Table Of ContentPublished by Institute of Physics Publishing for SISSA/ISAS
Received: November 07, 2001
Accepted: January 09, 2002
2 Finite Chern-Simons matrix model — algebraic
0 J
0
approach H
2
n E
a
J P
8
0
1
1
2 Larisa Jonke and Stjepan Meljanac
v (
0 Theoretical Physics Division, Rudjer Boˇskovi´c Institute
0 2
P.O. Box 180, HR-10002 Zagreb, Croatia
3
E-mail: [email protected],[email protected] 0
0
1
0
1
0 2
Abstract: We analyze the algebra of observables and the physical Fock space of the
/
h
finite Chern-Simons matrix model. We observe that the minimal algebra of observables )
t
-
p acting on that Fock space is identical to that of the Calogero model. Our main result is 0
e
the identification of the states in the l-th tower of the Chern-Simons matrix model Fock
h 0
: space and the states of the Calogero model with the interaction parameter ν = l+1. We
v 8
i describe quasiparticle and quasihole states in the both models in terms of Schur functions,
X
and discuss some non-trivial consequences of our algebraic approach.
r
a
Keywords: Chern-Simons Theories, Non-Commutative Geometry, Matrix Models,
Integrable Field Theories.
(cid:13)c SISSA/ISAS2008 http://jhep.sissa.it/archive/papers/jhep012002008/jhep012002008.pdf
Contents
1. Introduction 1
2. CS matrix model 2
2.1 Introduction — the physical Fock space 2
2.2 Algebraic structure of the CS matrix model 4
J
2.3 Relation to the Calogero model 5
H
3. Quasiparticles and quasiholes 7
E
4. Outlook and discussion 8 P
0
A. Calogero model 10
1
(
2
0
1. Introduction 0
2
It is well known that the canonical quantization of the system of particles in a strong mag-
)
neticfieldgivesanaturalrealization ofnon-commutativespace. Onecanspeculatewhether
0
it is possible to describe a real physical system — Quantum Hall fluid — using quantum
field theory on non-commutative plane. It was conjectured [1] that the Laughlin state of 0
electronsatafillingfraction1 1/kwasdescribedbythenon-commutativeversionoftheU(1)
8
Chern-Simonstheoryatlevelk. Thefieldsinthattheorywereinfinitematricescorrespond-
ing to an infinite number of electrons on infinite plane. Later, Polychronakos [2] proposed
a regularized version of the same model that could describe a finite number of electrons
localized on a plane. The complete minimal basis of exact wavefunctions for the theory at
an arbitrary level k and rank N was given in ref. [3], and the coherent-state representation
was analyzed in ref. [4]. Using the properties of the energy eigenvalues of the Calogero
model [5], an orthogonal basis for the Chern-Simons (CS) matrix model was identified [6].
Therelation between theCalogero modeland QuantumHall(QH) physicswas investi-
gated usingthe algebraic approach [7,8]and thecollective-field theory [9]. In ref.[7]it was
conjecturedandthenprovedinref.[8]thatonecouldmapanyonsinthelowestLandaulevel
into the Calogero model, using the complex representation of the S -extended Heisenberg
N
algebra underlyingtheCalogero model. Ontheotherhand,itwas showninref.[9]thatthe
correlation functionsoftheQHedgestateandtheCalogeromodelwererelatedfortheinte-
ger interaction parameter ν. Also, therelation between theCalogero modeland thematrix
1Actually, it was shown by Polychronakos that owing to the quantum effects the corresponding filling
fraction is 1/(k+1).
– 1 –
model was established [10]. Finally, an interesting link between non-commutative CS the-
ory and QH fluid was provided using branes in massive type-IIA string theory [11]. Taking
into consideration all these relations between the Calogero model, the matrix model and
QH physics, one hopes that this intricate network of connections between the apparently
different physical systems will provide useful insight into the underlying structure.
In this letter we analyze in detail the physical Fock space of the finite CS matrix
model. We observe that the minimal algebra of observables acting on that Fock space is
identicaltothatoftheCalogeromodel,butwestressthatthecompletealgebraicstructures
are different. Our main result is the identification of the states in the l-th tower of the
CS matrix model Fock space and the states of the Calogero model with the interaction J
parameter ν = l + 1. We discuss some mathematical and physical consequences of this
H
identification. Specialy, we describe quasiparticle and quasihole excitations in the CS
E
matrix model, and using the identification, the corresponding excitations in the Calogero
P
model. We also discuss a possible extension of the physical Fock space to include particles
with the fractional statistics. To make this paper self-contained, we add an appendix with 0
relevant expressions and results in the Calogero model [12, 13, 14]. 1
(
2. CS matrix model
2
2.1 Introduction — the physical Fock space 0
Let us start from the action proposed in ref. [2]: 0
2
B
S = dt Tr ε X˙ +i[A ,X ] X +2θA ωX2 +Ψ† iΨ˙ A Ψ . (2.1)
2 ab a 0 a b 0− a − 0 )
Z n (cid:16) (cid:17) o (cid:16) (cid:17)
Here, A and X , a = 1,2, are N N hermitean matrices and Ψ is a complex N-vector. 0
0 a
×
The eigenvalues of the matrices Xa represent the coordinates of electrons, A0 is a gauge 0
field, and Ψ acts like a boundary field. We choose the gauge A = 0 and impose the
0 8
equation of motion for A as a constraint:
0
iB[X ,X ]+ΨΨ† = Bθ. (2.2)
1 2
−
The trace part of eq. (2.2) gives
Ψ†Ψ = NBθ. (2.3)
Notice that the commutators have so far been classical matrix commutators. After quan-
tization, the matrix elements of X and the components of Ψ become operators satisfying
a
the following commutation relations:
†
Ψ ,Ψ = δ ,
i j ij
h i i
(X ) ,(X ) = δ δ . (2.4)
1 ij 2 kl B il jk
h i
ItisconvenienttointroducetheoperatorA= B/2(X +iX )anditshermiteanconjugate
1 2
A† obeying the following commutation relations:
p
† † †
A ,A = δ δ , [A ,A ]= A ,A = 0. (2.5)
ij kl il jk ij kl ij kl
h i h i
– 2 –
Then, one can write the hamiltonian of the model at hand as
N2 N2
H = ω +Tr(A†A) = ω + , (2.6)
A
2 2 N
(cid:18) (cid:19) (cid:18) (cid:19)
being the total number operator associated with A’s. Upon quantization, the con-
A
N
straints (2.2) become the generators of unitary transformations of both X and Ψ. The
a
trace part (2.3) demands that (the l.h.s. being the number operator for Ψ’s) Bθ l
≡
be quantized to an integer. The traceless part of the constraint (2.2) demands that the
wavefunction be invariant under SU(N) transformations, under which A transforms in the
adjoint2 and Ψ in the fundamental representation. J
Energyeigenstates willbeSU(N)singlets; generally, somelinearcombinationsofterms H
with at least lN(N 1)/2 A† operators and Nl Ψ† fields. Explicit expressions for the
− E
wavefunctions were written in ref. [3]:
P
N 0
Φ = (TrA†j)cjC†l 0 , (2.7)
| i | i 1
j=1
Y
(
where
2
C† εi1···iNΨ† (Ψ†A†) (Ψ†A†N−1) , (2.8)
≡ i1 i2··· iN
0
and A 0 = Ψ 0 = 0.
ij i
| i | i 0
The system contains N2+N oscillators coupled by N2 1 constraint equations in the
− 2
traceless part of eq. (2.2). Effectively, we can describe the system wih N +1 independent
oscillators. Therefore, the physical Fock space that consists of all SU(N)-invariant states )
can be spanned by N + 1 algebraically independent operators: Bn† TrA†n with n = 0
≡
1,2,...,N, and C†. The operators B† for k > N can be expressed as a homogeneous
k 0
† †
polynomial of total order k in B ,...,B , with constant coefficients which are common
{ 1 N} 8
to all operators A† [15]. Since
TrAkC†l 0 B C†l 0 = 0, k, l, (2.9)
k
| i ≡ | i ∀ ∀
the state C†l 0 0,l can be interpreted as a ground state — vacuum with respect to
| i ≡ | i
all operators B . Note that the vacuum is not normalized to one, i.e. 0,l 0,l = 1. The
k
h | i 6
whole physical Fock space can be decomposed into towers (modules) built on the ground
states with different l:
∞ ∞
FCS = FCS (l)= B†nk 0,l .
phys phys k | i
Xl=0 Xl=0nY o
Clearly, the states from different towers are mutually orthogonal.
2Note that as A transforms in the reducible representation (N2−1)+1, with the singlet B1 ≡ TrA,
one can introduce a pure adjoint representation as A¯ij = Aij − δijB1/N. This slightly modifies the
commutator (2.5), and completely decouples B1 from the Fock space. Physically, this correspond to the
separation of thecentre-of-mass coordinateas it hasbeendonefor theCalogero modelinref. [12]. Forthe
sake of simplicity, this will not be donehere.
– 3 –
The action of the hamiltonian (2.6) on the states in the physical Fock space is
N2 N
H B†nk 0,l = ω + kn +l B†nk 0,l , (2.10)
k | i 2 k 2 k | i
" #
k (cid:18) (cid:19)
Y X Y
and the ground-state energy is E (l) = ω[N2 + lN(N 1)]/2. Comparing this result
0
−
with the known one for the Calogero model (see eq. (A.5) in appendix A) we see that the
spectra of the two models are identical provided ν = l+1. This has already been noticed
in refs. [6, 10]. However, from the identification of spectra it only follows
N N
B†ni 0,l = B†nk 0 ,
i | i k | il+1 J
where the r.h.s. of this(cid:16)reYlation i(cid:17)s, in generaXl, a(cid:16)sYum of d(cid:17)ifferent terms of total order in H
N
the observables of the Calogero model. We use the same letter to denote observables in E
both models. In CS matrix model, B = TrAn and in the Calogero model, B = an,
n n i i P
but from the context it should be clear what B represents. The vacuum in the CS matrix
n P 0
model is denoted as 0,l and the vacuum in the Calogero model is denoted as 0 (see
ν
| i | i
appendix A). 1
(
2.2 Algebraic structure of the CS matrix model
2
Using [A ,B†] = n(A†n−1) , we find a general expression for the commutators between
ij n ij
0
observables:
0
m−1 n−1
[B ,B†] =n Tr(ArA†n−1Am−r−1) = m Tr(A†sAm−1A†n−s−1). (2.11) 2
m n
r=0 s=0
X X )
One can normally order the r.h.s. of eq. (2.11) using the recurrent relation
0
n−2
0
Tr(ArA†n−1Am−1)= Tr(Ar−1A†n−1Am)+ Tr(Ar−1A†s)Tr(A†n−s−2Am−r−1). (2.12)
8
s=0
X
With the formal mapping Tr(ArA†sAk) ara†sak, the relation (2.11) is identical to
→ i i i i
eq. (A.6) for the Calogero model. Also, the recurrent relation (2.12) has its counterpart
P
in the Calogero model with ν = 1, with the same formal mapping. In order to close
the algebra (2.11), one should include observables of the type Bα = Tr(A†i1Ai2A†i3 ),
m,n ···
where m is the number of A†’s and n is the number of A’s, in the trace. This algebra CS
BN
is a polynomial generalization of the Lie algebra. We stress that this algebra is larger than
the corresponding Cal algebra in the Calogero model described in detail in ref. [12], since
BN
in the CS matrix model we have more than one invariant of order (m,n), owing to matrix
multiplication. In other words, in the Calogero model there are more algebraic relations
connecting the trace invariants for fixed N.
In this paper we restrict ourselves to the minimal algebra including only observables
†
of the type B and B , defined with the following relations (including corresponding her-
n n
mitean conjugate relations):
n
[B ,[B ,[...,[B ,B†]...]] = n! i B , (2.13)
i1 i2 in n α I−n
α=1
Y
– 4 –
where I = n i and i ,...,i ,n = 1,2,...,N. These relations can be viewed as a
α=1 α 1 n
generalization of triple operator algebras defining para-Bose and para-Fermi statistics [16].
P
The identical successive commutators relations (2.13) holds for the observables acting on
the S -symmetric Fock space of the Calogero model [12]. The action of B on any state
N k
in the Fock space is of the form:
N−k
B B†ni 0,l = B†nj 0,l , = in k, (2.14)
k i | i j | i N i ≥
Yi X(cid:16)Y (cid:17) Xi
In order to calculate the precise form of the r.h.s. of eq. (2.14) we apply the hermitean
J
conjugate relation eq. (2.13) on the l.h.s. of eq. (2.14) shifting the operator B to the right,
k
at least for one place. We repeat this iteratively as long as the number of B†’s on the H
right from Bk is larger or equal to the index k. For k > ni, one directly calculates a E
finite set of relations from (2.5), the so-called generalized vacuum conditions. We show
P P
that the minimal set of generalized vacuum conditions needed to completely define the
0
representation of the algebra (2.13) on the Fock space is
1
†
B B 0,l = 2N(N +lN l)0,l ,
2 2| i − | i (
B B† 0,l = 3N[N2 +1+l(N 1)(2N 1)+l2(N 1)(N 2)]0,l = y 0,l ,
3 3| i − − − − | i | i 2
†2 † † †
B B 0,l = 54 (l+1)B B +[N +(N 2)l+y/27]B 0,l . (2.15)
3 3 | i { 1 2 − 3}| i 0
0
Namely, the operators B , B and hermitean conjugates play a distinguished role in the
2 3
algebra, since all other operators B , B† for n 4 can be expressed as successive commu- 2
n n
≥
tators (2.13), using only B2, B3 and their hermitean conjugates. Therefore, one can derive )
all other generalized vacuum conditions using (2.13) and (2.15).
0
The relations (2.13) and the generalized vacuum conditions (2.15) represent the min-
0
imal algebraic structure defining the complete physical Fock space representation. Using
8
these relations one can calculate the action of B on any state in the physical Fock space,
n
and calculate all matrix elements (scalar product) of the form 0,l ( Bni)( B†nj)0,l ,
h | i j | i
up to the norm of the vacuum.
Q Q
2.3 Relation to the Calogero model
Our main observation is that the generalized vacuum conditions for the CS matrix model
(2.15) and for the Calogero model (A.7) coincide for ν = l+1. This follows entirely from
the common algebraic structure (2.13) and the structure of the vacuum conditions. As
we have already said, the algebraic relations (2.13) and the generalized vacuum conditions
uniquely determine the action of observables on the Fock space, so we conclude that
B B†ni 0,l = B B†ni 0 , (2.16)
k i | i k i | il+1
(cid:16)Y (cid:17) (cid:16)Y (cid:17)
and that all scalar products in FCS can be identified with the corresponding matrix
phys
elements in the Fock space of the Calogero model:
0,l Bni B†nj 0,l = 0 Bni B†nj 0 . (2.17)
h | i j | i l+1h | i j | il+1
(cid:16)Y (cid:17)(cid:16)Y (cid:17) (cid:16)Y (cid:17)(cid:16)Y (cid:17)
– 5 –
Note that the vacuum is unique in both models, so we set 0,l = 0 , up to the norm.
l+1
| i | i
Now we can identify all states in FCS (l) with the corresponding states in FCal (l+1) as
phys symm
B†nk 0,l = B†nk 0 . (2.18)
k | i k | il+1
One can prove these results(cid:16)iYn differe(cid:17)nt ways. (cid:16)FYor exam(cid:17)ple, we can restrict ourselves to the
† † †
subspaceof theFock space generated by B ,B ,B and prove the relations (2.16), (2.17)
{ 1 2 3}
and (2.18) in this subspace, by straightforward calculation and by induction. Then, using
the algebraic relation (2.13) we simply generate the rest of the needed results on the
full (physical) Fock space. The other way to obtain our result is to notice that from
the equivalence of spectra it follows ν = l + 1. Also, from the fact that the algebraic J
relations (2.13) hold for both models we know that the states in the Calogero model and
H
the CS matrix model are identical, but we have no information on the relation between
E
l and ν. The combination of these two results leads again to the identification of the
states in FCS (l) and FCal (l +1), eq. (2.18). We point out that our algebraic proof of P
phys symm
eqs. (2.16), (2.17) and (2.18) relies only on the identical algebra (2.13) and the identical 0
minimal set of generalized vacuum conditions for both models, and not on the structure of 1
their hamiltonians.
(
This identification of the states in the CS matrix model with the corresponding states
2
in the Calogero model is non trivial, from both mathematical and physical point of view.
0
An important consequence of our analysis is that the operators a , defined in eq. (A.3),
i
can be interpreted as elements belonging to the spectrum of the matrix A. Namely, they 0
are solutions of the polynomial equation det A a = 0, i.e. α ak = 0, where
| − ·1N×N| k k 2
α are operators commuting among themselves and with a’s. Let us assume that there
k P )
are N different, mutually commuting solutions a ,...,a ; then TrAn = an. This pure
1 N i
0
algebraic relation should be applied to the ground states, according to our results:
P
nk 0
TrAn|0,li = ani|0il+1 = 0, and (TrA†k)nk|0,li = a†ik |0il+1. 8
!
i i
X Y Y X
†
Therefore, we conclude that, for consistency reasons, the operators a ,a have to satisfy
i j
†
[a ,a ]= (l+1)K , i= j.
i j − ij 6
Also, as a consequence of this identification, we generate an infinite number of non-
trivial identities in the following way. Using the relations valid for m n
≤
m
CS
B ,B† = c (1)Tr(A†n−lAm−l),
m n lmn
h i Xl=1
m N
Cal
B ,B† = c (ν) a†n−lam−l, (2.19)
m n lmn i i
h i Xl=1 Xi=1
where c (ν) are coefficients depending on all indices and ν, we find an infinite number
lmn
of identities:
[B ,B†]CS 0,l = [B ,B†]Cal 0 ,
m n | i m n | il+1
m
c (1)Tr(A†n−kAm−k)0,l = c (l+1)B† 0 . (2.20)
kmn | i mmn n−m| il+1
k=1
X
– 6 –
†
For example, from [B ,B ], n > 0:
2 n+2
N−2 [n/2]
(N k 1)( )kS = (N 1)m + m , (2.21)
− − − n−k,1k − (n) (n−r,r)
k=0 r=1
X X
where S is Schur function and m is symmetric monomial function, see ref. [17].
λ λ
3. Quasiparticles and quasiholes
Thelow-lyingexcitationsintheCSmatrixmodelcanbedescribedintermsofquasiparticles
J
and quasiholes [2]. We can identify these states in a way analogous to that for particles
H
and holes of a Fermi sea. Onequasiparticle obtained by exciting a “particle” at Fermi level
by energy amount ∆E = nω is E
P
π†(n)0,l = C†(l−1) εi1···iNΨ† (Ψ†A†N−2) (Ψ†A†N−1+n) 0 . (3.1)
| i × i1··· iN−1 iN| i 0
One quasihole excitation is obtained by creating a gap inside the QH droplet with energy
1
increase ∆E = (N k)ω
− (
χ†(k)0,l = C†(l−1) εi1···iNΨ† (Ψ†A†k−1) (Ψ†A†k+1) (Ψ†A†N) 0 . (3.2) 2
| i × i1··· ik ik+1··· iN| i
0
Observe that χ†(N 1) = π†(1). To see that the charge, i.e. the particle number of a
−
0
quasihole is quantized as 1/l, we simply observe that removing one particle (removing
(Ψ†A†k)l) from the ground state is equivalent to creation of l quasiholes. 2
)
P†(N)P(k)0,l = χ†l(k)0,l
| i | i 0
l
= C†(l−1) εi1···iNΨ† (Ψ†A†k−1) (Ψ†A†k+1) (Ψ†A†N) 0 .
× i1··· ik ik+1··· iN | i 0
h i
Wewishtowritetheexpressions(3.1) and(3.2)inourbasisofthephysicalFock space, 8
so we introduce the following functions:
εi1···iNΨ† (Ψ†A†N−2) (Ψ†A†N−1+n)
σ (B†,...,B† ) = i1··· iN−1 iN ,
n 1 N C†
εi1···iNΨ† (Ψ†A†k−1) (Ψ†A†k+1) (Ψ†A†N)
σ1N−k(B1†,...,BN† ) = i1··· ikC† ik+1··· iN .
Now, quasiparticle and quasihole excitations are written as π†(n)0,l = σ (B†,...,B† )
| i n 1 N
|0,liandχ†(k)|0,li = σ1N−k(B1†,...,BN† )|0,li, respectively. Inordertopreciselydetermine
thefunctionsσ ,weusethemappingfromtheCSmatrix modeltotheCalogero modeland
λ
†
the fact that the operators a , defined in eq. (A.3), belong to the spectrum of the matrix
i
A†. From the definition of Schur functions S , and using the notation of ref. [17], we have
λ
σ (B†,...,B† )0,l = S (a†,...,a† )0 = z−1B† B† 0 , (3.3)
n 1 N | i n 1 N | il+1 λ λ1··· λN| il+1
|λ|=n
X
σ1N−k(B1†,...,BN† )|0,li = S1N−k(a†1,...,a†N)|0il+1 = ελzλ−1Bλ†1···Bλ†N|0il+1.
|λ|=N−k
X
– 7 –
Here the summation goes over all partitions λ = (λ ,...,λ ), λ λ λ 0
1 N 1 2 N
≥ ≥ ··· ≥ ≥
of given weight λ = λ and lenght l(λ). The coefficients in relations (3.3) are z =
| | i i λ
imim !, wherem is thenumberof i’s in thepartition λ, andε = ( )|λ|−l(λ). Thesame
i i i P λ −
expression for the quasihole wave function in the Calogero model has been written down
P
in ref. [18] and, also, the norm of this state has been given:
k−1
N
0χ(k)χ†(k)0 = [1+νi].
ν ν
h | | i k
(cid:18) (cid:19)i=0
Y
The partition λ = (λ ,...,λ ) can be represented by the Young tableau with N rows,
1 N J
each having λ boxes. In the Calogero model, the quasiparticle π†(n) is then represented
i
H
by one row with n boxes and quasihole χ†(n) by a single column with N n boxes.
− E
In this representation it is evident that there is no fundamental distinction between
quasiparticles and quasiholes for finite N, since they obviously represent the same type of P
excitations in two different bases. Using the properties of Schur function we can write a 0
simple relation connecting these two bases:
1
n N n (
µ + +µ
S = ( )l+n S S = ( )l+n 1 ··· N Sµ1 SµN ,
n − 1i1 ··· 1il − µ ,...,µ 1 ··· 1N 2
Xl=1 i1,.X..,il=1 Xl=1 |Xµ|=l(cid:18) 1 N (cid:19)
0
and vice versa
0
n N n 2
µ + +µ
S1n = (−)l+n Si1···Sil = (−)l+n 1µ ,.·.·.·,µ N S1µ1···SNµN , )
Xl=1 i1,.X..,il=1 Xl=1 |Xµ|=l(cid:18) 1 N (cid:19)
0
where summations over iα and µ are subject to condition lα=1iα = Nj=1jµj = n. These 0
relations hold for σ funtions also, i.e. we can apply these relations in both models.
λ P P 8
4. Outlook and discussion
It is important to note that the whole picture is consistent only if the matrix A is not
†
diagonal. If we assume A = a δ , then [a ,a ] = δ , and that corresponds to the
ij i ij i j ij
l = 1 tower which does not exist in the CS matrix model (although the ν = l+1 = 0
−
case corresponds to N bosons in the Calogero model). There is also another way to
see this inconsistency. If we diagonalize the matrix A, the ground states factorize as
0,l ( Ψ†)l (a† a†)l and, generally, the vacuum conditions are not satisfied any
| i ∼ i i<j i − j
longer, ( an) (a† a†)l 0 = 0 for n l.
Qi i Q i − j | iν 6 ≤
SimiPlar arguQments can be applied to the states of the form ( ia†ik)nk (a†i −a†j)l|0iν
†
with [a ,a ] = δ , eqs. (11) and (12) in ref. [3], which correspond to the Laughlin states in
i j ij P Q
a Fock space. For l even, the above states are completely symmetric and can be written
as ( ia†ik)nk|0iν, and for any odd l, the states are ( ia†ik)nk (a†i −a†j)|0iν. Hence, all
l-even states reducetoaBosetower withl = 0, andalll-oddstates reducetoaFermitower
P P Q
with l = 1. Moreover, the vacuum conditions are not satisfied. So, there is no one-to-one
– 8 –
mappingbetween the CSmatrix modelstates and the above Fock-space states. This leaves
us with an unanswered question about the relation between QH physics and the finite CS
matrix model, as was also observed in ref. [19].
UsingthemappingfromFCS(l)toFCal (l+1)wecanshowthatthereexistsamapping
ph symm
from the free Bose oscillators b ,b ,...,b ,c to the observables in the CS matrix model
1 2 N 0
{ }
B ,B ,...,B ,C , and vice versa [13]. This mapping offers a natural orthogonal basis
1 2 N
{ }
in the physical Fock space (b†)nk 0,l , an alternative to the orthogonal basis in terms
{ k | i}
of Jack polynomials proposed in ref. [6]
Q
Also, the dynamical symmetry generators operating on degenerate states in a fixed
tower of states FCS(l) are the same as those found in the Calogero model with ν = l + J
ph
1 [12, 13]. Moreover, one can introduce additional generators acting between different
H
towers and so describe the larger dynamical symmetry operating on all degenerate states
E
in the CS matrix model.
P
The relation between the Calogero model and the physics of anyons in the lowest
Landau level has been described in ref. [8], and this relation is not restricted to the integer 0
values of the interaction parameter in the Calogero model. It would be interesting to 1
somehow enlarge this finite CS matrix model to describe also particles with fractional
(
statistics. This can be easily done in the Fock-space approach.
2
We can define a new SU(N) invariant operator D† = C†1/q such that = q ,
D C
N N 0
and being the number operators for D’s and C’s, respectively. The enlarged Fock
D C
N N
space FpChSy,sq is a space of all states of the type k(Bk†)nkD†l|0iν, for l integer. From 0
the action of the hamiltonian (2.6) on that Fock space we obtain ν = l/q +1. Also, the 2
Q
generalizedvacuumconditions(2.15)holdforanyrationalnumberl l/q,andthereforeall
)
→
identifications eqs.(2.16), (2.17) and(2.18) between theCSmatrixandtheCalogeromodel
0
aretrueforanyrationalν = l/q+1. ThisconstructionisanallowedextensionofFockspace
0
for rational ν, but we point out that the dynamical origin of such a picture is still missing.
In ref. [1] it was conjectured that the theory for QH fluid with the filling fraction n/k 8
would be level k U(n) non-commutative CS theory. In the string theory approach level k
U(n) non-commutative CS theory corresponds to the configuration of n D2-branes and 2k
D8-branes in a background B-field [11]. Even in this approach, the dynamical description
of the hierarchy structure and the mass gap of excitations in QH fluid is still missing.
In conclusion, we have discussed the Fock-space picture in detail and elucidated the
A-representation of the CS matrix model. We have precisely formulated the connection
between the states in the Fock spaces of the CS matrix and the Calogero model. We
stress that although the models have similar Fock spaces and a common minimal algebraic
structure, the complete algebraic structures are quite different. An important consequence
of our analysis is that the operators a , elements of S -extended Heisenberg algebra, can
i N
be interpreted as elements belonging to the spectrum of the matrix A.
Acknowledgments
We would like to thank D. Svrtan for useful discussions. This work was supported by the
MinistryofScienceandTechnologyoftheRepublicofCroatiaundercontractNo.00980103.
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