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arXiv:1002.2920v1 [hep-th] 15 Feb 2010

Superconformal Calogero models

as a gauged matrix mechanics⋆

Sergey Fedoruk

Bogoliubov Laboratory of Theoretical Physics, JINR,

141980 Dubna, Moscow region, Russia

fedoruk@theor.jinr.ru

Abstract

We present basics of the gauged superfield approach to constructing N-superconformal

multi-particle Calogero-type systems developed in arXiv:0812.4276, arXiv:0905.4951 and

arXiv:0912.3508. This approach is illustrated by the multi-particle systems possessing

SU(1,1|1) and D(2,1;α) supersymmetries, as well as by the model of new N=4 super-

conformal quantum mechanics.

————————————

⋆Talk at the Conference “Selected Topics in Mathematical and Particle Physics”, In Honor of 70-th 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.

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1Introduction

The celebrated Calogero model [1] is a prime example of an integrable and exactly solvable

multi–particle system. It describes the system of n identical particles interacting through an

inverse-square pair potential?

tions 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 mechanics [4]. Conformal properties of the Calogero model and supersymmetric

generalizations of the latter give possibilities to apply them in black hole physics, since the near–

horizon limits of the extreme black hole solutions in M-theory correspond to AdS2geometry,

having the same SO(1,2) isometry group. The analysis of physical fermionic degrees of freedom

in the black hole solutions of four- and five-dimensional supergravities shows that related d=1

superconformal systems must possess N=4 supersymmetry [5, 6, 7].

Superconformal Calogero models with N=2 supersymmetry were considered in [8, 9] and

with N=4 supersymmetry in [10, 11, 12, 13, 14, 15]. Unfortunately, a consistent Lagrange

formulations for n-particle Calogero model with N=4 superconformal symmetry for any n is

still lacking.

Recently, we developed a universal approach to superconformal Calogero models for an arbi-

trary number of interacting particles, including the N=4 models. It is based on the superfield

gauging of some non-abelian isometries of the d=1 field theories [16].

Our gauge model involves three matrix superfields. One is a bosonic superfield in the adjoint

representation of U(n). It carries physical degrees of freedom of superCalogero system. The

second superfield is in the fundamental (spinor) representation of U(n), and it is an auxiliary

one and is described by Chern–Simons mechanical action [17, 18]. The third matrix superfield

accommodates gauge “topological” supermultiplet [16]. N-extended superconformal symmetry

plays a very important 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].

a?=bg/(xa−xb)2, a,b = 1,...,n. Calogero model and its generaliza-

2Gauged formulation of Calogero model

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

?

where

∇X =˙X + i[A,X],

∇Z =˙Z + iAZ

S0= dt

?

Tr(∇X∇X) +i

2(¯Z∇Z − ∇¯ZZ) + cTrA

?

,(2.1)

∇¯Z =˙¯Z − i¯ZA .

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

(Xb

the matter, scalar and spinor fields, respectively. The n2“gauge fields” Ab

non–propagating ones in d=1 gauge theory. The second term in the action (2.1) is the Wess–

Zumino (WZ) term, whereas the third term is the standard Fayet–Iliopoulos (FI) one.

a(t),

a) = Xa

b, a,b = 1,...,n and complex commuting U(n)-spinor field Za(t),¯Za= (Za) present

a(t), (Ab

a) = Aa

bare

1

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The action (2.1) is invariant under the d=1 conformal SO(1,2) transformations:

δt = α, δXb

a=1

2˙ αXb

a, δZa= 0, δAb

a= − ˙ αAb

a,(2.2)

where constrained parameter ∂3

rameters of SO(1,2).

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

tα = 0 contains three independent infinitesimal constant pa-

X → gXg†,Z → gZ,A → gAg†+ i˙ gg†,(2.3)

where g(τ) ∈ U(n).

Let us demonstrate, in Hamiltonian formalism, that the gauge model (2.1) is equivalent to

the standard Calogero system.

The definitions of the momenta, corresponding to the action (2.1),

PX= 2∇X ,PZ=i

2¯Z ,

¯PZ= −i

2Z ,PA= 0(2.4)

imply the primary constraints

a)G ≡ PZ−i

2¯Z ≈ 0,

¯G ≡¯PZ+i

2Z ≈ 0;b) PA≈ 0(2.5)

and give us the following expression for the canonical Hamiltonian

H =1

4Tr(PXPX) − Tr(AT), (2.6)

where matrix quantity T is defined as

T ≡ i[X,PX] − Z·¯Z + 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 multipliers for these constraints.

Using canonical Poisson brackets [Xb

the Poisson brackets of the constraints (2.5a)

a,PXd

c]P=δd

aδb

c, [Za,Pb

Z]P=δb

a, [¯Za,¯PZ b]P=δa

b, we obtain

[Ga,¯Gb]P= −iδa

b.(2.9)

Dirac brackets for these second class constraints (2.5a) eliminates spinor momenta PZ,¯PZfrom

the phase space. The Dirac brackets for the residual variables take the form

[Xb

a,PXd

c]D= δd

aδb

c,[Za,¯Zb]D= −iδb

a.(2.10)

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

[Tb

a,Td

c]D= i(δd

aTb

c− δb

cTd

a)(2.11)

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

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In the notations

xa≡ Xa

a,pa≡ PXa

a

(no summation over a);xb

a≡ Xb

a,pb

a≡ PXb

a

for a ?= b

the constraints (2.7) take the form

Tb

a= i(xa− xb)pb

a− i(pa− pb)xb

a+ i

?

c

(xc

apb

c− pc

axb

c) − Za¯Zb≈ 0 for a ?= b,(2.12)

Ta

a= i

?

c

(xc

apa

c− pc

axa

c) − Za¯Za+ c ≈ 0 (no summation over a). (2.13)

The non-diagonal constraints (2.12) generate the transformations

δxb

a= [xb

a,ǫa

bTa

b]D∼ i(xa− xb)ǫa

b.

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

xb

a≈ 0.(2.14)

Then we introduce Dirac brackets for the constraints (2.12), (2.14) and eliminate xb

particular, the resolved expression for pb

a, pb

a. In

ais

pb

a= −

i

(xa− xb)Za¯Zb.(2.15)

The Dirac brackets of residual variables coincide with Poisson ones due to the resolved form of

gauge fixing condition (2.14).

After gauge-fixing (2.14), the constraints (2.13) become

Za¯Za− c ≈ 0(no summation over a)(2.16)

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

the gauge

Za−¯Za≈ 0.

The conditions (2.16) and (2.17) eliminate Zaand¯Zacompletely.

Finally, using the expressions (2.15) and the conditions (2.14), (2.16) we obtain the following

expression for the Hamiltonian (2.6)

??

which corresponds to the standard Calogero action [1]

?

a

(2.17)

H0=1

4Tr(PXPX) =1

4

a

(pa)2+

?

a?=b

c2

(xa− xb)2

?

,(2.18)

S0=dt

??

˙ xa˙ xa−

?

a?=b

c2

4(xa− xb)2

?

.(2.19)

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3

N=2 superconformal Calogero model

N=2 supersymmetric generalization of the system (2.1) is described by

• the even hermitian (n × n)–matrix superfield Xb

[the supermultiplets (1,2,1)];

a(t,θ,¯θ), (X)+= X, a,b = 1,...,n

• commuting chiral U(n)–spinor superfield Za(tL,θ), ¯ Za(tR,¯θ) = (Za)+, tL,R = t ± iθ¯θ

[the supermultiplets (2,2,0)];

• commuting n2complex “bridge” superfields bc

a(t,θ,¯θ).

N=2 superconformally invariant action of these superfields has the form

?

Here the covariant derivatives of the superfield X are

S2= dtd2θ

?

Tr?¯DX DX?+1

2¯ Z e2VZ − cTrV

?

.(3.1)

DX = DX + i[A,X],

¯DX =¯DX + i[¯A,X], (3.2)

D = ∂θ+ i¯θ∂t,

¯D = −∂¯θ− iθ∂t,

{D,¯D} = −2i∂t,

where the potentials are constructed from the bridges as

A = −iei¯b(De−i¯b),

¯A = −ieib(¯De−ib)(¯b ≡ b+).(3.3)

The gauge superfield prepotential Vb

a(t,θ,¯θ), (V )†= V , is constructed from the bridges as

e2V= e−i¯beib.(3.4)

The superconformal boosts of the N=2 superconformal group SU(1,1|1) ≃ OSp(2|2) have

the following realization:

δt = −i(η¯θ + ¯ ηθ)t,δθ = η(t + iθ¯θ),δ¯θ = η(t − iθ¯θ),(3.5)

δX = −i(η¯θ + ¯ ηθ)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) transfor-

mations:

• τ–transformations with the hermitian (n × n)–matrix parameter τ(t,θ,¯θ) ∈ u(n), (τ)+= τ;

• λ–transformations with complex chiral gauge parameters λ(tL,θ) ∈ u(n),¯λ(tR,θ) = (λ)+.

These U(n) transformations act on the superfields in the action (3.1) as

eib′= eiτeibe−iλ,e2V′= ei¯λe2Ve−iλ,(3.7)

X′= eiτX e−iτ,

Z′= eiλZ ,

¯ Z′=¯ Z e−i¯λ.(3.8)

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

X = e−ibX ei¯b,

X′= eiλXe−i¯λ,(3.9)

the action (3.1) takes the form

S2=

?

dtd2θ

?

Tr?¯DXe2VDXe2V?+1

2¯Z e2VZ − cTrV

?

(3.10)

where the covariant derivatives of the superfield X are

DX = DX + e−2V(De2V)X,

¯DX =¯DX − Xe2V(¯De−2V).(3.11)

For gauge λ–transformations we impose WZ gauge

V (t,θ,¯θ) = −θ¯θA(t).

Then, the action (3.10) takes the form

S2= S0+ SΨ

2,SΨ

2= −iTr

?

dt(¯Ψ∇Ψ − ∇¯ΨΨ)

(3.12)

where Ψ = DX| and

∇Ψ =˙Ψ + i[A,Ψ],

∇¯Ψ =˙¯Ψ + i[A,¯Ψ].

The bosonic core in (3.12) exactly coincides with the Calogero action (2.19).

Exactly as in pure bosonic case, residual local U(n) invariance of the action (3.12) eliminates

the nondiagonal fields Xb

supersymmetric generalization of the Calogero system are n bosons xa= Xa

Ψb

which are obtained from n2multiplets (1,2,1) by gauging procedure [16]. We can present it

by the plot:

a, a?=b, and all spinor fields Za. Thus, the physical fields in our N=2

aand 2n2fermions

a. These fields present on–shell content of n multiplets (1,2,1) and n2−n multiplets (0,2,2)

Xa

?

a= (Xa

a,Ψa

??

a,Ca

a)

?

(1,2,1)multiplets

Xb

?

a= (Xb

a,Ψb

??

a,Cb

a), a?=b

?

(1,2,1)multiplets

⇓

gauging

⇓

Xa

?

a= (Xa

a,Ψa

??

aand Bb

a,Ca

a)

?

aare auxiliary components of the supermultiplets. Thus,

(1,2,1)multiplets

interactΩb

?

a= (Ψb

a,Bb

??

a,Cb

a), a?=b

?

(0,2,2)multiplets

where the bosonic fields Ca

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].

a, Cb

5

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4

N=4 superconformal Calogero model

The most natural formulation of N=4,d=1 superfield theories is achieved in the harmonic

superspace [23] parametrized by

(t,θi,¯θk,u±

i) ∼ (t,θ±,¯θ±,u±

i),θ±= θiu±

i,

¯θ±=¯θiu±

i, i,k = 1,2.

Commuting SU(2)-doublets u±

u+iu−

by

i are harmonic coordinates [24], subjected by the constraints

i= 1. The N=4 superconformally invariant harmonic analytic subspace is parametrized

(ζ,u) = (tA,θ+,¯θ+,u±

i),tA= t − i(θ+¯θ−+ θ−¯θ+).

The integration measures in these superspaces are µH= dudtd4θ and µ(−2)

N=4 supergauge theory related to our task is described by:

A

= dudζ(−2).

• hermitian matrix superfields X(t,θ±,¯θ±,u±

i) = (Xb

a) subjected to the constraints

D++X = 0,

D+D−X = 0,(D+¯D−+¯D+D−)X = 0(4.1)

[the multiplets (1,4,3)];

• analytic superfields Z+(ζ,u) = (Z+

a) subjected to the constraint

D++Z+= 0(4.2)

[the multiplets (4,4,0)];

• the gauge matrix connection V++(ζ,u) = (V++b

a).

In (4.1) and (4.2) covariant derivatives are defined by

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

D++Z+= D++Z++ iV++Z+.

Also D+= D+,¯D+=¯D+and the connections in D−,¯D−are expressed through derivatives

of V++.

The N=4 superconformal model is described by the action

?

The tilde in? Z+denotes ‘hermitian’ conjugation preserving analyticity [24, 23].

integral transform (X0≡ Tr(X))

?

The real number α?=0 in (4.3) coincides with the parameter of N=4 superconformal group

D(2,1;α) which is symmetry group of the action (4.3). Field transformations under supercon-

formal boosts are (see the coordinate transformations in [23, 16])

Sα?=0

4

= −

1

4(1+α)

µHTr?X−1/α?+1

2

?

µ(−2)

A

V0? Z+Z++i

2c

?

µ(−2)

A

TrV++.(4.3)

The unconstrained superfield V0(ζ,u) is a real analytic superfield, which is defined by the

X0(t,θi,¯θi) =duV0

?tA,θ+,¯θ+,u±????

θ±=θiu±

i,¯θ±=¯θiu±

i

.

δX = −Λ0X,δZ+= ΛZ+,δV++= 0,(4.4)

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where Λ = 2iα(¯ η−θ+− η−¯θ+), Λ0= 2Λ − D−−D++Λ. It is important that just the superfield

multiplier V0in the action provides this invariance due to δV0= −2ΛV0(note that δµ(−2)

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

A

= 0).

X′= eiλXe−iλ,

Z+′= eiλZ+,V++ ′= eiλV++e−iλ− ieiλ(D++e−iλ), (4.5)

where λb

freedom (4.5) we choose the WZ gauge

a(ζ,u±) ∈ u(n) is the ‘hermitian’ analytic matrix parameter,?λ = λ. Using gauge

V++= −2iθ+¯θ+A(tA).(4.6)

Considering the case α= −1

auxiliary and gauge fields, we find that the action (4.3) has the following bosonic limit

?

a

a

where

(Sa)ij≡¯Za

(ˆS)ij≡

2(when D(2,1;α) ≃ OSp(4|2)) in the WZ gauge and eliminating

Sα=−1/2

4,b

=dt

??

˙ xa˙ xa+i

2

?

(¯Za

k˙Zk

a−˙¯Za

kZk

a) +

?

a?=b

Tr(SaSb)

4(xa− xb)2−nTr(ˆSˆS)

2(X0)2

?

,(4.7)

iZj

a,

?

a

?(Sa)ij−1

2δj

i(Sa)kk?.

The fields xaare “diagonal” fields in X = X|. The fields Zidefine first components in Z+,

Z+| = Ziu+

i. They are subject to the constraints

¯Za

iZi

a= c

∀a. (4.8)

These constraints are generated by the equations of motion with respect to the diagonal com-

ponents of gauge field A.

Using Dirac brackets [¯Za

we find that the quantities Safor each a form u(2) algebras

i,Zj

b]D= iδa

bδj

i, which are generated by the kinetic WZ term for Z,

[(Sa)ij,(Sb)kl]D= iδab

?δl

i(Sa)kj− δj

k(Sa)il?.

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

kinetic term for the field X = X|.

For α=0 it is necessary to modify the transformation law of X in the following way [16]

2, the action (4.3) yields non–trivial sigma–model type

δmodX = 2i(θk¯ ηk+¯θkηk).(4.9)

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

?

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 super Calogero model consid-

ered 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 supermultiplet.

Sα=0

4

= −1

4

µHTr?eX?+1

2

?

µ(−2)

A

? Z+Z++i

2c

?

µ(−2)

A

TrV++.(4.10)

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5D(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 mechanics.

Choosing 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,

?

Sf

= −i

(5.1)

Sb =dt

?

?

dt

˙ x˙ x +i

2

?¯ zk˙ zk−˙¯ zkzk?−α2(¯ zkzk)2

?¯ψk˙ψk−˙¯ψkψk?

4x2

− A?¯ zkzk− c??

+2

, (5.2)

+ 2α

?

dtψi¯ψkz(i¯ zk)

x2

3(1 + 2α)

?

dtψi¯ψkψ(i¯ψk)

x2

.(5.3)

The action (5.1) possesses D(2,1;α) superconformal invariance. Using the N¨ other proce-

dure, we find the D(2,1;α) generators. Quantum counterpart of them are

Qi= PΨi+ 2iαZ(i¯Zk)Ψk

X

+ i(1 + 2α)?ΨkΨk¯Ψi?

X

,(5.4)

¯Qi= P¯Ψi− 2iαZ(i¯Zk)¯Ψk

X

+ i(1 + 2α)?¯Ψk¯ΨkΨi?

X

, (5.5)

Si= −2XΨi+ tQi,

−2αZ(i¯Zk)Ψ(i¯Ψk)

¯Si= −2X¯Ψi+ t¯Qi.

− (1+2α)?ΨiΨi ¯Ψk¯Ψk?

(5.6)

H =1

4P2+α2(¯ZkZk)2+ 2¯ZkZk

4X2

X2

2X2

+(1 + 2α)2

16X2

, (5.7)

K = X2− t1

2{X,P} + t2H,

I1′1′= −iΨkΨk,

D = −1

4{X,P} + tH,(5.8)

Jik= i?Z(i¯Zk)+ 2Ψ(i¯Ψk)?,

The symbol ?...? denotes Weyl ordering.

It can be directly checked that the generators (5.4)–(5.9) form the D(2,1;α) superalgebra

?

[Tab,Tcd] = −i?ǫacTbd+ ǫbdTac?,

I2′2′= i¯Ψk¯Ψk,I1′2′= −i

2[Ψk,¯Ψk]. (5.9)

{Qai′i,Qbk′k} = −2ǫikǫi′k′Tab+ αǫabǫi′k′Jik− (1 + α)ǫabǫikIi′k′?

,(5.10)

(5.11)

[Jij,Jkl] = −i?ǫikJjl+ ǫjlJik?,

[Tab,Qci′i] = iǫc(aQb)i′i,

[Ii′j′,Ik′l′] = −i(ǫikIj′l′+ ǫj′l′Ii′k′), (5.12)

[Jij,Qai′k] = iǫk(iQai′j),[Ji′j′,Qak′i] = iǫk′(i′Qaj′)i

(5.13)

due to the quantum brackets

[X,P] = i,[Zi,¯Zj] = δi

j,

{Ψi,¯Ψj} = −1

2δi

j.(5.14)

In (5.10)-(5.13) we use the notation

T22= H, T11= K, T12= −D.

To find the quantum spectrum, we make use of the realization

Q21′i= −Qi, Q22′i= −¯Qi,Q11′i= Si, Q12′i=¯Si,

¯Zi= v+

i,Zi= ∂/∂v+

i

(5.15)

8

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for the bosonic operators where v+

realization of the odd operators

iis a commuting complex SU(2) spinor, as well as the following

Ψi= ψi,

¯Ψi= −1

2∂/∂ψi,(5.16)

where ψiare complex Grassmann variables.

The full wave function Φ = A1+ ψiBi+ ψiψiA2is subjected to the constraints

¯ZiZiΦ = v+

i

∂

∂v+

i

Φ = cΦ. (5.17)

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 function Φ(v+) is a homogeneous

polynomial in v+

iof 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

= v+

iB′

k1...kc−1v+k1...v+kc−1+ B′′

(ik1...kc)v+k1...v+kc.(5.20)

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

C2= T2+ αJ2− (1 + α)I2+i

4Qai′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)Rand

SU(2)L,

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

take the values listed in the Table

r0

ji

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

2sign(α)

c

2−1

2

0

B′′(c)

k

(x,v+)

|α|(c+1)+1

2

+1

2sign(α)

c

2+1

2

0

The fields B′

Ai′ = (A1,A2) form a doublet of SU(2)Lgenerated by Ii′k′.

Each of Ai′, B′

representations are eigenvectors of the generator R =

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

iand B′′

iform doublets of SU(2)Rgenerated by Jik, whereas the component fields

i, B′′

icarries a representation of the SU(1,1) group. Basis functions of these

1

2(a−1K + aH), where a is a constant

9

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6Outlook

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 aux-

iliary 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

(super)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 superconformal Calogero systems by canonical quantiza-

tion of systems (4.3) and (4.10).

• Obtaining the systems, constructed from mirror supermultiplets and possessing D(2,1;α)

symmetry, after use gauging procedures in bi-harmonic superspace [26].

• Obtaining other superextensions of the Calogero model distinct from the An−1type (re-

lated to the root system of SU(n) group), by applying the gauging procedure to other

gauge groups.

Acknowledgements

I thank the Organizers of Jiri Niederle’s Fest and the XVIII International Colloquium for the

kind hospitality in Prague. I would also like to thank my co-authors E. Ivanov and O. Lechten-

feld for a 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

Programs.

References

[1] F.Calogero, J.Math.Phys. 10 (1969) 2191; 10 (1969) 2197.

[2] M.A. Olshanetsky, A.M. Perelomov, Phys. Rept. 71, 313 (1981); 94, 313 (1983).

[3] A.P.Polychronakos, J.Phys.A:Math.Gen. 39 (2006) 12793.

[4] V.deAlfaro, S.Fubini, G.Furlan, Nuovo Cim. A34 (1976) 569.

[5] P.Claus, M.Derix, R.Kallosh, J.Kumar, P.K.Townsend, A.VanProeyen,

Phys. Rev. Lett. 81 (1998) 4553.

[6] G.W.Gibbons, P.K.Townsend, Phys. Lett. B454 (1999) 187.

[7] J.Michelson, A.Strominger, Commun. Math. Phys. 213 (2000) 1; JHEP 9909 (1999) 005;

A.Maloney, M.Spradlin, A.Strominger, JHEP 0204 (2002) 003.

10

Page 12

[8] D.Z.Freedman, P.F.Mende, Nucl.Phys. B344, 317 (1990).

[9] L.Brink, T.H.Hansson, M.A.Vasiliev, Phys. Lett. B286 (1992) 109;

L.Brink, T.H.Hansson, S.Konstein, M.A.Vasiliev, Nucl. Phys. B401 (1993) 591.

[10] N.Wyllard, J.Math.Phys. 41 (2000) 2826.

[11] S.Bellucci, A.Galajinsky, S.Krivonos, Phys.Rev. D68 (2003) 064010.

[12] S.Bellucci, A.V.Galajinsky, E.Latini, Phys.Rev. D71 (2005) 044023.

[13] A.Galajinsky, O.Lechtenfeld, K.Polovnikov, Phys. Lett. B643 (2006) 221;

JHEP 0711 (2007) 008; JHEP 0903 (2009) 113.

[14] S.Bellucci, S.Krivonos, A.Sutulin, Nucl.Phys. B805 (2008) 24.

[15] S.Krivonos, O.Lechtenfeld, K.Polovnikov, Nucl. Phys. B817 (2009) 265.

[16] F.Delduc, E.Ivanov, Nucl. Phys. B753 (2006) 211, B770 (2007) 179.

[17] L.Faddeev, R.Jackiw, Phys. Rev. Lett. 60 (1988) 1692;

G.V.Dunne, R.Jackiw, C.A.Trugenberger, Phys. Rev. D41 (1990) 661;

F.Roberto, R.Percacci, E.Sezgin, Nucl. Phys. B322 (1989) 255;

P.S.Howe, P.K.Townsend, Class. Quant. Grav. 7 (1990) 1655.

[18] A.P. Polychronakos, Phys. Lett. B266, 29 (1991).

[19] S.Fedoruk, E.Ivanov, O.Lechtenfeld, Phys. Rev. D79 (2009) 105015.

[20] S.Fedoruk, E.Ivanov, O.Lechtenfeld, JHEP 0908 (2009) 081.

[21] S.Fedoruk, E.Ivanov, O.Lechtenfeld, New D(2,1;α) Mechanics with Spin Variables,

arXiv:0912.3508 [hep-th].

[22] A. Gorsky, N. Nekrasov, Nucl. Phys. B414 (1994) 213.

[23] E.Ivanov, O.Lechtenfeld, JHEP 0309 (2003) 073.

[24] A.S.Galperin, E.A.Ivanov, V.I.Ogievetsky, E.S.Sokatchev, Harmonic Superspace,

Cambridge Univ. Press, 2001.

[25] A.P. Polychronakos, JHEP 0104 (2001) 011;

B. Morariu, A.P. Polychronakos, JHEP 0107 (2001) 006; Phys. Rev. D72 (2005) 125002.

[26] E. Ivanov, J. Niederle, Phys. Rev. D80 (2009) 065027.

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