Exact Yangian Symmetry in the Classical Euler-Calogero-Moser Model

11 Pages·0.15 MB·English

by  E. Billey

Exact Yangian Symmetry
in the classical Euler-Calogero-Moser Model
∗ ∗ ∗
E. Billey J. Avan O. Babelon
January 1994
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4 Abstract
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Wecomputether-matrixfor theellipticEuler-Calogero-Moser model. In thetrigonometric limit we

1 show that the model possesses an exact Yangian symmetry.
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PAR LPTHE 94-03
∗L.P.T.H.E.Universit´eParisVI(CNRSUA280),Box126,Tour16,1er ´etage,4placeJussieu,F-75252PARISCEDEX05
1 Introduction
The Euler-Calogero-Mosermodel wasdefined in[1,2]. In[3]we consideredthe rationalcaseandwe derived
the r-matrix. In this paper we are interested in its trigonometric and elliptic generalizations. In the elliptic
case we compute the r-matrix and show that the usual elliptic Calogero-Moser model and its r-matrix are
obtained by Hamiltonian reduction. In the trigonometric case we show that the current algebra symmetry
exhibited by Gibbons and Hermsen [1] in the rational case, is deformed into a Yangian symmetry algebra.
We consider a system of N particles on a line with pairwise interactions. The degrees of freedom consist
of the positions and momenta (p ,q ) and of antisymmetric additional variables (f = −f ) ,
i i i=1···N ij ji i,j=1···N
with the Poisson brackets
{p ,q } = δ (1)
i j ij
1
{f ,f } = (δ f +δ f +δ f +δ f ). (2)
ij kl il jk ki lj jk il lj ki
2
The Poisson brackets of the f just reproduce the O(N) Lie algebra. The Hamiltonian will be taken of the
ij
form
N N
1 1
H = p2− f f V(q ), q =q −q (3)
2 i 2 ij ji ij ij i j
i=1 i,j=1
X Xi6=j
with an even potential V(−x)=V(x).
The equations of motion are easily derived:
q˙ = p
i i
N
p˙ = f f V′(q )
i ij ji ij
j=1
Xj6=i
N
f˙ = f f [V(q )−V(q )].
ij ik jk ik jk
k=1
kX6=i,j
Such a system admits a Lax representation only for specific potentials. Indeed writing the following ansatz
for the Lax pair
N N
L(λ) = p e + l(q ,λ) f e (4)
i ii ij ij ij
i=1 i,j=1
X Xi6=j
N
M(λ) = m(q ,λ) f e (5)
ij ij ij
i,j=1
Xi6=j
where e is the N ×N matrix (e ) = δ δ and λ ∈ C is the spectral parameter, we find that the
ij ij kl ik jl
equations of motion can be written in the Lax form
L˙(λ)=[M(λ),L(λ)] (6)
if and only if the following equalities are satisfied:
∂
m(x,λ) = − l(x,λ)=−l′(x,λ) (7)
∂x
l′(x,λ) l(y,λ)−l′(y,λ) l(x,λ) = l(x+y,λ) [V(x)−V(y)] (8)
1
l(x) ∼ − when x→0. (9)
x
1
Eq.(8) is the famous functional equation of Calogero. Its general solution is [4, 5]
σ(x+λ)
l(x,λ)=− , V(x)=℘(x) (10)
σ(x) σ(λ)
whereσand℘areWeierstrassellipticfunctions,therelevantpropertiesofwhicharerecalledintheappendix.
TheellipticO(N)Euler-Calogero-Mosermodelispreciselydefinedbyeq.(3)withV(x)=℘(x)togetherwith
the Poisson brackets (1,2).
2 The r-matrix
From eq.(6) it follows that trLn(λ) is a set of conserved quantities. In particular
N
trL(λ)= p , trL2(λ)=2H +℘(λ).
i
i=1
X
The involution property of these quantities trLn(λ) will follow from the existence of an r-matrix which we
now calculate [6, 7]. Introducing the notations L (λ) = L(λ)⊗1 and L (λ) = 1⊗L(λ) we show that the
1 2
Poisson brackets of the Lax matrix elements can be recast as
{L (λ),L (µ)}=[r (λ,µ),L (λ)]−[r (µ,λ),L (µ)]. (11)
1 2 12 1 21 2
Following [3] we assume that r is of the form
N N N
r (λ,µ)=a(λ,µ) e ⊗e + b (λ,µ)e ⊗e + c (λ,µ)e ⊗e .
12 ii ii ij ij ji ij ij ij
i=1 i,j=1 i,j=1
X Xi6=j Xi6=j
Requiring that r (λ,µ) be independent of the p variables we obtain
12 i
b (λ,µ) = −b (µ,λ) (12)
ij ji
c (λ,µ) = c (µ,λ). (13)
ij ij
Moreover assuming that r (λ,µ) is independent of the f variables yields the following system:
12 ij
a(λ,µ) l(q ,λ)−b (λ,µ) l(q ,µ)+c (λ,µ) l(q ,µ)=−l′(q ,λ) (14)
ij ij ij ij ji ij
1
b (λ,µ) l(q ,λ)−b (λ,µ) l(q ,µ)=− l(q ,λ) l(q ,µ) (15)
ij jk ik jk ik ji
2
1
c (λ,µ) l(q ,λ)+c (λ,µ) l(q ,µ)= l(q ,λ) l(q ,µ) (16)
ij jk ik kj ik ij
2
1
c (λ,µ) l(q ,λ)+c (λ,µ) l(q ,µ)= l(q ,λ) l(q ,µ). (17)
ij ki kj ik kj ij
2
A solution to these equations is
1
a(λ,µ) = − [ζ(λ+µ)+ζ(λ−µ)]
2
1
b (λ,µ) = l(q ,λ−µ)
ij ij
2
1
c (λ,µ) = l(q ,λ+µ).
ij ij
2
Indeed substituting the preceding expressions in eq.(15,16,17)leads to the same relation:
l(q ,λ−µ) l(q ,λ)+l(q ,µ−λ) l(q ,µ)+l(q ,λ) l(q ,µ)=0
ij jk ki jk ik ji
2
which upon setting x = 1 (λ+q ), y = 1 (2µ−λ+q ), z = 1 (λ+q ) and t = 1 (−λ−q +q ) is
2 ij 2 ji 2 ji 2 ki kj
a direct consequence of relation (53). The expression for a(λ,µ) is then given by eq.(14), and is simplified
using eq.(51) and (55). Finally the r-matrix reads
N N
1 1
r (λ,µ) = l(q ,λ−µ)e ⊗e + l(q ,λ+µ)e ⊗e
12 ij ij ji ij ij ij
2 2
i,j=1 i,j=1
Xi6=j Xi6=j
N
1
− [ζ(λ+µ)+ζ(λ−µ)] e ⊗e . (18)
ii ii
2
i=1
X
3 The sl(N) model
The aboveO(N) model canbe obtainedfrom the more generalsl(N) model by a meanprocedure [8, 9, 10].
The sl(N) elliptic Euler-CalogeroMoser model is defined by the Hamiltonian
N N
1 1
H = p2− f f ℘(q ) (19)
2 i 2 ij ji ij
i=1 i,j=1
X Xi6=j
and the Poisson brackets
{p ,q } = δ (20)
i j ij
{f ,f } = δ f −δ f . (21)
ij kl jk il li kj
For this model we define a Lax matrix as
N N
L(λ)= (p −ζ(λ)f ) e + l(q ,λ) f e . (22)
i ii ii ij ij ij
i=1 i,j=1
X Xi6=j
The Hamiltonian is given by H = 1 dλ trL2(λ). A direct calculation gives
2 2iπλ
{L (λ),L (µR)} = [r (λ,µ),L (λ)]−[r (µ,λ),L (µ)]
1 2 12 1 21 2
N
− l′(q ,λ−µ) (f −f ) e ⊗e (23)
ij ii jj ij ji
i,j=1
Xi6=j
with the beautifully simple r-matrix
N N
r (λ,µ)=−ζ(λ−µ) e ⊗e + l(q ,λ−µ) e ⊗e . (24)
12 ii ii ij ij ji
i=1 i,j=1
X Xi6=j
At this point let us make two remarks:
• Because of the third term in the right member of eq.(23) the integrals of motion trLn(λ) are not in
involution. However we can restrict ourselves to the manifolds (f = constant) since trLn(λ)
ii i=1···N
Poisson-commute with f . On these manifolds trLn(λ) are in involution.
ii
• The r-matrix for the O(N) model eq.(24) is immediately seen to be of the form
1
rO(N) = (1+σ⊗1)rsl(N)
12 2 12
where σ is the involutive automorphism
σ :λne 7−→−(−λ)ne .
ij ij
This is typical of a mean construction.
3
Inthe followingwewillrestrictthef toasymplectic leafofthePoissonmanifold(21). Introducingvectors
ij
(ξ ) with ξ =(ξa)
i i=1···N i i a=1···r
(η ) with η =(ηa)
i i=1···N i i a=1···r
with the Poisson brackets
{ξa,ξb}=0, {ηa,ηb}=0, {ξa,ηb}=−δ δ , (25)
i j i j i j ij ab
we parametrize the f as follows:
ij
r
f =hξ |η i= ξaηa. (26)
ij i j i j
a=1
X
The phase space now becomes a true symplectic manifold.
4 The r-matrix of the elliptic Calogero model
We show here that the r-matrix for the elliptic Calogero model [6, 12] can be obtained from eq.(23) by a
Hamiltonian reduction procedure [8, 9, 10].
We choose r =1 in eq.(26). On the manifold f =ξ η acts an Abelian Lie group
ij i j
ξ −→λ ξ , η −→λ−1η . (27)
i i i i i i
RemarkthatthegroupactsonL(λ)asconjugationbythematrixdiag(λ ) andthereforealltheHamil-
i i=1···N
tonians trLn(λ) are invariant. Thus one can apply the method of Hamiltonian reduction. The infinitesimal
generator of this action is
N
H = ǫ f , λ =1+ǫ .
ǫ i ii i i
i=1
X
We fix the momentum by choosing
f =α.
ii
To compute the reduced Poisson brackets of the Lax matrix, we remark that the matrix
LCal(λ) = g−1L(λ) g with g =diag(ξ )
i i=1···N
N N
= [p −αζ(λ)] e +α l(q ) e (28)
i ii ij,λ ij
i=1 i,j=1
X Xi6=j
is invariant under the isotropy group G of α (which is the whole group itself since it is Abelian) and we
α
can compute the Poissonbrackets of its matrix elements directly. We find
{LCal(λ),LCal(µ)}=[rCal(λ,µ),LCal(λ)]−[rCal(µ,λ),LCal(µ)] (29)
1 2 12 1 21 2
with
1
rCal(λ,µ)=g−1g−1 r (λ,µ)−{g ,L (µ)}g−1+ [u ,L (µ)] g g
12 1 2 12 1 2 1 2 12 2 1 2
(cid:20) (cid:21)
where u ={g ,g }g−1g−1 is here equal to zero. Redefining
12 1 2 1 2
N
1
rCal(λ,µ)−→rCal(λ,µ)+ e ⊗e ,L (µ)
12 12 2α ii ii 2
" #
i=1
X
does not change eq.(29) and yields exactly the r-matrix found in [11, 12]
N N
1
rCal(λ,µ) = l(q ,λ−µ) e ⊗e + l(q ,µ) (e +e )⊗e
12 ij ij ji 2 ij ii jj ij
i,j=1 i,j=1
Xi6=j Xi6=j
N
−[ζ(λ−µ)+ζ(µ)] e ⊗e . (30)
ii ii
i=1
X
4
5 Yangian symmetry in the trigonometric case
The parametrization(26) of f introduces a sl(r) symmetry into the theory. The transformation
ij
r
ηa −→ uabηb
i i
b=1
X
r
ξa −→ (u−1)abξb
i i
b=1
X
leavesthef invariantandthereforealsotheHamiltonians. Thissymmetryisgeneratedbyasetofconserved
ij
currents
N
Jab = ξbηa. (31)
0 i i
i=1
X
It is remarkable that this current was shown, in the rational case [1], to be the first of a hierarchy building
a current algebra commuting with the Hamiltonian — and more generally with a subset of the commuting
Hamiltonians.
We now extend this result to the trigonometric case, and we will show that the hierarchy of currents
form a Yangian symmetry in this case. Taking the trigonometric limit (ω =∞ and ω =iπ) in the above
1 2 2
formulas, we see that the Lax matrix can be taken of the form
L(λ)=L −coth(λ)F (32)
0
with
N N N
L = p e − coth(q ) f e , F = f e . (33)
0 i ii ij ij ij ij ij
i=1 i,j=1 i,j=1
X Xi6=j X
By a straightforwardcalculation, or taking the limit of the elliptic case, we find
{L (λ),L (µ)} = [r0 ,L (λ)]−[r0 ,L (µ)]
1 2 12 1 21 2
1
− (1−coth(λ)coth(µ))([C,F ]−[C,F ])
1 2
2
N
1
− (f −f ) e ⊗e (34)
ii jj sinh2(q ) ij ji
i,j=1 ij
Xi6=j
where
N
r0 =− coth(q ) e ⊗e (35)
12 ij ij ji
i,j=1
Xi6=j
and C is the Casimir element of sl(N)
N
C = e ⊗e . (36)
ij ji
i,j=1
X
In spite of the unusual second term in eq.(34) the quantities trLn(λ) are still in involutionon the manifolds
Σ :(f =α) . Indeed,
α ii i=1···N
N
f −f
{trLn(λ),trLm(µ)} = n m ii jj [Ln−1(λ)] [Lm−1(µ)]
sinh2(q ) ij ji
i,j=1 ij
Xi6=j
n m
− (1−coth(λ)coth(µ)) tr Ln−1(λ)Lm−1(µ)[C,F −F ]
2 12 1 2 1 2
(cid:0) (cid:1)
5
and since tr ((1⊗A) C)=A, we obtain
2
N
f −f
{trLn(λ),trLm(µ)} = n m ii jj [Ln−1(λ)] [Lm−1(µ)]
sinh(q )2 ij ji
i,j=1 ij
Xi6=j
n m
− (1−coth(λ)coth(µ)) tr{Ln−1(λ)[Lm−1(µ),F]−Lm−1(µ)[Ln−1(λ),F]}.
2
If we notice that
L(λ)−L(µ)
F =−
coth(λ)−coth(µ)
we immediately get the involution property.
We consider now the subsettr(Ln)=tr(L +F)n of commuting Hamiltonians; notice that H belongs to
0
this subset, since H = 1trL2−α trL+ 1Nα2.
2 2
We introduce the following quantities:
Jab =tr(LnFab), a,b=1,···,r n=0,1,···,∞ (37)
n
where Fab is the N ×N matrix of elements
(Fab) =fab =ξbηa. (38)
ij ij i j
We define the generating functional of the currents Jab. It is the r×r matrix T(z) of elements
n
1 1 1 1
Tab(z)=− δ − Jab =− δ +tr Fab . (39)
2 ab zn+1 n 2 ab L−z
n∈N (cid:18) (cid:19)
X
Proposition. On the manifolds Σ we have the following two properties:
α
1. The currents Jab Poissoncommute with all the quantities of the form tr(Ln).
n
2. The generating functional T(z) satisfies the defining relation of a (classical) Yangian algebra:
{T(y)⊗, T(z)}=[R(y,z),T(y)⊗T(z)] (40)
with
r
Π
R(y,z)=−2 , Π= e ⊗e . (41)
ab ba
y−z
a,b=1
X
Proof. To prove this proposition we need the Poisson brackets
N
1
{L ,L } = [r0 ,L ]−[r0 ,L ]+ (f −f ) e ⊗e (42)
1 2 12 1 21 2 ii jj sinh2(q ) ij ji
i,j=1 ij
Xi6=j
{L ,Fab} = [−r0 +C,Fab] (43)
1 2 21 2
{Fab,Fcd} = (δ Fcb−δ Fad) C. (44)
1 2 ad 1 cb 2
Remark that the currents Jab and the Hamiltonians tr(Ln) are invariant under the symmetry
n
ξa −→λ ξa, ηa −→λ−1ηa.
i i i i i i
Therefore we can compute their Poisson brackets on the reduced phase space straightforwardly; restricting
ourselves to the manifolds f = α, the last term in eq.(42) vanishes, and we will systematically drop its
ii
contribution in intermediate calculations.
We emphasize that in eq.(42,43) the same r-matrix appears. Moreover it is the term [C,Fab] in eq.(43)
2
which is responsible for the quadratic form of eq.(40), as we shall see in what follows.
6
Introducing the generating functional H(z)=tr( 1 ) of the Hamiltonians tr(Ln) we compute
L−z
1 1 1 1 1 1 1 1
{ Fab, } = − r0 , Fab + r0 Fab ,
L −y 1 L −z L −z 12 L −z L −y 1 L −y 21 L −y 1 L −z
1 2 (cid:20) 2 2 1 (cid:21) (cid:20) 1 1 2 (cid:21)
1 1 1
+ [C,Fab] .
L −yL −z 1 L −z
1 2 2
Taking the trace we obtain
1 1
{Tab(y),H(z)}=tr Fab , =0.
L−y (L−z)2
(cid:18) (cid:20) (cid:21)(cid:19)
This proves the first part of the proposition. To prove the second part we evaluate
1 1 1 1 1
{ Fab, Fcd} = − r0 Fcd , Fab
L −y 1 L −z 2 L −z 12 L −z 2 L −y 1
1 2 (cid:20) 2 2 1 (cid:21)
1 1 1
+ r0 Fab , Fcd
L −y 21 L −y 1 L −z 2
(cid:20) 1 1 2 (cid:21)
1 1
+ δ Fcb−δ Fad C
L −yL −z ad 1 cb 2
1 2
1 1 (cid:0) 1 (cid:1) 1
+ [C,Fab] Fcd − [C,Fcd] Fab .
L −yL −z 1 L −z 2 2 L −y 1
1 2 (cid:26) 2 1 (cid:27)
Hence taking the trace we get
1 1
{Tab(y),Tcd(z)} = tr (δ Fcb−δ Fad)
ad cb
L−yL−z
(cid:18) (cid:19)
1 1 1 1 1 1
+tr Fcd , Fab −tr Fab , Fcd .
L−y L−z L−z L−z L−y L−y
(cid:18) (cid:20) (cid:21)(cid:19) (cid:18) (cid:20) (cid:21)(cid:19)
Using the cyclicity of the trace and
1 1 1 1 1
= −
L−yL−z y−z L−y L−z
(cid:18) (cid:19)
this becomes
1
{Tab(y),Tcd(z)} = δ (Tcb(y)−Tcb(z))−δ (Tad(y)−Tad(z))
ad cb
y−z
2 (cid:0) 1 1 1 1 (cid:1)
+ tr Fcd Fab− Fab Fcd .
y−z L−y L−z L−y L−z
(cid:18) (cid:19)
Remarking that
N
1 1 1 1
tr Fcd Fab = ξd ηc ξb ηa
L−y L−z L−y j k L−z l i
(cid:18) (cid:19) ijkl=1(cid:18) (cid:19)ij (cid:18) (cid:19)kl
X
N N
1 1
= ξd ηa ξb ηc
ij=1(cid:18)L−y(cid:19)ij j i kl=1(cid:18)L−z(cid:19)kl l k!
X X
 1  1
= Tad(y)+ δ Tcb(z)+ δ
ad cb
2 2
(cid:18) (cid:19)(cid:18) (cid:19)
we prove the result (40).2
7
The rational limit is obtained by applying the canonical transformation
1
p −→ p
i i
ǫ
q −→ ǫ q
i i
and sending ǫ to zero. In this limit
1
L −→ L
0 ǫ rational
1
r0 −→ r0 .
12 ǫ 12 rational
The Casimir term drops therefore from eq.(43), leaving us with a linear Poissonalgebra
1
{T(y)⊗, T(z)}=− [R(y,z),T(y)⊗1+1⊗T(z)] (45)
2
which is the result found by Gibbons and Hermsen.
6 Conclusion
TheEuler-Calogero-Mosermodelisbecomingmoreandmoreinteresting. Ontheonehandthecomputation
oftheclassicalr-matrixismadeconsiderablyeasierbytheexistenceoftheextravariablesf ,themorecom-
ij
plicatedr-matrix ofthe Calogero-MosermodelfollowingnaturallyfromaHamiltonianreductionprocedure.
Onthe otherhand,this modelexhibits anexactinfinite symmetry whichis justa currentalgebrasymmetry
in the rational case and becomes an exact Yangian symmetry in the trigonometric case. This structure is
very much reminiscent of the one discovered in [13]. Actually the two currents J and J (which generate
0 1
the full algebra) are identical in the two cases. Indeed, in our case we have
N N e−qij
Jab = p fab− f fab.
1 i ii sinh(q ) ij ji
i=1 i,j=1 ij
X Xi6=j
Setting (X )ab =fab, Θ =2 e2qj and using eq.(26) we can rewrite
i ii ij e2qi−e2qj
N N
Jab = p Xab− Θ (X X )ab.
1 i i ij i j
i=1 i,j=1
X Xi6=j
This is exactly the current found in [13]. In fact the model considered in [14, 15, 13] is a quantum version
of our model for a particular choice of orbit.
At this point two interesting problems arise. One is the understanding of the role of the r-matrix in the
quantization of these models. The other is the hypothetical extension of these results to the elliptic case,
which still remains quite mysterious.
Acknowledgements
We thank D. Bernard for discussions.
Appendix
The Weierstrass σ function of periods 2ω ,2ω is the entire function defined by
1 2
z z 1 z 2
σ(z)=z 1− exp + (46)
ω ω 2 ω
m,n6=0(cid:18) mn(cid:19) " mn (cid:18) mn(cid:19) #
Y
8
with ω =2mω +2nω . The functions ζ and ℘ are
mn 1 2
σ′(z)
ζ(z)= , ℘(z)=−ζ′(z), (47)
σ(z)
these functions having the symmetries
σ(−z)=−σ(z) , ζ(−z)=−ζ(z) , ℘(−z)=℘(z). (48)
Their behaviour at the neighbourhood of zero is
σ(z)=z+O(z5) , ζ(z)=z−1+O(z3) , ℘(z)=z−2+O(z2). (49)
Setting
σ(q+λ)
l(q,λ)=− (50)
σ(q) σ(λ)
it is easy to check that
l(−q,λ)=−l(q,−λ) , l′(q,λ)=l(q,λ) [ζ(λ+q)−ζ(q)]. (51)
We need several non trivial relations:
σ(λ−µ) σ(λ+µ)
− =℘(λ)−℘(µ), (52)
σ2(λ) σ2(µ)
σ(x−y)σ(x+y)σ(z−t)σ(z+t)+σ(y−z)σ(y+z)σ(x−t)σ(x+t)+σ(z−x)σ(z+x)σ(y−t)σ(y+t) =0, (53)
σ(2z) σ(x+y) σ(x−y)
=ζ(x+z)−ζ(x−z)+ζ(y−z)−ζ(y+z), (54)
σ(x+z) σ(x−z) σ(y+z) σ(y−z)
this last equation becoming, in terms of the l(q,λ) function,
l(q,λ) l(−q,λ−µ)
=ζ(λ)+ζ(µ−λ)+ζ(q)−ζ(µ+q). (55)
l(q,µ)
Choosing the periods ω =∞ and ω =iπ we obtain the hyperbolic case
1 2 2
z2 z 1 1
σ(z)=sinh(z)exp − , ζ(z)=coth(z)− , ℘(z)= + (56)
6 3 sinh2(z) 3
(cid:18) (cid:19)
and
sinh(λ+q) λ q
l(q,λ)=− exp − . (57)
sinh(λ)sinh(q) 3
(cid:18) (cid:19)
All these formulas were collected in [11].
References
[1] J. Gibbons and T. Hermsen, Physica 11D (1984) 337.
[2] S. Wojciechowski, Phys. Lett. 111A (1985) 101.
[3] E. Billey, J. Avan and O. Babelon, preprint hep-th/9312042.
[4] F. Calogero,Lett. Nuovo Cimento 13 (1975) 411.
[5] F. Calogero,Lett. Nuovo Cimento 16 (1976) 77.
[6] E.K.Sklyanin, “Onthe complete integrability ofthe Landau-Lifchitz equation.” preprintLOMI E-3-79
Leningrad 1979.
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