A Lie Algebra

Duality and Modularity in Elliptic Integrable Systems and Vacua of Gauge Theories

Antoine Bourget and Jan Troost Laboratoire de Physique Théorique1 Ecole Normale Supérieure 24 rue Lhomond, 75005 Paris, France

Abstract :

We study complexified elliptic Calogero-Moser integrable systems. We determine the value of the potential at isolated extrema, as a function of the modular parameter of the torus on which the integrable system lives. We calculate the extrema for low rank root systems using a mix of analytical and numerical tools. For we find convincing evidence that the extrema constitute a vector valued modular form for the congruence subgroup of the modular group. For and , the extrema split into two sets. One set contains extrema that make up vector valued modular forms for congruence subgroups (namely , and ), and a second set contains extrema that exhibit monodromies around points in the interior of the fundamental domain. The former set can be described analytically, while for the latter, we provide an analytic value for the point of monodromy for , as well as extensive numerical predictions for the Fourier coefficients of the extrema. Our results on the extrema provide a rationale for integrality properties observed in integrable models, and embed these into the theory of vector valued modular forms. Moreover, using the data we gather on the modularity of complexified integrable system extrema, we analyse the massive vacua of mass deformed supersymmetric Yang-Mills theories with low rank gauge group of type and . We map out their transformation properties under the infrared electric-magnetic duality group as well as under triality for with gauge algebra . We compare the exact massive vacua on to those found in a semi-classical analysis. We identify several intriguing features of the quantum gauge theories.

## 1 Introduction

Four-dimensional gauge theories accurately describe forces of nature. Since solving them is hard, we may revert to studying supersymmetric four-dimensional gauge theories, in which the power of holomorphy lends a helping hand. Twenty years ago, we realised how to solve for the low-energy effective action on the Coulomb branch of gauge theories in four dimensions [1, 2]. The solution techniques were soon recognised to lie close to those studied in integrable systems [3, 4]. It is the bridge between integrable models and supersymmetric gauge theories that we will further explore in this paper. We also attempt to reinforce both sides separately, and present results in a manner such that the contributions to these two domains may be read independently.

The link between supersymmetric gauge theories and integrable systems was useful in writing down the low-energy effective action for gauge theory, namely super Yang-Mills theory with gauge group , broken to supersymmetry by adding a mass term for one hypermultiplet. For the gauge group this program was completed in terms of a Hitchin integrable system with bundle over a torus with puncture [5]. The associated elliptic Calogero-Moser system permits generalisations to any root system, and allows for twists, which were used to provide Seiberg-Witten curves and differentials for theory with general gauge group [6]. The generalisation was non-trivial since the elegant technique of lifting to M-theory [7] is difficult to implement in the presence of orientifold planes (see e.g. [8, 9]), while the relevant generalised Hitchin integrable system has a gauge group which is related to the gauge group of the Yang-Mills theory in an intricate manner [10]. For a review of part of the history, see the lectures [11].

We will be interested in breaking supersymmetry further, from to by adding another mass term for the remaining chiral multiplet (providing us with three massive chiral multiplets of arbitrary mass). We will study this gauge theory with generic gauge group . With supersymmetry, we hope to calculate the effective superpotential at low energies exactly. For an adjoint mass deformation from to this was done in the original work [1] in certain cases. For and gauge group , the exact superpotential was proposed in [12] following the techniques of [1, 13]. The superpotential is the potential of the complexified elliptic Calogero-Moser integrable system associated to the root lattice of type . In [14] the exact superpotential for with more general gauge algebra was argued to be the potential of the twisted elliptic Calogero-Moser system with root lattice associated to the Lie algebra of the gauge group . See [15] for further generalizations to theories with twisted boundary conditions on .

In this paper, we wish to analyse the proposed exact superpotential in more detail. This involves a study of the properties of the isolated extrema of the complexified and twisted elliptic Calogero-Moser integrable system. The results are of independent interest, and we have therefore dedicated a first part of this paper to the study of the integrable systems per se.

The paper is structured as follows. In section 2, we review the relevant elliptic Calogero-Moser models. We pause to demonstrate a Langlands duality between the and type integrable systems. We then analyse the isolated extrema of the complexified potential of low rank integrable systems of and type, and their modular properties. We observe the strong connection to vector valued modular forms. The latter in turn provide a natural backdrop for integrality properties of integrable systems (see e.g. [16, 17, 18]). Section 2 is the technical heart of the paper, and we will lay bare many properties of the vector valued modular forms, using a combination of analytical work and extensive numerics. We will analytically describe the potential in certain classes of extrema. We also find sets of extrema that exhibit a monodromy in the interior of the fundamental domain. In these cases we are able to calculate the monodromy, as well as to provide extensive numerical data for the integer valued coefficients describing the value of the potential at the extrema.

Finally, in section 3, we reinterpret the results we obtained in terms of the physics of massive vacua of theories. We compare our results for the quantum theory on to semi-classical results for massive vacua and discuss electric-magnetic duality properties in the infrared under the modular group as well as the Hecke group. For , we also detail the action of the global triality symmetry on the massive vacua. We will encounter several interesting phenomena. We conclude in section 4 and argue that we have only scratched the surface of a broad field of open problems.

## 2 Elliptic Integrable Systems and Modularity

It is interesting to identify and study dynamical systems that are integrable. Often they form solvable subsectors of more complicated theories of even more physical interest. There exist one-dimensional models of particles with interactions that are integrable, and the Calogero-Moser models of our interest are one such class [19, 20, 21]. These models are associated to root systems of Lie algebras (amongst others). See e.g. [22, 23] for a review. Integrable systems are also known to have certain integrality properties. Namely, their minimal energy, frequencies of small oscillations as well as eigenvalues of Lax matrices are often expressible in terms of a series of integers [16, 17, 18].

In this section, we study properties of (twisted) elliptic Calogero-Moser systems. We analyse the complexified model, defined on a torus with modular parameter . In particular, we examine the extrema of the complexified potential, and exhibit their curious characteristics.

### 2.1 The Elliptic Calogero-Moser models

The member of the pyramid of Calogero-Moser integrable systems we concentrate on is the elliptic Calogero-Moser model. We concentrate on the models associated with a root system , as well as their twisted counterparts. These models have a Hamiltonian with rank variables, with canonical kinetic term, and a potential of the form:

 VΔ = g∑α∈Δ℘(α(X);ω1,ω2), (2.1)

where is the Weierstrass elliptic function on a torus with periods and and is a coupling constant. We choose the half-periods such that the imaginary part of the modular parameter is positive.2 The vector lives in the space dual to the root lattice of rank and the sum in the potential is over all the roots of the root system .3 The model is integrable for all Lie algebra root systems. The twisted elliptic Calogero-Moser model is defined in terms of twisted Weierstrass functions:

 ℘n(x;ω1,ω2) = ∑k∈Zn℘(x+kn2ω1;ω1,ω2), (2.2)

which are summed over shifts by fractions of periods (thus in effect modifying that period). We have a twisted elliptic Calogero-Moser model for all non-simply laced root systems and the value of is then given by the ratio of the length squared of the long versus the short roots. We will be interested in the twisted elliptic Calogero-Moser model with potential:

 VΔ,tw = gl∑αl∈Δl℘(αl(X);ω1,ω2)+gs∑αs∈Δs℘n(αs(X);ω1,ω2), (2.3)

where denote the long and the short roots in the root system , and and are two coupling constants. We will concentrate on the root systems , , and corresponding to the classical algebras , , and . We allow complex values for the components of the vector (i.e. ).

#### The symmetries of the potential

Let us discuss in detail the symmetries of the twisted elliptic Calogero model that act on the set of variables . We first observe that the Weyl group action leaves invariant the scalar product and that the root system is Weyl invariant.4 This implies that the Weyl group action on leaves the potential invariant. Secondly, we note that the outer automorphisms of the Lie algebra, which correspond to symmetries of the Dynkin diagram, also leave the set of roots and the scalar product invariant. Therefore, outer automorphisms as well form a symmetry of the model.

Moreover, the periodicities of the model in the two directions of the torus are as follows. By the definition of the dual weight, or co-weight lattice, we have that for all roots . This implies that shifts of by , namely shifts by periods times co-weights, leave the potential invariant.

To discuss the periodicity in the direction, we concentrate for simplicity on the algebras and , and normalize their long roots to have length squared two. We then have that for a long root and a weight , the equation holds while for a short root of the or algebras we have , for all weights . As a consequence, the periodicity in the (twisted) direction is the lattice where is the weight lattice. The group of all symmetries is a semi-direct product of the lattice shifts, the Weyl group as well as the outer automorphism group.

### 2.2 Langlands Duality

Beyond the many features of these integrable systems already discussed in the literature, the first supplementary property that will be pertinent to our study of isolated extrema, is their behaviour under an inversion of the modular parameter . We therefore briefly digress in this subsection to discuss a few of the details of the duality. Models associated to simply laced Lie algebras map to themselves under the modular S-transformation . This is easily confirmed using the transformation rule (B.3) of the Weierstrass function under modular transformations. We do have a non-trivial Langlands or short-long root duality between the twisted elliptic Calogero-Moser model of B-type and the twisted model of C-type. In order to exhibit the duality, we make the potential for the (twisted) theory more explicit:5

 VB = bl[∑i

and for the theory as well:

 VC = cs[∑i

We have chosen a standard parameterisation of the vector as well as the root systems, and we have assigned half-periods to the B-system and to the C-system. We have also made explicit the twisted Weierstrass functions with twisting index , which is the ratio of lengths squared of the long and short roots. To demonstrate the duality between these models, we use the elliptic function identities (B.5) to manipulate the potential such that it becomes of the form of the potential:

 VB = bl[∑i

In the last equality, we used the modular transformation rule (C.3) for a combination of second Eisenstein series. We observe that the end result (2.5) can be identified with the potential (2.4), provided we match parameters as follows:

 ω′1=2ω2ω′2=−ω1yi=xi cs=blcl=4bs, (2.6)

and we allow for a -dependent shift of the potential that invokes the second Eisenstein series . These identifications imply a duality (which we will denote ) between the modular parameters of the and -type integrable systems:

 τB ≡ −12τC. (2.7)

In the following, we will be interested in and models in which the ratio of the long to short root coupling constants is equal to two, i.e. we put and .6 It is important that this relation is compatible with the duality map (2.6). We rewrite the identity of the potentials for this specific ratio of parameters:

 ∑i

and the integrable system duality can be summarised as:

 VB(xi,τ)=12τ2VC(xi2τ,−12τ)+π2r23[2E2(2τ)−E2(τ)], (2.8)

when we use the rescaling (B.2). The duality may be viewed as a standard Langlands duality. We went through its detailed derivation since the -dependent shift in the duality transformation (2.8) is important for later purposes.

#### Langlands duality at rank two

There is a further special case of low rank which is of particular interest to us in the following. The and type Lie algebras of rank two are identical: . If we apply the duality of and type potentials to this special case, we derive that the following transformations leave the potential invariant:

 ω′1=2ω2ω′2=−ω1 c′=2b x′2−x′1=2x1x′1+x′2=2x2. (2.9)

If we parameterise the potential in terms of the modular parameter , the duality transformation for reads:

 Vso(5)(x1,x2,τ)=12τ2Vso(5)(x1+x22τ,x1−x22τ,−12τ)+4π23[2E2(2τ)−E2(τ)]. (2.10)

In summary, we derived a Langlands duality between and type (twisted) elliptic Calogero-Moser models. The resulting identities captured in equations (2.8) and (2.10) and the shifts appearing in these duality transformations will be useful. We return to the more general discussion of the integrable systems, and in particular their extrema.

### 2.3 Integrable Models at Extrema

There have been many studies of classical integrable models at equilibrium. These have uncovered remarkable properties, like the integrality of the minimum of the potential and of the frequencies of small oscillations around the minimum, amongst others (see e.g. [16, 17, 18]). We will analyse the potential of certain elliptic integrable systems evaluated at generalised equilibrium positions. We show that they give rise to interesting vector valued modular forms as well as more general non-analytic modular vectors. Modularity provides a more conceptual way of understanding the integrality properties of the integrable system. This rationale then continues to hold for the integrable systems that can be obtained from the elliptic Caloger-Moser systems by limiting procedures (e.g. the trigonometric models). Thus, studying elliptic integrable systems, depending on a modular parameter, is found to have an additional pay-off.

It is known that -type integrable systems often have simpler properties than do the integrable systems associated with other root systems. As a relevant example, let us quote the fact that the (real) Calogero-Moser (Sutherland) system with trigonometric potential of -type has equally spaced equilibrium positions along the real axis, while the -type potentials have minima associated to zeroes of Jacobi polynomials [17], which satisfy known relations [25], but are not known explicitly in general. The elliptic Calogero-Moser systems that we examine show a similar dichotomy. Extrema of the (complex) elliptic -model are equally spaced. This fact leads to relatively easily constructable values for the potential at extrema, for any rank [5, 12, 26]. For the -type models that we study in this paper, much less is known, and we need to combine numerical searches with analytic approaches to determine the extremal values of the potential, for low rank cases.

To be more precise, we will be interested in extrema of the complexified potential, satisfying:7

 ∂XiV(Xj)=0∀i, (2.11)

and we moreover demand that at the extremum (2.11) the function

 r∑i=1|∂XiV(Xj)|2 (2.12)

not posses any flat directions.8

Recall that the group of symmetries acting on the variables were a lattice group of translations, the Weyl group as well as the outer automorphisms of the Lie algebra. Using these symmetries, we will introduce a notion of equivalence on the variables . We will consider the vector to be identified by the periodicities of the model. The periodicity in the direction is given by the weight lattice , while in the direction it is the co-weight lattice . Furthermore, we will consider extrema that are related by the action of the Weyl group of the Lie algebra to be equivalent. By contrast, outer automorphisms are taken to be global symmetries of the problem. When the global symmetry group is broken by a given extremum, the global symmetries will generate a set of degenerate extrema.

### 2.4 The Case Ar=su(r+1)

The extrema of the elliptic Calogero-Moser model of type have been studied in great detail, mostly in the context of supersymmetric gauge theory dynamics (see e.g. [5, 12, 26]). Firstly, we remark that in this case, the equivalence relations that follow from the periodicity of the potential as well as the Weyl symmetry group of the Lie algebra are straightforwardly implemented. We use the parameterisation of simple roots in terms of orthogonal vectors , and the fundamental weights then read , with weight lattice spanned by the vectors . We can parameterise the coordinates of our integrable system by a vector living in the dual to the root space (and ). The Weyl group acts by permuting the components . We can shift one of the components to zero by convention. The equivalence under shifts by fundamental weights is identical to the toroidal periodicity relations for the individual coordinates . The inequivalent extrema of the potential (satisfying the additional condition (2.12) of non-flatness) are then argued to correspond one-to-one to sublattices of order of the torus with modular parameter [5, 12]. These extrema are classified by two integers and satisfying that is a divisor of and . The number of extrema is equal to the sum of the divisors of . The outer automorphism of acts trivially on the minima, since it acts by permutation, combined with a sign flip for all , which leaves a sublattice ankered at the origin invariant.

The value of the potential at one of these extrema is (with a given choice of coupling constant):

 VAn−1(τ) = n324(E2(τ)−pqE2(pqτ+kq)). (2.13)

Under the action on the torus modular parameter , the sublattices of order of the torus are permuted into each other (in a way that depends intricately on the integer ). The permutation of the sublattices also entails the permutation of the values (2.13) at these extrema under . The list of extremal values of the elliptic Calogero-Moser model therefore form a vector valued modular form (see e.g. [27, 28, 29]) of weight two under the group . The associated representation of the modular group is a representation in terms of permutations specified by the action on sublattices of order . One can identify a subgroup of the modular group under which a given component of the vector-valued modular form is invariant, and then use minimal data to fix it [30].

In summary, the extrema of the Calogero-Moser model of type are under analytic control. The positioning of the extrema can be expressed linearly in terms of the periods of the model, and the vector valued modular form of extremal values for the potential has an automorphy factor that can be characterised by sublattice permutation properties. The extremal values are generalised Eisenstein series of weight two under congruence subgroups of the modular group.

### 2.5 The B,c,d Models

For other algebras, we are at the moment only able to study low rank cases. From the analysis, it is clear that crucial simplifying properties of the case are absent. Nevertheless, generic features of the case persist in a subclass of extrema, in that we find vector-valued modular forms as extremal values for the potential. We also find a class of extremal values that exhibit new features.

To describe in detail which extrema are considered to be equivalent, we must discuss the equivalence relations that we mod out by for the and root systems individually.

#### Dr=so(2r)

For the case, we can parameterise the roots as (for ) and . We put and imply that the relation holds. The equivalence of the vector under shifts proportional to the weight lattice implies that each variable lives on a torus with modular parameter . It moreover identifies the vector with the vector shifted by a half-period in each variable simultaneously. The Weyl group is , and acts by permutation of the components , as well as the sign change of an even number of them. The outer automorphism group (for ) is equal to and acts as . For , the global symmetry group is triality.

#### Br=so(2r+1)

For , the roots are (for ) and . We recall that the periodicity is the weight lattice in the direction (due to the twist), and the co-weight lattice in the direction. Thus, we can shift components of the vector by periods, or all components simultaneously by a half period in the direction. In the direction, we allow shifts of the individual components by periods. The Weyl group acts by combinations of permutations and any sign flip of the coordinates.

#### Cr=sp(2r)

The roots are (for ) and .9 We can shift components of by half-periods in the direction, while in the direction, we can allow shifts by any period, as well as a half-period shift of all simultaneously. The Weyl group allows any permutation and sign flip of the coordinates. The equivalence relations and symmetries in the and cases, beyond permutation symmetries and toroidal periodicity, are summarized in the table:

Br Individual Xi→−Xi Collective Xi→Xi+ω1 Individual Xi→−Xi and Xi→Xi+ω1 Collective Xi→Xi+ω2 Even number of sign flips Xi→−Xi Collective Xi→Xi+ω1 and Xi→Xi+ω2 Global symmetries : Z2 generically and S3 for D4.

Armed with this detailed knowledge about the equivalence of configurations, we programmed a numerical search for isolated extrema. In the following subsections, we list the results we found by root system. For simply laced root systems we studied the elliptic Caloger-Moser model, while results for non-simply laced root systems correspond to the twisted elliptic Calogero-Moser model with a coefficient for the short root term which is equal to one half the coefficient in front of the long root terms (as described below equation (2.7)).

### 2.6 The Case C2=sp(4)=so(5) and Vector Valued Modular Forms

Since the root system is the first example of our series, we provide a detailed discussion. We discuss the positions of the isolated extrema, the series expansions relevant to the potential at these extrema, the action of the duality group, as well as the identification of the relevant vector valued modular forms.

#### The positions of the extrema

For the Lie algebra we found 7 isolated extrema of the potential. We provide their positioning at in figure 1. We have drawn in bold the positions of the extrema as well as their opposites, in a fundamental cell of the torus.10

These numerical results were found using a Mathematica program, which was written around the built-in function FindMinimum. Careful programming augments the precision of the algorithm to at least two hundred digits. The most costly part of the algorithm is the random search for extrema. Indeed, the intricate landscape drawn by the potential can hide extrema. We gave a drawing of the position of the numbered extrema on the torus with modular parameter . The positions of the extrema for other values of the modular parameter can be reached by interpolation. We have analytic control over a few extra properties of the extrema. E.g. if we follow extremum 1 to , we find that the equilibrium positions are given by where are the zeroes of the Jacobi polynomial . The first extremum, which we label 1, lies on the real axis and is the equilibrium position of the real integrable system. The extremum 2 lies on the imaginary axis, while extrema 3 and 4 are then approximately obtained by applying the transformation . The extrema 5 and 6 are Langlands duals of extrema 3 and 4. It is easy to deduce from the potential that the positions of the extrema generically behave non-linearly as a function of .

#### Series expansions of the extrema

By numerically evaluating the extrema of the potential for a range of values of the modular parameter , we are able to write the extrema as an expansion in terms of a power of the modular parameter . The extremal values can be written in terms of the series:

 A0(q) = 124+q+q2+4q3+q4+6q5+4q6+8q7+q8+13q9+6q10+12q11 (2.14) +4q12+14q13+8q14+… A1(q) = 1+48q+828q2+8064q3+109890q4+1451520q5+11198088q6+141212160q7 +1666682811q8+9413050176q9+145022264892q10+1838450006784q11 +11103941590326q12+138638111404032q13+… A2(q) = 2+48q+576q2+9792q3+99576q4+743904q5+13146624q6+115737984q7 +1015727364q8+14338442448q9+102050482176q10+935515738944q11 +12532363069968q12+122390111091744q13+… A3(q) = 13216+7q+541q2+24508q3+939669q4+19944842q5+764752180q6 +21016537080q7+905672825157q8+38827071780859q9+827503353279726q10+… A4(q) = 1+148q+7446q2+154344q3+5100349q4+352720380q5+10627587582q6 +166124184888q7+5419843397586q8+294399334337124q9+… A5(q) = −1216+29q+431q2+80468q3−231081q4+94846414q5+1301490428q6 (2.15) +90560563752q7−529100109849q8+93349951292249q9+….

The integer coefficients have been determined up to an accuracy of at least . For the first order terms, the accuracy can be up to . In terms of these series, the potential in extremum number 1, on the real axis is (with a given choice of normalisation):

 V1=144π2A3(q27). (2.16)

The potential in the other extrema are:

 V2 = −12π2(83A0(q)+(2q)1/3A1(q/9)+(2q)2/3A2(q/9)) V3 = −12π2(83A0(q)+(2q)1/3e2πi/3A1(q/9)+(2q)2/3e4πi/3A2(q/9)) V4 = −12π2(83A0(q)+(2q)1/3e4πi/3A1(q/9)+(2q)2/3e2πi/3A2(q/9)), (2.17)

and

 V5,6 = 72π2(A5(q27)±i√q27A4(−q27)) V7 = 483π2A0(q). (2.18)

The growth properties of these series, as well as the fact that we are dealing with a physical system living on a torus suggests turning these numerical data into an analytic understanding, based on the theory of modular forms. In the following, we show that this is possible for the rank 2 root system .

#### The extrema as modular forms of the Hecke group and the Γ0(4) subgroup

We need to introduce a few groups related to the modular group. We already noted the duality transform for the -type twisted Calogero-Moser system under the map (see equation (2.7)). For the Lie algebra, which is identical to the Lie algebra, this transformation maps the integrable system to itself (up to a dependent shift of the potential and an overall factor – see equation (2.10)). The map also maps the integrable system to itself. Together, these transformations generate the action of a Hecke group dubbed on the modular parameter . This group contains a subgroup which is a congruence subgroup of the modular group . Generators of the group can be chosen to be the matrices:

 T : (1101) U : (1041). (2.19)

The action of these matrices on coincides with the action of the elements and of the Hecke group. For more information on Hecke groups and associated modular forms see e.g. the lectures [31].

The extremal values of the potential may therefore form a vector valued modular form with respect to the Hecke group , and as a consequence also with respect to the congruence subgroup of the modular group , since we expect extrema to be at most permuted and/or rescaled under the group. Here, we assume analyticity in the interior of the fundamental domain. We will mostly exploit the group in the following, since the literature on the subject of modular forms with respect to congruence subgroups is abundant. For starters, we determine the action of the operations and on the vector of extremal values of the twisted Calogero-Moser potential:

 T : ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1000000000100001000000010000000001000001000000001⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, S2 : ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝010000−2100000−2000010−2000001−2001000−2000100−2000000−1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (2.20)

See figure 2 for a summary of the action of the duality group. To this information, we add the last column in the matrix , which originates in the shift of the potential under Langlands duality. From these data, we easily calculate the action of the generator on the vector valued modular form:

 U : ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0000100010000000100000001000000001010000000000001\par⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (2.21)

We thus find the action of on the vector valued modular form, and we observe the following pattern: there is one entry (the seventh) which is an ordinary modular form of weight under , and there are two sets of three components (namely and ) that mix under . Thus, our vector valued modular form of dimension seven splits into a singlet and a sextuplet. Concentrating on the ordinary modular form of weight , we have that it is a linear combination of Eisenstein series defined by:

 E2,N(τ) = E2(τ)−NE2(Nτ). (2.22)

Indeed, the dimension of the space of modular forms of is two, and it is spanned by and . We thus only need two Fourier coefficients to fix the entire modular form, and we find that:

 A0(q) = −124E2,2(τ)=148(θ43+θ44)(τ) (2.23) V7 = π23(θ43+θ44)(τ). (2.24)

We then have a slew of consistency checks on all the other integers that we determined numerically (see (2.14)). These thirteen checks work out. We do therefore claim that the result (2.24) is exact. This is a simple example illustrating our methodology.

Next, we consider the triplet consisting of the components . We find three eigenvectors of , with eigenvalues corresponding to the cubic roots of unity. The eigenvector with eigenvalue is also mapped to itself under the transformation, and forms again a modular form of weight under . It is indeed proportional to :

 V2+V3+V4=−2π2(θ43+θ44)(τ). (2.25)

The other two eigenvectors, we raise to the power three, such that they become invariant under the -transformation. These forms belong to the space of weight six modular forms. The dimension of this vector space is (see theorem 3.5.1 in [32] with and ), and it consists of three Eisenstein series, and one cusp form. A basis for these vector spaces is given by:

 E16 = −1252E6(τ) (2.26) E26 = −1252E6(2τ) (2.27) E46 = −1252E6(4τ) (2.28) S6 = η(q2)12, (2.29)

where is the Eisenstein series of weight six, and is the -function, also recorded in appendix C. We need four coefficients to fix the eigenvectors in terms of this basis and we find (using the notation ) :

 (V2+ω3V3+ω23V4)3 = −23328π6(E16−E26−2S6) (2.30) (V2+ω23V3+ω3V4)3 = −23328π6(E16−E26+2S6). (2.31)

The consistency checks using the numerics work out.

For the second triplet, we diagonalise first, and proceed very analogously as above, except that we have to take a higher power for the second combination to find a modular form of weight 12 with respect to . We find the relations:

 (V1+ω3V5+ω23V6)3+(V1+ω23V5+ω3V6)3 = 5832π6(E16(q)−64E26) ((V1+ω3V5+ω23V6)3−(V1+ω23V5+ω3V6)3)2 = 136048896π12η(q)24. (2.32)

Note that the sum of all potentials is necessarily a modular form with weight 2 of . Indeed, this sum is equal to (as follows from the identity ).

#### A remark on a manifold of extrema

There are also branches of extrema, namely, non-isolated extrema. These too, we expect to behave well under a modular subgroup. Although this was not the focus of our investigation, we did find numerical evidence for a manifold of extrema at which the potential takes the covariant value .

#### Summary

In summary, we have full analytic control over the value of the potential for all isolated extrema of the twisted Calogero-Moser integrable system. We have found a vector valued modular form of weight two of , and we were able to explicitly express its seven components in terms of ordinary modular forms of . The vector valued septuplet splits into a singlet modular form and a sextuplet vector valued modular form. The plot will thicken at higher rank.

### 2.7 The Case D4=so(8) and the Point of Monodromy

At this stage, we choose to present our results on the rank four model first, since they are simpler than those on the non-trivial rank three cases to be presented in subsection 2.8. The model is simply laced and we therefore expect the ordinary modular group to play the leading role. The integrable system exhibits a global symmetry group that permutes the three satellite simple roots of the Dynkin diagram of . We will refer to the permutation group as triality. We turn to the enumeration and classification of the extrema of the potential. We found 34 extrema. These are listed and labelled in appendix D.1. If we mod out by the global symmetry group, we are left with 20 extrema. The latter fall into multiplets of the duality group of size and . We discuss these multiplets in the following paragraphs.

#### The singlet

There is a singlet under and duality as well as triality. It has zero potential: .

#### The triplet

There is also a triplet under the duality group, labelled , and the dualities act as:

 T=⎛⎜⎝100001010⎞⎟⎠S=⎛⎜⎝010100001⎞⎟⎠.

The relations and are satisfied. We note that in these extrema, the positions belong to the lattice generated by and . For this multiplet,T-duality acts geometrically.

We would like to deduce again from the and matrices and from the known first coefficients of the series expansions (see appendix D.1) the exact expressions of the potentials in these extrema. The functions are expected to transform well under some congruence subgroup of the modular group. Note that the sum of the three functions must be a full-fledged modular form – indeed, the sum vanishes. A brute force strategy leading to the identification of the appropriate congruence subgroup is the following. We decompose the generators of congruence subgroups 11 in terms of a product of and operations. We evaluate the product using the representation at hand (here matrices) and check whether it is trivial for every generator.

It turns out that the subgroup acts trivially on the extremal potentials. Hence all the potentials , and belong to . This space has dimension 2, and it is the set of linear combinations of the three Eisenstein functions associated to the three vectors of order 2 in which have the property that the sum of the three coefficients vanishes. (See appendix C for details and conventions). Matching a few coefficients, we find that

 V2 = 12(2G2,2[01]−G2,2[11]−G2,2[10]) V3 = 12(−G2,2[01]−G2,2[