N=4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F , which obey a system of partial nonlinear differential equations generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F . The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide A n and B 3 examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.
We are investigating systems of n+1 identical point particles with unit mass whose motion on the real line is governed by the Hamiltonian
H = 12 pipi + VB(x1,...,xn+1) (1.1)
and subject to the canonical quantization relations
[xi,pj]=iδij. (1.2)
together with
D=−14(xipi+pixi)
and K=12xixi, (1.3)
this Hamiltonian realizes an so(1, 2) conformal algebra
[D,H]=−iH, [H,K] = 2iD, [D,K]=iK (1.4)
if the potential is homogeneous of degree −2,.
(Xi∂i+2) VB=0. (1.5)
When demanding also permutation and translation invariance as well as admitting only two-body forces, the solution is uniquely given by the Calogero potential,
VB=∑i<jg2(xi−xj)2 (1.6)
Let us extend the algebra from to the super algebra with central charge C by enlarging the set of generators
(H,D,K)→(H,D,K,Qα,Sα,Ja,C)
With α=1,2,
a=1,2,3,
(Qα)†=Q¯¯¯α, (Sα)†=S¯¯α
And imposing the nonvanishing (anti)commutators:
[D,H]=-iH [H,K]=2iD
[D,K]=+iK [Ja,Jb]=i∈abcJc
{Qα,Q¯¯¯β}=2Hδαβ
{QαS¯¯β}=+2i(σa)αβJa−2Dδαβ−iCδaβ
{Sα,S¯¯β}=2Kδαβ
{Q¯¯¯α,Sβ}=−2i(αa)βα Ja−2Dδβα+iCδβα
[D,Qα]=−i2Qα
[D,Sα]=+i2Sα
[K,Qα]=+ iSα
[H,Sα]=− iQα
[Ja,Qα]=−12(σa)αβQβ
[Ja,Sα]=−12(σa)αβSβ
[D,Q¯¯¯α]=−i2Q¯¯¯α
[D,S¯¯α]=+i2S¯¯α
[K,Q¯¯¯α]=+ iS¯¯α
[H,S¯¯α]=− iQ¯¯¯α
[Ja,Q¯¯¯α]=12Q¯¯¯β(σa)βα
[Ja,S¯¯α]=12S¯¯β(σa)βα
To realize this algebra one must pair the bosonic coordinates xi
with fermionic partners ψiα
and ψ¯¯iα=ψiα†
with i = 1 , . . ., n +1 and α = 1 , 2 subject to
{ψiα,ψjβ}=0, {ψ¯¯iα,ψ¯¯iβ}=0, {ψiα,ψ¯¯jβ}=δαβδij. (2.1)
Surprisingly, the non-interacting generator candidates
Q0α=piψiα, Q¯¯¯α0=piψ¯¯iα and S0α=xiψiα, S¯¯α0=xiψ¯¯iα, (2.2)
H0=12pipi, D0=−14(xipi+pixi), K0=12xixi, J0a=12ψ¯¯iα(σa)αβψiβ (2.3)
fail to obey the su (1, 1|2) algebra, and hence interactions are needed! Their simplest implementation changes only
Qα=Q0α−i [S0α,V] and H=H0+V, (2.4)
Just requiring the invention of a potential V(x,ψ,ψ¯¯).
A minimal ansatz to close the su (1, 1|2) algebra reads [1,2]
V=VB(x)−Uij(x)⟨ψiαψ¯¯jα⟩+14Fijkl(x)⟨ψiαψjαψ¯¯kβψ¯¯lβ⟩ (2.5)
Where ⟨....⟩
denotes symmetric (Weyl) ordering. The coefficient functions Uij
and Fijkl
are totally symmetric and homogeneous of degree -2. With this, the super symmetry generators in (2.4) become
Qα=(pj−ixiUij(x)) ψjα−i2xiFijkl(x)⟨ψjβψkβψ¯¯Iα⟩. (2.6)
Inserting the minimal V ansatz (2.5) into the su (1, 1|2) algebra and demanding its closure produces conditions on Uij
and Fijkl
First, one learns that
Uij=∂i∂jU and Fijkl=∂i∂j∂k∂lF, (3.1)
introducing two scalar prepotentials U and F . Second, these prepotentials are subject to the “structure equations” [1,2]
(∂i∂k∂pF) (∂p∂l∂jF)=(∂i∂l∂pF),
xi∂i∂j∂kF=−δjk (3.2)
∂i∂jU−(∂i∂j∂kF) ∂kU=0,
xi∂iU=−C. (3.3)
The quadratic equation for F is the famous WDVV equation [3,4]. The relation below it (linear in U) resembles a covariant constancy equation, and we label it as the “flatness condition”. Its consistency implies the WDVV equation contracted with ∂jU.
Both the WDVV equation and the flatness condition trivialize when contracted withxi
. Finally, the two right equations are homogeneity properties for F and U . One of their consequences is
xiFijkl=−∂j∂k∂lF and xiUij=−∂jU. (3.4)
The one for F may be integrated twice to
(xi∂i−2)F=−12xixi. (3.5)
Clearly, there is the redundancy of adding a quadratic polynomial to F and a constant to U . The third outcome of the su (1, 1|2) algebra is
VB=12(∂iU) (∂iU)+ℏ28(∂i∂j∂kF) (∂i∂j∂kF) (3.6)
Where we have reinstalled ~ to exhibit the quantum part in VB
.
In case of vanishing central charge, C=0
, a partial solution consists in putting U ≡ 0
. Since U does not enter in (3.2), the natural strategy is to firstly solve the WDVV equation and secondly turn on a flat U in this F background.
The homogeneity conditions (xi∂i−2)F=−12xixiund xi∂iU=−C
are solved by [1]
F=−12∑αfαα(x)2ln |α(x)|+Fhom
and U=−∑αgα ln |α(x)|+Uhom,
(4.1)
Where Fhom
and Uhom
are arbitrary homogeneous functions of degree −2 and 0, respectively. The sums run over a set of real covectors α (not indexed!) with values α(x)=αixi,
which are subject to the constraints
∑αfαα(x2)=xixi=:R2
and ∑αgα=C.
(4.2)
The coefficients fα
are essentially fixed by (4.2) and (if positive) may be absorbed into a rescaling of α, while the gα
will emerge as coupling constants which, however, may be frozen to zero. One may rewrite the expressions (4.1) as
F=−12R2 ln R+Fhom
and U=−C ln R+U′hom
(4.3)
or linearly combine (4.1) and (4.3) with coefficents adding to one.
Due to the generality of Fhom
, we are currently unable to solve the WDVV equation (3.2) with (4.1) or (4.3), except for F(')hom=0
. Even then, the nonlinearity of (3.2) restricts the linear combinations to
F=−12∑αfα(x)2 ln |α(x)|
or F=+12∑αfαα(x)2 ln |α(x)|−R2 ln R,
(4.4)
and imposes [5,6]
∑α,βfαfβα⋅βα(x)β(x)(α∧β)⊗2=0
with (α∧β)⊗2ijkl=(αiβj−αjβi) (αkβl−αlβk).
(4.5)
Thus, let us limit ourselves to the ansatz (4.4) and try to turn on U . Even this is too difficult in general, so let us drop the homogeneous pieces in (4.1) and (4.3) and just combine the inhomogeneous parts. Then, the flatness condition (3.3) rules out all ‘ R ’ terms in F or U and demands
∑β(gβ1β(x)−fβ∑αgαα⋅βα(x)) 1β(x)β⊗β=0
with (β⊗β)ij=βiβj,
(4.6)
While the bosonic potential reads
VB=12∑α,βα⋅βα(x)β(x)(gαgβ+ℏ24fαfβ(α⋅β)2). (4.7)
Because the equations decouple for mutually orthogonal sets of covectors, it suffices to take {α} as being indecomposable. In particular, it is convenient in translation-invariant models to decouple the center of mass αcom(x)=∑ixi,
reducing the bosonic configuration space from Rn+1 to Rn
. Note that this alters
R2=xixi
to R2=(xi−1n+1αcom(x))2=1n+1[n∑ixixi−2∑j<lxkxl].
(4.8)
Partial results are known for n ≤ 3 [1,2,7-10], but the case n = 2 is special since the WDVV equation is empty then, which admits many extra solutions.
It was shown by Martini and Gragert [5] and extended by Veselov [6] that the set Φ+ of positive roots of any simple Lie algebra solves the left equation in (4.5). Let us normalize the long and short roots as
α⋅α=2
for α∈ϕ+L
and β⋅β=2r
for β∈ϕ+S
with r=1,2,3.
(5.1)
Recalling that
∑α∈ϕ+α⊗α=h∨1
and ∑α∈ϕ+2α⋅αα⊗α=h1
(5.2)
are determined by the Coxeter number h and the dual Coxeter number , the left condition in (4.2) becomes
1=∑αfαα⊗α=fL∑α∈ϕ+Lα⊗α+fs∑α∈ϕ+α⊗α=fLrh∨−hr−11+fsh−h∨r−11, (5.3)
which is solved by the one-parameter family
fL=1h∨+(hh∨−1)t
and fs=1h∨−(r−hh∨)t
for t∈R.
(5.4)
It is not hard to see that the subset of roots belonging to any plane spanned by a short root β and its string (α,α+β,α+2β,...,α+rβ)
through a long root α makes the double sum in the left equation of (4.5) already vanish. Since the full double sum decomposes into contributions of such planes, we get a prepotential solution (Figure 1)
F(t)=12(fL∑α∈ϕ+L+fS∑α∈ϕ+S)α(x)2 ln |α(x)| (5.5)=−12h∨∑α∈ϕ+α(x)2 ln |α(x)|−t2⎡⎣(hh∨−1)∑α∈ϕ+L−(r−hh∨)∑α∈ϕ+S⎤⎦ α(x) ln |α(x)|
which is unique only for simply-laced Lie algebras. Note that |fL|
and |fS|
may be absorbed into a rescaling of α but their signs cannot, and so the non-simply-laced solution generalizes the t=0
one found before [5,6] by adding to it a concrete Fhom
.
Let us give two examples, with n + 1 and 3 particles, respectively:
An⊕A1: {α(x)}={xi−xj, ∑ixi | 1≤i<j≤n+1}
and fα=1n+1,
(5.6)
G2⊕A1:{α(x)}= {13√(xi + xj−2xk), 13√(xi−xj),x1+x2+x3∣∣(i,j,k) cyclic} (5.7)and fL=14+12t, fs=14-32t, fcom=13.
The Weyl groups of the simple Lie algebras can be extended by the non-crystallographic Coxeter groups H 4 (60 positive roots), H 3 (15 positive roots) and I2(p)
( p positive roots), which also clear the WDVV equation [6]. The dihedral groups I2(p)
with fα=1p
cover all rank-two root systems, including A1⊗A1,A2,BC2
and G2
for p = 2, 3, 4 and 6, respectively, upon rescaling of α (Figure 2).
U = 0 Solutions: Deformed Root Systems
The Lie-algebra root systems are only the tip of an iceberg of WDVV solutions. It has been shown [7,9] that certain deformations of them retain the WDVV property. Let us rephrase some examples in our terminology.
The three positive roots of A2
may be rearranged as the edges of an equilateral triangle. Consider now a deformation of this triangle, keeping the incidence relation α+β−λ=0.
The homogeneity condition (4.2) (and therefore also the WDVV equation) is easily solved by fα=∣∣β⋅γ4A2∣∣
and cyclic permutations, where A denotes the area of the triangle.
If we try the same idea on the A 3 system, we obtain the six edges of a regular tetrahedron and deform to encounter the five-dimensional moduli space of tetrahedral shapes (modulo scale). Again, the homogeneity condition (4.2) has a unique solution, fα
but now the WDVV equation enforces the three conditions (Figure 3)
α⋅α′=0, β⋅β′=0, γ⋅γ′=0
(6.1)
on the skew edge pairs. These relations restrict the above moduli space to the three-dimensional subspace of orthocentric tetrahedra (modulo scale), with
fα=∣∣β⋅γ β′⋅γ′36V2∣∣
and fα′=∣∣β⋅γ β′⋅γ36V2∣∣
and cyclic (6.2)
where V is the volume. Alternatively, we may implement the conditions (6.1) by picking three non-coplanar covectors, say α′,β′
and γ′,
scaling them such that
α′⋅β′ = β′⋅γ′ = γ′⋅α′ = 1
(6.3)
And employing the three-dimensional vector product in fixing the remaining three covectors via
α=α′ × (β′ × γ′)=β′−γ′, β = β′ × (γ′ × α′) = γ′−α′, γ = γ′ × (α′ × β′) = α′−β′.
(6.4)
With these data one gets 6V = α′⋅(β′ × γ′)
as well as
fα=α′⋅α′−136V2
and fα′=(β′⋅β′−1)(γ′⋅γ′−1)36 V2
and cyclic.(6.5)
In fact, this strategy generalizes to orthocentric n-simplices as n-parametric deformations of the regular n-simplex generated by the 12n(n+1)
positive roots of An,
with
fα=∣∣β⋅γ β′⋅γ′β′′⋅γ′′ ... β(n−2)⋅γ(n−2)∣∣(n! V)2
etc (6.6)
The orthocentricity derives from the WDVV equation by a the following dimensional reduction argument. Take nˆixi→∞
for some fixed convector nˆ
. Then, any factor 1α(x)
in the WDVV equation (4.5) vanishes unless α⋅nˆ=0
, which amounts to a reduction of the covector set {α}
to its intersection with the hyper plane orthogonal to nˆ
. This process may be iterated until only covectors laying in a plane α∧β
spanned by two covectors α
and β survive. This situation admits two possibilities: Either the α and β are concurrent, in which case another covector α + β or α − β completes a triangle satisfying the WDVV equation, or else α and β are skew, in which case there is no further covector in their plane and WDVV demands orthogonality (Figure 4).
The B 3 root system provides another example. Four copies of the 3 short and 6 long positive roots can be assembled into the edge set of a truncated cube. We deform this polyhedron to
{α(x)}={d1x1, d2x2, d3x3; c3(c2x1±c1x2), c1(c3x2±c2x3), c2(c1x3±c3x1)}
(6.7)
with ci,di ∈ R,
, retaining the ‘incidence relations’ of a truncated cuboid. For c2:=c20+c21+c22+c23
and
{fα}= { c20+c21−c22−c23c2d21, c20−c21+c22−c23c2d21, c20−c21−c22+c23c2d21; 1c2c23, 1c2c21, 1c2c22}
(6.8)
we satisfy the homogeneity condition (4.2), i.e. ∑αfαα⊗α = 1
. The relevant combinations fα−−√ α
depend only on the three ratios ci/c0
. The rigid B 3 root system with (5.4) occurs for c1 = c2 = c3 = 1
and c20=2−3t1+t.
The case of C3
is very similar.
Finally, let us present an example based on weights rather than roots, namely a deformation of the B3
representation i.e. the vector plus spin or weights. For the 3 positive ‘vector’ and 4 positive ‘spin or’ covectors we take
α(x)=d1x1, β(x)=d2x2, γ(x)=d3x3; α+β+γ2, α−β−γ2, −α+β−γ2, −α−β+γ2
with di ∈ R,
keeping the relations between vector and spin or weights. For d2:=d21+d22+d23
and
fα=−d21+d22+d23d2d21, fβ=d21−d22+d23d2d22, fγ = d21+d22−d23d2d23
and fspinor=2d2
(6.10)
We obey (4.2) and achieve a two-parameter deformation of the original weight system at c1=c2=c3.
The corresponding polyhedron, whose edges are built from 4 copies of the vector and 6 copies of the spin or weights, is a (in homogeneously scaled) rhombic dodecahedron with the faces dissected into triangles (Figure 5).
It is important to realize that all examples fulfill the WDVV equation, because the above dimensional reduction argument applies. The crucial properties are the mutual orthogonality of non-concurrent non-parallel edges as well as the incidence relations, which ‘sew’ the triangles together into a polyhedron. Yet, these properties are only necessary but not sufficient. Finally we remark that all our examples are part of a larger moduli space of n=3
families of WDVV solutions [7,9].
U≠ 0
Solutions: No-Go ‘Theorem’ For N > 2
Recall that, for turning on
U=−∑αgα ln |α(x)|
With ∑αgα=C
(7.1)
In a given F background determined by {α,fα}
, we need to solve the flatness condition (4.6). In principle, we may modify (4.6) by adding a homogeneous term Uhom
to the prepotential above, but let us postpone this option for the time being. Then, matching the coefficients of the double poles in (4.6) requires that
either gβ=0
or else β⋅βfβ=1
for each convector β (7.2)
In the undeformed irreducible root-system solutions, the Weyl group identifies the fα
and gα
coefficients for all roots of the same length. Hence, besides the fL
and fs
values in (5.4) we have couplings gL
and gs
for a number pL
and ps
of long and short positive roots, respectively b . This simplifies the trace of (5.3) to
n=∑αα⋅αfα=2fLpL+2rfsps→pL+ps=n
if gL,gs≠0.
(7.3)
Since the total number pL+ps
of positive roots always exceeds n (except for A⊕n1
), we are forced to put either gs=0
or gL=0
. Therefore, all simply-laced root systems are ruled out! For the r>1
root systems, we get
either gs=0, gL=g −→−−−(7.2)(7.3) fs=r2n−pLps≤0, fL=12
(7.4)
or gs=g, gL=0 −→−−−(7.2)(7.3) fs=r2, fL=12n−pspL≤0.
(7.5)
We see that in the non-simply-laced one-parameter family (5.4) there is always one member which obeys (7.4) or (7.5). For it, we must still check the remainder of (4.6),
∑α≠βgαfβα⋅βα(x)β(x)β⊗β=0.
(7.6)
Even though its trace is always satisfied, the traceless part is violated for any nontrivial root system with the data (7.4) or (7.5). Hence, there do not exist U solutions of the standard form (7.1) for any Coxeter root system. Perhaps this no-go result may be overcome by adding suitable Uhom
contributions. Certainly it can be avoided for n=2
because in this case (4.4) may be relaxed (see below). Finally, we have not yet studied the flatness conditions for the deformed root systems of the previous section.
U ≠ 0
Solutions: Dihedral Solutions for n = 2
As mentioned before, the case of three particles with translation invariance, i.e. n=2
, is special for the absence of the WDVV equation. In fact, it is easy to see that any set {α}
of covectors can be made to obey the left condition in (4.2) with suitably chosen fα
. To study concrete examples, we look at the most symmetric cases, namely the dihedral root systems mentioned earlier.
It is crucial that we take advantage of the freedom at n = 2 to add ‘radial terms’ in our ansatz:
F=−12∑αfαα(x)2 ln |α(x)|−12fRR2 ln R→∑αfαα⊗α=(1−fR)1, (8.1)
U=−∑αgα ln |α(x)|−gR ln R → ∑αgα=C−gR. (8.2)
The flatness condition then reduces to (7.2) and the trace of (7.6) plus the relation gR+(C−gR)fR=0
. It is obeyed for the I2(p)
system if (geven, godd)=:(gS,gL)
when p is even and if gα=:g ∀α
p is odd. Turning on g couplings for all covectors fixes α⋅αfα=1,
and so we obtain
p=2(1−fR) → gR=p−2pC → 2pC=∑αgα= {p2(gS+gL) for p evenp g for p odd
(8.3)
In order to ease the interpretation as three-particle systems, we embed the relative-motion configuration space R2
into R3 ∍ (x1,x2,x3)
and rotate such that αcom=(1,1,1).
for identical particles we require invariance under permutations (x1,x2,x3)→(xπ1,xπ2,xπ3)
of the full three-body coordinates. This limits p to multiples of 3. The ‘radial coordinate’ then becomes
R2=13={(x12)2+(x23)2+(x31)2}=23{(x1)2+(x2)2+(x3)2−x1x2−x2x3−x3x1}
(8.4)
For illustration we explicitly display the I2(p)
solutions based on (8.1) and (8.2) for the first few values of p .
p=2¯¯¯¯¯¯¯¯: A1⊕A1
model fR=0, gR=0, →gS+gL=C
α(x)α⋅α−−−−√∈{12–√(x1−x2), 16–√(x1+x2−2x3)}
VB=g2s+ℏ24(x1−x2)2+3(g2L+ℏ24)(x1+x2−2x3)2
p=3¯¯¯¯¯¯¯¯: A2
model fR=−12, gR=13C → g=29C
α(x)α⋅α−−−−√∈{12–√(x1−x2), 12–√(x1−x3), 12–√(x2−x3)}
VB=(g2+ℏ24) (1(x1−x2)2+1(x2−x3)2+1(x3−x1)2)+58(9g2−ℏ2)1R2
p=4¯¯¯¯¯¯¯¯: BC2
model fR=−1, gR=12C →gs+gL=14C
α(x)α⋅α√∈{x1−x22√, x1+x2−2x36√, τx1+τ¯x2−x33√, −τ¯x1+τx2−x33√}
With {τ=12(3–√+1)τ¯=12(3–√+1)
VB=g2S+ℏ24(x1−x2)2+3(g2S+ℏ24)(x1+x2−2x3)2+32(g2L+ℏ24)(τx1−τ¯x2−x3)2+32(g2L+ℏ24)(τx1 + τ¯x2−x3)2+6(gS+gL)2−32ℏ2R2
p=6¯¯¯¯¯¯¯¯: G2
model fR=−2, gR=−23C → gS+gL=19C
α(x)α⋅β−−−−√∈{x1−x22–√,x1−x32–√,x2−x32–√, 2x1+x2−x36–√,x1+x2−2x36–√,−x1+2x2−x36–√}
VB=g2S+ℏ24(x1−x2)2+3(g2L+ℏ24(x1+x2−2x3)2+
Cyclic +36(gS+gL)2−4ℏ2R2
p=12¯¯¯¯¯¯¯¯¯: I2 (12)
model fR=−5, gR=56C→gS+gL=136C
α(x)α⋅α−−−−√∈⎧⎩⎨⎪⎪x1−x22√,x1−x32√,x2−x32√,2x1−x2−x36√,x1+x2−2x36√,−x2+2x2−x36√,τx1−x2−τ¯x33√,τx1−τ¯x2−x33√,x1+τ¯x2−τx33√,τ¯x1+x2−τx33√,−τ¯x1+τx2−x33√,−x1+τx2−τ¯x33√⎫⎭⎬⎪⎪
VB=g2S+ℏ24(x1−x2)2+3(g2S+ℏ24(x1+x2−2x3)2+32(g2L+ℏ24)(τx1x2−τ¯x3)2+32(g2L+ℏ24(τx1−τ¯x2−x3)2+ cyclic +630(gS+gL)2−352ℏ2R2
Finally, let us investigate the effect of adding to U a homogeneous piece Uhom
for obtaining Utot=U+Uhom
. At n=2
, all we have to solve is the trace of the flatness condition,
∂⋅∂Uhom+∑αfαα⋅αα⋅∂a(x)Uhom=0
besides x1∂iUhom=0.
(9.1)
It is convenient to pass to polar coordinates on R2
via (x1,x2)=(Rcosϕ,Rsinϕ)
. In the dihedral class I2(p)
, the sum over the roots can be performed, and the flatness conditions for U and for Uhom
is solved by
U(R,ϕ)=−C ln R−{g ln |cos(pϕ)| for p oddgS ln ∣∣cos(p2ϕ)∣∣ +gL ln ∣∣sin(p2ϕ)∣∣ for p even
(9.1)
Uhom(ϕ)=λp ln ∣∣tan(p2ϕ+δ)∣∣ ln ∣∣tan(p2ϕ+δ)∣∣
with δ={π4 for p odd0 for p even
(9.2)
After lifting to the full configuration space R3
as in Figure 6, we arrive at
∂iUtot=−∑αgααiα(x)−p−2pCxiR2+λ⎛⎝⎜x2−x3x3−x1x1−x2⎞⎠⎟Rp−2∏α(α(x))−1
(9.3)
For the A 2 model as the simplest example, one gets
F=−14((x12)2 ln ∣∣x12∣∣+(x23)2 ln ∣∣x23∣∣+(x31)2 ln ∣∣x31∣∣)+14R2 ln R
(9.4)
∇→Utot=[x12x23x31]−1⎛⎝⎜[λR−g(x31−x12)]x23[λR−g(x12−x23)]x31[λR−g(x23−x31)]x12⎞⎠⎟−32gR−12⎛⎝⎜x1x2x3⎞⎠⎟,
(9.5)
VtotB=(g2+23λ2+ℏ24)(1(x12)2+1(x23)2+1(x31)2) +58(9g2−ℏ2)R−2+λgR(x12−x23)(x23−x31)(x31−x12)(x12x23x31)2,
(9.6)
with R2=13{(x12)2+(x23)2}
. A pure Calogero potential is possible only for g=0=ℏ
.
Acknowledgements
The author is grateful to Anton Galajinsky and Kirill Polovnikov for a very fruitful collaboration. His work is partially supported by the Deutsche Forschungsgemeinschaft.