Superconformal Calogero models as a gauged matrix mechanics

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;\alpha) supersymmetries, as well as by the model of new N=4
superconformal quantum mechanics.

Full text

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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.
2

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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≈ 0for 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)
4