometry, there are various sign ambiguities in the possible definitions.

The chosen simple roots $\{\alpha _1,\ldots ,\alpha _l\}$ form a basis of ${\mathfrak {t}}^\vee ,$ and we let $\{e_1,\ldots , e_l\}$ stand for the corresponding dual basis of ${\mathfrak {t}}$ (consisting of fundamental coweights).

One can then set $\lambda _i= e_i\otimes _{\mathbb {C}}\varvec{\alpha }_i,$ a global section of ${\mathfrak {t}}\otimes _{\mathbb {C}}\pi ^*K_X\subseteq {\mathfrak {t}}\otimes _{\mathbb {C}}\pi ^*\Omega ^1_M,$ and write the Liouville form and the 2-form as

$$\begin{aligned} \lambda = \sum _{i=1}^l \lambda _i=\begin{bmatrix} \varvec{\alpha }_1\\ \varvec{\alpha }_2\\ \vdots \\ \varvec{\alpha }_l\\ \end{bmatrix},\ \omega = - \sum _{i=1}^l e_i\otimes _{\mathbb {C}}d\varvec{\alpha }_{i}= \begin{bmatrix} -d\varvec{\alpha }_1\\ -d\varvec{\alpha }_2\\ \vdots \\ - d\varvec{\alpha }_l\\ \end{bmatrix}. \end{aligned}$$

(24)

Finally, if we choose local coordinates as in Eq. (17), we obtain for the pullback of $\lambda$ and $\omega$ to $\Delta \times {\mathfrak {t}}$

$$\begin{aligned} (\phi ^{-1})^*\lambda = \sum _{i=1}^n e_i\otimes _{\mathbb {C}}\alpha _i dz = \begin{bmatrix} \alpha _1 dz \\ \vdots \\ \alpha _l dz\\ \end{bmatrix},\ (\phi ^{-1})^*\omega = \sum _{i=1}^l e_i\otimes _{\mathbb {C}}dz\wedge d\alpha _i = \begin{bmatrix} dz\wedge d\alpha _1\\ \vdots \\ dz\wedge d\alpha _l\\ \end{bmatrix}. \end{aligned}$$

3 Background: two results

3.1 A result of Hurtubise and Markman

We begin with the special case of a result of Hurtubise and Markman [16, Proposition 2.11] mentioned in the introduction. We spell out some of the details of their argument for this special case.

Proposition 3.1

For each generic $b\in {\mathcal {B}},$ the pullback of global sections via $p_b,$ followed by the isomorphism (14) and contraction with $\omega$ induces an isomorphism

(25)

or,  using the choice of invariant polynomials $\{I_k\},$ an isomorphism

$$\begin{aligned} H^0\left( X, \bigoplus _{k=1}^l K_X^{d_k}\right) \simeq H^0\left( {\widetilde{X}}_b, \bigoplus _{k=1}^l p_b^*K_X^{d_k}\right) ^W \simeq H^0\left( {\widetilde{X}}_b, K_{{\widetilde{X}}_b}^{\oplus l}\right) ^W. \end{aligned}$$

Thus, the isomorphism $\beta$ (25) is a composition of two maps. The first one is pullback (adjunction) $\textbf{g}\longmapsto p_b^*\textbf{g},$ for $\textbf{g}\in {\mathcal {B}}=H^0(X,{\varvec{U}}).$ The second one is the map on global sections, induced by the map of bundles

$$\begin{aligned} N_{{\widetilde{X}}} \longrightarrow {\mathfrak {t}}\otimes _{\mathbb {C}}K_{{\widetilde{X}}} \end{aligned}$$

(26)

$$\begin{aligned} s\longmapsto \left. \omega ({\widetilde{s}},\ ) \right| _{{\widetilde{X}}}, \end{aligned}$$

where ${\widetilde{s}}$ is a lift of s to a section of $T_M.$ One may denote this map simply by $\lrcorner \ \omega$ (contraction with $\omega$), but should keep in mind the restriction to ${\widetilde{X}}.$

The proof of Proposition 3.1 relies on a dimension count, combined with good understanding of the bundle map (26) and the induced map on fibres at $m\in {\widetilde{X}}.$ For that, the cases when m is not a ramification point and when it is one should be considered separately. Notice that if m is not a ramification point, then $T_{{\widetilde{X}},m}\nsubseteq \pi ^{-1}(\pi (m))={\mathfrak {t}}\otimes _{\mathbb {C}}K_{X,p(m)},$ while $T_{{\widetilde{X}},m}\subseteq \pi ^{-1}(\pi (m))$ if m is a ramification point.

So let us choose a point $m\in M$ and consider the fibre of $\pi :M\rightarrow X,$ passing through m. We set $L:= \pi ^{-1}(\pi (m))={\mathfrak {t}}\otimes _{\mathbb {C}}K_{X,\pi (m)}\subseteq M,$ and write $N_L$ for the normal bundle of the vector space $L\subseteq M.$

Using the local description of $\omega ,$ we obtain that $\lrcorner \ \omega$ fits in the following diagram:

Since $N_L= T_{X,\pi (m)}\otimes _{\mathbb {C}}{\mathcal {O}}_L$ and $T_L=L\otimes _{\mathbb {C}}{\mathcal {O}}_L= {\mathfrak {t}}\otimes _{\mathbb {C}}K_{X,\pi (m)}\otimes _{\mathbb {C}}{\mathcal {O}}_L,$ there is a canonical trivialisation $T_L\otimes N_L={\mathfrak {t}}\otimes _{\mathbb {C}}{\mathcal {O}}_L.$

Using the normal sequence for $L\subseteq M,$ one obtains:

Lemma 3.1

The map $\lrcorner \ \omega$ induces a trivialisation $T_L\otimes N_L\simeq _\omega {\mathfrak {t}}\otimes _{\mathbb {C}}{\mathcal {O}}_L,$ which coincides up to sign with the canonical trivialisation. That is, 

Thus, in particular, $\lrcorner \omega$ induces, for any $m\in L,$ a W-equivariant isomorphism $T_{L,m}\otimes N_{L,m}\simeq {\mathfrak {t}}.$ A similar result is stated, in a much more general setup, in [16][Theorem 2.8 (5)].

Lemma 3.2

Consider a point $m\in {\widetilde{X}}$ that is not a ramification point of $p: {\widetilde{X}}\rightarrow X.$ The map on fibres,  induced by the bundle map (26) is an isomorphism

This is again a local calculation, using the explicit form of $\omega .$ Notice that since m is not a ramification point, the composition

is an isomorphism. However, at ramification points the behaviour of $\lrcorner \omega$ is different. In fact, at such points the map (26) is not an isomorphism of bundles if $l>1,$ as is clear from the next Lemma.

Lemma 3.3

Let $m\in {\widetilde{X}}$ be a ramification point of $p:{\widetilde{X}}\rightarrow X.$ Then

commutes.

This result is shown by a local calculation, which in turn boils down to a linear-algebraic result, using the explicit form of $\omega .$ It is also stated in [16, Lemma 2.10].

Proof of Proposition 3.1

The map $\beta ,$ i.e., (25) is a composition of two maps, both of which are injective. Indeed, $H^0(X,{\varvec{U}})\hookrightarrow H^0({\widetilde{X}}, p^*{\varvec{U}}),$ and the image is contained in $H^0({\widetilde{X}}, p^*{\varvec{U}})^W.$ Furthermore, Lemma 3.2 implies that the map (26) induces an injection on global sections, $H^0({\widetilde{X}}, {\mathfrak {t}}\otimes _{\mathbb {C}}p^*K_X/W)\hookrightarrow H^0({\widetilde{X}}, {\mathfrak {t}}\otimes _{\mathbb {C}}K_{{\widetilde{X}}}),$ and it preserves W-invariant sections. Finally, by Serre duality (and the fact that taking duals commutes with taking invariants), we get

$$\begin{aligned} {\mathcal {B}}= H^0(X,{\varvec{U}})\hookrightarrow H^0({\widetilde{X}}, p^*{\varvec{U}})^W\hookrightarrow H^0({\widetilde{X}}, {\mathfrak {t}}\otimes _{\mathbb {C}}K_{{\widetilde{X}}})^W\simeq H^1\left( {\widetilde{X}},{\mathfrak {t}}\otimes _{\mathbb {C}}{\mathcal {O}}_{{\widetilde{X}}}\right) ^{W\vee }. \end{aligned}$$

But $H^1\left( {\widetilde{X}},{\mathfrak {t}}\otimes _{\mathbb {C}}{\mathcal {O}}_{{\widetilde{X}}}\right) ^{W}$ is the tangent space of the generalised Prym variety, and by the complete integrability of the Hitchin system, its dimension equals the dimension of the base ${\mathcal {B}}.$ Hence both injections are isomorphisms, and so is their composition. $\square$

3.2 A result of Hertling, Hoevenaars and Posthuma

We introduced earlier a certain weight-1 ${\mathbb {Z}}$-VHS $({\mathcal {V}}, \nabla ^{GM}, {\mathcal {V}}_{\mathbb {Z}}, {\mathcal {V}}^\bullet , S)$ on ${\mathscr {B}}\subseteq {\mathcal {B}}.$

The bundle of lattices ${\mathcal {V}}_{\mathbb {Z}}$ was defined as $R^1p_*^W(\Lambda ),$ where $p: {\mathcal {X}}\rightarrow {\mathscr {B}}$ is the universal cameral cover, and the vector bundle ${\mathcal {V}}={\mathcal {V}}_{\mathbb {Z}}\otimes _{\mathbb {Z}}{\mathcal {O}}_{{\mathscr {B}}}\simeq R^1p_*^W({\mathfrak {t}}\otimes _{\mathbb {C}}p^*{\mathcal {O}}_{{\mathscr {B}}}).$ The relative holomorphic Poincaré Lemma gives a quasi-isomorphism $p^{-1}{\mathcal {O}}_{{\mathscr {B}}}\simeq _{quis}\Omega ^\bullet _{{\mathcal {X}}/{\mathscr {B}}},$ leading to ${\mathcal {V}}\simeq R^1p_*^W({\mathfrak {t}}\otimes _{\mathbb {C}}\Omega ^\bullet _{{\mathcal {X}}/{\mathscr {B}}}).$

The Hodge bundles, as for geometric VHS, are determined by the naive filtration of $\Omega ^\bullet _{{\mathcal {X}}/{\mathscr {B}}},$ see [26, §10.2] and [27, §5.1]. In our case of weight one, $\Omega ^{\bullet \ge 1}_{{\mathcal {X}}/{\mathscr {B}}}[-1]\subseteq \Omega ^\bullet _{{\mathcal {X}}/{\mathscr {B}}}$ determines a subbundle ${\mathcal {V}}^1\subseteq {\mathcal {V}},$ as $R^1p_*^W({\mathfrak {t}}\otimes _{\mathbb {C}}\Omega ^{\bullet \ge 1}_{{\mathcal {X}}/{\mathscr {B}}}[-1])\simeq R^0p_*^W({\mathfrak {t}}\otimes _{\mathbb {C}}\Omega ^1_{{\mathcal {X}}/{\mathscr {B}}}).$

The Gauss–Manin connection $\nabla ^{GM}:{\mathcal {V}}\rightarrow {\mathcal {V}}\otimes \Omega ^1_{{\mathscr {B}}}$ can be defined in either topological or holomorphic terms. The topological description relies on Ehresmann’s theorem, i.e., on the $C^\infty$-local triviality of $p: {\mathcal {X}}\rightarrow {\mathscr {B}}.$ In this case, the homotopy-invariance of de Rham cohomology implies that ${\mathcal {V}}_{\mathbb {Z}}$ is a locally constant sheaf and $\nabla ^{GM}$ can be described by a Cartan–Lie formula. For geometric VHS this is described, e.g., in [26, §8.2].

The holomorphic description of $\nabla ^{GM}$ is discussed in [13, §8], following [18], see also [4] and [27, §5.1]. The Koszul–Leray filtration on $\Omega ^\bullet _{{\mathcal {X}}}$ gives rise to a spectral sequence, for which $(E_1^{\bullet ,0},d_1)$ is identified with $(\Omega ^\bullet _{{\mathscr {B}}}({\mathcal {V}}),\nabla ^{GM}).$

One has the following result.

Theorem 3.1

[13, Proposition 8.2] The isomorphisms $\nabla ^{GM}\varvec{\lambda }_{SW}$ and $\beta$ (25) coincide. That is, 

$$\begin{aligned} \nabla ^{GM}_\xi \varvec{\lambda }_{SW}=\beta (\xi ), \end{aligned}$$

for all tangent vectors $\xi \in T_b{\mathscr {B}}$ and all $b\in {\mathscr {B}}.$

The result is proved by an explicit hypercohomology calculation, using the Čech resolution of the relative de Rham complex $(\Omega ^\bullet _{{\mathcal {X}}/{\mathscr {B}}},d).$

4 Proof of Theorem A

We now turn to the proof of our main result, Theorem A. Recall that in the statement of the theorem we use the algebra homomorphism $\iota$ from Eq. (22), so $\iota (D\textbf{I})^{-1}$ is a global meromorphic section of ${\underline{End}} \left( {\mathbb {C}}^l \otimes _{\mathbb {C}}\bigoplus _{k\ge 0}\pi ^*K_X^k \right)$ with poles along the zeros of $\det \iota D\textbf{I}.$ That is, the homogeneous polynomials $\partial _i I_j\in \text {Sym}^{d_j-1}({\mathfrak {t}}^\vee )$ are considered as global sections of $\pi _b^*K_X^{d_j-1},$ or, after restriction to ${\widetilde{X}}_b,$ as sections of $p_b^*K_X^{d_j-1}.$

Using Cramer’s formula and the fact that $\iota$ is an algebra homomorphism, we can rewrite the right side of (6) as a linear combination of (restrictions of) $\lambda _i$ with coefficients of the kind

$$\begin{aligned} \left. \frac{\det \left[ \iota \partial _1 \textbf{I},\ldots , p_b^*\textbf{g}, \ldots ,\iota \partial _l \textbf{I}\right] }{\varvec{\alpha }_i \det \iota D\textbf{I}} \right| _{{\widetilde{X}}_b}\in {\mathcal {K}}({\widetilde{X}}_b), \end{aligned}$$

i.e., global meromorphic functions on ${\widetilde{X}}_b,$ since both the numerator and the denominator belong to $H^0\left( {\widetilde{X}}_b, p_b^*K_X^{\sum _i d_i - l+ 1} \right) .$

Now we prove Theorem A. Let us fix $\textbf{g}\in T_b{\mathcal {B}}=H^0(X,\bigoplus _i K_X^{d_i})$ and denote by $\textbf{s}\in H^0({\widetilde{X}}_b, {\mathfrak {t}}\otimes _{\mathbb {C}}K_{{\widetilde{X}}_b})$ the image of $\textbf{g}$ under the isomorphism (5). Let us also denote by ${\widetilde{\textbf{s}}}$ the section from the right hand side of Eq. (6), i.e.,

$$\begin{aligned} {\widetilde{\textbf{s}}} = - \left. \sum _{i=1}^l \frac{\left( \iota (D\textbf{I})^{-1} \cdot \pi ^{*}\textbf{g}\right) _i}{\varvec{\alpha }_i}\lambda _i \right| _{{\widetilde{X}}_b}. \end{aligned}$$

This is a meromorphic section of ${\mathfrak {t}}\otimes _{\mathbb {C}}K_{{\widetilde{X}}_b}$ with poles at most along the ramification of $p_b:{\widetilde{X}}_b\rightarrow X.$ We are going to prove that $\textbf{s}={\widetilde{\textbf{s}}}.$ We use Theorem 3.1 and the representation of the isomorphism $\beta$ from Eq. (25) is a composition of two maps.

As a first step, we show that

$$\begin{aligned} \left. \textbf{s}\right| _{{\widetilde{X}}_b\backslash \text {Ram}(p_b) } = \left. {\widetilde{\textbf{s}}}\right| _{{\widetilde{X}}_b\backslash \text {Ram}(p_b) }. \end{aligned}$$

(27)

For that we restrict the cameral cover to the complements of the ramification and branch divisors

$$\begin{aligned} p_b: {\widetilde{X}}_b\backslash \text {Ram}(p_b) \longrightarrow X\backslash \text {Bra}(p_b) \end{aligned}$$

and choose $U\subseteq X\backslash \text {Bra}(p_b),$ biholomorphic to an open disk (via $\psi :U\rightarrow \Delta$). In this case, ${\widetilde{X}}_U \subseteq {\widetilde{X}}\cap \left( \det \iota D\textbf{I}\ne 0\right)$ has |W| (analytic) connected components, each isomorphic to U,  labelled by the different Weyl chambers

$$\begin{aligned} {\widetilde{X}}_U={\widetilde{X}}_U^1\coprod \ldots \coprod {\widetilde{X}}_U^{|W|}. \end{aligned}$$

We choose (an analytic) local coordinate z on U and use z (i.e., its pullback $p_b^*z$) as a coordinate on ${\widetilde{X}}_U\subseteq {\widetilde{X}}_b\backslash \text {Ram}(p_b).$

Then, setting $\varvec{\gamma }$ for the coordinate vector of $p_b^*\textbf{g},$

$$\begin{aligned} (\phi ^{-1})^*\varvec{\gamma }= \begin{pmatrix} -\varvec{\beta }'&{} \left. D\textbf{I}\right| _{\phi ({\widetilde{X}}_U)}\\ \end{pmatrix} \begin{pmatrix} 0\\ \left. (D\textbf{I})\right| ^{-1}_{\phi ({\widetilde{X}}_U)}((\phi ^{-1})^*\varvec{\gamma })\\ \end{pmatrix}\in {\mathcal {O}}_\Delta ^{\oplus l}(\Delta ), \end{aligned}$$

i.e., we obtain a lift ${\widetilde{\varvec{\gamma }}}$ of $\varvec{\gamma }$

$$\begin{aligned} (\phi ^{-1})^*{\widetilde{\varvec{\gamma }}}= \sum _{i=1}^l (\left. (D\textbf{I})^{-1}\right| _{\phi ({\widetilde{X}}_U)}(\phi ^{-1})^*\varvec{\gamma })_i \frac{\partial }{\partial \alpha _i}\in \Gamma \left( \phi ({\widetilde{X}}_U),\left. (\phi ^{-1})^*T_M\right| _{{\widetilde{X}}_U}\right) . \end{aligned}$$

(28)

Note that the expression for ${\widetilde{\varvec{\gamma }}}$ is well-defined on ${\widetilde{X}}_U$: away from ramification, we can solve locally-analytically for $\alpha _i$ in terms of z,  so $(D\textbf{I})^{-1},$ when restricted to a connected component of $\phi ({\widetilde{X}}_U),$ is actually a section of $\text {End}({\mathbb {C}}^l)\otimes _{\mathbb {C}}{\mathcal {O}}^{an}_{\Delta }.$

Then, using the lift ${\widetilde{\textbf{g}}}$ from Eq. (28), we obtain

$$\begin{aligned} \left. {\widetilde{\textbf{g}}}\lrcorner \omega \right| _{{\widetilde{X}}_U}= & {} - \phi ^*\left( \sum _{i=1}^l \left( \left. (D\textbf{I})\right| _{\phi ({\widetilde{X}}_U)}^{-1}\cdot \varvec{\gamma }\right) _i \otimes e_i\otimes [dz] \right) \\{} & {} - \sum _{i=1}^l \left. \frac{\phi ^*\left( \left. (D\textbf{I})\right| _{\phi ({\widetilde{X}}_U)}^{-1}\cdot \varvec{\gamma }\right) _i}{\alpha _i} \lambda _i\right| _{{\widetilde{X}}_U}, \end{aligned}$$

as $(\phi ^{-1})^*\lambda _i= \alpha _i [dz]\otimes e_i.$ We write [dz] rather than dz since the cotangent sheaf of ${\widetilde{X}}_U$ is a quotient of $\left. \Omega ^1_{M_U}\right| _{{\widetilde{X}}_U}.$ This is precisely the expression for ${\widetilde{\textbf{s}}}$ from Eq. (6), written locally.

Having shown (27), we now note that the sheaf of meromorphic sections of a holomorphic vector bundle on a smooth curve is trivial (see e.g. [12, p.76]; see also [25, Lemma 31.25.3]). As two meromorphic functions that coincide away from a finite set of points are equal, Eq. (27) shows that $\textbf{s}= {\widetilde{\textbf{s}}}.$ Since the two sections $\textbf{s}$ and ${\widetilde{\textbf{s}}}$ are equal, and $\textbf{s}$ is known to be W-invariant, so is ${\widetilde{\textbf{s}}}.$ $\square$

5 Examples

5.1 $SL_2({\mathbb {C}})$

For completeness, we start with the simplest case of $G=SL_2({\mathbb {C}}).$ The Cartan subalgebra ${\mathfrak {t}}$ of diagonal traceless $2\times 2$ matrices is identified with ${\mathbb {C}}$ via $\alpha (A)=A_{11}$ and we take the ${\mathbb {Z}}/2{\mathbb {Z}}$-invariant polynomial $I=\det ,$ i.e., $I(\alpha )=-\alpha ^2.$ The cameral (and spectral) curve ${\widetilde{X}}_b\subseteq \text {tot }K^2_X$ has equation $\varvec{\alpha }^2=\pi ^*b,$ for $b\in {\mathcal {B}}=H^0(X,K_X^2).$ Then, for generic b,  the isomorphism (5)

is given by

$$\begin{aligned} \textbf{g}\longmapsto \left( \nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}\right) _b= \left. \frac{\pi ^*\textbf{g}}{2\varvec{\alpha }^2}\varvec{\lambda }_{SW}\right| _{{\widetilde{X}}_b} = - \left. \frac{\pi ^*\textbf{g}}{2\sigma }\lambda \right| _{{\widetilde{X}}_b}= \left. \frac{\pi ^*\textbf{g}}{2\varvec{\lambda }_{SW}}\right| _{{\widetilde{X}}_b} \end{aligned}$$

(29)

where $\sigma =-\varvec{\alpha }^2\in H^0(M,\pi ^*K_X^2)$ is the tautological section (10) of $\textbf{U}= K_X^2$ and $\varvec{\lambda }_{SW}=\varvec{\alpha }$ is the Liouville (Seiberg–Witten) form.

5.2 $SL_3({\mathbb {C}})$

Consider $G=SL_3({\mathbb {C}}),$ with the standard choices of Borel (upper-triangular) and Cartan (diagonal) subgroups. Here ${\mathfrak {t}}\subseteq {\mathfrak {s}}{\mathfrak {l}}_3({\mathbb {C}})$ is the subspace of diagonal traceless $3\times 3$ matrices and $W=S_3.$ If we set $\alpha _1(A)=A_{11}-A_{22},$ $\alpha _2(A)=A_{22}-A_{33}$ (two simple positive roots), then we can choose the invariant polynomials to be

$$\begin{aligned} \left| \begin{array}{l} I_1(\alpha _1,\alpha _2)= \alpha _1^2 +\alpha _1\alpha _2 +\alpha _2^2\\ I_2(\alpha _1,\alpha _2)= -2\alpha _1^3-3\alpha _1^2\alpha _2 +3\alpha _1\alpha _2^2+ 2\alpha _2^3.\\ \end{array} \right. \end{aligned}$$

(30)

In fact, these are $I_1(A)= -3(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33})$ and $I_2(A)=-27\det A.$

Consequently, the cameral curve ${\widetilde{X}}_b,$ corresponding to a generic section $b=(b_1,b_2)\in {\mathcal {B}}= H^0(X,K_X^2)\oplus H^0(X,K_X^3)$ is cut out in $M=\text {tot }(K_X^2\oplus K_X^3)$ by the equations

$$\begin{aligned} \left| \begin{array}{l} \varvec{\alpha }_1^2+\varvec{\alpha }_1\varvec{\alpha }_2+\varvec{\alpha }_2^2=\pi ^*b_1\\ -2\varvec{\alpha }_1^3-3\varvec{\alpha }_1^2\varvec{\alpha }_2+3\varvec{\alpha }_1\varvec{\alpha }_2^2+2\varvec{\alpha }_2^3= \pi ^*b_2\\ \end{array} \right. \end{aligned}$$

(31)

and

$$\begin{aligned} \left( \nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}\right) _b=\left. \frac{1}{\det \iota D\textbf{I}} \begin{bmatrix} 3\varvec{\alpha }_1^2- 6\varvec{\alpha }_1\varvec{\alpha }_2 - 6\varvec{\alpha }_2^2 &{} 2\varvec{\alpha }_2 + \varvec{\alpha }_1\\ -6\varvec{\alpha }_1^2 - 6\varvec{\alpha }_1\varvec{\alpha }_2 + 3\varvec{\alpha }_2^2 &{} - 2\varvec{\alpha }_1 -\varvec{\alpha }_2 \\ \end{bmatrix} \begin{bmatrix} g_1\\ g_2\\ \end{bmatrix}\right| _{{\widetilde{X}}_b}, \end{aligned}$$

where

$$\begin{aligned} \det \iota D\textbf{I}= 27 \varvec{\alpha }_1\varvec{\alpha }_2 (\varvec{\alpha }_1+\varvec{\alpha }_2). \end{aligned}$$

5.3 $G_2$

It is well-known [17, p.103] that the $G_2$ root system can be embedded in the $B_3$ root system—and that in fact, this can be done in a way that simple roots of the former are expressed as linear combinations of simple roots of the latter. An explicit description of such an embedding can be obtained by extending the calculations in [1, §4], but we do not need this now. Using this embedding, we can take the ${\mathfrak {g}}_2$ Cartan subalgebra ${\mathfrak {t}}$ to consist of diagonal matrices of the form $h=\text {diag}(-a-b,-a,-b,0,b,a,a+b),$ for $a,b\in {\mathbb {C}}.$ Two simple roots $\alpha _1,$ $\alpha _2\in {\mathfrak {t}}^\vee$ are, e.g., b and $a-b,$ i.e.,

$$\begin{aligned} \alpha _1\left( h\right) = h_{55},\ \alpha _2\left( h\right) = h_{66}-h_{55}. \end{aligned}$$

The six positive roots are then $\alpha _1,$ $\alpha _2,$ $\alpha _1+\alpha _2,$ $2\alpha _1+\alpha _2,$ $3\alpha _1+\alpha _2,$ $3\alpha _1+2\alpha _2.$ The characteristic polynomial of $h\in {\mathfrak {t}}$ is

$$\begin{aligned} \det (h-\lambda E_7)= -\lambda ^7+ \lambda ^5 2I_1(h) -\lambda ^3 I_1^2(h) + \lambda I_2(h), \end{aligned}$$

where, if we use $\alpha _1$ and $\alpha _2$ as coordinates on ${\mathfrak {t}},$ we have for the invariants

$$\begin{aligned} \left| \begin{array}{l} I_1(\alpha _1,\alpha _2)= 3\alpha _1^2+3\alpha _1\alpha _2+\alpha _2^2\\ I_2(\alpha _1,\alpha _2)= 4\alpha _1^6+12\alpha _1^5\alpha _2 + 13\alpha _1^4\alpha ^2_2+ 6 \alpha _1^3\alpha _2^3+ \alpha _1^2\alpha _2^4.\\ \end{array} \right. \end{aligned}$$

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The eigenvalues of a matrix from ${\mathfrak {g}}_2\subseteq {\mathfrak {s}}{\mathfrak {o}}_7$ are $0, \pm \lambda _1,\pm \lambda _2,\pm \lambda _3,$ $\sum _{i=1}^3 \lambda _i=0.$ The two invariants are, respectively, $\frac{1}{2}(\lambda _1^2+\lambda _2^2+\lambda _3^2)$ and $(\lambda _1\lambda _2\lambda _3)^2.$

Consequently, the cameral curve ${\widetilde{X}}_b,$ corresponding to a generic section $(b_1,b_2)\in {\mathcal {B}}= H^0(X,K_X^2)\oplus H^0(X,K_X^6)$ is cut out in $M=\text {tot }(K_X^2\oplus K_X^6)$ by the equations

$$\begin{aligned} \left| \begin{array}{l} 3\varvec{\alpha }_1^2+3\varvec{\alpha }_1\varvec{\alpha }_2+\varvec{\alpha }_2^2=\pi ^*b_1\\ 4\varvec{\alpha }_1^6+12\varvec{\alpha }_1^5\varvec{\alpha }_2 + 13\varvec{\alpha }_1^4\varvec{\alpha }^2_2+ 6 \varvec{\alpha }_1^3\varvec{\alpha }_2^3+ \varvec{\alpha }_1^2\varvec{\alpha }_2^4= \pi ^*b_2.\\ \end{array} \right. \end{aligned}$$

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Now, identifying the adjoint quotient $\chi : {\mathfrak {t}}\rightarrow {\mathfrak {t}}/W$ with $\textbf{I}=(I_1,I_2):{\mathbb {C}}^2\rightarrow {\mathbb {C}}^2,$ we obtain that under the isomorphism from Theorem A a section $\textbf{g}=(g_1,g_2)^T\in H^0(X,K_X^2)\oplus H^0(X,K_X^6)=T_b{\mathscr {B}}$ is mapped to $\left( \nabla ^{GM}_\textbf{g}\varvec{\lambda }_{SW}\right) _b,$ i.e.,

$$\begin{aligned} \left. \frac{1}{\det \iota D\textbf{I}} \begin{bmatrix} -2\varvec{\alpha }_1^2(6\varvec{\alpha }_1^3+13\varvec{\alpha }_1^2\varvec{\alpha }_2+9\varvec{\alpha }_1\varvec{\alpha }_2^2+2\varvec{\alpha }_2^3) &{} 3\varvec{\alpha }_1 + 2\varvec{\alpha }_2\\ 2\varvec{\alpha }_1(12\varvec{\alpha }_1^4+30\varvec{\alpha }_1^3\varvec{\alpha }_2+26\varvec{\alpha }_1^2\varvec{\alpha }_2^2+9\varvec{\alpha }_1\varvec{\alpha }_2^3+\varvec{\alpha }_2^4) &{} - 6\varvec{\alpha }_1 -3\varvec{\alpha }_2 \\ \end{bmatrix} \begin{bmatrix} \pi ^*g_1\\ \pi ^*g_2\\ \end{bmatrix}\right| _{{\widetilde{X}}_b}, \end{aligned}$$

where

$$\begin{aligned} \det \iota D\textbf{I}= -2\varvec{\alpha }_1\varvec{\alpha }_2(\varvec{\alpha }_1+\varvec{\alpha }_2)(2\varvec{\alpha }_1+\varvec{\alpha }_2)(3\varvec{\alpha }_1+\varvec{\alpha }_2)(3\varvec{\alpha }_1+2\varvec{\alpha }_2). \end{aligned}$$

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We see that in all examples $\det D\textbf{I}$ is a constant multiple of the product of all positive roots. In fact, this follows from a classical result of Steinberg [24]. Hence $(\det D\textbf{I})^2$ is proportional to the discriminant ${\mathfrak {D}}$ of ${\mathfrak {g}}$—the product of all roots. Being W-invariant, the discriminant can be expressed as a polynomial in the generators of ${\mathbb {C}}[{\mathfrak {t}}]^W$—here, $I_1$ and $I_2.$ We sketch a possible way of obtaining this expression without too much brute force. Using the embedding ${\mathfrak {g}}_2\subseteq {\mathfrak {s}}{\mathfrak {o}}_7,$ we can identify $\alpha _1$ and $\alpha _2$ as $\lambda _1$ and $\lambda _2-\lambda _1,$ up to reordering $\lambda _i$’s. Consequently,

$$\begin{aligned} {\mathfrak {D}}= & {} \alpha _1^2(\alpha _1+\alpha _2)^2(2\alpha _1+\alpha _2)^2\alpha _2^2(3\alpha _1+ \alpha _2)^2(3\alpha _1+2\alpha _2)^2\nonumber \\= & {} \lambda _1^2\lambda _2^2\lambda _3^2(\lambda _1-\lambda _2)^2(\lambda _1-\lambda _3)^2(\lambda _2-\lambda _3)^2. \end{aligned}$$

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The product of the first three terms is $I_2.$ The product of the last three terms is a polynomial of degree 6,  and hence must be a linear combination of $I_1^3$ and $I_2,$ and so one checks immediately that

$$\begin{aligned} {\mathfrak {D}}= I_2(4I_1^3-27 I_2). \end{aligned}$$

Consequently, the restriction of $\iota {\mathfrak {D}}$ to the universal cameral cover ${\mathcal {X}}$ is the pull-back of a section of ${\mathcal {O}}_{{\mathscr {B}}}\otimes H^0(X,K_X^{12}),$ namely,

$$\begin{aligned} {\mathscr {B}}\ni (b_1,b_2)\longmapsto b_2(4b_1^3-27b_2). \end{aligned}$$

It is the Lie derivative of this section that enters the Balduzzi–Pantev formula and its generalization [2, 3].

We refer the reader to the beautiful papers [15, 19] for additional details on the $G_2$-Hitchin system, including Langlands duality and the description of Hitchin fibres.