Elsevier

Journal of Algebra

Volume 551, 1 June 2020, Pages 301-341
Journal of Algebra

Exterior powers of the adjoint representation and the Weyl ring of E8

https://doi.org/10.1016/j.jalgebra.2020.01.020Get rights and content

Abstract

I derive explicitly all polynomial relations in the character ring of E8 of the form χke8pk(χ1,,χ8)=0, where ke8 is an arbitrary exterior power of the adjoint representation and χi is the ith fundamental character. This has simultaneous implications for the theory of relativistic integrable systems, Seiberg–Witten theory, quantum topology, orbifold Gromov–Witten theory, and the arithmetic of elliptic curves. The solution is obtained by reducing the problem to a (large, but finite) dimensional linear problem, which is amenable to an efficient solution via distributed computation.

Introduction

Let G be a complex, simple, simply-connected Lie group of rank r and VRepC(G) an element of its representation ring. We may view the latter, upon taking characters VχV, as the Weyl ring of Ad-invariant regular functions on G, or equivalently, as the ring of W(G)-invariant regular functions on the Cartan torus, where W(G) is the Weyl group of G. It is a basic fact in Lie theory that this is a polynomial ring over the integers, RepC(G)Z[χ1,,χr], where χj, j=1,,r denotes the character of the jth fundamental representation of G.

In this paper we will be concerned with a special instance of the following

Problem 1.1

Given a finite-dimensional representation V of G, find polynomials pkVZ[x1,,xr], k=1,,dimCV, such thatdetV(gμ1)=kpkV(χ1(g),,χr(g))(μ)dimVkZ[χ1(g),,χr(g)][μ], for all gG and k=1,,dimCV. Equivalently, given an arbitrary exterior power of V, determine the corresponding polynomial relations in Rep(G) of the form0=χkVpkV(χ1,,χr).

In other words, Problem 1.1 asks to find explicit expressions for characteristic polynomials in a given representation V (alternatively, of antisymmetric characters of V) in terms of polynomials in the fundamental characters. For example, if G=SLN(C) and V=Vω1 is the defining representation of SLN(C), we have simplypkV(χ1,,χr)=χk since Vωk=kVω1.

Let d0(V) denote the dimension of the zero-weight space of V. A case of particular importance for applications is when the characteristic polynomial detV(gμ1) (respectively: detV(gμ1)(1μ)d0(V)) in (1.1) is irreducible over Rep(G): this amounts to V being a minuscule (respectively: quasi-minuscule) irreducible representation. In the quasi-minuscule setting, Problem 1.1 is computationally easy for most Dynkin types and quasi-minuscule representations (see [3], [4]), with one single, egregious exception: this is G=E8 and V=e8, which is of formidable complexity. The purpose of the present paper is to present a solution of this exceptional case, which had previously been announced in [4, Appendix A].

As it stands, Problem 1.1 is of purely representation theoretic character. At the same time, my motivation for looking at it is mostly extrinsic in nature, and it is eminently geometrical: there are indeed six different classes of questions in Geometry and Mathematical Physics that are simultaneously answered by giving an explicit solution to Problem 1.1, and in particular to the case when (G,V)=(E8,e8), as follows.

Let T denote the maximal torus of G. A central object in the theory of algebraically complete integrable systems is the datum of a rational mapL:Cg×PG from the product of a 2r-complex algebraic symplectic variety P and a fixed smooth curve Cg with h1,0(Cg)=g to G, called the Lax map. Associated to L is the family of spectral curves C[L] where

  • B=χ(T) is the image of the torus under the fundamental regular characters χi, i=1,,r; the natural co-ordinates u on B, uiχi(g) for gG, give a co-ordinate chart on T;

  • for fixed u=(ui=χi(L))i, Σu is the compact Riemann surfaceΣu=V(detV(Lμ1)) given by the smooth completion (normalisation of the projective closure) of the algebraic curve in A1×Cg given by the vanishing locus of the characteristic polynomial of L at fixed u.

More explicitly,detV(Lμ1)=k=0dimV(μ)dimVkχkVLMCg[μ;u1,,ur] where MCg is the ring of meromorphic functions on Cg. A Hamiltonian dynamical system can then be defined on P in terms of isospectral flows on L; in particular, any Ad-invariant function of L is an integral of motion, and a maximal involutive set of these is given by the fundamental characters ui(L). For fixed u, the resulting flow is linear on the Picard group of the irreducible components Σu, and the dynamics is independent of V so long as V1 [8], [19], [24], [25]: so for simplicity, we may assume V to be minuscule (resp. quasi-minuscule for G=E8), which entails that Σu (resp. the reduced component of Σu) is generically irreducible.

A central example is given by the periodic relativistic Toda chain of type G: in this case CgP1, and in an affine co-ordinate λ on the base P1 the Lax map satisfies [3], [4]λui(L)=δiitop(11λ2), where Vitop is the top dimensional fundamental representation. This means that (1.7) becomesdetV(Lμ1)=k=0dimV(μ)dimVkχkVLZ[μ,λ±;u1,,ur] where the last step requires expanding χkV as a polynomial in χi from the solution of Problem 1.1, and using (1.8). A complete presentation for the family of spectral curves (1.5), and the complete solution for the dynamics of the underlying integrable model, follows thus from solving Problem 1.1 for the given pair (G,V).

The Toda systems of the previous section have a central place in the study of supersymmetric quantum field theories [15], [23], [26]. In particular, constructing explicitly the family of spectral curves (1.5) encodes the solution of the low energy effective dynamics for N=1 supersymmetric gauge theories with no hypermultiplets on R1,3×S1: in this setting, the fundamental characters ui are the semiclassical, Weyl-invariant gauge parameters and the full effective action up to two derivatives in the supercurvature of the gauge field, including all-order instanton corrections, is recovered via period integrals of logμdlogλ on Σu. I refer the reader to [3], [4], [26] for a fuller discussion of the link between the relativistic Toda chain and this class of five-dimensional quantum field theories.

The case G=E8 has been outstanding since the solution of the Wilsonian dynamics of the theory was proposed shortly after the celebrated work of Seiberg–Witten [23], [26], [29]; the same considerations for the relativistic Toda chain recast this problem into solving Problem 1.1 for (E8,e8).

Let g be a simple complex Lie algebra g of type An, Dn or En. Let Vg be the defining representation for g=sl(n+1), the vector representation for g=so(2n), and the 27E6, 56E7 and 248E8=e8 for g=e6,7,8 respectively, and denote by Γg<SU(2) the finite order subgroup of SU(2) of the same Dynkin type of g associated to g by the classical McKay correspondence. Considerations about large N duality in gauge theory led [3], [4] to propose that the family of curves (1.5) for (G,V)=(exp(g),Vg) determines the asymptotics of the Witten–Reshetikhin–Turaev invariant of spherical 3-space forms Mg=S3/Γg via the Chekhov–Eynard–Orantin topological recursion [5], [11], [12]. In particular, the case Γe8=I120 being the binary icosahedral group gives the distinguished case of the Poincaré integral homology sphere Me8=Σ(2,3,5). The all-order asymptotic expansion of the quantum invariants of the Poincaré sphere is fully determined by periods of logμdlogλ and the topological recursion on the spectral curves (1.5) for the G-relativistic Toda chain – and therefore, ultimately, by solving Problem 1.1 for (G,V)=(E8,e8).

Let OP1(1) denote the tautological bundle on the complex projective line, and let Xg[O(1)P12/Γg] be the fibrewise quotient stack of Tot(O(1)O(1)P1) by the action of the finite group Γg as in Section 1.2.3. The Gromov–Witten potential of Xg is a formal generating series of virtual counts of twisted stable curves to the orbifold Xg, to all genera, degrees, and insertions of twisted cohomology classes [1]. It was proposed and non-trivially checked in [3], [4] that the full Gromov–Witten potential of Xg coincides with the Chern–Simons partition function of Mg; therefore the full curve counting information on Xg is encoded into (1.5), and solved by Problem 1.1, as an instance of 1-dimensional mirror symmetry for Calabi–Yau manifolds.

A suitable degeneration2 of the family (1.5) was proved in [4] to provide a new 1-dimensional mirror for the orbifold quantum co-homology of the orbifold projective line

, or, equivalently [28], to the Frobenius manifold structure on the orbits of the affine Weyl group of the same ADE type of Γg [9]. This allowed [4] to solve the long-standing problem [9] of determining flat co-ordinates for the Saito metric for all ADE types, and gave a higher genus reconstruction theorem by the Chekhov–Eynard–Orantin topological recursion as a bonus. Once more, the central tool in the theorem is the closed-form presentation of the family of spectral curves (1.5), and therefore, the solution of Problem 1.1 in all ADE types; particularly, (G,V)=(E8,e8).

A subject of particular interest in arithmetic geometry is the construction of extensions of the rationals with Galois group equal to the full Weyl group of E8 [30], [32]. By a theorem of Shioda [30, Thm .7.2], the Galois action on the extension arises from the action of the full Weyl group of E8 on the Mordell–Weil lattice of a non-isotrivial elliptic curve E over Q(t), and in turn, from the datum of a degree 240, monic integer polynomial ΨZ[x] whose splitting field has W(E8) as Galois group: a few explicit instances of these polynomials were constructed in [18], [21], [31]. Solving Problem 1.1 for (G,V)=(E8,e8) gives manifestly a Z8-worth of candidate such integer polynomials3 upon specialising χiZ: and for generic integral values [21], the resulting polynomial has the full Weyl group of E8 as Galois group.4 The solution of Problem 1.1 generates in this way an infinite wealth of examples of multiplicatively excellent families of elliptic surfaces (in the sense of [21]) of type E8.

I briefly describe here the strategy employed to solve Problem 1.1 for (G,V)=(E8,e8), which was announced in my previous work [4, Appendix C]. In its crudest terms, what we would like to achieve amounts to enforcingχke8=pk(χ1,,χ8)Z[χ1,,χ8] for an unknown polynomial pk as an identity of regular functions on the Cartan torus – that is, as an identity between integral Laurent polynomials. However, a direct calculation of pk in this vein is unviable because of the exceptional complexity of E8, even for small values of k. We find a workaround that breaks up the task of determining pk in (1.10) into three main steps:

    Casimir bound and finite-dimensional reduction

    An a priori bound on the number of monomials appearing in pk in (1.10) holds. The set of allowed monomials has cardinality 106, and (1.10) reduces then to a finite-dimensional linear problem of the same rank upon taking sufficiently many numerical “sampling points” gT and evaluating χke8(g) and χi(g) at g.

    Partition of the monomial set

    For generic sampling sets, the resulting linear problem is size-wise three orders of magnitude beyond the reach of practical calculations. There exist however special choices of the sampling set such that the original linear problem is equivalent to 4×103 linear sub-systems of size varying from one to 3×103. This can be realised by constructing the sampling set numerically using Newton–Raphson inversion, evaluating derivatives of characters in exponentiated linear co-ordinates on the torus up to sufficiently high order, and establishing rigorous analytic bounds to perform an exact integer rounding.

    Partition of the sampling set and distributed computing

    The problem can then be solved effectively on a computer: most of the runtime comes from the construction of the linear subsystems, by carrying out the generation of the sampling set and the evaluation of derivatives of χi and χke8, and it is in the order of a few years. On the other hand the calculation can be effectively parallelised by a suitable segmentation of the sampling set, and subsequent distribution among different processor cores: this allowed to reduce in our computer implementation the total absolute clock-time taken by the entire computation to about six weeks on a small departmental cluster. The result is available as a computer package at http://tiny.cc/E8Char, and a description of the individual files is given in Appendix A.

The last two points indicate quite clearly that this project had a very substantial computational component to it. I discuss rather diffusely the details of its concrete implementation in Sections 2.3 and 3.

The paper is organised as follows. In Section 2 we review the problem and explain the Casimir bound that reduces Problem 1.1 to a finite-dimensional, generically dense linear problem of rank 106; the Section ends with the central statement (Theorem 2.4) that suitable choices of sampling sets lead to a block-reduction of this linear problem to 4×103 linear sub-systems over the rationals by suitable partitions of the set of admissible monomials; both the reduction and the solution of these linear problems can be performed effectively by parallel computation. This statement is justified in Section 3, which occupies the main body of the paper: after reviewing general computational strategies in Section 2.3.1, I describe the partition of the original problem in Section 3.1, and the exact computation of the reduced linear problems using semi-numerical methods and analytic bounds for exact integer roundings in Sections 3.2–3.4; additional details of the computer implementation are given in Section 3.5. Finally, Section 4 contains some applications to the construction of explicit integral polynomials with Galois group Weyl(E8); the reader is referred to [4] for further applications of the results obtained here to integrable systems (E8 relativistic Toda lattices), E8 Seiberg-Witten theory, quantum invariants of the Poincaré sphere, and Frobenius manifolds on orbits of the affine Weyl group of type E8. Use and abuse of notation throughout the text will be in accordance with Table 1.

I am grateful to G. Borot and A. Klemm for useful discussions on related topics, and to A. D'Andrea, C. Bonnafé, M. Coti Zelati, D. Craven, S. Goodwin and T. Weigel for taking the time to answer my questions at various stages of progress of this work. I am indebted to F. David–Collin and especially B. Chapuisat and A. Thomas for patiently dealing with my incessant requests for libraries and resources to be installed, respectively, on the Omega departmental cluster at the Institut Montpelliérain Alexander Grothendieck in Montpellier, and the Maths Compute Cluster at Imperial College London, where most of the debugging, tests, and parallel calculations leading up to the results of this work were carried out. Partial computing support from the interfaculty HPC@LR computing cluster Muse at the University of Montpellier is also acknowledged. I made extensive use of the GNU C libraries GMP, MPFR, and MPC [10], [13], [16] for the use of arbitrary precision floating-point arithmetics over the complex numbers, and of the FLINT C library [17] for multi-precision arithmetics over the integers and the rationals; C source code for all calculations is available upon request. This research was partially supported by the ERC Grant no. 682603 (PI: T. Coates) and the EPSRC Fellowship grant EP/S003657/1.

Section snippets

Partition of the monomial set

The key point we explained at the end of the previous section is that while generic choices of sampling sets in Method 1 give rise to large, dense linear systems, there are special choices of the numerical values of Q(ι)TE8 that reduce the calculation of Nι(k) to the solution of a large number of linear problems of much smaller size. We explore this idea in detail in this section.

Letφ:I{0,1}8ι(θ(ι1),,θ(ι8)) where for nNθ(n)={1n00n=0, and let Iϕ=φ1(ϕ) for ϕ{0,1}8. The map φ maps an

Applications

A big chunk of the “Applications” section of this paper is really [4]; the polynomial character decompositions found here were used there to construct the spectral curves of the E8ˆ-relativistic Toda chain (Section 1.2.1) and the Seiberg–Witten curves of E8 minimallly supersymmetric Yang–Mills theory on R1,3×S1 (Section 1.2.2), as well as to prove an all-genus version of the Gopakumar–Ooguri–Vafa correspondence (Section 1.2.3) for the Poincaré sphere, and to provide a mirror theorem for the

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