1 Introduction

spectral curves


I provide an explicit construction of spectral curves for the affine relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system.
I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg–Witten theory, Gromov–Witten theory of orbifolds of the resolved conifold, and Chern–Simons theory to establish a version of the B-model Gopakumar–Vafa correspondence for the Lê–Murakami–Ohtsuki invariant of the Poincaré integral homology sphere to all orders in . On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau–Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type introduced by Dubrovin–Zhang (equivalently, the orbifold quantum cohomology of the type- polynomial orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov–Eynard–Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury–Scott to ADE -orbifolds, and a mirror of the Dubrovin–Zhang construction for all Weyl groups and choices of marked roots.


equationEq.Eqs. \crefnameeqnarrayEq.Eqs. \crefnameconjConjectureConjectures \crefnamedefnDefinitionDefinitions \crefnamelemLemmaLemmas \crefnamethmTheoremTheorems \crefnameclaimClaimClaims \crefnamermkRemarkRemarks \crefnamepropPropositionPropositions \crefnamesectionSectionSections \crefnameappendixAppendixAppendices \crefnamecorCorollaryCorollaries \crefnamefigureFigureFigures \crefnametableTableTables \crefnameexampleExampleExamples


1. Introduction

Spectral curves have been the subject of considerable study in a variety of contexts. These are moduli spaces of complex projective curves endowed with a distinguished pair of meromorphic abelian differentials and a marked symplectic subring of their first homology group; such data define (one or more) families of flat connections on the tangent bundle of the smooth part of moduli space. In particular, a Frobenius manifold structure on the base of the family, a dispersionless integrable hierarchy on its loop space, and the genus zero part of a semi-simple CohFT are then naturally defined in terms of periods of the aforementioned differentials over the marked cycles; a canonical reconstruction of the dispersive deformation (resp. the higher genera of the CohFT) is furthermore determined by through the topological recursion of [Eynard:2007kz].

The one-line summary of this paper is that I offer two constructions (related to Points (II) and (IV) below) and two isomorphisms (related to Points (III), (V) and (VI)) in the context of spectral curves with exceptional gauge symmetry of type .

1.1. Context

Spectral curves are abundant in several problems in enumerative geometry and mathematical physics. In particular:

  1. in the spectral theory of finite-gap solutions of the KP/Toda hierarchy, spectral curves arise as the (normalised, compactified) affine curve in given by the vanishing locus of the Burchnall–Chaundy polynomial ensuring commutativity of the operators generating two distinguished flows of the hierarchy; the marked abelian differentials here are just the differentials of the two coordinate projections onto the plane. In this case, to each smooth point in moduli space with fibre a smooth Riemann surface there corresponds a canonical theta-function solution of the hierarchy depending on times, and the associated dynamics is encoded into a linear flow on the Jacobian of the curve;

  2. in many important cases, this type of linear flow on a Jacobian (or, more generally, a principally polarised Abelian subvariety thereof, singled out by the marked basis of 1-cycles on the curve) is a manifestation of the Liouville–Arnold dynamics of an auxiliary, finite-dimensional integrable system. Coordinates in moduli space correspond to Cauchy data – i.e., initial values of involutive Hamiltonians/action variables – and flow parameters are given by linear coordinates on the associated torus;

  3. all the action has hitherto taken place at a fixed fibre over a point in moduli space; however additional structures emerge once moduli are varied by considering secular (adiabatic) deformations of the integrals of motions via the Whitham averaging method. This defines a dynamics on moduli space which is itself integrable and admits a -function; remarkably, the logarithm of the -function satisfies the big phase-space version of WDVV equations, and its restriction to initial data/small phase space defines an almost-Frobenius manifold structure on the moduli space;

  4. from the point of view of four dimensional supersymmetric gauge theories with eight supercharges, the appearance of WDVV equations for the Whitham -function is equivalent to the constraints of rigid special Kähler geometry on the effective prepotential; such equivalence is indeed realised by presenting the Coulomb branch of the theory as a moduli space of spectral curves, the marked differentials giving rise to the the Seiberg–Witten 1-form, the BPS central charge as the period mapping on the marked homology sublattice, and the prepotential as the logarithm of the Whitham -function;

  5. in several cases, the Picard–Fuchs equations satisfied by the periods of the SW differential are a reduction of the GKZ hypergeometric system for a toric Calabi–Yau variety, whose quantum cohomology is then isomorphic to the Frobenius manifold structure on the moduli of spectral curves. What is more, spectral curve mirrors open the way to include higher genus Gromov–Witten invariants in the picture through the Chekhov–Eynard–Orantin topological recursion: a universal calculus of residues on the fibres of the family , which is recursively determined by the spectral data. This provides simultaneously a definition of a higher genus topological B-model on a curve, a higher genus version of local mirror symmetry, and a dispersive deformation of the quasi-linear hierarchy obtained by the averaging procedure;

  6. in some cases, spectral curves may also be related to multi-matrix models and topological gauge theories (particularly Chern–Simons theory) in a formal expansion: for fixed ’t Hooft parameters, the generating function of single-trace insertion of the gauge field in the planar limit cuts out a plane curve in . The asymptotic analysis of the matrix model/gauge theory then falls squarely within the above setup: the formal solution of the Ward identities of the model dictates that the planar free energy is calculated by the special Kähler geometry relations for the associated spectral curve, and the full expansion of connected multi-trace correlators is computed by the topological recursion.

A paradigmatic example is given by the spectral curves arising as the vanishing locus for the characteristic polynomial of the Lax matrix for the periodic Toda chain with particles. In this case (I) coincides with the theory of -gap solutions of the Toda hierarchy, which has a counterpart (II) in the Mumford–van Moerbeke algebro-geometric integration of the Toda chain by way of a flow on the Jacobian of the curves. In turn, this gives a Landau–Ginzburg picture for an (almost) Frobenius manifold structure (III), which is associated to the Seiberg–Witten solution of pure gauge theory (IV). The relativistic deformation of the system relates the Frobenius manifold above to the quantum cohomology (V) of a family of toric Calabi–Yau threefolds (for , this is ), which encodes the planar limit of Chern–Simons–Witten invariants on lens spaces in (VI).

1.2. What this paper is about

A wide body of literature has been devoted in the last two decades to further generalising at least part of this web of relations to a wider arena (e.g. quiver gauge theories). A somewhat orthogonal direction, and one where the whole of (I)-(VI) have a concrete generalisation, is to consider the Lie-algebraic extension of the Toda hierarchy and its relativistic counterpart to arbitrary root systems associated to semi-simple Lie algebras, the standard case corresponding to . Constructions and proofs of the relations above have been available for quite a while for (II)-(IV) and more recently for (V)-(VI), in complete generality except for one, single, annoyingly egregious example: , whose complexity has put it out of reach of previous treatments in the literature. This paper grows out of the author’s stubborness to fill out the gap in this exceptional case and provide, as an upshot, some novel applications of Toda spectral curves which may be of interest for geometers and mathematical physicists alike. As was mentioned, the aim of the paper is to provide two main constructions, and prove two isomorphisms, as follows.

Construction 1:

The first construction fills the gap described above by exhibiting closed-form expressions for arbitrary moduli of the family of curves associated to the relativistic Toda chain of type for its sole quasi-minuscule representation – the adjoint. This is achieved in two steps: by determining the dependence of the regular fundamental characters of the Lax matrix on the spectral parameter, and by subsequently computing the polynomial character relations in the representation ring of (viewed as a polynomial ring over the fundamental characters) corresponding to the exterior powers of the adjoint representation. The last step, which is of independent representation-theoretic interest, is the culmination of a computational tour-de-force which in itself is beyond the scope of this paper, and will find a detailed description in [E8comp]; I herein limit myself to announce and condense the ideas of [E8comp] into the 2-page summary given in \crefsec:charE8, and accompany this paper with a Mathematica package1 containing the solution thus achieved. As an immediate spin-off I obtain the generating function of the integrable model (in the language of [Fock:2014ifa]) as a function of the basic involutive Hamiltonians attached to the fundamental weights, and a family of spectral curves as its vanishing locus. In the process, this yields a canonical set of integrals of motion in involution in cluster variables and in Darboux co-ordinates for the integrable system on a special double Bruhat cell of the coextended Poisson–Lie loop group , which, by analogy with the case of -series, I call “the relativistic Toda chain”, and whose dynamics is solved completely by the preceding construction.

Construction 2:

The previous construction gives the first element in the description of the spectral curve – a family of plane complex algebraic curves, which are themselves integrals of motion. The next step determines the three remaining characters in the play, namely the two marked Abelian differentials and the distinguished sublattice of the first homology of the curves; this goes hand in hand with the construction of appropriate action–angle variables for the system. The ideology here is classical [MR1013158, MR533894, MR815768, MR1401779, MR1397059] in the non-relativistic case, and its adaptation to the relativistic setting at hand is straightforward: I identify the phase space of the Toda system with a fibration over the Cartan torus of (times ) by Abelian varieties, which are Prym–Tyurin sub-tori of the spectral curve Jacobian. These are selected by the curve geometry itself, due to an argument going back to Kanev [MR1013158], and the Liouville–Arnold flows linearise on them. The Hamiltonian structure inherited from the embedding of the system into a Poisson–Lie–Bruhat cell translates into a canonical choice of symplectic form on the universal family of Prym–Tyurins, and it pins down (up to canonical transformation) a marked pair of Abelian third kind differentials on the curves.

Altogether, the family of curves, the marked 1-forms, and the choice of preferred cycles lead to the assignment of a set of Dubrovin–Krichever data (\crefdefn:dk) to the family of spectral curves. Armed with this, I turn to some of the uses of Toda spectral curves in the context of Fig. 1.

5d SW theory Chern-Simons/WRTinvariant of B-model: spectral curveof relativistic TodaA-model on
Figure 1. Duality web for the B-model on Toda spectral curves
Isomorphism 1:

Toda spectral curves have long been proposed to encode the Seiberg–Witten solution of pure gluodynamics in four dimensional Minkowski space [Gorsky:1995zq, Martinec:1995by], as well as of its higher dimensional parent theory on [Nekrasov:1996cz] in the relativistic case. From the physics point of view, Constructions 1-2 provide the Seiberg–Witten solution for minimal, five-dimensional supersymmetric Yang–Mills theory on ; and as the latter should be related to (twisted) curve counts on an orbifold of the resolved conifold by the action of the binary icosahedral group , the same construction provides a conjectural 1-dimensional mirror construction for the orbifold Gromov–Witten theory of these targets, as well as to its large Chern–Simons dual theory on the Poincaré sphere [Gopakumar:1998ki, Ooguri:1999bv, Aganagic:2002wv, Borot:2015fxa]. I do not pursue here the proof of either the bottom horizontal (SW/integrable systems correspondence) or the diagonal (mirror symmetry) arrow in the diagram of Fig. 1, although it is highlighted in the text how having access to the global solution on its Coulomb branch allows to study particular degeneration limits of the solution corresponding to superconformal (maximally Argyres–Douglas) points where mutually non-local dyons pop up in the massless spectrum, and limiting versions of mirror symmetry for the Toda curves in Isomorphism 2 below are also considered. What I do prove instead is a version of the vertical arrow, completing results in a previous joint paper with Borot [Borot:2015fxa]: namely, that the Chern–Simons/Reshetikhin–Turaev–Witten invariant of restricted to the trivial flat connection (the Lê–Murakami–Ohtsuki invariant), as well as the quantum invariants of fibre knots therein in the same limit and for arbitrary colourings, are computed to all orders in from the Chekhov–Eynard–Orantin topological recursion on a suitable subfamily of relativistic Toda spectral curves. As in [Borot:2015fxa], the strategy resorts to studying the trigonometric eigenvalue model associated to the LMO invariant of the Poincaré sphere at large and to prove that the planar resolvent is one of the meromorphic coordinate projections of a plane curve in , which is in turn shown to be the affine part of the spectral curve of the relativistic Toda chain.

Isomorphism 2:

I further consider two meaningful operations that can be performed on the spectral curve setup of Construction 1-2. The first is to take a degeneration limit to the leaf where the natural Casimir function of the affine Toda chain goes to zero; this corresponds to the restriction to degree-zero orbifold invariants on the top-right corner of Fig. 1, and to the perturbative limit of the 5D prepotentials of the bottom-right corner. The second is to replace one of the marked Abelian integrals with their exponential; this is a version of Dubrovin’s notion of (almost)-duality of Frobenius manifolds [MR2070050].

I conjecture and prove that the resulting spectral curve provides a 1-dimensional Landau–Ginzburg mirror for the Frobenius manifold structure constructed on orbits of the extended affine Weyl group of type by Dubrovin and Zhang [MR1606165]. Their construction depends on a choice of simple root, and the canonical choice they take matches with the Frobenius manifold structure on the Hurwitz space determined by our global spectral curve. This extends to the first (and most) exceptional case the LG mirror theorems of [Dubrovin:2015wdx] for the classical series; and it opens the way to formulate a precise conjecture for how the general case, encompassing general choices of simple roots in the Dubrovin–Zhang construction, should receive an analogous description in terms of Toda spectral curves for the corresponding Poisson–Lie group and twists thereof by the action of a Type I symmetry of WDVV (in the language of [Dubrovin:1994hc]). Restricting to the simply-laced case, this gives a mirror theorem for the quantum cohomology of ADE orbifolds of ; our genus zero mirror statement then lifts to an all-genus statement by virtue of the equivalence of the topological recursion with Givental’s quantisation for R-calibrated Frobenius manifolds. This provides a version, for the ADE series, of statements by Norbury–Scott [MR3268770, DuninBarkowski:2012bw, MR3654104] for the Gromov–Witten theory of .

The two constructions and two isomorphisms above will find their place in \crefsec:E8chain, 3, 4 and 5 respectively. I have tried to give a self-contained exposition of the material in each of them, and to a good extent the reader interested in a particular angle of the story may read them independently (in particular \crefsec:applI,sec:applII).


I would like to thank G. Bonelli, G. Borot, A. D’Andrea, B. Dubrovin, N. Orantin, N. Pagani, P. Rossi, A. Tanzini, Y. Zhang for discussions and correspondence on some of the topics touched upon in this paper, and H. Braden for bringing [MR1182413, MR1401779, MR1668594] to my attention during a talk at SISSA in 2015. For the calculations of \crefsec:charE8 and [E8comp], I have availed myself of cluster computing facilities at the Université de Montpellier (Omega departmental cluster at IMAG, and the HPC@LR centre Thau/Muse inter-faculty cluster) and the compute cluster of the Department of Mathematics at Imperial College London. I am grateful to B. Chapuisat and especially A. Thomas for their continuous support and patience whilst these computations were carried out. This research was partially supported by the ERC Consolidator Grant no. 682603 (PI: T. Coates).

2. The and relativistic Toda chain

I will provide a succinct, but rather complete account of the construction of Lax pairs for the relativistic Toda chain for both the finite and affine root system. This is mostly to fix notation and key concepts for the discussion to follow, and there is virtually no new material here until \crefsec:speccurve. I refer the reader to [Fock:2014ifa, Williams:2012fz, MR1993935, Reyman:1979ru, Olshanetsky:1981dk] for more context, references, and further discussion. I will subsequently move to the explicit construction of spectral curves and the action-angle map for the affine chain in \crefsec:speccurve,sec:actangl.

2.1. Notation

I will start by fixing some basic notation for the foregoing discussion; in doing so I will endeavour to avoid the uncontrolled profileration of subscripts “8” related to throughout the text, and stick to generic symbols instead (such as for the Lie group, for its Lie algebra, and so on). I wish to make clear from the outset though that whilst many aspects of the discussion are general, the focus of this section is on alone; the attentive reader will notice that some of its properties, such as simply-lacedness, or triviality of the centre, are implicitly assumed in the formulas to follow.

Let then denote the complex simple Lie algebra corresponding to the Dynkin diagram of type (Fig. 2). I will write for the corresponding simply-connected complex Lie group, for the maximal torus (the exponential of the Cartan algebra ), and for the Weyl group. I will also write for the set of simple roots (see e.g. (B.1)), and , , , to indicate respectively the full root system, the non-vanishing roots, the zero roots, and the negative/positive roots; the choice of splitting determines accordingly Borel subgroups intersecting at . Each Borel realises as a disjoint union of double cosets , the double Bruhat cells of . The Euclidean vector space is a vector subspace of with an inner product structure given by the dual of the Killing form; in particular, is the Cartan matrix (B.3). For a weight in the lattice , I will write for the parabolic subgroup stabilised by ; the action of on weights is the restriction of the coadjoint action on ; since in our case, the weight lattice is isomorphic to the root lattice . Corresponding to the choice of , Chevalley generators for will be chosen satisfying


Accordingly, the correponding time- flows on lead to Chevalley generators , for the Lie group. Finally, I denote by the representation ring of , namely the free abelian group of virtual representations of (i.e. formal differences), with ring structure given by the tensor product; this is a polynomial ring over the integers with generators given by the irreducible -modules having as their highest weights, where .

Figure 2. The Dynkin diagrams of type and, superimposed in red, type ; roots are labelled following Dynkin’s convention (left-to-right, bottom-to-top). The numbers in blue are the Dynkin labels for each vertex – for the non-affine roots, these are the components of the highest root in the -basis.

Most of the notions (and notation) above carries through to the setting of the Kac–Moody group2 where is the (necessarily untwisted, for ) affine Lie algebra corresponding to . In this case we adjoin the highest (affine) root as in (B.2), leading to the Dynkin diagram and Cartan matrix in Fig. 2 and (B.4). Elements are linear -differential polynomials in the spectral parameter ; namely, , with the pointwise multiplication rule leading to


The Chevalley generators for the simple Lie group are then lifted to , with the Dynkin labels as in Fig. 2, and extended to include where


with and the Lie algebra generators corresponding to the highest (lowest) roots – i.e. the only non-vanishing iterated commutators of order of (), .

2.2. Kinematics

Consider now the 16-dimensional symplectic algebraic torus

with Poisson bracket


Semi-simplicity of amounts to the non-degeneracy of the bracket, so that is symplectic.

There is an injective morphism from to a distinguished Bruhat cell of , as follows. Notice first that carries an adjoint action by the Cartan torus which obviously preserves the Borels, and therefore, descends to an action on the double cosets of the Bruhat decomposition. Consider now Weyl group elements where is the ordered product of the eight simple reflections in . The corresponding cell has dimension 16 [Fock:2014ifa], and it inherits a symplectic structure from , as I now describe. Recall that the latter carries a Poisson structure given by the canonical Belavin–Drinfeld–Olive–Turok solution of the classical Yang–Baxter equation [MR674005, Olive:1983mw]:


with given by


Since is a trivial Poisson submanifold, inherits a Poisson structure from the parent Poisson–Lie group. Consider now the (Lax) map


Then the following proposition holds.

Proposition 2.1 (Fock–Goncharov, [Mr2263192]).

is an algebraic Poisson embedding into an open subset of .

Similar considerations apply to the affine case. In with exponentiated linear co-ordinates and log-constant Poisson bracket


consider the hypersurface , where are the Dynkin labels of Fig. 2. Since , is not symplectic anymore, unlike the simple Lie group case above; in particular, the regular function


is a Casimir of the bracket (2.8), and it foliates symplectically. As before, there is a double coset decomposition of indexed by pairs of elements of the affine Weyl group , and a distinguished cell labelled by the element corresponding to the longest cyclically irreducible word in the generators of . Projecting to trivial central (co)extension


induces a Poisson structure on the projections of the cells (and in particular ), as well as their quotients by the adjoint action of the Cartan torus, upon lifting to the loop group the Poisson–Lie structure of the non-dynamical r-matrix (2.5). I will write for the resulting Poisson manifold; and we have now that [Fock:2014ifa]

Consider now the morphism


It is instructive to work out explicitly the form of the loop group element corresponding to ; we have


where in moving from the first to the second line we have expanded as a linear differential operator and grouped together all the multiplicative shifts, and then used that on , which gives indeed an element with trivial co-extension. The same line of reasoning of \crefprop:fg shows that is a Poisson monomorphism.

2.3. Dynamics

For functions , the Poisson bracket (2.5) reads, explicitly,


where (resp. ) denotes the left (resp. right) invariant vector field generated by . Then a complete system of involutive Hamiltonians for (2.5) on , and any Poisson Ad-invariant submanifold such as , is given by Ad-invariant functions on the group – or equivalently, Weyl-invariant functions on . This is a subring of generated by the regular fundamental characters


where is the irreducible representation having the fundamental weight as its highest weight. In the affine case the same statements hold, with the addition of the central Casimir in (2.9). The Lax maps (2.7), (2.11) then pull-back this integrable dynamics to the respective tori and . Fixing a faithful representation (say, the adjoint), the same dynamics on and takes the form of isospectral flows [MR1995460, Sec. 3.2-3.3]


where is the expression of the Weyl-invariant Laurent polynomial in terms of power sums of the eigenvalues of , and denotes the projection to the positive Borel.

2.4. The spectral curve

We henceforth consider the affine case only. Since (2.16) is isospectral, all functions of the spectrum of are integrals of motion. A central role in our discussion will be played by the spectral invariants constructed out of elementary symmetric polynomials in the eigenvalues of , for the case in which is the adjoint representation, that is, is the minimal-dimensional non-trivial irreducible representation of . I write


for the characteristic polynomial of in the adjoint, thought of as a 2-parameter family of maps . It is clear by (2.16) that is an integral of motion for all , and so is therefore the plane curve in given by its vanishing locus .

We will be interested in expanding out the flow invariant (2.17) as an explicit polynomial function of the basic integrals of motion (2.14). I will do so in two steps: by determining the dependence of (2.14) on the spectral parameter when in (2.12) and (2.14), and by computing the dependence of on the basic invariants (2.14). We have first the following

Lemma 2.2.

, are Laurent polynomials in , which are constant except for . In particular, there exist functions such that


with and .

Proof (sketch).

The proof follows from a lengthy but straightforward calculation from (2.12). Since we are looking at the adjoint representation, explicit matrix expressions for the Chevalley generators (2.1) can be computed by systematically reading off the structure constants in (2.1), the full set of which for all the generators of the algebra is determined from the canonical assignment of signs to so-called extra-special pairs of roots reflecting the ordering of simple roots within (see [MR2047097] for details). The resulting matrix in (2.12), with coefficients depending on , is moderately sparse, which allows to compute power sums of its eigenvalues efficiently. We can then show from a direct calculation that (2.18) holds for the relations in


which are an easy consequence of the decomposition into irreducibles of , , and their tensor powers for ,3 and the use of Newton identities relating power sum polynomials (i.e. traces of powers) to elementary/complete symmetric polynomials (i.e. antisymmetric/symmetric traces). A little more work is required to show that is constant for ; this uses the more complicated character relations (2.25)-(2.26) of \crefsec:charE8. The final result is (2.18). ∎

It is immediately seen from (2.18) that are involutive, independent integrals of motion; they are equal to the fundamental Hamiltonians (2.14) for , and for they are a linear combination of and the Casimir . Denote now by the image of under the map . It is clear from (2.17) and (2.18) that factors through and a map given by the decomposition of the characteristic polynomial into fundamental characters:


where the reality of the adjoint representation has been used. Here is the polynomial relation of formal characters


evaluated at the group element . For fixed and , the vanishing locus of the characteristic polynomial is a complex algebraic curve in ; I shall write for the variety of parameters this polynomial will depend on. Even though is irreducible, the curve is reducible since is. Indeed, conjugating to an element in the Cartan torus, , we have


For a general representation , we would obtain as many irreducible components as the number of Weyl orbits in the weight system. When , and for this case alone, we have only one non-trivial orbit, as well as eight trivial orbits corresponding to the zero roots. I will factor out the trivial component corresponding to zero roots by writing .

Definition 2.1.

For , let be the normalisation of the projective closure of . We call the corresponding family of plane curves ,


the family of spectral curves of the relativistic Toda chain in the adjoint representation. In (2.23), are the points added in the compactification of (see \crefrmk:ptsinf,tab:ptsinf below) and are the sections marking them.

As is known in the more familiar setting of , and as we will discuss in \crefsec:actangl, spectral curves are a key ingredient in the integration of the Toda flows. Knowledge of the spectral curves is encoded into knowledge of the character relations (2.21), which grant access to the explicit form of the polynomial to spectral curves for arbitrary moduli : the description of the spectral curves is then reduced to the purely representation-theoretic problem of determining these relations.

In view of this, denote , . What we are looking for are explicit polynomials


where the index runs over a suitable finite set , , and . Since what we are ultimately interested in is the reduced characteristic curve , it suffices to compute (and hence ) for .

Claim 2.3 ([E8comp]).

We determine for all , .

This is the result of a series of computer-assisted calculations, of independent interest and whose details will appear elsewhere [E8comp], but for which I provide a fairly comprehensive summary in \crefsec:charE8. For the sake of example, we obtain for the first few values of ,