Towards a classification of charge-3 monopoles with symmetry

Abstract

We classify all possible charge-3 monopole spectral curves with non-trivial automorphism group and within these identify those with elliptic quotients. By focussing on elliptic quotients the transcendental constraints for a monopole spectral curve become ones regarding periods of elliptic functions. We construct the Nahm data and new monopole spectral curves with $D_6$ and $V_4$ symmetry, the latter based on an integrable complexification of Euler’s equations, and for which energy density isosurfaces are plotted. Extensions of our approach to higher charge and hyperbolic monopoles are discussed.

1 Introduction

In this letter, we shall classify and construct new charge-3 ${\text {SU}}(2)$ monopoles; throughout we will work just with this gauge group. To describe this work further we first recall some necessary background. Following the success of the ADHM construction of instanton solutions to the self-dual Yang–Mills equations Nahm introduced his eponymous equations to solving the reduction of the self-dual Yang–Mills equations $F = \star D\phi$ which describes monopoles; alternately these reduced equations arise as the BPS limit of a Yang–Mills–Higgs system. Here a (Euclidean BPS) monopole is the data $(A, \phi )$ where A is a connection of a ${\text {SU}}(2)$ principal bundle over ${\mathbb {R}}^3$ with associated curvature F, and $\phi$ (the Higgs field) is a section of the adjoint bundle. These satisfy the equation $F = \star D\phi$ as well as the boundary conditions that as $r \rightarrow \infty$ (i) $\left|\phi \right| = 1 - \frac{k}{2r} + {\mathcal {O}}(r^{-2})$, where $\left|\phi \right|^2=-\frac{1}{2} {{\,\textrm{Tr}\,}}(\phi ^2)$ is the norm coming from the Killing form on ${\mathfrak {s}}{\mathfrak {u}}(2)$ and r is the Euclidean distance from origin; (ii) $\frac{\partial \left|\phi \right|}{\partial \Omega } = {\mathcal {O}}(r^{-2})$ and $\Omega$ is a solid angle; (iii) $\left|D\phi \right| = {\mathcal {O}}(r^{-2})$. The (positive) integer k fixes the topological class of the data and is called the charge. Nahm’s modification of the ADHM construction then sought a triplet of $k\times k$ matrices of a real parameter $s\in [0,2]$ such that these satisfied (i) the Nahm equations

$$\begin{aligned} \frac{dT_i}{ds} = \frac{1}{2} \sum _{j,k=1}^3 \epsilon _{ijk} [T_j,T_k], \end{aligned}$$

(1)

together with (ii) the $T_i(s)$ are regular for all $s \in (0,2)$ and with simple poles at $s=0,2$ if $k>1$, whose residues form an irreducible k-dimensional representation of ${\text {SU}}(2)$; and (iii) $T_i(s) = -T_i^\dagger (s), \, T_i(s) = T^T_i(2-s)$. Hitchin in his seminal work [21] introduced yet a third description of monopoles, the spectral curve, and proved the equivalence of all three descriptions. Here the spectral curve ${\mathcal {C}} \subset T{\mathbb {P}}^1 \overset{\pi }{\rightarrow }{\mathbb {P}}^1$ is a compact algebraic curve with no multiple components of genus $(k-1)^2$ such that (i) ${\mathcal {C}}$ is real with respect to a (to be given) anti-holomorphic involution $\tau$; (ii) there is a family of line bundles ${\mathcal {L}}^{s}$ on ${\mathcal {C}}$ such that ${\mathcal {L}}^2$ is trivial and ${\mathcal {L}}(k-1):={\mathcal {L}}\otimes \pi ^*{\mathcal {O}}_{{\mathbb {P}}^1}(k-1)$ is real; and (iii) $H^0({\mathcal {C}}, {\mathcal {L}}^s(k-2))=0$ for all $s \in (0,2)$. Introducing coordinates $\zeta , \eta$ on $T{\mathbb {P}}^1$ corresponding to the base ${\mathbb {P}}^1$ and fibre, respectively, then ${\mathcal {L}}^{s} \rightarrow T{\mathbb {P}}^1$ (and by restriction, to ${\mathcal {C}}$) is the (holomorphic) line bundle defined by $\exp (s\eta /\zeta )$ with $\eta /\zeta \in H^1(T{\mathbb {P}}^1, {\mathcal {O}})$. Note that $T{\mathbb {P}}^1$ arises here because it is the mini-twistor space of oriented geodesics in Euclidean 3-space [20]. The action of $\tau$ is $(\zeta , \eta ) \mapsto (-1/{\bar{\zeta }}, -{\bar{\eta }}/{\bar{\zeta }}^2)$, and so a generic spectral curve satisfying the reality constraints may be written as the zero set of a polynomial

$$\begin{aligned} 0 = P(\zeta , \eta ):= \eta ^k + \sum _{r=1}^{k} p_{2r}(\zeta ) \eta ^{k-r}, \quad p_{2r}(\zeta ) = (-1)^r \zeta ^{2r} \overline{p_{2r}(-1/{\bar{\zeta }})}, \end{aligned}$$

where $p_{2r}$ is a polynomial of degree 2r in $\zeta$. In what follows, we shall refer to Nahm data as matrices $\{T_i\}$ satisfying Nahm’s three constraints and a monopole spectral curve as a curve ${\mathcal {C}}$ satisfying Hitchin’s three constraints.

Our understanding of monopoles, the self-duality equations and Nahm’s equation have developed greatly in the 40 years since [21]. The moduli space of monopoles of a given charge has attracted much attention, and a rational map description [14] allows different insights and facilitates numerical solutions; integrable systems techniques have also been brought to the fore [11, 17]. Yet despite this progress very few monopole spectral curves have been found in the intervening period owing to the transcendentality of the Hitchin conditions (see [8]). While monopoles of charge 1 and 2 are well-understood (for a review, see [12]) little progress has been made for higher charges. In all cases known the use of symmetry to simplify the conditions has been required; in nearly all of these we may quotient by the group of symmetries to an elliptic curve. Motivated by this history our first result here is to classify all possible charge-3 monopole spectral curves by their automorphism groups and within these identify those with elliptic quotients. From these we will then construct new monopole spectral curves with $D_6$ and $V_4$ symmetry. In Sect. 2 we prove:

Theorem 1.1

Let ${\mathcal {C}} \subset T{\mathbb {P}}^1$ be a charge-3 monopole spectral curve with $H \le {{\,\textrm{Aut}\,}}({\mathcal {C}})$ such that the quotient genus $g({\mathcal {C}}/H) = 1$. Then, up to an automorphism of $T{\mathbb {P}}^1$, the curve is given by the vanishing of one of the following 5 forms:

• 1. $\eta ^3 + \eta [(a+ib) \zeta ^4 + c \zeta ^2 + (a-ib)] + [(d+ie)\zeta ^6 + (f+ig) \zeta ^4 - (f-ig) \zeta ^2 - (d-ie)]$,

• 2. $\eta ^3 + \eta [a(\zeta ^4 + 1) + b\zeta ^2] + ic\zeta (\zeta ^4-1)$,

• 3. $\eta ^3 + a \eta \zeta ^2 + ib\zeta (\zeta ^4 - 1)$,

• 4. $\eta ^3 + a\eta \zeta ^2 + b(\zeta ^6 - 1)$,

• 5. $\eta ^3 + ia\zeta (\zeta ^4 - 1)$,

where $a, b, c, d, e, f, g \in {\mathbb {R}}$.

Our result does not itself guarantee the existence of monopole spectral curves in these families, and the classes intersect (for example, 5 is a special case of 3). In previous works monopole spectral curves of the form 3 and 5 have been understood in [26] and [22] as corresponding to charge-3 twisted line scattering and the tetrahedrally-symmetric monopole, respectively, while one special case of the form 2 was understood in [25] as the class of inversion-symmetric monopoles, with another in [21] as the axially-symmetric 3-monopole. Curves of the form 2 had been observed in [23, (3.71)], but the Hitchin constraints were only imposed for a restricted subset. In Theorem 3.2 we determine the general monopole spectral curve and Nahm data in class 4 and in Theorem 4.3 the same for class 2. This provides the necessary data in order to plot energy density isosurfaces following [24]¹; Fig. 1 gives for example a previously unknown $V_4$ configuration. Along with the class of charge-3 monopoles described via an implicit condition in [9], these form all the charge-3 monopole spectral curves currently known, which fit together as shown in Fig. 2 for some parameter values. Figure 3 shows the relations between the symmetry groups of the curves.

Fig. 1

Surface of constant energy density ${\mathcal {E}}=0.18$ for the $V_4$ monopole given by the parameters (see Theorem 4.3) $m=0.6$, $\alpha = -2.0$, ${{\,\textrm{sgn}\,}}= 1$

Our approach is as follows. In Sect. 2 of the paper we will prove Theorem 1.1. Once we have the automorphism groups of interest we will take the procedure introduced in [22] and developed in [24,25,26] and apply this to the relevant symmetry. This procedure is recalled in Appendix B where Nahm’s equations for case 4 are reduced to the Toda equations before further reduction is described in the text. Similarly in Appendix C Nahm’s equations for the $V_4$ symmetric case are determined. This yields a complex extension of the Euler equations. We then show how these equations are solved in terms of elliptic functions on the quotient elliptic curve, first in Sect. 3 for the $D_6$-symmetric monopole and then in Sect. 4 for the $V_4$-symmetric monopole. Here the rationale for focussing on elliptic quotients is most evident: the transcendental constraints implicit in the works of Hitchin and Ercolani-Sinha become ones regarding periods of elliptic functions. We relegate to Appendix A a number of properties of elliptic and related functions used in the text and proofs of some statements requiring these. We will not deal with the remaining $C_2$-symmetric case here discussing this further in Sect. 5 which is a conclusion.

Fig. 2

Known charge-3 spectral curves and their relations. We do not specify the constraints on the parameters

Fig. 3

Automorphism groups of known charge-3 spectral curves and their relations, presented as G or $H \le G$ where G is the full automorphism group and H is the subgroup quotienting to an elliptic curve when it exists

2 Classifying curves by automorphism group

In this section, we determine the charge-3 monopole spectral curves we shall focus on, beginning with minimal restrictions and gradually imposing these.

A monopole spectral curve is a compact algebraic curve ${\mathcal {C}}$ lying in Euclidean mini-twistor space ${{\mathbb {M}}}{{\mathbb {T}}}$, the space of oriented lines in Euclidean 3-space. If the direction of the oriented line is given by $\zeta$, an affine coordinate of $[\zeta _0:\zeta _1]\in {\mathbb {P}}^1$, and $\eta \in {\mathbb {C}}$ describes the point in the plane perpendicular to this through which the line passes then we have $\eta \partial _\zeta \in T{\mathbb {P}}^1 \cong {{\mathbb {M}}}{{\mathbb {T}}}$. A generic charge-k monopole spectral curve may then be written as the zero set of a polynomial $0 = \eta ^k + \sum _{r=1}^{k} p_{2r}(\zeta _0,\zeta _1) \eta ^{k-r},$ where $p_{2r}$ is a homogeneous polynomial of degree 2r in $\zeta _0$, $\zeta _1$; equivalently a polynomial of degree 2r in $\zeta$. Now $T{\mathbb {P}}^1$ is non-compact and two compactifications of this are common, either by inclusion in the (singular) weighted projective space ${\mathbb {P}}^{1,1,2}= \{(\zeta _0,\zeta _1, \eta )\in {\mathbb {C}}^3{\setminus }\{0\} \,|\, (\zeta _0,\zeta _1,\eta )\sim (\lambda \zeta _0,\lambda \zeta _1,\lambda ^2\eta ), \ \lambda \in {\mathbb {C}}^*\},$ or (as by Hitchin) in the Hirzebruch surface ${\mathbb {F}}_2$. We adopt the former view and note that the singular point [0 : 0 : 1] does not lie in $T{\mathbb {P}}^1$ and hence on ${\mathcal {C}}$. Next, via the Veronese embedding, we have $\iota :{\mathbb {P}}^{1,1,2} \hookrightarrow {\mathbb {P}}^{3}$, $\iota ([\zeta _0:\zeta _1: \eta ])= [\zeta _0^2:\zeta _0\zeta _1:\zeta _1^2: \eta ]$. Under this a homogeneous polynomial of degree 2r becomes a homogeneous polynomial of degree r in the new coordinates and ${\mathbb {P}}^{1,1,2}$ becomes a cone, a quadric, over the cone point $\iota ([0:0:1])$ (see, for example, [37, §8.2.11]). Thus a monopole spectral curve may be viewed as the complete intersection of a quadric cone and a degree-k hypersurface in ${\mathbb {P}}^{3}$. This is known to be a curve of genus $(k-1)^2$ (see for example [18, Exercise V.2.9]) which is non-hyperelliptic for $k\ge 3$ ( [18, Exercise IV.5.1]). For $k=3$ we then have that ${\mathcal {C}}$ is a non-hyperelliptic genus-4 curve.

In 1895 Wiman [39] classified all non-hyperelliptic genus-4 curves by their automorphism group and gave explicit defining equations for these. Wiman’s classification had two families: curves arose either as the intersections of a cubic surface and non-singular quadric in ${\mathbb {P}}^{3}$, or as the intersection of a cubic surface and quadric cone in ${\mathbb {P}}^{3}$. Thus charge-3 monopole spectral curves with automorphism group must lie in Wiman’s second family. (The two rulings of the non-singular quadric of Wiman’s first family lead to projections from the curve to ${\mathbb {P}}^{1}\times {\mathbb {P}}^{1}$, which is relevant for spectral curves of hyperbolic monopoles; this will be developed elsewhere.) We note that although ${{\,\textrm{Aut}\,}}({\mathbb {P}}^{3})={{\,\textrm{PGL}\,}}(4,{\mathbb {C}})$ differs from ${{\,\textrm{Aut}\,}}({\mathbb {P}}^{1, 1, 2})\cong {\mathbb {C}}^{3}\rtimes \left( {{\,\textrm{GL}\,}}(2,{\mathbb {C}}))/\{\pm {{\,\textrm{Id}\,}}_2\} \right)$ (with the natural action of ${{\,\textrm{GL}\,}}(2,{\mathbb {C}})$ on $(\zeta _0,\zeta _1)$ inducing that on $(\zeta _0^2,\zeta _0\zeta _1,\zeta _1^2)$), Wiman, in determining his normal forms, considered only those transformations of ${{\,\textrm{Aut}\,}}({\mathbb {P}}^{3})$ that preserved the cone, and so his normal forms include all possible charge-3 monopole spectral curves. In Table 1, we give those curves in Wiman’s classification which lie on a cone presenting² these in terms of a curve given by the vanishing of a polynomial P(x, z). We also write down their full automorphism group $G:= {{\,\textrm{Aut}\,}}({\mathcal {C}})$ and the corresponding signature $c:= c_G = (g_0; c_1, \dots , c_r)$ giving the quotient genus $g_0 = g({\mathcal {C}}/G)$ and the ramification indices $c_i$ of the quotient map ${\mathcal {C}} \rightarrow {\mathcal {C}}/G$ (see [30]). These have been calculated with the help of the information available from [29]. We make some remarks about Table 1.

• The label $D_n$ refers to the dihedral group of order 2n, following the convention [29].

• Wiman’s parameters are to be understood as generic: there may exist specific values of the parameters for which the automorphism group is larger than that indicated.

• Wiman provides a form where the $z^2$ term is always zero, equivalent to centring the monopole.

• All the curves given are irreducible, so we can only find reducible spectral curves as limiting members of the above families.

• The completeness of the above data on signatures and elliptic quotients is reliant on the completeness of the data of the LMFDB.

• We recognise the curve with $C_3 \times S_4$ symmetry as corresponding to the tetrahedrally-symmetric monopole.

Table 1 Potential charge-3 monopole spectral curves with nontrivial automorphism group and those (with subgroups) quotienting to genus 1

Not all curves on the list will yield monopoles spectral curves; for example by the following result.

Proposition 2.1

([10]) There are only two curves in the family $\eta ^3 + \chi (\zeta ^6 + b \zeta ^3 + 1)=0$, $\chi , b \in {\mathbb {R}}$, that correspond to BPS monopoles; these are tetrahedrally-symmetric monopole spectral curves.

2.1 Moduli space dimension

We now also briefly discuss another aspect of this group theoretic approach that can be used to identify particularly tractable monopole curves.

There is a moduli space $N_k$ of charge-k monopoles up to gauge transform, which one typically enlarges by a phase to $M_k$ which has a natural action of the Euclidean group ${\text {E}}(3)$ and circle group U(1). Associated with this is the submanifold of (strongly-) centred charge-k monopoles $M_k^0 \subset M_k$ with action of the orthogonal group ${\text {O}}(3)$ which parametrises monopoles up to gauge transform with fixed centre [4, 31]. The spectral curves corresponding to such monopoles have $a_1(\zeta )=0$, while the corresponding Nahm data have ${{\,\textrm{Tr}\,}}(T_i)=0$. $M_k^0$ is a totally geodesic manifold of real dimension $4(k-1)$, and given $G \le {\text {O}}(3)$ we may consider the submanifold of the moduli space of G-invariant strongly-centred monopoles $(M_k^0)^G$, which is also totally geodesic [6]. This we shall distinguish from $(M_k^0)^{[G]}$ which is the moduli space of monopoles invariant under an element of the conjugacy class of G inside ${\text {SO}}(3)$; clearly $(M_k^0)^G \subset (M_k^0)^{[G]}$ and when G is discrete we have $\dim _{\mathbb {R}} (M_k^0)^G = \dim _{\mathbb {R}} (M_k^0)^{[G]} - \dim _{\mathbb {R}} {\text {SO}}(3)$. As scattering of magnetic monopoles is approximated by geodesic motion in the moduli space [34], if one can find $G \le {\text {O}}(3)$ for which $\dim (M_k^0)^G = 1$, it is known that the corresponding 1-parameter family of monopoles corresponds to a scattering process. We may use group theory to help identify such families via the following result.

Proposition 2.2

For discrete G, $\dim _{{\mathbb {R}}} (M_3^0)^G \le 3g_0 - 3 + r$.

Proof

[30, Lemma 3.1] gives that the complex dimension of each component of the locus of equivalence classes of genus $g \ge 2$ curves admitting an action of a group isomorphic to G with signature c is (provided it is non-empty) $\delta (g, G, c):= 3(g_0-1)+r = \dim _{{\mathbb {C}}} {\mathcal {M}}_{g_0, r}$ the moduli space of genus $g_0$ curves with r marked points. The ${\text {SO}}(3)$ action on $(M_3^0)^{[G]}$ is trivial on the moduli space of curves because it induces a birational isomorphism. The result then follows as the ${\text {SO}}(3)$ orbits of the moduli space of monopoles will form a component of this locus, hence³$\dim _{\mathbb {R}} (M_3^0)^G = \dim _{\mathbb {R}} (M_3^0)^{[G]} - \dim _{\mathbb {R}} {\text {SO}}(3) \le \dim _{\mathbb {R}} \left( {\mathcal {M}}_{g_0, r}^\tau \right)$, and using the fact from Teichmüller theory that $\dim _{\mathbb {R}} \left( {\mathcal {M}}_{g_0, r}^\tau \right) = \dim _{\mathbb {C}}{\mathcal {M}}_{g_0, r}$ [16, §3.1]. $\square$

Remark 2.3

Note in the above we could have used $H \le G$ and its corresponding signature, but this would have given a weaker bound as $\delta (g, G, c_G) \le \delta (g, H, c_H)$ [30].

The remarkable fact about this is that the bound depends on the signature only. The curves of [22], with automorphism group H of order $2k(k-1)$ for $k=3, 4, 6$, have signature $c_H = (1;k-1)$ [22, Proposition 4]. For these curves and all known monopole spectral curves of charge 3 we have $\dim (M_k^0)^H = \delta (g, H, c_H) - 1$, and hence one might conjecture that this is always true in the case $g_0=1$. A calculation for the case of inversion-symmetric monopoles considered in [6] for which there is a $C_2$ action $\eta \mapsto -\eta$ shows that such a conjecture would certainly not be true for all $g_0$.

2.2 Genus-1 reductions

Table 1 gives us a list of putative spectral curves with symmetry before we have imposed the further constraints of Hitchin. We know from [21] that Nahm’s equations correspond to a linear flow in the Jacobian of the corresponding spectral curve ${\mathcal {C}}$; the direction of this linear flow is given by the Ercolani-Sinha vector ${{\varvec{U}}}$ [17]. Braden [7] has shown that when we have a symmetry group G we may be able to reduce to the quotient curve ${\mathcal {C}} \overset{\pi }{\rightarrow }\ {\mathcal {C}}':={\mathcal {C}}/G$ and reduced Ercolani-Sinha vector ${\varvec{U}}'$ when ${{\varvec{U}}} = \pi ^*{\varvec{U}}'$. For example charge-k monopoles with $C_k$ symmetry reduce to questions about a genus-$(k-1)$ hyperelliptic curve. The $k=3$ case was studied in [9].

Notwithstanding the attendant simplifications, the list of Table 1 is too long for the purposes of this letter and we require a further criterion to reduce this. Here we adopt the following: does the genus-4 spectral curve (assumed with real structure) quotient (either by ${{\,\textrm{Aut}\,}}({\mathcal {C}})$ or a subgroup) to an elliptic curve? The rationale for this is that the remaining of Hitchin’s conditions are most straightforwardly answered for elliptic curves; equivalently the Ercolani-Sinha constraint becomes one on the real period of an elliptic curve. There are also a number of curves known with this property [22, 24,25,26].

Thus we seek curves ${\mathcal {C}}$ with real structure from Wiman’s list for which there exists $H \le {{\,\textrm{Aut}\,}}({\mathcal {C}})$ such that $g({\mathcal {C}}/H)=1$. Here we may use the database of [29] which has enumerated all the possible H and the corresponding signatures for genus-4 curves. We may then use our knowledge of the explicit forms of the curves to match up these cases, which leaves us with the reduced list in the final two columns of Table 1. As previously noted, the $H = A_4$ case corresponds to the tetrahedrally-symmetric monopole [22], and the $H=C_4$ case has already been solved in [26]. We also see that the cases $H= S_3$ and $H=C_6$ arise from the same curve, indicating that the curve has two distinct quotients to an elliptic curve.

In the following sections we will investigate in more detail the two new cases $H=C_6$ (or equivalently $H=S_3$) with full automorphism group $G=D_6$, and $H = V_4$ (with full automorphism group $G=V_4$); we do not treat the $C_2$ case here. We will begin with the $D_6$ case which is both illustrative and simpler, though ultimately the new solutions and their scattering family are less interesting.

Before turning to these however we may complete the proof of Theorem 1.1. With the exception of the $H=C_3$ curve, imposing reality on the curves with groups H listed in Table 1 yields the curves of Theorem 1.1 (and in the same order). Note not all $M \in {{\,\textrm{GL}\,}}_2({\mathbb {C}})/\left\langle -1 \right\rangle \le {{\,\textrm{Aut}\,}}({\mathbb {P}}^{1, 1, 2})$ will commute with the action of $\tau$, but $S \in SU(2)$ will. One may use Schur decomposition to write $M = S T S^{-1}$ for some $S \in SU(2)$, T an upper-triangular matrix, and so when imposing reality on Wiman’s normal forms one should consider the orbits under upper-triangular matrices. The only real forms present in the orbit of the $G=H=C_3$ family have $a=b=0$ in the corresponding defining equation P; the resulting curve then lies in the family described by Proposition 2.1. Only the tetrahedrally-symmetric monopole within this family quotients to an elliptic curve and by a rotation this may written as $\eta ^3 + ia\zeta (\zeta ^4 - 1)=0$, the final entry of the Theorem. We have thus established Theorem 1.1.

3 $D_6$ monopoles

To understand the spectral curves of this section it helps to first begin with the general centred $C_k$ invariant spectral curve ${\mathcal {C}}$ (with reality imposed),

$$\begin{aligned} \eta ^k+ \alpha _2\eta ^{k-2}\zeta ^2+ \alpha _3\eta ^{k-3}\zeta ^3+ \cdots + \alpha _{k-1}\eta \,\zeta ^{k-1}+ \alpha _k\zeta ^{k}+\beta [ \zeta ^{2k}+(-1)^{k}]=0,\nonumber \\ \end{aligned}$$

(2)

where $\alpha _k,\beta \in {\mathbb {R}}$. This is invariant under $s: (\eta ,\zeta )\rightarrow (\omega \eta ,\omega \zeta )$, $\omega =\exp (2\pi \imath /k)$ with $C_k=\left\langle s \right\rangle$. The work of [7] shows (2) is the unbranched cover of the hyperelliptic curve

$$\begin{aligned} y^2&=(x^k+ \alpha _2 x^{k-2}+ \alpha _3 x^{k-3}+ \cdots + \alpha _{k})^2 -(-1)^k 4\beta ^2, \end{aligned}$$

(3)

where $x=\eta /\zeta$ and $y=\beta [ \zeta ^{k} - (-1)^k\zeta ^{-k}]$. The curve (2) also has the symmetry $t:(\zeta , \eta ) \mapsto (-1/\zeta , -\eta /\zeta ^2)$ and $G=\left\langle s,t \right\rangle =D_k$ is the full automorphism group. The transformation t corresponds to a reflection $\mathbf {\underline{v}}\rightarrow {{\,\textrm{diag}\,}}(1,-1,1)\mathbf {\underline{v}}$ in ${\text {O}}(3,{\mathbb {R}})$ and it becomes the hyperelliptic involution $t:(x,y)\rightarrow (x,-y)$ on the quotient curve. For $k=3$ we are describing the curve in Table 1 with full automorphism group $G=S_3$ ($=D_3$ in [29] notation). Further, the work of [7] shows that Nahm’s equations for every charge-k monopole with $C_k$ rotational symmetry are equivalent to the $A_{k-1}^{(1)}$⁴ affine Toda equations (in real Flaschka variables)

$$\begin{aligned} a_i^\prime = \frac{1}{2}a_i (b_i - b_{i+1}), \quad b_i^\prime = a_i^2 - a_{i-1}^2, \end{aligned}$$

(4)

where i is taken mod k, and we use${}^\prime$ to denote $\frac{d}{ds}$. These equations may also be found in other ways. In Appendix B we describe how taking the $C_k$ invariant polynomials $Q_i = \zeta _0^i \zeta _1^i, i=1, \dots , k$ and $Q_{k+1} = \zeta _0^k - \zeta _1^k$ as the inputs to the procedure of [22] we obtain equations (4).

In general the solutions to the Toda system (4) linearise on the genus-$(k-1)$ Jacobian of the curve (3). For $k=3$ this was the approach taken in [9] where a family of monopoles including the tetrahedrally-symmetric monopole was investigated. Are simplifications possible? In the remainder of this section, we shall show that for $k=3$ a one-parameter family of elliptic Nahm data exists with $C_3$ symmetry. As this is contrary to results in the literature we begin with four results that lead to such an elliptic reduction before determining the parameters that yield Nahm data. Three particular points in the family will be identified before concluding with a description of the scattering described by the family.

3.1 Four lessons

For $k=3$ equations (4) take the form (with $a_0\equiv a_3$, $b_0\equiv b_3$)

$$\begin{aligned} \begin{aligned} a_0^\prime =&\frac{1}{2}a_0 ( b_3 - b_1),&a_1^\prime =&\frac{1}{2} a_1 (b_1 - b_2),&a_2^\prime =&\frac{1}{2} a_2 ( b_2 - b_3),\\ b_1^\prime =&a_1^2 - a_0^2,&b_2^\prime =&a_2^2 - a_1^2,&b_3^\prime =&a_0^2 - a_2^2, \end{aligned} \end{aligned}$$

(5)

coming from Nahm matrices derived in Appendix B given by

$$\begin{aligned} T_1&= \frac{1}{2}\begin{pmatrix} 0 &{}\quad a_1 &{}\quad -a_0 \\ -a_1 &{}\quad 0 &{}\quad a_2 \\ a_0 &{}\quad -a_2 &{}\quad 0 \end{pmatrix},&T_2 = \frac{1}{2i} \begin{pmatrix} 0 &{}\quad a_1 &{}\quad a_0 \\ a_1 &{}\quad 0 &{}\quad a_2 \\ a_0 &{}\quad a_2 &{}\quad 0 \end{pmatrix}, \nonumber \\ T_3&= \frac{-i}{2} \begin{pmatrix} b_1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad b_2 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad b_3 \end{pmatrix}. \end{aligned}$$

(6)

Here, we find the constants

$$\begin{aligned} \alpha _2 {=} b_{1} b_{2} {+} b_{1} b_{3} {+} b_{2} b_{3} {+} a_{0}^{2} {+} a_{1}^{2} {+} a_{2}^{2}, \alpha _3 {=} b_{1} b_{2} b_{3} {+} b_{1} a_{2}^{2} {+} b_{2} a_{0}^{2} {+} b_{3} a_{1}^{2}, \quad \beta = a_0 a_1 a_2, \end{aligned}$$

for the (centred) spectral curve

$$\begin{aligned} \eta ^3 + \alpha _2 \eta \zeta ^2 + \alpha _3 \zeta ^3 + \beta (\zeta ^6 - 1)=0 \end{aligned}$$

which covers

$$\begin{aligned} y^2=(x^3+ \alpha _2 x+ \alpha _3)^2 + 4\beta ^2. \end{aligned}$$

(7)

We have 6 differential equations, 6 variables and three conserved quantities.

3.1.1 Direct simplification

Appendix B shows that we may use the constants $\alpha _2,\alpha _3, 0 = \sum b_i$ to eliminate the $b_i$, resulting in the equations

$$\begin{aligned} 0 {=} \sum _{i=0}^2 a_i^2 {-} \alpha _2 {-} \frac{1}{3}(d_1^2 {+} d_1 d_2 {+} d_2^2), \ 0 {=} a_1^2 d_2 {-} a_2^2 d_1 {+} \alpha _3 {+} \frac{1}{3}\alpha _2(d_1 - d_2) {+} \frac{1}{27}(d_1 - d_2)^3, \end{aligned}$$

where we have introduced $d_i = {2a_i^\prime }/{a_i}$. Using $\beta = a_0 a_1 a_2$ to eliminate $a_0$ we then have two nonlinear ODE’s in two variables, the maximal reduction one can achieve with generic $\alpha _i$ and $\beta$. The appendix shows further that if $a_1^2 = a_2^2$ additional simplification is possible; and that we may consistently set $a_1^2 - a_2^2 = 0$ provided $b_2 a_1^2 = 0$. As $b_2^\prime = a_2^2 - a_1^2$, this means we can consistently set $a_1^2 = a_2^2$ and $b_2 = 0$. Making these restrictions we find $\alpha _3=0$ and that we reduce to one equation

$$\begin{aligned} a_1^2\left( {2\frac{da_1}{ds}}\right) ^2 = \beta ^2 + 2 a_1^6 - \alpha _2 a_1^4. \end{aligned}$$

Upon setting $u=a_1^2$ this becomes

$$\begin{aligned} \left( {\frac{du}{ds}}\right) ^2 = \beta ^2 + 2 u^3 - \alpha _2 u^2, \end{aligned}$$

(8)

to which we shall return. We record that the j-invariant of the associated elliptic curve $y^2=\beta ^2 + 2 u^3 - \alpha _2 u^2$ is $16\alpha _2^6/(\beta ^2[\alpha _2^3-27 \beta ^2] )$.

3.1.2 Sutcliffe’s ansatz

Some time ago, in the context of Seiberg-Witten theory, Sutcliffe [35] gave an ansatz for charge-k cyclically symmetric monopoles in terms of affine Toda theory. With $a_j=\gamma e^{(q_j-q_{j+1})/2}$, $b_j=q_j'$ equations (4) follow from a Hamiltonian⁵$H=\frac{1}{2} \sum _{j=1}^{k} b_j^2-\sum _{j=0}^{k-1} a_j^2 =\frac{1}{2} \sum _{j=1}^{k} p_j^2-\gamma ^2\sum _{j=0}^{k-1} e^{q_j-q_{j+1}}.$ (The constant $\gamma$ here is to account for the constant $\prod _{i=0}^{k-1}a_i=\gamma ^k=(-1)^{k-1}\beta$.) Sutcliffe showed that for $k=2$ Nahm data could be constructed, but for $k=3$ although he could solve the equations he couldn’t find solutions with the correct pole behaviour. The solution was obtained from the infinite chain solution as follows. We have from

$$\begin{aligned} \left( \ln a_j^2\right) ''=-a_{j-1}^2+2 a_{j}^2-a_{j+1}^2 \end{aligned}$$

and the standard elliptic function identity for the Weierstrass $\wp$-function

$$\begin{aligned} \frac{d^2}{du^2}\,\ln [ \wp (u)-\wp (v) ]= -\wp (u+v) +2 \wp (u)-\wp (u-v) \end{aligned}$$

that with $u= j u_0+t+t_0$ and $v=u_0$ then

$$\begin{aligned}{} & {} \frac{d^2}{dt^2}\,\ln [ \wp (j u_0+t+t_0)-\wp (u_0) ]=- \wp ([j+1] u_0+t+t_0) \\{} & {} \quad +2 \wp (j u_0+t+t_0) -\wp ([j-1] u_0+t+t_0) \end{aligned}$$

and we may identify $a_j=\wp (j u_0+t+t_0)- \wp (u_0)$. This yields the solution for the infinite chain and we must still impose periodicity to obtain a solution. Imposing periodicity yields (for $k=3$) that $a_j=\wp (2j K/3 +t)-\wp (2K/3)$ which is equivalent to the solution of [35] which is given in Jacobi elliptic functions.⁶ Now the ansatz employed here forces only one of the $a_j$ to be singular at any point, and this means the pole condition on the Nahm matrices cannot be satisfied. If we are to find an alternative solution that does indeed yield a monopole then this would suggest that one appropriate route would be to pick a simplification which forces multiple variables to have poles simultaneously. Such is the case when $a_1^2=a_2^2$ found previously.

3.1.3 Imposing symmetry on Nahm matrices

We next show that $a_1^2=a_2^2$ follows from the symmetry⁷$\mathbf {\underline{v}}\rightarrow A\mathbf {\underline{v}}:= {{\,\textrm{diag}\,}}(1,-1,-1)\mathbf {\underline{v}}$ in ${\text {SO}}(3,{\mathbb {R}})$. From [31, Equation (8.169)] we know that the conditions for the Nahm matrices to be symmetric under A are that

$$\begin{aligned} T_1 = CT_1 C^{-1}, \quad -T_2 = C T_2 C^{-1}, \quad -T_3 = C T_3 C^{-1}, \end{aligned}$$

for some constant invertible matrix C. Recalling the form of the $T_i$ from (6) we see that as $T_3$ is diagonal and traceless the only way to achieve the invariance under A is if at least one of the $b_i$ is 0 and C permutes the other two. By conjugating with a permutation matrix we can without loss of generality pick $b_2 = 0$ so $b_1 = -b_3$ which gives that the generic C is $C = \left( \begin{array}{lll} 0 &{}\quad 0 &{}\quad a \\ 0 &{}\quad b &{}\quad 0 \\ c &{}\quad 0 &{}\quad 0 \end{array}\right)$. Picking a generic a, b, c we get

$$\begin{aligned} a_0(a+c) = a_1 a - a_2 b = a_1 c + a_2 b = a_1 b + a_2 c = a_2 a - a_1 b = 0. \end{aligned}$$

To avoid having an $a_i = 0$ we required $a= -c$, and so these reduce to

$$\begin{aligned} a_1 a - a_2 b = 0 = a_1 b - a_2 a, \end{aligned}$$

and consequently $(a/b)^2=1$ and $a_1 =\pm a_2$ yielding the desired $a_1^2=a_2^2$. Note that this also means $\alpha _3 = 0$.

3.1.4 Reduction of the spectral curve; folding

In order to get the curve with $D_6$ symmetry of Table 1, we must set $\alpha _3=0$. We have seen that for $k=3$ this is a consequence of the symmetry $r:(\zeta ,\eta )\rightarrow (1/\zeta ,-\eta /\zeta ^2 )$. For general k this means we keep only the even terms of (2),

$$\begin{aligned} \eta ^k+ \alpha _2\eta ^{k-2}\zeta ^2+ \alpha _4\eta ^{k-4}\zeta ^4+ \cdots +\beta [ \zeta ^{2k}+(-1)^{k}]=0. \end{aligned}$$

(9)

The full automorphism group of this curve is $D_{k}\times C_2$; for $k=3$ this is the curve with full automorphism group $D_6\cong D_3\times C_2$ that we are interested in. Setting $x=\eta /\zeta$ in (9) we have

$$\begin{aligned} x^k+ \alpha _2 x^{k-2}+ \alpha _4 x^{k-4}+ \cdots + \alpha _k+\beta [ \zeta ^{k}+\zeta ^{-k}]&=0,{} & {} k\ \text {even},\\ x^k+ \alpha _2 x^{k-2}+ \alpha _4 x^{k-4}+ \cdots + \alpha _{k-1} x+\beta [ \zeta ^{k}-\zeta ^{-k}]&=0,{} & {} k\ \text {odd}. \end{aligned}$$

If $y=\beta [ \zeta ^{k} - (-1)^k\zeta ^{-k}]$ then $r:(x,y)\rightarrow (-x, (-1)^{k-1}y)$; thus y is invariant under r only for k odd, in which case it will be a function on the quotient curve $\hat{{\mathcal {C}}}/\left\langle s,r \right\rangle$; for k-even $v=xy$ is invariant. Thus we have curves

$$\begin{aligned} v^2&=x^2(x^k+ \alpha _2 x^{k-2}+ \alpha _4 x^{k-4}+ \cdots + \alpha _k)^2 -4\beta ^2 x^2{} & {} k\ \text {even},\\ y^2&=(x^k+ \alpha _2 x^{k-2}+ \alpha _4 x^{k-4}+ \cdots + \alpha _{k-1} x)^2 +4\beta ^2{} & {} k\ \text {odd}. \end{aligned}$$

Setting $k=2l$ or $k=2l-1$ for the even and odd cases of the curves then with $u=x^2$ we have these curves covering 2 : 1 the curves

$$\begin{aligned} v^2&=u(u^l+ \alpha _2 u^{l-1}+ \alpha _4 u^{l-2}+ \cdots + \alpha _k)^2 -4\beta ^2u{} & {} k\ \text {even}, \end{aligned}$$

(10)

$$\begin{aligned} y^2&=u(u^{l-1}+ \alpha _2 u^{l-2}+ \alpha _4 u^{l-3}+ \cdots + \alpha _{k-1})^2 +4\beta ^2{} & {} k\ \text {odd}. \end{aligned}$$

(11)

The first has genus l and the second has genus $l-1$. Under the cyclic transformation, it was shown in [7] that

$$\begin{aligned} \frac{\eta ^{k-2}d\zeta }{\partial _\eta P}=\pi ^*\left( -\frac{1}{k} \frac{x^{k-2}dx}{y}\right) \end{aligned}$$

for the curve (3) and we observe that this differential is invariant under r for k both even and odd. Further

$$\begin{aligned} \frac{x^{k-2}dx}{y}={\left\{ \begin{array}{ll} \dfrac{x^{2l-2}dx}{y}=\dfrac{x^{2l-2}du}{2xy}= \dfrac{u^{l-1}du}{2v},\\ \dfrac{x^{2l-3}dx}{y}=\dfrac{x^{2l-4}du}{2y}=\dfrac{u^{l-2}du}{2y}. \end{array}\right. } \end{aligned}$$

In each case we obtain the maximum degree in u differential on the corresponding hyperelliptic curve and the work of [7] tells us the Ercolani-Sinha vector, if it exists, will reduce to one on the quotient curve.

In particular the $k=3$ curve $y^2=(x^3+ \alpha _2 x)^2 + 4\beta ^2$ covers the elliptic curve ${\mathcal {E}}={\mathcal {C}}/H$,

$$\begin{aligned} y^2 =u(u+\alpha _2)^2+4\beta ^2, \end{aligned}$$

with $H=\left\langle s,r \right\rangle \cong S_3$. The j-invariant of this curve is $j_{\mathcal {E}}=16\alpha _2^6/(\beta ^2[\alpha _2^3-27 \beta ^2] )$, the value observed earlier. We note that the genus-2 curve also covers the elliptic curve ${\mathcal {E}}'={\mathcal {C}}/H'$,

$$\begin{aligned} w^2 =u^2(u+\alpha _2)^2+4\beta ^2, \end{aligned}$$

where now $H'=\left\langle s,rt \right\rangle \cong C_6$ with $w=xy$ the invariant coordinate. Because $\pi ^*(du/(2w))=dx/y$ does not pull back to the differential appearing in the Ercolani-Sinha constraint we cannot solve the Hitchin constraints in terms of ${\mathcal {E}}^\prime$. We record that the curve is in general distinct $j_{{\mathcal {E}}'}={\left( \alpha _2^{4}+48 \beta ^{2}\right) ^{3}}/\left( {\beta ^{4} \left[ \alpha _2^{4}+64 \beta ^{2}\right] }\right)$. We have that ${{\mathcal {E}}}$ and ${{\mathcal {E}}'}$ are the two quotients identified in Table 1.

We remark that the reduction of the spectral curve we have just described may be understood directly in terms of the Toda equations and ‘folding’. For the $k=3$ case at hand set $e^{\rho _i}:=a_i^2=\beta ^{2/3}e^{q_i-q_{i+1}}$ (so that $a_0a_1a_2=\beta$) and again take $b_j=q_j'$ and Hamiltonian $H=\frac{1}{2} \sum _{j=1}^{k} b_j^2-\sum _{j=0}^{k-1} a_j^2 =\frac{1}{2} \sum _{j=1}^{k} p_j^2-\beta ^{2/3}\sum _{j=0}^{k-1} e^{q_j-q_{j+1}}.$ Then the Toda equations take the form

$$\begin{aligned} \rho _i^{\prime \prime }&= 2e^{\rho _i} - e^{\rho _{i-1}} - e^{\rho _{i+1}} ={\overline{K}}_{ij} e^{\rho _j} \end{aligned}$$

where ${\overline{K}}_{ij}$ is the extended Cartan matrix of $A_2$. Folding [33] corresponds to the action $\rho _i\rightarrow \rho _{\sigma (i)}$ by a diagram automorphism $\sigma$ of the extended Dynkin diagram: this retains integrability and here corresponds to identifying $\rho _1=\rho _2:=\rho _{12}$, equivalently $a_1^2=a_2^2$. Using $e^{\rho _{0}}=\beta ^2 e^{-2\rho _{12}}$ the equations of motion $\rho _{12}''=e^{\rho _{12}}-e^{\rho _{0}}$ and $\rho _0''=2(e^{\rho _{0}}-e^{\rho _{12}})$ reduce to the one equation,

$$\begin{aligned} \rho _{12}''=e^{\rho _{12}}-\beta ^2 e^{-2\rho _{12}}, \end{aligned}$$

the ODE reduction of the Bullough-Dodd equation, a known integrable equation. This may be directly integrated. With $u=e^{\rho _{12}}$ we obtain precisely (8). More generally we are seeing the reduction by folding $A_{2l-1}^{(1)}\rightarrow C_l^{(1)}$ for $k=2l$ even, and $A_{2[l-1]}^{(1)} \rightarrow A_{2[l-1]}^{(2)}$ for $k=2l-1$ odd, both coming from an order-2 symmetry of the Dynkin diagram.

3.2 Solving for Nahm data

A number of different arguments lead us then to an elliptic reduction of the Toda equations for $k=3$ with corresponding ODE (8). The aim of this subsection is to show that from this ODE Nahm data can be constructed. In doing so we will use properties of hypergeometric functions, and we lay out some of these details in Appendix A.

We have seen that the reduction leads to $a_1^2=a_2^2$ and $b_2=0$. In continuing to solve for the Nahm data one finds that the choice of sign of $a_2$ relative to $a_1$ does not affect the ability to impose the Hitchin constraints. Indeed, changing the choice of sign merely corresponds to changing the sign of $\beta$, and again as we will see this does not restrict the spectral curve. As such we take $a_2 = - a_1$ in what follows. Now setting ${\tilde{u}} = u - \frac{\alpha _2}{6}$ and ${{\tilde{s}}}= s/\sqrt{2}$ we may transform (8) into standard Weierstrass form with solution

$$\begin{aligned} {\tilde{u}} = \wp \left( (s - s_0)/\sqrt{2}; g_2, g_3\right) , \end{aligned}$$

where $g_2 = \frac{\alpha _2^2}{3}$ and $g_3 = \frac{\alpha _2^3}{27} - 2 \beta ^2$. Here we assume $\Delta :=g_2^3 -27 g_3^2=4\beta ^2(\alpha _2^3-27 \beta ^2)\ne 0$ to avoid nonsingularity, commenting on the singular limits at the appropriate junctures. The j-invariant of the elliptic curve is as we have already seen

$$\begin{aligned} j = 1728 \frac{g_2^3}{g_2^3 - 27 g_3^2} = \frac{16 \alpha _2^6}{\beta ^2 (\alpha _2^3 - 27 \beta ^2)}. \end{aligned}$$

To be Nahm data we require that the Nahm matrices have a pole at $s=0$ which can be achieved by setting $s_0 = 0$. We can then express all the Flaschka variables as

$$\begin{aligned} a_1&= \pm \sqrt{\wp (s/\sqrt{2}; g_2, g_3) + \frac{\alpha _2}{6}},&a_2&= - a_1,&a_0&= \frac{\beta }{a_1 a_2}, \end{aligned}$$

(12)

$$\begin{aligned} b_1&= \pm \sqrt{2 a_1^2 + a_0^2 - \alpha _2},&b_2&= 0,&b_3&= -b_1. \end{aligned}$$

(13)

We have some signs of the square roots to set above.

• (i) Using that, around $s=0$, $\wp (s/\sqrt{2}; g_2, g_3) \sim 2\,s^{-2} \Rightarrow a_1^2 \sim \frac{2}{s^2}$, we have $a_0 \sim \frac{\beta s^2}{2}$. The ODE for $a_0^\prime$, with $b_3 = -b_1$, gives

  $$\begin{aligned} b_1 = - \frac{a_0^\prime }{a_0} \sim -\frac{(\beta s)}{(\beta s^2/2)} = -\frac{2}{s}. \end{aligned}$$

  This requires us to take the negative square root for $b_1$ around $s=0$. We will want residues at $s=2$, and it will turn out by applying similar analysis that we need the positive root around $s=2$. These swap over when $b_1=0$, which corresponds to $a_1^\prime = 0$. As we see later this must happen at $s=1$. Alternatively one can see this from the observation that $a_1$ is even about $s=1$ by a judicious choice of period, and so $b_1 = \frac{2 a_1^\prime }{a_1}$ is odd about the same point.

• (ii) The sign of $a_1$ is a free choice, and does not affect the geometry of the monopole, hence in what follows below we always take the positive sign.

The corresponding Nahm matrices (6) have residues at $s=0$ given by

$$\begin{aligned} R_1 =\frac{1}{\sqrt{2}} \left( \begin{array}{rrr} 0 &{}\quad 1 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 1 &{}\quad 0 \end{array}\right) , \quad R_2 = \frac{i}{\sqrt{2}} \left( \begin{array}{rrr} 0 &{}\quad -1 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 \end{array}\right) , \quad R_3 =i \left( \begin{array}{rrr} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -1 \end{array}\right) \end{aligned}$$

which yield a 3-dimensional irreducible representation.

Next we require a simple pole at $s=2$ again forming a 3-dimensional irreducible representation. There are two ways to achieve a residue at $s=2$:

• (i) have that $2/\sqrt{2} = \sqrt{2}$ is in the lattice corresponding to the values $g_2, g_3$, or

• (ii) have that around $s=2$, $\wp (s/\sqrt{2}; g_2, g_3) \sim -\frac{\alpha _2}{6} + {\mathcal {O}}(s-2)$.

These correspond to having $a_1$ and $a_0$ be singular at $s=2$, respectively. (Because of the constant $\beta$ they cannot both be singular.) One can check that the second condition would give a reducible representation at $s=2$ (as again only one of the $a_i$ have a pole here) and so we discount it.

Focussing then on the first condition, one way to fix the real period of the associated lattice is to invert the j-invariant of the elliptic curve corresponding to $g_2, g_3$ to give the period $\tau$. Here this is most readily achieved by solving the quadratic (for example, see [5])

$$\begin{aligned} 4\alpha (1-\alpha )&= \frac{1728}{j} = 108 (\beta ^2/\alpha _2^3) \left[ 1 - 27(\beta ^2/\alpha _2^3) \right] , \end{aligned}$$

for which we see the two solutions are $\alpha = \frac{27 \beta ^2}{\alpha _2^3}$, and $1-\alpha$. The corresponding normalised period is

$$\begin{aligned} \tau = \tau (\alpha ):= i \frac{{}_2 F_1 (1/6,5/6, 1; 1-\alpha )}{{}_2 F_1 (1/6,5/6, 1; \alpha )}. \end{aligned}$$

Some analytic properties of this function we need are given in Appendix A.

Remark 3.1

If we had taken the other root $\alpha$ in the numerator of the hypergeometric function then this would give the period $-1/\tau$.

As we want the lattice corresponding to $g_2, g_3$ to be $\sqrt{2}{\mathbb {Z}} + \sqrt{2}\tau {\mathbb {Z}}$, we get the transcendental equations

$$\begin{aligned} \frac{1}{3}\alpha _2^2 = \frac{1}{4} g_2(1, \tau ), \quad \frac{1}{27} \alpha _2^3 - 2 \beta ^2 = \frac{1}{8} g_3 \left( 1, \tau \right) . \end{aligned}$$

For any given value of $\alpha \in (0,1)$, let $\alpha _2^2 = \frac{3}{4} g_2(1, \tau )$. We then have two equations defining $\beta$:

$$\begin{aligned} \beta ^2 = \frac{\alpha \alpha _2^3}{27}, \quad \beta ^2 = \frac{1}{2} \left[ \frac{1}{27} \alpha _2^3 - \frac{1}{8} g_3(1, \tau ) \right] . \end{aligned}$$

To have a valid solution we must have that the two equations are consistent with each other, which one can check (see Appendix A) is equivalent to ${\text {sgn}}(g_3(1, \tau )) = {\text {sgn}}(\alpha _2){\text {sgn}}(1-2\alpha )$. A consideration of the information given about $\tau$ and $g_3$ in Appendix A tells us that we only get solutions in the region $\alpha \in [0,1]$, where $\alpha = 0, 1$ really correspond to the limits $\lim _{\epsilon \rightarrow 0^+} \epsilon , 1-\epsilon$, respectively.

In order to exclude the possibility of other poles of the Nahm matrices in the region $s \in (0, 2)$, it is necessary that for all $s \in (0,2)$

$$\begin{aligned} \wp (s/\sqrt{2}; g_2, g_3) + \frac{\alpha _2}{6} > 0. \end{aligned}$$

We know that (i) $\wp$ takes its minimum at $s=1$; (ii) that the minimum value is the most-positive root of the corresponding cubic $4 \wp ^3 - g_2 \wp - g_3 = 0$; (iii) that this root is positive [13, §23.5]. Therefore there are no other poles in (0, 2). Further as $\alpha _2\ne 0$ has the same sign as $\alpha$, then $\alpha _2 > 0$ so $a_1^2 > 0$. Therefore we know that $a_1$ is always real, and hence so are all the Flaschka variables, thus giving all the Nahm variables being real as desired.

The remaining condition required for valid Nahm data is that $T_i(s) = T_i(2-s)^T$. The nature of the Weierstrass $\wp$ is such that $\wp ((2-s)/\sqrt{2}; g_2, g_2) = \wp (s/\sqrt{2}; g_2, g_3)$, so we automatically have that $a_1(2-s) = a_1(s)$, $a_0(s) = a_0(2-s)$. Moreover, because of the change in the sign of the square root giving $b_1$ at $s=1$, we have that $b_1(2-s) = - b_1(s)$. Taken together these ensure the desired symmetry of the Nahm matrices and we have a one-parameter family of new solutions.

As such, we have now proven the following theorem.

Theorem 3.2

Given $\alpha \in [0,1]$, define

$$\begin{aligned} \tau = \tau (\alpha ) = i \frac{{}_2 F_1 (1/6,5/6, 1; 1-\alpha )}{{}_2 F_1 (1/6,5/6, 1; \alpha )}. \end{aligned}$$

Solving

$$\begin{aligned} \frac{1}{3}\alpha _2^2 = \frac{1}{4} g_2(1, \tau ), \quad \frac{1}{27} \alpha _2^3 - 2 \beta ^2 = \frac{1}{8} g_3 \left( 1, \tau \right) , \end{aligned}$$

with ${\text {sgn}}(\alpha _2) = {\text {sgn}}(\alpha )$ yields a monopole spectral curve with $D_{6}$ symmetry

$$\begin{aligned} \eta ^3 + \alpha _2 \eta \zeta ^2 + \beta (\zeta ^6 - 1) = 0. \end{aligned}$$

Moreover, the Nahm data is given explicitly in terms of $\wp$-functions by (6) and (12).

3.3 Distinguished curves

Having solved for general $\alpha \in [0,1]$ we now investigate the special values of $\alpha = 0, 1/2, 1$.

3.3.1 $\alpha = 0^+$

The limit $\alpha \rightarrow 0$ corresponds to $\tau \rightarrow +i \infty$, and we have using the asymptotic expansion of the Eisenstein series that $g_2(1, \tau ) \rightarrow \frac{4\pi ^4}{3}$, $g_3(1, \tau ) \rightarrow \frac{8 \pi ^6}{27}$, so $\alpha =0$ is indeed a solution with $\beta =0$, $\alpha _2 = \pi ^2$. This recreates the well known axially-symmetric monopole with spectral curve $\eta (\eta ^2 + \pi ^2\zeta ^2) = 0$ [20, 21].

If we had $\beta =0$ from the beginning (and so $\Delta =0$, and for $\alpha _2\ne 0$ then $\alpha =0$), we would have found a singular elliptic curve

$$\begin{aligned} 4 {\tilde{u}}^3 - \frac{1}{3}\alpha _2^2 {\tilde{u}} - \frac{1}{27}\alpha _2^3 = 4 \left( {\tilde{u}} + \frac{\alpha _2}{6} \right) ^2 \left( {\tilde{u}} - \frac{\alpha _2}{3} \right) , \end{aligned}$$

with solution to the corresponding ODE (using known integrals) given by

$$\begin{aligned} {\tilde{u}}&= \frac{\alpha _2}{3} + \frac{\alpha _2}{2} \tan ^2 \left[ \frac{\sqrt{\alpha _2}}{2}(s - s_0) \right] , \quad a_1 = \sqrt{\frac{\alpha _2}{2}} \sec \left[ \frac{\sqrt{\alpha _2}}{2}(s - s_0) \right] . \end{aligned}$$

We could then manufacture the right residue at $s=0$ by having $s_0 = \frac{\pi }{2} \cdot \frac{2}{\sqrt{\alpha _2}}$. To get the correct periodicity, we would require that $\frac{\pi }{2} = \frac{\sqrt{\alpha _2}}{2} (2 - s_0)$ and consequently that $\alpha _2 = \pi ^2$ again giving the axially-symmetric monopole.

3.3.2 $\alpha = 1^-$

To get this limit, we use $\tau (1^-) = -1/\tau (0^+)$, so

$$\begin{aligned} g_2(1, \tau (1^-)) = g_2(1, -1/\tau (0^+)) = \tau (0^+)^4 g_2(1, \tau (0^+)) = \frac{1}{\tau (1^-)^4} \frac{4 \pi ^4}{3}, \end{aligned}$$

and likewise for $g_3$. Solving gives

$$\begin{aligned} \alpha _2 \sim -\left( \frac{\pi }{\tau } \right) ^2, \quad \beta \sim \pm \frac{i}{3 \sqrt{3}}\left( \frac{\pi }{\tau } \right) ^3, \end{aligned}$$

or equivalently writing $\tau = i \epsilon$ for $0 < \epsilon \ll 1$,

$$\begin{aligned} \alpha _2 \sim 3\left( \frac{\pi }{\sqrt{3}\epsilon } \right) ^2, \quad \beta \sim \pm \left( \frac{\pi }{\sqrt{3}\epsilon } \right) ^3, \end{aligned}$$

The corresponding spectral curve thus factorises as

$$\begin{aligned} 0&= \eta ^3 + 3\left( \frac{\mp \pi }{\sqrt{3}\epsilon } \right) ^2\eta \zeta ^3 - \left( \frac{\mp \pi }{\sqrt{3}\epsilon } \right) ^3(\zeta ^6 - 1), \\&= \left[ \eta - \left( \frac{\mp \pi }{\sqrt{3}\epsilon } \right) (\zeta ^2 - 1) \right] \left[ \eta - \left( \frac{\mp \pi }{\sqrt{3}\epsilon } \right) (\omega \zeta ^2 - \omega ^2) \right] \left[ \eta - \left( \frac{\mp \pi }{\sqrt{3}\epsilon } \right) (\omega ^2\zeta ^2 - \omega ) \right] . \end{aligned}$$

This corresponds to three well-separated 1-monopoles on the vertices of an equilateral triangle in the x, y-plane with side length $\frac{\pi }{\epsilon }$ [22]. As $\epsilon$ tends to zero these three vertices tend to the point $\infty$, the singular degeneration to the cuspidal elliptic curve with $\Delta =0$ and $\alpha =1$.

3.3.3 $\alpha = 1/2$

In this case $\tau =i$, and the lattice is the square lattice. The values of $g_2, g_3$ for this lattice are known explicitly [13, 23.5.8], giving the equations

$$\begin{aligned} \frac{1}{3}\alpha _2^2 = \frac{1}{4} \frac{\Gamma (1/4)^8}{16 \pi ^2}, \quad \frac{1}{27} \alpha _2^3 - 2 \beta ^2 = 0 \Rightarrow \alpha _2 = \frac{\sqrt{3}\Gamma (1/4)^4}{8 \pi }, \quad \beta = \pm \frac{\Gamma (1/4)^6}{32(\sqrt{3}\pi )^{3/2}}. \end{aligned}$$

Remark 3.3

The coefficients seen here are the same, up to a sign, as those of a distinguished monopole found in [26]. This is no accident, but arises because the square lattice is behind the distinguished “twisted figure-of-eight" monopole, as we show later in Sect. 4.2.

3.4 Scattering

To complete our understanding of these monopoles we discuss the corresponding scattering. This has already been described using the rational map approach in [36]. The $D_6$-symmetric monopoles described here corresponds to the prismatic subgroup $D_{3h}$ of ${\text {O}}(3)$: this confines the monopoles to lie in a plane, and thus any scattering observed must be planar. Note for each value of $\alpha \ne 0$ there are two choices of $\beta$ from the defining equations, and these two branches coalesce where $\beta =0 \Leftrightarrow \alpha = 0$. This gives us a view of scattering from $\alpha =1$ with three initially well-separated 1-monopoles with a choice of sign. They move inwards along the axes of symmetry of the corresponding equilateral triangle through $\alpha = 0$ where the 3-monopoles instantaneously takes the configuration of the axially-symmetric monopole. Here we change branch (i.e. sign of $\beta$), and move back out to $\alpha = 1$ where now because of the change of sign these three well-separated 1-monopoles are deflected by $\pi /3$ radians. Note that as with the planar scattering of 2-monopoles [3], because of symmetry one cannot associate a given in-going monopole with an out-going one but rather interpret the scattering process as the the 3 in-going monopoles splitting into thirds which then recombine to form the out-going monopoles.

4 $V_4$ monopoles

4.1 Solving for Nahm data

For our curve with $V_4$ symmetry the generators of the automorphism group are $(\zeta ,\eta ) \mapsto (-\zeta , -\eta )$ and $(\zeta , \eta ) \mapsto (-1/\zeta , \eta /\zeta ^2)$; equivalently these correspond to the rotations ${{\,\textrm{diag}\,}}(-1,-1,1)$ and ${{\,\textrm{diag}\,}}(-1,1,-1)$ whose product is the earlier r. If we impose further the involution $(\zeta , \eta ) \mapsto (\zeta , -\eta )$ as a symmetry (the composition of inversion with the anti-holomorphic involution) we restrict to the case of the inversion-symmetric 3-monopoles known in [25]. Here they solve for Nahm matrices given in terms of 3 real-valued functions $f_i$ satisfying $f_1^\prime = f_2 f_3$ (and cyclic), with the corresponding spectral curve being

$$\begin{aligned} \eta ^3 + \eta \left[ \left( f_1^2 - f_2^2 \right) (\zeta ^4 + 1) + (2f_1^2 + 2f_2^2 - 4 f_3^3)\zeta ^2 \right] = 0. \end{aligned}$$

We find in Appendix C that the same procedure, now without imposing the extra symmetry, yields Nahm matrices in terms of 3 complex-valued functions satisfying⁸

$$\begin{aligned} \bar{f_1}^\prime = f_2 f_3 \quad \text {(and cyclic)}, \end{aligned}$$

(14)

with the corresponding spectral curve being

$$\begin{aligned} \eta ^3 + \eta \left[ a(\zeta ^4 + 1) + b \zeta ^2 \right] + c\zeta (\zeta ^4 - 1) = 0, \end{aligned}$$

(15)

where

$$\begin{aligned} a=\left|f_1 \right|^2 - \left|f_2 \right|^2,\quad b= 2\left|f_1 \right|^2 + 2\left|f_2 \right|^2 - 4 \left|f_3 \right|^3,\quad c= 2(f_1 f_2 f_3 - \bar{f_1} \bar{f_2} \bar{f_3}). \end{aligned}$$

The Nahm matrices are given by

$$\begin{aligned} T_1&= \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -\bar{f_1} \\ 0 &{}\quad f_1 &{}\quad 0 \end{pmatrix},&T_2&= \begin{pmatrix} 0 &{}\quad 0 &{}\quad f_2 \\ 0 &{}\quad 0 &{}\quad 0 \\ -\bar{f_2} &{}\quad 0 &{}\quad 0 \end{pmatrix},&T_3&= \begin{pmatrix} 0 &{}\quad -\bar{f_3} &{}\quad 0 \\ f_3 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{pmatrix}. \end{aligned}$$

(16)

Remark 4.1

We observe that equations (14) come from the Poisson structure $\left\{ f_i, \bar{f_j} \right\} = \delta _{ij}$, with Hamiltonian $c/2= f_1 f_2 f_3 - \bar{f_1} \bar{f_2} \bar{f_3}$. This complex extension of the Euler equations is integrable.

Remark 4.2

We have not fully used up the gauge symmetry available to us. Namely, if we conjugate the $T_i$ by $U = {\text {diag}}(u_1, u_2, u_3)$ where $u_j = e^{i \phi _j}$ and $\sum \phi _j = 0$, we get

$$\begin{aligned} f_1 \mapsto u_3 u_2^{-1} f_1, \quad f_2 \mapsto u_1 u_3^{-1} f_2, \quad f_3 \mapsto u_2 u_1^{-1} f_3, \end{aligned}$$

which preserves the form of the equations.

A consequence of this remark and the form (16) is that for the $T_i$ to have residues which form an irreducible representation of ${\mathfrak {s}}{\mathfrak {u}}(2)$ it is sufficient for the $f_i$ to have simple poles at $s=0,2$.

In order to find a solution we note that $a_{ij} = \left|f_i \right|^2 - \left|f_j \right|^2$ and $c = 2(f_1 f_2 f_3 - \bar{f_1} \bar{f_2} \bar{f_3})$ are now constants. As c is imaginary it will be useful to introduce ${\tilde{c}}:=-ic$. Setting $F = |f_1|$ we have

$$\begin{aligned} (F^\prime )^2&= \left\{ \left[ \left( f_1 \bar{f_1} \right) ^{1/2} \right] ^\prime \right\} ^2 = \left\{ \frac{1}{2} (f_1 \bar{f_1})^\prime (f_1 \bar{f_1})^{-1/2} \right\} ^2 = \frac{1}{4} (f_1 f_2 f_3 + \bar{f_1} \bar{f_2} \bar{f_3})^2 F^{-2}, \\&= \frac{1}{4} F^{-2} \left[ (c/2)^2 + 4 |{f_1}|^2 |{f_2}|^2 |{f_3}|^2 \right] , \\&= \frac{1}{4} F^{-2} \left[ (c/2)^2 + 4 F^2 (F^2 - a_{12}) (F^2 + a_{31}) \right] , \end{aligned}$$

and so with $G = F^2$ we get

$$\begin{aligned} (G^\prime )^2 = \frac{1}{4}c^2 + 4G(G - a_{12})(G + a_{31}), \end{aligned}$$

which then has solutions in terms of elliptic functions. In terms of the coefficients of the spectral curve we already have $a_{12} = a$, and we can moreover find $a_{31} = \frac{-1}{4}(b +2a)$, so we can rewrite the equation as

$$\begin{aligned} ({\tilde{G}}^\prime )^2 = 4 {\tilde{G}}^3 - g_2 {\tilde{G}} - g_3, \end{aligned}$$

(17)

where ${\tilde{G}} = G - \frac{b + 6a}{12}$, $g_2 = a^2 + \frac{b^2}{12}$, and $g_3 = \frac{b(b^2 - 36a^2)}{216} + \frac{1}{4}{\tilde{c}}^2$. Then ${{\tilde{G}}}=\wp$, the Weierstrass $\wp$-function. The j-invariant for this elliptic curve is

$$\begin{aligned} j&= 1728 \frac{g_2^3}{g_2^3 - 27 g_3^3} = \frac{(12a^2 + b^2)^3}{\left( a^6 - \frac{1}{2}a^4 b^2 + \frac{1}{16}a^2 b^4 + \frac{9}{4}a^2 b {\tilde{c}}^2 - \frac{1}{16}b^3 {\tilde{c}}^2 - \frac{27}{16}{\tilde{c}}^4 \right) }, \end{aligned}$$

(18)

which is precisely that of the quotient of (15) by the $V_4$ symmetry. We also note that the pull-back of the invariant differential of this quotient is exactly that needed when discussing the Ercolani-Sinha constraint.

Before going on to solve this completely, let’s recall what remains to be shown to get a monopole spectral curve (i.e. to have our Nahm matrices satisfy all the conditions to give Nahm data). We need to have that the $\wp$-function associated with the above elliptic curve has real period 2, but we will be able to impose this by tuning the coefficients. Also, as the right-hand side of

$$\begin{aligned} \wp =|f_1|^2-\frac{b+6a}{12} \end{aligned}$$

(19)

is always real this requires $\wp$ to be real and so to be taken on a rectangular or rhombic lattice. Also for reality we need that

$$\begin{aligned} G(s) = \wp (s) + \frac{b+6a}{12}, \quad G(s)-a_{12} = \wp (s) + \frac{b-6a}{12}, \quad G(s)+a_{31} = \wp (s) - \frac{b}{6} \end{aligned}$$

are always positive. Once we have achieved this we will have regularity in the region (0, 2), and so get the right pole structure. The final condition is symmetry about $s=1$, which is enforced on the $\left|f_j \right|$ (because $\left|f_j \right| \sim \sqrt{\wp }$), and so the remaining Nahm constraint $T_j(s) = T_j(2-s)^T$ becomes simply $f_j(s) = - \bar{f_j}(2-s)$: that is we require $\arg f_j(s) = \pm \pi - \arg f_j(2-s)$.

Indeed writing $f_j = \left|f_j \right| e^{i \theta _j}$ we can work out the equations for the angles, using

$$\begin{aligned} f_j^\prime = \left( \left|f_j \right|^\prime + i \theta _j^\prime \left|f_j \right| \right) e^{i \theta _j} = \left( \frac{\left|f_j \right|^\prime }{\left|f_j \right|} + i \theta _j^\prime \right) f_j \Rightarrow \theta _j^\prime = \frac{1}{i}\left[ \frac{f_j^\prime }{f_j} - \frac{\left|f_j \right|^\prime }{\left|f_j \right|} \right] = \frac{-{\tilde{c}}}{4 \left|f_j \right|^2}. \end{aligned}$$

(20)

The $\theta _j$ are thus strictly monotonic (unless ${\tilde{c}}=0$, in which case they are constant), and symmetry about $s=1$ of $\left|f_j \right|$ then necessitates that $\theta _j(s) - \theta _j(1)$ is antisymmetric about $s=1$.

We also have that

$$\begin{aligned} {\tilde{c}} = 4 \left|f_1 \right| \left|f_2 \right| \left|f_3 \right| \sin (\theta _1 + \theta _2 + \theta _3) = \sqrt{{\tilde{c}}^2 + 4(G^\prime )^2} \sin (\theta _1 + \theta _2 + \theta _3). \end{aligned}$$

At $s=1$ where $G^\prime (s)=0$ we need $\sin (\theta _1 + \theta _2 + \theta _3) = 1$, and by our gauge freedom we can choose $\theta _1(1)=\pi /2=\theta _2(1)$ and so $\theta _3(1) = -\pi /2$, thus enforcing our condition of symmetry about $s=1$. We then see that the anti-symmetry of $\theta _j(s)-\theta _j(1)$ about $s=1$ enforces the remaining reality condition. We also note that as $|f_j(s)|^2=\wp (s)-c_j:=\wp (u)-\wp (v_j)$ for appropriate s and $v_j = \int _{\infty }^{c_j} \left[ 4 u^3 - g_2 u - g_3 \right] ^{-1/2} du$ we have [28, (6.14.6)]

$$\begin{aligned} \int \frac{du}{\wp (u)-\wp (v)}= \frac{1}{\wp '(v)}\left[ 2 u \zeta (v) +\ln \frac{\sigma (u-v)}{\sigma (u+v)}\right] , \end{aligned}$$

(21)

allowing us to find the $\theta _j(s)$ explicitly which is done in (22) in Appendix A.3.

It remains to fix the real period of the corresponding elliptic curve. We describe two methods. The first makes use of the Jacobi elliptic functions to express the lattice invariants in terms of complete elliptic functions [1, §18.9]. We explain this in Appendix A.4 in which, by showing that we may fix the real period, establishes the following theorem.

Theorem 4.3

Given $\alpha \in {\mathbb {R}}$, $m \in [0,1]$, and ${{\,\textrm{sgn}\,}}= \pm 1$, define $g_2, g_3$ by $g_2 = 12 \left( K(m)^2/3 \right) ^2 q_1(m)$, $g_3 = 4\left( {K(m)^2}/{3} \right) ^3(2\,m-1)q_2(m)$, where

$$\begin{aligned} q_1(m) = \left\{ \begin{array}{cc} 1 - m + m^2 &{} {{\,\textrm{sgn}\,}}=1, \\ 1 - 16 m + 16 m^2 &{} {{\,\textrm{sgn}\,}}=-1,\end{array} \right. \quad q_2(m) = \left\{ \begin{array}{cc} (m-2)(m+1) &{}\quad {{\,\textrm{sgn}\,}}=1, \\ 2(32m^2 - 32m - 1) &{}\quad {{\,\textrm{sgn}\,}}=-1.\end{array} \right. \end{aligned}$$

If m is such that $g_2>0$ and the polynomial $(4-2\alpha )x^3 - g_2 x - g_3$ has a real root $x_*$ with $\left|x_* \right|<\sqrt{g_2/3}$ and ${\text {sgn}}(x_*) = -{\text {sgn}}(\alpha )$, then we may solve

$$\begin{aligned} a^2 + \frac{b^2}{12} = g_2, \quad \frac{b(b^2 - 36a^2)}{216} + \frac{{\tilde{c}}^2}{4} = g_3 \end{aligned}$$

for $a, b, {\tilde{c}} \in {\mathbb {R}}$ taking $\alpha = \frac{-27 {\tilde{c}}^2}{b^3}$. Then

$$\begin{aligned} \eta ^3 + \eta \left[ a(\zeta ^4 + 1) + b\zeta ^2 \right] + i {\tilde{c}}\zeta (\zeta ^4-1) = 0 \end{aligned}$$

is a monopole spectral curve with $V_4$ symmetry. Moreover the Nahm data is given explicitly in terms of elliptic functions by (16), (19) and (22).

A second approach to fixing the correct real period to give Nahm data is to invert the j-invariant (18) as done in the earlier $D_6$ case. Though we are unable to invert in terms of a single rational $\alpha$ as with the $D_6$-symmetric monopole, we may use [15, (4)] which gives

$$\begin{aligned} \tau = i\left[ \frac{2\sqrt{\pi }}{\Gamma (7/12)\Gamma (11/12)} \frac{{}_2 F_1 (1/12,5/12, 1/2; x)}{{}_2 F_1 (1/12,5/12, 1; 1-x)} - 1 \right] , \end{aligned}$$

where $x = 1 - \frac{1728}{j} = \frac{(1 - 2\alpha - 3\gamma )^2}{(1+\gamma )^3}$, with $\alpha = - \frac{27 {\tilde{c}}^2}{b^3}$, $\gamma = \frac{12a^2}{b^2}$. One may then fix the real period of the lattice, which will give solutions consistent with the definition of x for some range of the parameters $\alpha , \gamma$. We investigate one particular restriction of this kind in Sect. 4.2. We remark that [23] solved the associated Nahm data only for the (one-parameter) case $\Delta =0$ in which the elliptic curve degenerates and has trigonometric solutions.

4.2 $D_4$ monopoles

In [26] a subfamily of (15) with $D_4$ symmetry was studied. To the existing $V_4$ symmetries is appended the order-4 element $(\zeta ,\eta )\mapsto (i\zeta , -i\eta )$ (corresponding to the composition of inversion with a rotation of $\pi /2$ in the xy-plane). This symmetry then requires $a=0$. By a dimension argument we expect the j-invariant inversion to yield a geodesic 1-parameter family for the enlarged symmetry group, and this was the case considered in [26] where the $C_4$ quotient yields an elliptic curve. Placing this curve in our $V_4$ family allows us a different approach to this family of curves. The restriction $a=0$ means that $\frac{1728}{j} = 4\alpha (1-\alpha )$ with $\alpha = - \frac{27 {\tilde{c}}^2}{b^3}$ and we can then fix the real period via the same approach as for the $D_6$ monopole. The equations we get are

$$\begin{aligned} \frac{b^2}{3} = \frac{1}{4}g_2(1, \tau ), \quad \frac{b^3}{27} + 2{\tilde{c}}^2 = \frac{1}{8} g_3(1, \tau ), \end{aligned}$$

with these being consistent with the definition of $\alpha$ provided ${\text {sgn}}(g_3(1, \tau )) = {\text {sgn}}(b){\text {sgn}}(1-2\alpha )$. To also have that ${\tilde{c}}$ is real, we must have ${\text {sgn}}(b) = - {\text {sgn}}(\alpha )$ and hence our consistency condition is ${\text {sgn}}(g_3(1, \tau )) = -{\text {sgn}}(\alpha ){\text {sgn}}(1-2\alpha )$. We thus have solutions in the region $\alpha \in (0,1/2)$ if ${\text {sgn}}(g_3(1, \tau )) <0$, which requires $\tau = -1/\tau (\alpha )$. We can extend this to $\alpha \in (1/2, 1)$ still taking $\tau = -1/\tau (\alpha )$. Moreover, for $\alpha < 0$, we require $g_3(1, \tau ) >0$, which can be achieved taking $\tau = \tau (\alpha )$. Finally, for $\alpha >0$, we require ${\text {sgn}}(g_1(1, \tau )) >0$, achievable with $\tau = -1/\tau (\alpha )$. As such the parameter region in this case is the whole of ${\mathbb {R}}$. A case-by-case consideration shows that G, $G - a_{12}$, $G + a_{31}$ are always positive on the interval [0, 2], so we do indeed get Nahm data as desired.

As with the $D_6$-symmetric monopoles we may identify special values of $\alpha$ and the curves they give. A similar analysis gives those found in [26], namely

• $\alpha =\pm \infty$ gives the tetrahedrally-symmetric monopole,

• $\alpha = 0^+, 0^-$ gives three well-separated 1-monopoles and the axially-symmetric monopole, respectively,

• $\alpha =1/2$ gives the “twisted figure-of-eight" monopole. Note $\alpha =1/2$ corresponds to the square lattice we saw as distinguished for the $D_6$ monopole.

We additionally see the curve with $\alpha =1$ as distinguished in our parametrisation, which gives the curve

$$\begin{aligned} \eta ^3 - \pi ^2 \eta \zeta ^2 \pm \frac{i}{\sqrt{27}} \pi ^3 \zeta (\zeta ^4-1) = 0. \end{aligned}$$

In terms of the parameters $a, \epsilon$ of [26], this curve is given by $a = 2\sqrt{2}$, $\epsilon =-1$.

4.2.1 Scattering

As such we can now understand our scattering as starting at $\alpha =0^+$ with three well-separated 1-monopoles. As $\alpha$ increases to $\infty$ we have to pick a choice of ${\tilde{c}}$ continuously (though there is no specific choice at $\alpha = 0^+$ as the map $\zeta \mapsto -\zeta$ which swaps the choice of ${\tilde{c}}$ is a symmetry of our well separated configuration), and we pass through two distinguished curves, arriving at the tetrahedrally-symmetric monopoles in one orientation. We match that to $\alpha = -\infty$ taking the tetrahedrally-symmetric monopole with the same orientation there, allowing $\alpha$ to then increase up to $0^-$ where it takes the configuration of the axially-symmetric monopole. Here the two branches of ${\tilde{c}}$ coalesce, we change branch and do the process in reverse.

5 Conclusion

In this letter, we have begun systematising the classification of charge-3 monopole spectral curves with automorphisms, providing an exhaustive list of candidate curves; we nevertheless expect this list to contain curves that do not correspond to monopole spectral curves. We have also identified how one may use group theory to identify the subset of these candidates that quotient to an elliptic curve. This was done because such curves are amenable to the construction of Nahm matrices in terms of elliptic functions using the procedure of [22]. Here the imposition of Hitchin’s conditions (or equivalently those of Ercolani-Sinha) reduces to questions about the real periods of elliptic functions. Having provided new candidate spectral curves we solved for the Nahm data in two new cases, those of $D_6$ and $V_4$ symmetry. The latter led us to an integrable system (14) that may be viewed as the complexified Euler equations. Given Nahm matrices and the corresponding group action what is not yet clear is how to methodically extract from the resulting coupled ODE’s the relevant elliptic equations; providing such an understanding would simplify the construction of the solutions to Nahm’s equations from the spectral data. This is the reason for our not treating the $C_2$-symmetric monopoles here: in this case we have 13 coupled ODE’s with 7 conserved quantities.

One can generalise to higher charge several of the viewpoints put forward in this paper. We have seen that compactifying mini-twistor space in ${\mathbb {P}}^{1,1,2}$ and then looking at its image in ${\mathbb {P}}^3$ a possible charge-k spectral curve is represented by the intersection of the cone and a degree-k hypersurface. There may be value in this viewpoint for providing a candidate list of monopole spectral curves in higher charge. Further, the methods used to calculate the group-signature pairs giving elliptic quotients in genus 4 extends to higher genus, and so may be used to provide candidate spectral curves potentially amenable to solutions in terms of elliptic functions at higher charges. At present, these data have not been computed in the LMFDB, and so a first step would be the tabulation of those results. In the event that such a computation produced too extensive a list we suggest restricting to the case where $\delta (g, G, c)=1, 2$, for which we expect any corresponding monopole spectral curves to be either isolated points in the moduli space or to correspond to geodesic motion, respectively, as we conjectured.

Finally, the geometry we introduced here may have applications for the understanding of spectral curves of hyperbolic monopoles. Spectral curves corresponding to hyperbolic monopoles live in the mini-twistor space of hyperbolic space, which is isomorphic to ${\mathbb {P}}^1 \times {\mathbb {P}}^1$, and specifically charge-k hyperbolic monopoles are bidegree-(k, k) curves in this surface [3]. As ${\mathbb {P}}^1 \times {\mathbb {P}}^1$ is isomorphic to the non-singular quadric in ${\mathbb {P}}^3$, and bidegree-(3, 3) curves in this correspond to the other class of non-hyperelliptic curves classified by Wiman, our work highlights the potential of classifying certain hyperbolic monopole spectral curves.

Appendices

Appendix A: Hypergeometric functions and lattice invariants

We gather here some of properties of elliptic and related functions used in the text and prove those statements noted in the text. We follow the conventions of [13, Chapter 23]. First we recall the Weierstrass-$\wp$ function is defined by

$$\begin{aligned} \wp '^2=4\wp ^3-g_2\wp -g_3=4(\wp -e_1)(\wp -e_2)(\wp -e_3). \end{aligned}$$

Here $g_k=g_k(\omega ,\omega ^\prime )$ are defined by the lattice $\Lambda :=2\omega {\mathbb {Z}}+2\omega ^\prime {\mathbb {Z}}$. Let $\tau =\omega ^\prime /\omega$. We have

• (i) $g_k(\lambda \omega , \lambda \omega ^\prime ) = \lambda ^{-2k} g_k(\omega , \omega ^\prime )$,

• (ii) given $\left( \begin{array}{ll} a &{}\quad b \\ c &{}\quad d\end{array}\right) \in {{\,\textrm{SL}\,}}(2, {\mathbb {Z}})$, $g_k(1, (a\tau +b)(c \tau + d)^{-1}) = (c \tau + d)^{2k} g_k(1, \tau )$,

• (iii) $\lim _{{\text {Im}}\tau \rightarrow \infty } g_2(1, \tau ) = \frac{4 \pi ^4}{3}$, $\lim _{{\text {Im}}\tau \rightarrow \infty } g_3(1, \tau ) = \frac{8 \pi ^6}{27}$.

• (iv) When $\tau = i$, $g_2(1, \tau ) = \frac{\Gamma (1/4)^8}{256 \pi ^2}$, $g_3(1, \tau ) = 0$.

• (v) When $\tau = e^{2\pi i/3}$, $g_2(1, \tau ) = 0$, $g_3(1, \tau ) = \frac{\Gamma (1/3)^{18}}{(2\pi )^6}$.

Our parametrization of the Nahm matrices requires us to know the reality properties of the $\wp$-function. We have [27, Theorem 3.16.2]

$$\begin{aligned} g_2, g_3 \in {\mathbb {R}} \Leftrightarrow \forall z \in {\mathbb {C}}, \ \wp ({\bar{z}}; g_2, g_3) = \overline{\wp (z; g_2, g_3)} \Leftrightarrow \Lambda = {\overline{\Lambda }}. \end{aligned}$$

Lattices for which $\Lambda = {\overline{\Lambda }}$ are called real lattices and they fall into two classes; rectangular lattices ($\omega \in {\mathbb {R}}$, $\omega ^\prime \in i{\mathbb {R}}$), and rhombic lattices (${\overline{\omega }} = \omega ^\prime$). The rhombic lattices correspond to $\tau$ being on the boundary of the fundamental domain of the ${{\,\textrm{SL}\,}}(2, {\mathbb {Z}})$ action on the upper half plane while the rectangular lattices correspond to $\tau$ on the imaginary axis with $\Im (\tau )\ge 1$. When restricted to rectangular or rhombic lattices we can say more about the values of $g_2(1, \tau )$ and $g_3(1, \tau )$. This is done by relating the $g_k$ to the roots $e_i$ of the corresponding cubic equation by

$$\begin{aligned} g_2(1, \tau ) = 2(e_1^2 + e_2^2 + e_3^2), \quad g_3(1, \tau ) = -4 e_1 e_2 e_3. \end{aligned}$$

• (i) On a rectangular lattice have $e_i \in {\mathbb {R}}$ so $g_2 > 0$; further, $g_3 > 0$ if $\left|\tau \right| > 1$, $g_3< 0$ if $\left|\tau \right|<1$.

• (ii) On a rhombic lattice, $e_1 \in {\mathbb {R}}$, $e_2 = {\bar{e}}_3$, and ${\text {sgn}}(e_1) = {\text {sgn}}(g_3)$.

1.1 A.1 Properties of $\tau (\alpha )$

In order to use the process of j-invariant inversion to impose the correct periodicity constraints and to give the limiting behaviours noted in the text enabling us to find certain distinguished monopoles we require the properties of

$$\begin{aligned} \tau = \tau (\alpha ) = i \frac{{}_2 F_1 (1/6,5/6, 1; 1-\alpha )}{{}_2 F_1 (1/6,5/6, 1; \alpha )}. \end{aligned}$$

This is multi-valued when $\alpha < 0$ [13, 15.2.3], with a principal branch $\tau _p$ and second branch $\tau _p + 1$, but for our purposes this difference will not not be important. The specific properties we require are that

• (i) $\forall \alpha \in (0,1),$ $\tau (\alpha ) \in i {\mathbb {R}}_{>0}$,

• (ii) $\tau ( 0^+) = +i\infty$, $\tau (1/2)=i$, $\tau (1^-) = 0$,

• (iii) $\forall \alpha < 0$, ${\text {Re}}(\tau (\alpha )) \equiv 1/2$ mod 1,

• (iv) $\tau (-\infty ) = e^{2\pi i/3}$, $\tau (0^-) = \frac{1}{2} + i \infty$.

Evaluated at the specific $\tau (\alpha )$ above we find that

$$\begin{aligned} {\text {sgn}}(g_3(1, \tau (\alpha ))) = \left\{ \begin{array}{cc} 1 &{}\quad \alpha < 1/2, \\ -1 &{}\quad \alpha \in (1/2, 1). \end{array}\right. \end{aligned}$$

Here we provide the necessary definitions and proofs.

We can understand the behaviour of $\tau$ using known results about hypergeometric functions (see for example [1, §15]). First, in the region $\alpha \in (0,1)$, we may use the series expression for ${}_2 F_1 (a, b, c; z)$ when $\left|z \right| < 1$:

$$\begin{aligned} {}_2 F_1 (a, b, c; z) = \sum _{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}, \end{aligned}$$

where $(a)_n$ is the rising Pochhammer symbol

$$\begin{aligned} (a)_n = \left\{ \begin{array}{cc} 1 &{}\quad n=0, \\ a(a+1)\dots (a+n-1) &{}\quad n \ge 1. \end{array}\right. \end{aligned}$$

This means we have (i) that for all $\alpha \in (0,1)$, $\tau (\alpha ) \in i{\mathbb {R}}_{>0}$. This is important as it makes the lattice rectangular, which forces the Weierstrass $\wp$-function to be real on the real axis [13, §23.5]. Moreover, as ${}_2 F_1 (a, b, c; z)$ is increasing in $z\in (0,1)$, ${\text {Im}}\tau (\alpha )$ is strictly decreasing in $\alpha$. We can calculate the limits to be

$$\begin{aligned} \tau ( 0^+) = +i\infty , \quad \tau (1^-) = 0, \end{aligned}$$

so giving (ii). We may use [1, 15.3.10] which says that when $\left|1-z \right|<1,$ $\left|\arg (1-z) \right| < \pi$,

$$\begin{aligned}{} & {} {}_2 F_1 (a, b, a+b; z) = \frac{\Gamma (a+b)}{\Gamma (a) \Gamma (b)} \sum _{n=0}^\infty \frac{(a)_n(b)_n}{(n!)^2} \\{} & {} \qquad \qquad \left[ 2 \psi (n+1) - \psi (a+n) - \psi (n+b) - \log (1-z) \right] (1-z)^n, \end{aligned}$$

where $\psi$ is the digamma function, to understand exactly this limiting behaviour, namely that the divergence is logarithmic. We can also highlight a special value in this regions, namely $\tau (1/2) = i$.

For $\alpha \not \in [0,1]$ we no longer have that $\tau$ lies on the imaginary axis, and we would thus need to get a rhombic lattice (that is ${\text {Re}}\tau = 1/2$) for the reality of $\wp$. Numerical tests suggest that while this happens for $\alpha < 0$ for $\alpha > 1$ we instead get ${\text {Re}}(-1/\tau ) = 1/2$. Indeed we may use [13, 15.10.29] to say⁹

$$\begin{aligned} {}_2 F_1 (1/6,5/6, 1 ; 1-\alpha )&= e^{5 \pi i/6} \frac{\Gamma (1) \Gamma (1/6)}{\Gamma (1) \Gamma (1/6)}{}_2 F_1 (1/6,5/6, 1 ; \alpha ) \\&\quad + e^{-\pi i/6} \frac{\Gamma (1) \Gamma (1/6)}{\Gamma (5/6)\Gamma (1/3)} \alpha ^{-1/6} {}_2 F_1 (1/6,1/6, 1/3 ; 1/\alpha ), \\&= e^{5 \pi i/6}{}_2 F_1 (1/6,5/6, 1 ; \alpha ) \\&\quad + (-\alpha )^{-1/6} \frac{ \Gamma (1/6)}{\Gamma (5/6)\Gamma (1/3)} {}_2 F_1 (1/6,1/6, 1/3 ; 1/\alpha ), \end{aligned}$$

and hence when $\alpha < 0$ (and taking the principal branch of the hypergeometric function) we get

$$\begin{aligned} \tau (\alpha ) = i \left[ e^{5\pi i/6} + T(\alpha ) \right] \end{aligned}$$

with

$$\begin{aligned} T(\alpha ) = (-\alpha )^{-1/6} \frac{ \Gamma (1/6)}{\Gamma (5/6)\Gamma (1/3)} \frac{{}_2 F_1 (1/6,1/6, 1/3; 1/\alpha )}{{}_2 F_1 (1/6,5/6, 1; \alpha )} \in {\mathbb {R}}. \end{aligned}$$

This means ${\text {Re}}(\tau (\alpha )) \equiv 1/2 \mod 1$, which yields (iii).

To get the asymptotics as $\alpha \rightarrow -\infty$, we use [1, 15.3.7]

$$\begin{aligned} {}_2 F_1 (a, b, c; z)&= \frac{\Gamma (c) \Gamma (b-a)}{\Gamma (b)\Gamma (c-a)} (-z)^{-a} {}_2 F_1 (a, a+1-c, a+b-1; z^{-1}) \\&\quad + \frac{\Gamma (c) \Gamma (a-b)}{\Gamma (a)\Gamma (c-b)} (-z)^{-b} {}_2 F_1 (b, b+1-c, b+a-1; z^{-1}). \end{aligned}$$

Taking $\alpha = - \epsilon ^{-1}$, this gives that as $\epsilon \rightarrow 0^+$,

$$\begin{aligned} {}_2 F_1 (a, b, c; -\epsilon ^{-1}) \sim \frac{\Gamma (2/3)}{\Gamma (5/6)^2} \epsilon ^{1/6}, \quad {}_2 F_1 (a, b, c; 1+\epsilon ^{-1}) \sim \frac{\Gamma (2/3)}{\Gamma (5/6)^2} (-\epsilon )^{1/6}, \end{aligned}$$

and so $\tau (-\infty ) = e^{2 \pi i /3} = \frac{-1}{2} + \frac{i\sqrt{3}}{2}$. To get the remaining asymptotics of (iv), as $\alpha \rightarrow 0^-$ we write $\alpha = -\epsilon$. Then

$$\begin{aligned} {}_2 F_1 (a, b, c; -\epsilon ) \sim 1, \quad {}_2 F_1 (a, b, c; 1+\epsilon ) \sim \frac{-\Gamma (1)}{\Gamma (1/6)\Gamma (5/6)}\log (-\epsilon ) = \frac{-1}{2\pi }(i \pi + \log \epsilon ), \end{aligned}$$

and so $\tau (0^-) = \frac{1}{2} + i \infty$.

To get the asymptotics as $\alpha \rightarrow 1^+$ we recognise that $\tau (1-\alpha ) = -1/\tau (\alpha )$ and so $-1/\tau (1^+) = \frac{1}{2} + i \infty$. Finally to get the asymptotics as $\alpha \rightarrow \infty$ we do the same, so $-1/\tau (\infty ) = \frac{1}{2} + \frac{i\sqrt{3}}{2}$.

1.2 A.2 Check of consistency

We see that $\alpha _2$ must be the same sign as $\alpha$ to get $\beta \in {\mathbb {R}}$. Moreover, as $g_2(1,\tau )>0$ because $\alpha _2$ is real, we can check that

$$\begin{aligned} \frac{1}{2} \left[ \frac{1}{27} \alpha _2^3 - \frac{1}{8} g_3(1, \tau ) \right]&= \frac{1}{2} \left[ \frac{{\text {sgn}}(\alpha _2)}{27} \left( 3 g_2(1,\tau )/4 \right) ^{3/2} - \frac{1}{8} g_3(1, \tau ) \right] , \\&= \frac{{\text {sgn}}(\alpha _2) g_2(1,\tau )^{3/2}}{16 \sqrt{27}} \left[ 1 - \frac{{\text {sgn}}(g_3(1, \tau ))}{{\text {sgn}}(\alpha _2)}\sqrt{\frac{27 g_3^2}{g_2^3}} \right] , \\&= \frac{{\text {sgn}}(\alpha _2)\left( 4 \alpha _2^2/3 \right) ^{3/2}}{16 \sqrt{27}} \left[ 1 - \frac{{\text {sgn}}(g_3(1, \tau ))}{{\text {sgn}}(\alpha _2)}\left( 1 - \frac{1728}{j} \right) ^{1/2} \right] , \\&= \frac{\alpha _2^3}{2 \times 27} \left[ 1 - \frac{{\text {sgn}}(g_3(1, \tau ))}{{\text {sgn}}(\alpha _2)}\left( 1 - 4 \alpha (1-\alpha ) \right) ^{1/2} \right] , \\&= \frac{\alpha _2^3}{2 \times 27} \left[ 1 - \frac{{\text {sgn}}(g_3(1, \tau ))}{{\text {sgn}}(\alpha _2){\text {sgn}}(1-2\alpha )}(1-2\alpha ) \right] , \\&= \frac{\alpha \alpha _2^3}{27} \quad \text { if } \quad {\text {sgn}}(g_3(1, \tau )) = {\text {sgn}}(\alpha _2){\text {sgn}}(1-2\alpha ). \end{aligned}$$

Hence the two equations are consistent, provided the stated sign condition holds, or if $\alpha = 0$.

1.3 A.3 The theta integration

We note that $|f_j(s)|^2=\wp (s)-c_j:=\wp (s)-\wp (v_j)$ does not fix the sign of $v_j$ for $\wp (\pm v_j)=c_j$. We fix the sign as follows. First observe that

$$\begin{aligned} ({\tilde{G}}^\prime (s))^2=({G}^\prime (s))^2= \frac{1}{4}c^2 + 4[\wp (s)-\wp (v_1)][\wp (s)-\wp (v_2)][\wp (s)-\wp (v_3)], \end{aligned}$$

and so $\wp ^{\prime \,2}(v_i)=c^2/4$; we fix the sign so that $\wp ^{\prime }(v_i)=c/2=i{\tilde{c}}/2$. Further consider the elliptic function $\wp ^{\prime }(s)-c/2$ with three zeros (at $s\in \{v_1,v_2,v_3\}$) and three poles (at $s=0$). Then with the base of the Abel-Jacobi map at $s=0$ (as is standard) we have that $\sum _i v_i$ is a lattice point. Also observe that

$$\begin{aligned} \zeta (v_i)+\zeta (v_j)=\zeta (v_i+v_j). \end{aligned}$$

We find from (20, 21) that

$$\begin{aligned} \theta _i(s):= \theta _i(1)+ i \left[ s \zeta (v_j) +\frac{1}{2}\ln \frac{\sigma (s-v_j)\sigma (1+v_j)}{\sigma (s+v_j)\sigma (1-v_j)} \right] , \end{aligned}$$

(22)

where $\theta _i(1)$ is a constant of integration and chosen as described in the text. Then $\theta _i(-s)-\theta _i(1)=-\left[ \theta _i(s)-\theta _i(1)\right]$ is anti-symmetric as required. Using the Legendre relation we find that $\sin (\theta _1+\theta _2+\theta _3)$ is periodic in s as required for consistency.

1.4 A.4 Restrictions on elliptic function parameters

Here we prove Theorem 4.3. Given the discriminant $\Delta$ for the cubic defining $\wp$ [1, §18.9] gives equations for the lattice invariants in terms of complete elliptic functions. With our earlier definitions, $g_2 = 12 \left( K(m)^2/3 \right) ^2 q_1(m)$ and $g_3 = 4\left( {K(m)^2}/{3} \right) ^3(2\,m-1)q_2(m)$ then with ${{\,\textrm{sgn}\,}}=\mathop {\textrm{sgn}}(\Delta )$ we have for $\Delta > 0$

$$\begin{aligned} g_2 = 12\left( \frac{K^2}{3 \omega _1} \right) ^2\left( 1 - m + m^2 \right) , \quad g_3 = 4\left( \frac{K^2}{3 \omega _1} \right) ^3 (m-2)(2m-1)(m+1); \end{aligned}$$

whereas for $\Delta < 0$

$$\begin{aligned} g_2 = 12\left( \frac{K^2}{3 \omega _2} \right) ^2\left( 1 - 16m + 16m^2 \right) , \quad g_3 = 8\left( \frac{K^2}{3 \omega _2} \right) ^3 (2m-1) (32m^2 - 32m - 1). \end{aligned}$$

Here $m=k^2 \in (0,1)$ is the argument of K, the underlying lattice has periods $2 \omega$, $2\omega ^\prime$, $\omega _1 = \omega$ and $\omega _2 = \omega + \omega ^\prime$. Fixing 2 as a period of the lattice, and that the lattice is real, sets $\omega _1 = 1$ for $\Delta >0$ and $\omega _2=1$ for $\Delta <0$. Observe that for ${{\,\textrm{sgn}\,}}(\Delta )=\pm 1$ that $g_2(m)$ takes it’s minimum value at $m=1/2$ while for $m\in (0,1/2)$ we have $g_3(m)>0$.

Our elliptic curve gave the equations $a^2 + \frac{b^2}{12} = g_2$, $\frac{b(b^2 - 36a^2)}{216} + \frac{1}{4}{\tilde{c}}^2 = g_3$. These equations are underdetermined, but we may substitute for $a^2$ and take $\alpha = {-27 {\tilde{c}}^2}/{b^3}$ to find

$$\begin{aligned} (4-2\alpha ) {\tilde{b}}^3 - g_2 {\tilde{b}} - g_3 = 0, \end{aligned}$$

(23)

where ${\tilde{b}} = \frac{b}{6}$. The discriminant of this cubic is

$$\begin{aligned} \Delta _\alpha (m) = 4(4-2\alpha ) g_2^3 - 27(4-2\alpha )^2 g_3^2 = 4(4-2\alpha )\left[ g_2^3 -27(1-\alpha /2)g_3^2 \right] . \end{aligned}$$

Note $\Delta _0 = \Delta$. For a given generic value of $\alpha$ in some region we may solve (23), determining b, ${\tilde{c}}$ and a in turn.

In order to get Nahm data, we require that b, ${\tilde{c}}$, and a are real. We know that this cubic has real coefficients, and so there will always be a real root of the cubic. To get reality of ${\tilde{c}}$, we need that this real root ${\tilde{b}}$ satisfies ${\text {sgn}}({\tilde{b}}) = -{\text {sgn}}(\alpha )$, and for reality of a we need $\left|{\tilde{b}} \right| \le g_2/3$. Necessary conditions to find such solutions are as follows.

First consider $\Delta > 0$. Then $g_2 > 0$ and $g_3$ is monotonically decreasing for $m\in (0,1)$ with ${\text {sgn}}(g_3) = -{\text {sgn}}(m - 1/2)$. We have the following properties:

• (i) If $\alpha > 2$ the discriminant $\Delta _\alpha (m)<0$. Then (23) has one real root whose sign is opposite that of $g_3$. Now ${\text {sgn}}({\tilde{b}}) = -{\text {sgn}}(\alpha )<0$ is opposite that of $g_3$; hence we require $g_3>0$ and so $m\in (0,1/2)$.

• (ii) For $\alpha \in (0,2)$ the discriminant $\Delta _\alpha (m)>0$ upon comparison with $\Delta =g_2^3 - 27 g_3^2 > 0$. Then, because the sum of the roots is zero, they cannot all be the same sign.

• (iii) When $\alpha < 0$, from the derivative of the cubic we know it will have a local maxima and minima at ${\tilde{b}} = \pm \sqrt{\frac{g_2}{3(4-2\alpha )}}$; it is the minima when the sign is positive. Recalling that we require a root with sign ${{\,\textrm{sgn}\,}}{\tilde{b}} = -{{\,\textrm{sgn}\,}}\alpha = 1$, the local minima must be non-positive, and the value at this ${\tilde{b}}$ is $-\frac{2}{3}{\tilde{b}}g_2 - g_3$. As the value at this minima is monotonically increasing for $m>1/2$, and negative at $m=1/2$, then the value at the minima is negative for all $m < m_*$, the value for which is it zero. Solving, one gets the condition $\Delta _\alpha (m_*)=0$, taking the root greater than 1/2.

Therefore necessary conditions for a real root of the right sign to exist for $\Delta >0$ are that

• if $\alpha > 2$, $m < 1/2$,

• if $\alpha \in (0, 2)$, any m is valid

• if $\alpha < 0$, $m <m_2(\alpha )$, where $m_2$ is the root of $\Delta _\alpha (m)=0$ in (1/2, 1).

To get Nahm data we require that this real root is bounded in magnitude by $\sqrt{g_2/3}$, with the case that it is equal corresponding to $a=0$, i.e. to the $D_4$ monopoles of [26]. Figures showing these parameter regions are given in Fig. 4.

In the case $\Delta < 0$, in order to get real roots of the right sign one analogously gets restrictions on m relative to $\alpha$ such that

• if $\alpha < 0$, $m < 1/2$,

• if $\alpha \in (0,2)$, $m > 1/2$ or $m < m_1(\alpha )$, defined to be the root $<1/2$ of the polynomial $\Delta _\alpha (m)=0$.

• if $\alpha > 2$, $m < m_2(\alpha )$, now defined to be the root $>1/2$ of the polynomial $\Delta _\alpha (m)=0$.

Fixing the size of the root in this case requires more work, complicated by the fact that $g_2$ is real only if $\left|m-1/2 \right| > \sqrt{3}/4$. Using explicitly formulas for the roots ${\tilde{b}}$ from Cardano’s formula one can achieve explicit bounds, but here we omit these. In practice, when using this approach to plot monopoles, numerical methods can be used to find the appropriate m region for a given $\alpha$, as done to generate Fig. 4.

Fig. 4

Valid parameter regions for $V_4$ monopoles, with the subset corresponding to $D_4$ monopoles highlighted

Note that, for certain admissible $\alpha , m$ there may be two possible monopoles because two roots of the cubic defining b satisfy the required conditions. Numerical investigations indicates that this phenomenon only occurs for $\Delta > 0$. We plotted two examples of this, seen in Fig. 5 to investigate the difference in the associated monopoles. These regions stitch together so as to make the moduli space connected.

Fig. 5

Comparison of two pairs of monopoles with equal values of $\alpha , m$. taking ${\mathcal {E}}=0.17$

Appendix B: Initial computation of $D_6$ Nahm matrices

Here we will take the procedure introduced in [22] and developed in [24,25,26] and apply this to the $C_k$ symmetric monopole. In brief, this procedure uses representation theory to construct Nahm matrices with a given symmetry group $G \le {\text {SO}}(3)$. This is done by establishing an isomorphism between the ${\text {SO}}(3)$-representation space ${\mathbb {R}}^3 \otimes {\mathfrak {s}}{\mathfrak {u}}(k)$ containing the matrix triples and a ${\text {SO}}(3)$-representation space described by homogeneous bivariate polynomials acted on naturally via the isomorphism ${\text {SO}}(3) \cong {\text {PSU}}(2)$. On the latter the action of the symmetry group is well understood, and so one can take a polynomial Q invariant under G and return via the isomorphism a triple of matrices $(S_i)$ also invariant under the symmetry. The number of possible matrix triples one can get from the input polynomial is determined by the representation theory, and there is always a triple $(\rho _i)$ invariant under the action of ${\text {SO}}(3)$ corresponding to the polynomial 1. The matrices returned by this isomorphism need not be anti-Hermitian, but they can be fixed to be so.

We describe the procedure for general k before applying this in the $k=3$ context. The steps of the procedure are:

• (1) Take the input polynomials to be $Q_l = \zeta _0^l \zeta _1^l, i=l, \dots , k$ and $Q_{k+1} = \zeta _0^{2k} - \zeta _1^{2k}$.

• (2) Construct the invariant matrix triples $(\rho _i), (S_i^{(j)})$ corresponding to the polynomials 1, $Q_l$, respectively, and scale them so they are all anti-Hermitian. The degree of the input polynomial d determines how many direct summands of the $(d+1)$-dimensional irreducible representation space of $\text {SO} (3)$ there are in ${\mathbb {R}}^3 \otimes {\mathfrak {g}}{\mathfrak {l}}(k)$ when decomposed into irreducibles. There are $3k-2$ invariant S-vectors and so variables $y_j$ associated with them; together with $(\rho _i)$ and the associated variable x we have $3k-1$ variables.

• (3) Set now $T_i = x \rho _i + \sum _j S_i^{(j)}$, where $x, y_j$ are real functions and so $T_i$ is anti-Hermitian.

• (4) Diagonalise the matrix $T_3$ with a unitary matrix U whose columns are the normalised eigenvectors of $T_3$, that is construct $U^{-1} T_3 U$. As $T_3$ is anti-hermitian and linear in the invariant vectors, the diagonal entries which are the eigenvalues will be pure-imaginary, linear in $\left\{ x, y_j \right\}$.

• (5) Now from [7], conjugating by the same unitary matrix will give

  $$\begin{aligned} U^{-1} (T_1 + i T_2)U = \sum _{j=1}^k \alpha _j E_{j, j+1}, \quad U^{-1} (T_1 - i T_2)U = \sum _{j=1}^k -{\bar{\alpha }}_j E_{j+1, j}, \end{aligned}$$

  for some $\alpha _1, \dots \alpha _n \in {\mathbb {C}}$.

• (6) Writing $\alpha _j = r_j e^{i\phi _j}$ (as generically $\alpha _j \ne 0$) and solving

  $$\begin{aligned} \phi _j + \theta _{j+1} - \theta _j&= 0,\; j=1, \dots k-1, \quad \sum _j \theta _j = 0, \end{aligned}$$

  for $\theta _1, \dots , \theta _k \in {\mathbb {R}}$, then conjugating by the unitary matrix $D = {\text {diag}}(e^{i\theta _1}, \dots , e^{i \theta _k})$ preserves the form of $T_3$, but acts to make each $\alpha _j$ real for $j=1, \dots , k-1$ (as it multiplies $\alpha _j$ by $e^{i(\theta _{j+1}-\theta _j)}$). The effect on $\alpha _k$ is to multiply this by $e^{i(\theta _1 - \theta _k)} = e^{i(\phi _1 + \cdots + \phi _{k-1})} = \prod _{j=1}^{k-1} (\alpha _j/r_j)$. After quotienting by this action the number of independent variables we have is $3k-2 + 1 - (k-1) = 2k$.

We now apply the algorithm in the case $k=3$. This yields Nahm matrices

$$\begin{aligned} T_1&= \left( \begin{array}{rrr} 0 &{}\quad i y_{0} &{}\quad i y_{1} + y_{5} + i y_{6} \\ i y_{0} &{}\quad 0 &{}\quad -2 x - y_{2} + i y_{3} \\ i y_{1} - y_{5} + i y_{6} &{}\quad 2 x + y_{2} + i y_{3} &{}\quad 0 \end{array}\right) , \\ T_2&= \left( \begin{array}{rrr} i y_{0} &{}\quad 0 &{}\quad 2 x + y_{2} - i y_{3} \\ 0 &{}\quad -i y_{0} &{}\quad i y_{1} + y_{5} + i y_{6} \\ -2 x - y_{2} - i y_{3} &{}\quad i y_{1} - y_{5} + i y_{6} &{}\quad 0 \end{array}\right) , \\ T_3&= \left( \begin{array}{rrr} i y_{1} + i y_{4} - \frac{2}{3} i y_{6} &{}\quad -2 x + 2 y_{2} &{}\quad 0 \\ 2 x - 2 y_{2} &{}\quad i y_{1} + i y_{4} - \frac{2}{3} i y_{6} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -2 i y_{1} + i y_{4} + \frac{4}{3} i y_{6} \end{array}\right) . \end{aligned}$$

with accompanying ODEs in 8 real variables

$$\begin{aligned} x^\prime&= 2 x^{2} - \frac{1}{3} y_{0}^{2} - \frac{5}{6} y_{1}^{2} - \frac{1}{2} y_{2}^{2} + \frac{1}{6} y_{3}^{2} + \frac{1}{6} y_{5}^{2} + \frac{5}{6} y_{6}^{2}, \\ y_0^\prime&= -4 x y_{0} + 4 y_{0} y_{2}, \\ y_1^\prime&= -4 x y_{1} - \frac{16}{5} y_{1} y_{2} - \frac{6}{5} y_{3} y_{5} - \frac{6}{5} y_{2} y_{6}, \\ y_2^\prime&= \frac{2}{3} y_{0}^{2} - \frac{4}{3} y_{1}^{2} - 2 x y_{2} - y_{2}^{2} - \frac{1}{3} y_{3}^{2} - \frac{1}{3} y_{5}^{2} - y_{1} y_{6} + \frac{1}{3} y_{6}^{2}, \\ y_3^\prime&= 2 x y_{3} - 2 y_{2} y_{3} - 3 y_{1} y_{5} + 2 y_{5} y_{6}, \\ y_4^\prime&= 0,\\ y_5^\prime&= -3 y_{1} y_{3} + 2 x y_{5} - 2 y_{2} y_{5} + 2 y_{3} y_{6}, \\ y_6^\prime&= -\frac{9}{5} y_{1} y_{2} + \frac{6}{5} y_{3} y_{5} + 6 x y_{6} + \frac{6}{5} y_{2} y_{6}. \end{aligned}$$

The associated spectral curve is

$$\begin{aligned} \eta ^3 + \alpha _1 \eta ^2 \zeta + \alpha _2 \eta \zeta ^2 + \alpha _3 \zeta ^3 + \beta \zeta ^6 - {\bar{\beta }}=0 \end{aligned}$$

where

$$\begin{aligned} \alpha _1&= -6 y_{4}, \\ \alpha _2&= 4 y_{0}^{2} - 8 y_{1}^{2} + 48 x y_{2} - 12 y_{2}^{2} + 4 y_{3}^{2} + 12 y_{4}^{2} + 4 y_{5}^{2} + 24 y_{1} y_{6} - \frac{4}{3} y_{6}^{2}, \\ \alpha _3&= -160 x^{2} y_{1} + 16 y_{0}^{2} y_{1} + 8 y_{1}^{3} + 128 x y_{1} y_{2} - 40 y_{1} y_{2}^{2} - 8 y_{1} y_{3}^{2} - 8 y_{0}^{2} y_{4} \\&\quad + 16 y_{1}^{2} y_{4} - 96 x y_{2} y_{4} + 24 y_{2}^{2} y_{4} - 8 y_{3}^{2} y_{4} - 8 y_{4}^{3} - 32 x y_{3} y_{5} + 32 y_{2} y_{3} y_{5} \\&\quad - 8 y_{1} y_{5}^{2} - 8 y_{4} y_{5}^{2} - \frac{32}{3} y_{0}^{2} y_{6} - \frac{128}{3} y_{1}^{2} y_{6} - 32 x y_{2} y_{6} + 80 y_{2}^{2} y_{6} \\&\quad + \frac{16}{3} y_{3}^{2} y_{6} - 48 y_{1} y_{4} y_{6} + \frac{16}{3} y_{5}^{2} y_{6} + 24 y_{1} y_{6}^{2} + \frac{8}{3} y_{4} y_{6}^{2} + \frac{16}{27} y_{6}^{3}, \\ \beta&= -16 x^{2} y_{0} + 4 y_{0} y_{1}^{2} - 16 x y_{0} y_{2} - 4 y_{0} y_{2}^{2} + 8 j y_{0} y_{1} y_{3} - 4 y_{0} y_{3}^{2} - 16 i x y_{0} y_{5} \\&\quad - 8 i y_{0} y_{2} y_{5} + 4 y_{0} y_{5}^{2} + 8 y_{0} y_{1} y_{6} + 8 i y_{0} y_{3} y_{6} + 4 y_{0} y_{6}^{2}. \end{aligned}$$

In order to make the variables real we have imposed the anti-Hermiticity condition required of the Nahm matrices at the beginning, by making the invariant vectors corresponding to each variable anti-Hermitian.

We may consistently set $y_3 = 0 = y_5$, which we may view as using the conjugation action of diagonal matrices ${\text {diag}}(e^{i\theta _1}, e^{i\theta _2}, e^{i\theta _3})$, $\theta _1 + \theta _2 + \theta _3 = 0$. This leaves us with the $2 \times 3 = 6$ real variables we would expect to have from the corresponding Toda. Note that because $\alpha _1^\prime =0$, the centre of mass of the Toda system is already fixed. Moreover, we may centre to consistently set $y_4=0$, and so we now have the equations in the remaining 5 variables as

$$\begin{aligned} x^\prime&= 2 x^{2} - \frac{1}{3} y_{0}^{2} - \frac{5}{6} y_{1}^{2} - \frac{1}{2} y_{2}^{2} + \frac{5}{6} y_{6}^{2}, \\ y_0^\prime&= -4 x y_{0} + 4 y_{0} y_{2}, \\ y_1^\prime&= -4 x y_{1} - \frac{16}{5} y_{1} y_{2} - \frac{6}{5} y_{2} y_{6}, \\ y_2^\prime&= \frac{2}{3} y_{0}^{2} - \frac{4}{3} y_{1}^{2} - 2 x y_{2} - y_{2}^{2} - y_{1} y_{6} + \frac{1}{3} y_{6}^{2}, \\ y_6^\prime&= -\frac{9}{5} y_{1} y_{2} + 6 x y_{6} + \frac{6}{5} y_{2} y_{6}, \end{aligned}$$

with conserved quantities

$$\begin{aligned} \alpha _2&= 4 y_{0}^{2} - 8 y_{1}^{2} + 48 x y_{2} - 12 y_{2}^{2} + 24 y_{1} y_{6} - \frac{4}{3} y_{6}^{2}, \\ \alpha _3&= -160 x^{2} y_{1} + 16 y_{0}^{2} y_{1} + 8 y_{1}^{3} + 128 x y_{1} y_{2} - 40 y_{1} y_{2}^{2} \\&\qquad - \frac{32}{3} y_{0}^{2} y_{6} - \frac{128}{3} y_{1}^{2} y_{6} - 32 x y_{2} y_{6} + 80 y_{2}^{2} y_{6} \\&\qquad + 24 y_{1} y_{6}^{2} + \frac{16}{27} y_{6}^{3}, \\ \beta&= -16 x^{2} y_{0} + 4 y_{0} y_{1}^{2} - 16 x y_{0} y_{2} - 4 y_{0} y_{2}^{2} \\&\qquad + 8 y_{0} y_{1} y_{6} + 4 y_{0} y_{6}^{2}. \end{aligned}$$

At this stage the resulting ODEs are somewhat opaque and we may use the connection to Toda to clarify. Following the steps of the procedure from [7] outlined earlier we may put the Nahm Lax pair in Toda form, namely with

$$\begin{aligned} T_1 + i T_2&= \left( \begin{array}{rrr} 0 &{}\quad -2 \sqrt{2} x - \sqrt{2} y_{1} - \sqrt{2} y_{2} - \sqrt{2} y_{6} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 2 \sqrt{2} x - \sqrt{2} y_{1} + \sqrt{2} y_{2} - \sqrt{2} y_{6} \\ 2 y_{0} &{}\quad 0 &{}\quad 0 \end{array}\right) , \\ T_1 - i T_2&= \left( \begin{array}{rrr} 0 &{}\quad 0 &{}\quad -2 y_{0} \\ 2 \sqrt{2} x + \sqrt{2} y_{1} + \sqrt{2} y_{2} + \sqrt{2} y_{6} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -2 \sqrt{2} x + \sqrt{2} y_{1} -\sqrt{2} y_{2} + \sqrt{2} y_{6} &{}\quad 0 \end{array}\right) , \\ -2iT_3&= \left( \begin{array}{rrr} -4 x + 2 y_{1} + 4 y_{2} -\frac{4}{3} y_{6} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -4 y_{1} + \frac{8}{3} y_{6} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 4 x + 2 y_{1} -4 y_{2} -\frac{4}{3} y_{6} \end{array}\right) . \end{aligned}$$

This gives us variables

$$\begin{aligned} \begin{aligned} a_0&= 2 y_0, \quad a_1 = -2 \sqrt{2} x - \sqrt{2} y_{1} - \sqrt{2} y_{2} - \sqrt{2} y_{6}, \quad \\&a_2 = 2 \sqrt{2} x - \sqrt{2} y_{1} + \sqrt{2} y_{2} - \sqrt{2} y_{6}, \\ b_1&= 4 x - 2 y_{1} - 4 y_{2} + \frac{4}{3} y_{6}, \quad b_2 = 4 y_{1} - \frac{8}{3} y_{6}, \quad \\&b_3 = -4 x - 2 y_{1} + 4 y_{2} + \frac{4}{3} y_{6}. \end{aligned} \end{aligned}$$

These variables are the Flaschka coordinates for the periodic Toda system. (Any 6-tuple satisfying $\sum _i b_i = 0$ gives valid $x, y_j$.) In these new variables we have

$$\begin{aligned} \begin{aligned} a_0^\prime =\,&\frac{1}{2}a_0 ( b_3 - b_1),\quad a_1^\prime = \frac{1}{2} a_1 (b_1 - b_2), \quad a_2^\prime = \frac{1}{2} a_2 ( b_2 - b_3),\\ b_1^\prime =\,&a_1^2 - a_0^2,\quad b_2^\prime = a_2^2 - a_1^2,\quad b_3^\prime = a_0^2 - a_2^2,\quad \end{aligned} \end{aligned}$$

(24)

together with the constants

$$\begin{aligned}&\alpha _2 = b_{1} b_{2} + b_{1} b_{3} + b_{2} b_{3} + a_{0}^{2} + a_{1}^{2} + a_{2}^{2}, \quad \\&\alpha _3 = b_{1} b_{2} b_{3} + b_{1} a_{2}^{2} + b_{2} a_{0}^{2} + b_{3} a_{1}^{2}, \quad \beta = a_0 a_1 a_2. \end{aligned}$$

At this stage we have 6 variables and 3 constraints. One could in principle solve these explicitly using the fact that the flow linearises on the Jacobian of the associated hyperelliptic curve as in [38, Theorem 5.1]. Are simplifications possible? We may use Gröbner bases in Sage to utilise the constants $\alpha _2,\alpha _3, 0 = \sum b_i$ to eliminate the $b_i$, and we get the equations described in the text,

$$\begin{aligned} 0&= \sum _{i=0}^2 a_i^2 - \alpha _2 - \frac{1}{3}(d_1^2 + d_1 d_2 + d_2^2), \\ 0&= a_1^2 d_2 - a_2^2 d_1 + \alpha _3 + \frac{1}{3}\alpha _2(d_1 - d_2) + \frac{1}{27}(d_1 - d_2)^3, \end{aligned}$$

where we have introduced $d_i = \frac{2a_i^\prime }{a_i}$. This in principle is the maximal reduction one can achieve with the variables provided when the $\alpha _i$ and $\beta$ are generic.

One simplification which can be achieved is by attempting to make the second equation a polynomial in $d_1 - d_2$. To do this we would need $a_1^2 = a_2^2$. We can calculate that

$$\begin{aligned} \frac{d}{ds}(a_1^2 - a_2^2)&= 2\left[ a_1 \left( \frac{1}{2} a_1 (b_1 - b_2) \right) - a_2 \left( \frac{1}{2} a_2 ( b_2 - b_3) \right) \right] , \\&= a_1^2 (b_1 - b_2) - a_2^2 (b_2 - b_3), \\&= a_1^2 (b_1 - 2b_2 + b_3) + (a_1^2 - a_2^2)(b_2 - b_3), \\&= -3 b_2 a_1^2 + (a_1^2 - a_2^2)(b_2 - b_3). \end{aligned}$$

Hence we can consistently set $a_1^2 - a_2^2 = 0$ provided $b_2 a_1^2 = 0$. As $b_2^\prime = a_2^2 - a_1^2$, this means we can consistently set $a_1^2 = a_2^2$ and $b_2 = 0$. Making these restrictions we can now eliminate the one remaining equation to find

$$\begin{aligned} 0 = a_0^2 + 2 a_1^2 - \alpha _2 - d_1^2 \Rightarrow a_1^2 \left( 2\frac{da_1}{ds} \right) ^2 = \beta ^2 + 2 a_1^6 - \alpha _2 a_1^4. \end{aligned}$$

Appendix C: Initial computation of $V_4$ Nahm matrices

Taking the polynomials $\zeta _0 \zeta _1 (\zeta _0^4-\zeta _1^4)$, $\zeta _0^2 \zeta _1^2$, and $\zeta _0^4 + \zeta _1^4$ as the inputs to the procedure of outlined in Appendix B gives the ODES in the six real-valued variables

$$\begin{aligned} x^\prime&= 2 x^{2} - \frac{1}{6} y_{0}^{2} + \frac{1}{2} y_{1}^{2} - \frac{1}{2} y_{2}^{2} + \frac{1}{6} y_{3}^{2} - \frac{1}{2} y_{4}^{2},&y_2^\prime&= \frac{1}{3} y_{0}^{2} + y_{1}^{2} - 2 x y_{2} - y_{2}^{2} - \frac{1}{3} y_{3}^{2} - y_{1} y_{4}, \\ y_0^\prime&= -2 x y_{0} + 2 y_{0} y_{2} + 2 y_{1} y_{3} + y_{3} y_{4},&y_3^\prime&= 2 y_{0} y_{1} + 2 x y_{3} - 2 y_{2} y_{3} + y_{0} y_{4},\\ y_1^\prime&= 2 x y_{1} + 2 y_{1} y_{2} + \frac{2}{3} y_{0} y_{3} - y_{2} y_{4},&y_4^\prime&= -2 y_{1} y_{2} + \frac{2}{3} y_{0} y_{3} - 4 x y_{4}, \end{aligned}$$

with the corresponding spectral curve

$$\begin{aligned} {\mathcal {C}}: \quad \eta ^3 + \eta \left[ a(\zeta ^4 + 1) + b \zeta ^2 \right] + c\zeta (\zeta ^4 - 1) = 0, \end{aligned}$$

where

$$\begin{aligned} a&= 8 x y_{0} + 4 y_{0} y_{2} - 4 y_{1} y_{3} + 4 y_{3} y_{4}, \\ b&= 4 y_{0}^{2} - 12 y_{1}^{2} + 48 x y_{2} - 12 y_{2}^{2} + 4 y_{3}^{2} - 24 y_{1} y_{4}, \\ c&= -8 i y_{0}^{2} y_{1} - 8 i y_{1}^{3} + 48 i x y_{1} y_{2} + 24 i y_{1} y_{2}^{2} - 16 i x y_{0} y_{3} + 16 i y_{0} y_{2} y_{3} \\&\qquad + 8 i y_{1} y_{3}^{2} + 48 i x^{2} y_{4} - 4 i y_{0}^{2} y_{4} + 12 i y_{1}^{2} y_{4} - 12 i y_{2}^{2} y_{4} + 4 i y_{3}^{2} y_{4} - 4 i y_{4}^{3}. \end{aligned}$$

The full Nahm matrices are

$$\begin{aligned} T_1&= \begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -\bar{f_1} \\ 0 &{}\quad f_1 &{}\quad 0 \end{pmatrix},&T_2&= \begin{pmatrix} 0 &{}\quad 0 &{}\quad f_2 \\ 0 &{}\quad 0 &{}\quad 0 \\ -\bar{f_2} &{}\quad 0 &{}\quad 0 \end{pmatrix},&T_3&= \begin{pmatrix} 0 &{}\quad -\bar{f_3} &{}\quad 0 \\ f_3 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{pmatrix}. \end{aligned}$$

where the $f_i$ are given by

$$\begin{aligned} f_1&= 2x + y_0 - iy_1 + y_2 + iy_3 + iy_4, \\ f_2&= 2x - y_0 - iy_1 + y_2 - iy_3 + iy_4, \\ f_3&= 2x + 2iy_1 - 2y_2 + iy_4, \end{aligned}$$

One can check that setting $y_1 = y_3 = y_4 = 0$ is consistent, and corresponds to the inversion symmetric case. Note the condition on the residues of the Nahm data now become that the residue of each $f_i$ at the poles is 1.