ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

Superconformal Calogero Models as a Gauged Matrix Mechanics

S. Fedoruk

Abstract

We present basics of the gauged superfield approach to constructing the N -superconformal multi-particle Calogero-type
systems developed in arXiv:0812.4276, arXiv:0905.4951 and arXiv:0912.3508. This approach is illustrated by multi-particle
systems possessing SU(1, 1|1) and D(2, 1;α) supersymmetries, as well as by the model of newN = 4 superconformal quantum
mechanics.

1 Introduction
The celebrated Calogero model [1] is a prime exam-
ple of an integrable and exactly solvable multi-particle
system. It describes the system of n identical parti-
cles interacting through an inverse-square pair poten-
tial

∑
a�=b

g/(xa − xb)
2, a, b = 1, . . . , n. The Calogero

model and its generalizations provide deep connections
of various branches of theoretical physics and have a
wide range of physical and mathematical applications
(for a review, see [2, 3]).
An important property of the Calogero model is

d = 1 conformal symmetry SO(1, 2). Being multi-
particle conformal mechanics, this model, in the two-
particle case, yields the standard conformal mechan-
ics [4]. Conformal properties of the Calogero model
and the supersymmetric generalizations of the latter
give possibilities to apply them in black hole physics,
since the near-horizon limits of extreme black hole
solutions in M -theory correspond to AdS2 geometry,
having the same SO(1, 2) isometry group. Analysis of
the physical fermionic degrees of freedom in the black
hole solutions of four- and five-dimensional supergrav-
ities shows that related d = 1 superconformal systems
must possess N = 4 supersymmetry [5, 6, 7].
Superconformal Calogero models with N = 2 su-

persymmetry were considered in [8, 9] and with N = 4
supersymmetry in [10, 11, 12, 13, 14, 15]. Unfortu-
nately, consistent Lagrange formulations for the n-
particle Calogero model with N = 4 superconformal
symmetry for any n is still lacking.
Recently, we developed a universal approach to su-

perconformal Calogero models for an arbitrary number
of interacting particles, including N = 4 models. It is
based on the superfield gauging of some non-abelian
isometries of d = 1 field theories [16].
Our gauge model involves three matrix superfields.

One is a bosonic superfield in the adjoint representa-
tion of U(n). It carries the physical degrees of free-
dom of the superCalogero system. The second super-
field is in the fundamental (spinor) representation of

U(n) and is described by Chern-Simons mechanical
action [17, 18]. The third matrix superfield accommo-
dates the gauge “topological” supermultiplet [16]. N -
extended superconformal symmetry plays a very im-
portant role in our model. Elimination of the pure
gauge and auxiliary fields gives rise to Calogero-like
interactions for the physical fields.
The talk is based on the papers [19, 20, 21].

2 Gauged formulation of the
Calogero model

The renowned Calogero system [1] can be described
by the following action [18, 22]:

S0 =
∫
dt

[
Tr (∇X∇X)+

i

2
(Z̄∇Z −∇Z̄Z) + cTrA

]
,

(2.1)

where

∇X = Ẋ + i[A, X ],

∇Z = Ż + iAZ ∇Z̄ = ˙̄Z − iZ̄A .

The action (2.1) is the action of U(n), d = 1 gauge the-
ory. The hermitian n×n-matrix field Xb

a(t), (Xb
a) =

Xa
b , a, b = 1, . . . , n and the complex commuting U(n)-
spinor field Za(t), Z̄a = (Za) present the matter,
scalar and spinor fields, respectively. The n2 “gauge
fields” Ab

a(t), (Ab
a) = Aa

b are non-propagating ones
in d = 1 gauge theory. The second term in the ac-
tion (2.1) is the Wess-Zumino (WZ) term. The third
term is the standard Fayet-Iliopoulos (FI) term.
The action (2.1) is invariant under the d = 1 con-

formal SO(1, 2) transformations:

δt = α, δXb
a =
1
2

α̇Xb
a,

δZa = 0, δAb
a = −α̇Ab

a ,
(2.2)

where the constrained parameter ∂3t α = 0 contains
three independent infinitesimal constant parameters of
SO(1, 2).

Talk at the Conference “Selected Topics in Mathematical and Particle Physics”, In Honor of 70th Birthday of Jiri Niederle, 5–7 May
2009, Prague and at the XVIII International Colloquium “Integrable Systems and Quantum Symmetries”, 18–20 June 2009, Prague,
Czech Republic.

23



Acta Polytechnica Vol. 50 No. 3/2010

The action (2.1) is also invariant with respects to
the local U(n) invariance

X → gXg†, Z → gZ, A→ gAg† + iġg† , (2.3)

where g(τ) ∈ U(n).
Let us demonstrate, in Hamiltonian formalism,

that the gauge model (2.1) is equivalent to the stan-
dard Calogero system.
The definitions of the momenta, corresponding to

the action (2.1),

PX = 2∇X , PZ =
i

2
Z̄ ,

P̄Z = −
i

2
Z , PA = 0

(2.4)

imply the primary constraints

a) G ≡ PZ −
i

2
Z̄ ≈ 0 , Ḡ ≡ P̄Z +

i

2
Z ≈ 0 ;

b) PA ≈ 0
(2.5)

and give us the following expression for the canonical
Hamiltonian

H =
1
4
Tr (PXPX)− Tr (AT ) , (2.6)

where matrix quantity T is defined as

T ≡ i[X, PX]− Z · Z̄ + cIn . (2.7)

The preservation of the constraints (2.5b) in time
leads to the secondary constraints

T ≈ 0 . (2.8)

The gauge fields A play the role of the Lagrange mul-
tipliers for these constraints.
Using canonical Poisson brackets [Xb

a, PX
d
c ]P =

δd
aδb

c, [Za, P b
Z
]
P
= δb

a, [Z̄
a, P̄Z b]P = δa

b , we obtain the
Poisson brackets of the constraints (2.5a)

[Ga, Ḡb]P = −iδa
b . (2.9)

Dirac brackets for these second class constraints (2.5a)
eliminate spinor momenta PZ, P̄Z from the phase space.
The Dirac brackets for the residual variables take the
form

[Xb
a, PX

d
c ]D = δd

aδb
c , [Za, Z̄b]

D
= −i δb

a . (2.10)

The residual constraints (2.8) T = T+ form the
u(n) algebra with respect to the Dirac brackets

[T b
a , T d

c ]D = i(δd
aT b

c − δb
cT

d
a ) (2.11)

and generate gauge transformations (2.3). Let us fix
the gauges for these transformations.
In the notations

xa ≡ Xa
a , pa ≡ PX

a
a (no summation over a) ;

xb
a ≡ Xb

a , pb
a ≡ PX

b
a for a �= b

the constraints (2.7) take the form

T b
a = i(xa − xb)pb

a − i(pa − pb)xb
a + (2.12)

i
∑

c

(xc
apb

c − pc
axb

c)− ZaZ̄b ≈ 0 for a �= b ,

T a
a = i

∑
c

(xc
apa

c − pc
axa

c )− ZaZ̄a + c ≈ 0 (2.13)

(no summation over a) .

The non-diagonal constraints (2.12) generate the
transformations

δxb
a = [x

b
a, εa

bT a
b ]D ∼ i(xa − xb)εa

b .

Therefore, in case of the Calogero-like condition
xa �=xb, we can impose the gauge

xb
a ≈ 0 . (2.14)

Then we introduce Dirac brackets for the con-
straints (2.12), (2.14) and eliminate xb

a, pb
a. In par-

ticular, the resolved expression for pb
a is

pb
a = −

i

(xa − xb)
ZaZ̄

b . (2.15)

The Dirac brackets of residual variables coincide with
Poisson ones due to the resolved form of the gauge
fixing condition (2.14).
After gauge-fixing (2.14), the constraints (2.13) be-

come

ZaZ̄
a − c ≈ 0 (no summation over a) (2.16)

and generate local phase transformations of Za. For
these gauge transformations we impose the gauge

Za − Z̄a ≈ 0 . (2.17)

The conditions (2.16) and (2.17) eliminate Za and Z̄a

completely.
Finally, using the expressions (2.15) and the condi-

tions (2.14), (2.16) we obtain the following expression
for the Hamiltonian (2.6)

H0 =
1
4
Tr (PXPX) =

1
4

⎛⎝∑
a

(pa)2 +
∑
a�=b

c2

(xa − xb)2

⎞⎠ ,
(2.18)

which corresponds to the standard Calogero action [1]

S0 =
∫
dt

[∑
a

ẋaẋa −
∑
a�=b

c2

4(xa − xb)2

]
. (2.19)

3 N = 2 superconformal
Calogero model

N = 2 supersymmetric generalization of the sys-
tem (2.1) is described by
• the even hermitian (n × n)-matrix superfield
X b

a (t, θ, θ̄), (X )+ = X , a, b = 1, . . . , n [supermulti-
plets (1,2,1)];

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Acta Polytechnica Vol. 50 No. 3/2010

• commuting chiral U(n)–spinor superfield
Za(tL, θ), Z̄a(tR, θ̄) = (Za)+, tL,R = t ± iθθ̄
[supermultiplets (2,2,0)];

• commuting n2 complex “bridge” superfields
bc
a(t, θ, θ̄).
The N = 2 superconformally invariant action of

these superfields has the form

S2 =
∫
dt d2θ

[
Tr

(
D̄X DX

)
+

1
2
Z̄ e2VZ − cTrV

]
.

(3.1)

Here the covariant derivatives of the superfield X are

DX = DX +i[A,X ] , D̄X = D̄X +i[Ā,X ] , (3.2)

D = ∂θ + iθ̄∂t , D̄ = −∂θ̄ − iθ∂t , {D, D̄} = −2i∂t ,

where the potentials are constructed from the bridges
as

A = −i eib̄(De−ib̄) ,

Ā = −i eib(D̄e−ib) (b̄ ≡ b+) .
(3.3)

The gauge superfield prepotential V b
a (t, θ, θ̄), (V )

† =
V , is constructed from the bridges as

e2V = e−ib̄ eib . (3.4)

The superconformal boosts of the N = 2 supercon-
formal group SU(1, 1|1) � OSp(2|2) have the following
realization:

δt = −i(ηθ̄ + η̄θ)t ,

δθ = η(t+ iθθ̄) , δθ̄ = η(t− iθθ̄) ,
(3.5)

δX = −i(ηθ̄ + η̄θ)X , δZ = 0 ,

δb = 0 , δV = 0 .
(3.6)

Its closure with N = 2 supertranslations yields the full
N = 2 superconformal invariance of the action (3.1).
The action (3.1) is invariant also with respect to

the two types of the local U(n) transformations:
• τ -transformations with the hermitian (n×n)-matrix
parameter τ(t, θ, θ̄) ∈ u(n), (τ)+ = τ ;
• λ–transformations with complex chiral gauge param-
eters λ(tL, θ) ∈ u(n), λ̄(tR, θ) = (λ)+.
These U(n) transformations act on the superfields in
the action (3.1) as

eib′ = eiτ eibe−iλ , e2V
′
= eiλ̄ e2V e−iλ , (3.7)

X ′ = eiτ X e−iτ , Z ′ = eiλZ , Z̄ ′ = Z̄ e−iλ̄ . (3.8)

In terms of τ -invariant superfields V , Z and new
hermitian (n× n)-matrix superfield

X = e−ib X eib̄ , X ′ = eiλ X e−iλ̄ , (3.9)

the action (3.1) takes the form

S2 =
∫
dt d2θ

[
Tr

(
D̄X e2V DX e2V

)
+

1
2
Z̄ e2VZ − cTrV

] (3.10)

where the covariant derivatives of the superfieldX are

DX = DX+ e−2V (De2V )X ,

D̄X = D̄X−X e2V (D̄e−2V ) .
(3.11)

For gauge λ-transformations we impose the WZ
gauge

V (t, θ, θ̄) = −θθ̄A(t) .

Then, the action (3.10) takes the form

S2 = S0 + SΨ2 ,

SΨ2 = −iTr
∫
dt (Ψ̄∇Ψ−∇Ψ̄Ψ)

(3.12)

where Ψ = DX| and

∇Ψ = Ψ̇ + i[A,Ψ] , ∇Ψ̄ = ˙̄Ψ + i[A, Ψ̄] .

The bosonic core in (3.12) exactly coincides with the
Calogero action (2.19).
Exactly as in the pure bosonic case, residual lo-

cal U(n) invariance of the action (3.12) eliminates the
nondiagonal fields Xb

a, a �=b, and all spinor fields Za.
Thus, the physical fields in our N = 2 supersymmet-
ric generalization of the Calogero system are n bosons
xa = Xa

a and 2n
2 fermions Ψb

a. These fields present
the on-shell content of n multiplets (1,2,1) and n2−n
multiplets (0,2,2) which are obtained from n2 multi-
plets (1,2,1) by the gauging procedure [16]. We can
present it by the plot:

X a
a = (X

a
a ,Ψa

a, Ca
a )︸ ︷︷ ︸

(1,2,1)multiplets

X b
a = (X

b
a,Ψb

a, Cb
a), a �=b︸ ︷︷ ︸

(1,2,1)multiplets

⇓ gauging ⇓

X a
a = (X

a
a ,Ψa

a, Ca
a )︸ ︷︷ ︸

(1,2,1)multiplets

interact Ωb
a = (Ψ

b
a, Bb

a, Cb
a), a �=b︸ ︷︷ ︸

(0,2,2)multiplets

where the bosonic fields Ca
a , C

b
a and Bb

a are auxiliary
components of the supermultiplets. Thus, we obtain
some new N = 2 extensions of the n-particle Calogero
models with n bosons and 2n2 fermions as compared to
the standard N = 2 superCalogero with 2n fermions
constructed by Freedman and Mende [8].

4 N = 4 superconformal
Calogero model

The most natural formulation of N = 4, d = 1 su-
perfield theories is achieved in the harmonic super-
space [23] parametrized by

(t, θi, θ̄
k, u±

i ) ∼ (t, θ±, θ̄±, u±
i ) ,

θ± = θiu±
i , θ̄± = θ̄iu±

i , i, k = 1, 2.

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Acta Polytechnica Vol. 50 No. 3/2010

Commuting SU(2)-doublets u±
i are harmonic coordi-

nates [24], subjected by the constraints u+iu−
i = 1.

The N = 4 superconformally invariant harmonic ana-
lytic subspace is parametrized by

(ζ, u) = (tA, θ+, θ̄+, u±
i ), tA = t−i(θ+θ̄−+θ−θ̄+) .

The integration measures in these superspaces are
μH = du dt d

4θ and μ
(−2)
A = du dζ(−2).

The N = 4 supergauge theory related to our task
is described by:
• hermitian matrix superfields X(t, θ±, θ̄±, u±

i ) =
(Xb

a) subjected to the constraints

D++X = 0, D+D− X = 0,

(D+D̄
−
+ D̄

+
D−)X = 0

(4.1)

[multiplets (1,4,3)];
• analytic superfields Z+(ζ, u) = (Z+a ) subjected to
the constraint

D++Z+ = 0 (4.2)

[multiplets (4,4,0)];
• the gauge matrix connection V ++(ζ, u) =
(V ++b

a).
In (4.1) and (4.2) the covariant derivatives are defined
by

D++X = D++X+ i [V ++, X],

D++Z+ = D++Z+ + i V ++Z+.

Also D+ = D+, D̄
+
= D̄+ and the connections in D−,

D̄
−
are expressed through derivatives of V ++.
The N = 4 superconformal model is described by

the action

Sα�=0
4 = − 1

4(1 + α)

∫
μH Tr

(
X−1/α

)
+ (4.3)

1
2

∫
μ
(−2)
A V0 Z̃+Z+ +

i

2
c

∫
μ
(−2)
A TrV ++ .

The tilde in Z̃+ denotes ‘hermitian’ conjugation pre-
serving analyticity [24, 23].
The unconstrained superfield V0(ζ, u) is a real ana-

lytic superfield, which is defined by the integral trans-
form (X0 ≡ Tr (X))

X0(t, θi, θ̄
i) =∫

duV0
(
tA, θ+, θ̄+, u±) ∣∣∣

θ±=θiu±
i

, θ̄±=θ̄iu±
i

.

The real number α�=0 in (4.3) coincides with
the parameter of the N = 4 superconformal group
D(2, 1;α) which is symmetry group of the action (4.3).
Field transformations under superconformal boosts
are (see the coordinate transformations in [23, 16])

δX = −Λ0X , δZ+ = ΛZ+,

δV ++ = 0 ,
(4.4)

where Λ = 2iα(η̄−θ+−η−θ̄+), Λ0 = 2Λ−D−−D++Λ.
It is important that just the superfield multiplier V0
in the action provides this invariance due to δV0 =
−2ΛV0 (note that δμ

(−2)
A = 0).

The action (4.3) is invariant under the local U(n)
transformations:

X ′ = eiλXe−iλ, Z+′ = eiλZ+,

V ++ ′ = eiλ V ++ e−iλ − i eiλ(D++e−iλ),
(4.5)

where λb
a(ζ, u±) ∈ u(n) is the ‘hermitian’ analytic ma-

trix parameter, λ̃ = λ. Using gauge freedom (4.5) we
choose the WZ gauge

V ++ = −2i θ+θ̄+A(tA). (4.6)

Considering the case α = −1
2
(when D(2, 1;α) �

OSp(4|2)) in the WZ gauge and eliminating auxiliary
and gauge fields, we find that the action (4.3) has the
following bosonic limit

S
α=−1/2
4,b =

∫
dt

{∑
a

ẋaẋa +
i

2

∑
a

(Z̄a
k Żk

a − ˙̄Za
kZk

a ) +

∑
a�=b

Tr(SaSb)
4(xa − xb)2

− nTr(ŜŜ)
2(X0)2

⎫⎬⎭ , (4.7)

where

(Sa)ij ≡ Z̄a
i Zj

a, (Ŝ)ij ≡
∑

a

[
(Sa)ij −

1
2
δj
i (Sa)kk

]
.

The fields xa are “diagonal” fields in X = X|. The
fields Zi define first components in Z+, Z+| = Ziu+i .
They are subject to the constraints

Z̄a
i Zi

a = c ∀ a . (4.8)

These constraints are generated by the equations of
motion with respect to the diagonal components of
gauge field A.
Using Dirac brackets [Z̄a

i , Zj
b ]D = iδa

b δj
i , which are

generated by the kinetic WZ term for Z, we find that
the quantities Sa for each a form u(2) algebras

[(Sa)i
j , (Sb)k

l]D = iδab

{
δl
i(Sa)k

j − δj
k(Sa)i

l
}

.

Thus modulo center-of-mass conformal potential (up
to the last term in (4.7)), the bosonic limit (4.7) is none
other than the integrable U(2)-spin Calogero model in

the formulation of [25, 3]. Except for the case α = −1
2
,

the action (4.3) yields non-trivial sigma-model type ki-
netic term for the field X =X|.
For α = 0 it is necessary to modify the transfor-

mation law of X in the following way [16]

δmodX = 2i(θkη̄k + θ̄kηk) . (4.9)

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Acta Polytechnica Vol. 50 No. 3/2010

Then the D(2, 1;α = 0) superconformal action reads

Sα=0
4 = −1

4

∫
μH Tr

(
eX

)
+ (4.10)

1
2

∫
μ
(−2)
A Z̃+Z+ + i

2
c

∫
μ
(−2)
A TrV ++ .

The D(2, 1;α = 0) superconformal invariance is not
compatible with the presence of V in the WZ term
of the action (4.10), still implying the transformation
laws (4.4) for Z+ and for V ++ . This situation is
quite analogous to what happens in the N = 2 su-
per Calogero model considered in Sect. 3, where the
center-of-mass supermultiplet Tr(X) decouples from
the WZ and gauge supermultiplets. Note that the
(matrix)X supermultiplet interacts with the (column)
Z supermultiplet in (3.1) and (4.10) via the gauge su-
permultiplet.

5 D(2, 1;α) quantum mechanics

The n = 1 case of the N = 4 Calogero-like model (4.3)
above (the center-of-mass coordinate case) amounts to
a non-trivial model of N = 4 superconformal mechan-
ics.
Choosing the WZ gauge (4.6) and eliminating the

auxiliary fields by their algebraic equations of motion,
we obtain that the action takes the following on-shell
form

S = Sb + Sf , (5.1)

Sb =
∫
dt

[
ẋẋ+

i

2

(
z̄kżk − ˙̄zkzk

)
− (5.2)

α2(z̄kzk)2

4x2
−A

(
z̄kzk − c

) ]
,

Sf = −i

∫
dt

(
ψ̄kψ̇k − ˙̄ψkψk

)
+ (5.3)

2α
∫
dt

ψiψ̄kz(iz̄k)

x2
+

2
3
(1 + 2α)

∫
dt

ψiψ̄kψ(iψ̄k)

x2
.

The action (5.1) possesses D(2, 1;α) superconfor-
mal invariance. Using the Nöther procedure, we find
the D(2, 1;α) generators. The quantum counterparts
of them are

Qi = PΨi + 2iα
Z(iZ̄k)Ψk

X
+ (5.4)

i(1 + 2α)
〈ΨkΨkΨ̄i〉

X
,

Q̄i = P Ψ̄i − 2iα
Z(iZ̄k)Ψ̄k

X
+ (5.5)

i(1 + 2α)
〈Ψ̄kΨ̄kΨi〉

X
,

Si = −2XΨi + tQi, S̄i = −2XΨ̄i + t Q̄i . (5.6)

H =
1
4

P 2 + α2
(Z̄kZk)2 + 2Z̄kZk

4X2
− (5.7)

2α
Z(iZ̄k)Ψ(iΨ̄k)

X2
−

(1 + 2α)
〈ΨiΨi Ψ̄kΨ̄k〉
2X2

+
(1 + 2α)2

16X2
,

K = X2 − t
1
2
{X, P}+ t2H ,

D = −1
4
{X, P}+ tH ,

(5.8)

Jik = i
[
Z(iZ̄k) + 2Ψ(iΨ̄k)

]
, I1

′1′ = −iΨkΨk ,

I2
′2′ = iΨ̄kΨ̄k , I1

′2′ = − i

2
[Ψk, Ψ̄k] . (5.9)

The symbol 〈. . .〉 denotes Weyl ordering.
It can be directly checked that the genera-

tors (5.4)–(5.9) form the D(2, 1;α) superalgebra

{Qai′i,Qbk′k} = −2
(
εikεi′k′

Tab + (5.10)

αεabεi′k′
Jik − (1 + α)εabεikIi

′k′
)

,

[Tab,Tcd] = −i
(
εacTbd + εbdTac

)
, (5.11)

[Jij ,Jkl] = −i
(
εikJjl + εjlJik

)
, (5.12)

[Ii
′j′ , Ik

′l′ ] = −i
(
εikIj

′l′ + εj′l′Ii
′k′)

,

[Tab,Qci′i] = iεc(aQb)i′i, (5.13)

[Jij ,Qai′k] = iεk(iQai′j),

[Ji′j′ ,Qak′i] = iεk′(i′Qaj′)i

due to the quantum brackets

[X, P ] = i , [Zi, Z̄j ] = δi
j ,

{Ψi, Ψ̄j} = −
1
2

δi
j .

(5.14)

In (5.11)–(5.14) we use the notation Q21
′i = −Qi,

Q22
′i = −Q̄i, Q11

′i = Si, Q12
′i = S̄i, T22 = H,

T11 = K, T12 = −D.
To find the quantum spectrum, we make use of the

realization

Z̄i = v+i , Zi = ∂/∂v+i (5.15)

for the bosonic operators where v+i is a commuting
complex SU(2) spinor, as well as the following realiza-
tion of the odd operators

Ψi = ψi, Ψ̄i = −
1
2

∂/∂ψi , (5.16)

where ψi are complex Grassmann variables.
The full wave function Φ = A1 +ψiBi +ψiψiA2 is

subjected to the constraints

Z̄iZ
iΦ = v+i

∂

∂v+i
Φ = cΦ. (5.17)

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Acta Polytechnica Vol. 50 No. 3/2010

Table 1

r0 j i

A
(c)
k′ (x, v+)

|α|(c+ 1) + 1
2

c

2
1
2

B
′(c)
k (x, v+)

|α|(c+ 1) + 1
2

− 1
2
sign(α)

c

2
− 1
2
0

B
′′(c)
k (x, v+)

|α|(c+ 1) + 1
2

+
1
2
sign(α)

c

2
+
1
2
0

Requiring the wave function Φ(v+) to be single-valued
gives rise to the condition that positive constant c is
integer, c ∈ Z. Then (5.17) implies that the wave func-
tion Φ(v+) is a homogeneous polynomial in v+i of the
degree c:

Φ = A
(c)
1 + ψiB

(c)
i + ψiψiA

(c)
2 , (5.18)

A
(c)
i′ = Ai′,k1...kcv

+k1 . . . v+kc , (5.19)

B
(c)
i = B

′(c)
i +B

′′(c)
i = (5.20)

v+i B′
k1...kc−1v

+k1 . . . v+kc−1 +

B′′
(ik1...kc)v

+k1 . . . v+kc .

On the physical states (5.17), (5.18) the Casimir
operator takes the value

C2 =T2 + αJ2 − (1 + α) I2 +
i

4
Qai′iQai′i =

α(1 + α)(c+ 1)2/4 . (5.21)

On the same states, the Casimir operators of the
bosonic subgroups SU(1, 1), SU(2)R and SU(2)L,

T2 = r0(r0 − 1) , J2 = j(j+1) , I2 = i(i+1) ,

take the values listed in the Table 1.
The fields B′

i and B′′
i form doublets of SU(2)R gen-

erated by Jik , whereas the component fields Ai′ =
(A1, A2) form a doublet of SU(2)L generated by I

i′k′
.

Each of Ai′ , B′
i, B′′

i carries a representation of
the SU(1,1) group. Basis functions of these rep-
resentations are eigenvectors of the generator R =
1
2

(
a−1K+ aH

)
, where a is a constant of the length

dimension. These eigenvalues are r = r0 + n, n ∈ N.

6 Outlook
In [19, 20, 21], we proposed a new gauge approach
to the construction of superconformal Calogero-type
systems. The characteristic features of this approach
are the presence of auxiliary supermultiplets with WZ
type actions, the built-in superconformal invariance
and the emergence of the Calogero coupling constant

as a strength of the FI term of the U(1) gauge (su-
per)field.
We see continuation of the researches presented in

the solution of some problems, such as
• An analysis of possible integrability properties of
new superCalogero models with finding-out a role
of the contribution of the center of mass in the
case of D(2, 1;α), α�=0, invariant systems.

• Construction of quantum N = 4 superconfor-
mal Calogero systems by canonical quantization
of systems (4.3) and (4.10).

• Obtaining the systems, constructed from mirror
supermultiplets and possessing D(2, 1;α) symme-
try, after use gauging procedures in bi-harmonic
superspace [26].

• Obtaining other superextensions of the Calogero
model distinct from the An−1 type (related to the
root system of the SU(n) group), by applying the
gauging procedure to other gauge groups.

Acknowledgement

I thank the Organizers of Jiri Niederle’s Fest and the
XVIII International Colloquium for the kind hospital-
ity in Prague. I would also like to thank my co-authors
E. Ivanov and O. Lechtenfeld for fruitful collaboration.
I acknowledge a support from the RFBR grants 08-02-
90490, 09-02-01209 and 09-01-93107 and grants of the
Heisenberg-Landau and the Votruba-Blokhintsev Pro-
grams.

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Sergey Fedoruk
E-mail: fedoruk@theor.jinr.ru
Bogoliubov Laboratory of Theoretical Physics, JINR
141980 Dubna, Moscow region, Russia

29