Superconformal Calogero models as a gauged matrix mechanics

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.


Introduction
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 a =b g/(x a − x b ) 2 , a, b = 1, ..., n. 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 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 nearhorizon limits of the extreme black hole solutions in M-theory correspond to AdS 2 geometry, 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 arbitrary 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.

Gauged formulation of Calogero model
The renowned Calogero system [1] can be described by the following action [18,22]: The action (2.1) is the action of U(n), d=1 gauge theory. The hermitian n×n-matrix field X b a (t), (X b a ) = X a b , a, b = 1, . . . , n and complex commuting U(n)-spinor field Z a (t),Z a = (Z a ) present the matter, scalar and spinor fields, respectively. The n 2 "gauge fields" A b a (t), (A b a ) = A a b are 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.
The action (2.1) is also invariant with respects to the local U(n) invariance 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), imply the primary constraints and give us the following expression for the canonical Hamiltonian where matrix quantity T is defined as The preservation of the constraints (2.5b) in time leads to the secondary constraints The gauge fields A play the role of the Lagrange multipliers for these constraints.
, we obtain the Poisson brackets of the constraints (2.5a) Dirac brackets for these second class constraints (2.5a) eliminates spinor momenta P Z ,P Z from the phase space. The Dirac brackets for the residual variables take the form (2.10) The residual constraints (2.8) T = T + form u(n) algebra with respect to the Dirac brackets and generate gauge transformations (2.3). Let us fix the gauges for these transformations.
In the notations the constraints (2.7) take the form The non-diagonal constraints (2.12) generate the transformations Therefore, in case of Calogero-like condition x a =x b , we can impose the gauge Then we introduce Dirac brackets for the constraints (2.12), (2.14) and eliminate x b a , p b a . In particular, the resolved expression for p b a is 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 Z aZ a − c ≈ 0 (no summation over a) (2.16) and generate local phase transformations of Z a . For these gauge transformations we impose the gauge The conditions (2.16) and (2.17) eliminate Z a andZ a completely. Finally, using the expressions (2.15) and the conditions (2.14), (2.16) we obtain the following expression for the Hamiltonian (2.6) (2. 19) 3 N = 2 superconformal Calogero model N = 2 supersymmetric generalization of the system (2.1) is described by [the supermultiplets (2, 2, 0)]; • commuting n 2 complex "bridge" superfields b c a (t, θ,θ). N = 2 superconformally invariant action of these superfields has the form Here the covariant derivatives of the superfield X are where the potentials are constructed from the bridges as The gauge superfield prepotential V b a (t, θ,θ), (V ) † = V , is constructed from the bridges as The superconformal boosts of the N = 2 superconformal group SU(1, 1|1) ≃ OSp(2|2) have the following realization: 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,θ) ∈ u(n), (τ ) + = τ ; • λ-transformations with complex chiral gauge parameters λ(t L , θ) ∈ u(n),λ(t R , θ) = (λ) + . These U(n) transformations act on the superfields in the action (3.1) as In terms of τ -invariant superfields V , Z and new hermitian (n × n)-matrix superfield the action (3.1) takes the form where the covariant derivatives of the superfield X are For gauge λ-transformations we impose WZ gauge Then, the action (3.10) takes the form 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 X b a , a =b, and all spinor fields Z a . Thus, the physical fields in our N = 2 supersymmetric generalization of the Calogero system are n bosons x a = X a a and 2n 2 fermions Ψ b a . These fields present on-shell content of n multiplets (1,2,1) and n 2 −n multiplets (0,2,2) which are obtained from n 2 multiplets (1,2,1) by gauging procedure [16]. We can present it by the plot: where the bosonic fields C a a , C b a and B b 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 2n 2 fermions as compared to the standard N = 2 superCalogero with 2n fermions constructed by Freedman and Mende [8].

The action (4.3) is invariant under the local U(n) transformations:
where λ b a (ζ, u ± ) ∈ u(n) is the 'hermitian' analytic matrix parameter, λ = λ. Using gauge freedom (4.5) we choose the WZ gauge 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 The fields x a are "diagonal" fields in X = X|. The fields Z i define first components in Z + , These constraints are generated by the equations of motion with respect to the diagonal components of gauge field A. Using Dirac brackets [Z a i , Z j b ] D = iδ a b δ j i , which are generated by the kinetic WZ term for Z, we find that the quantities S a for each a form u(2) algebras 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 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] δ mod X = 2i(θ kη k +θ k η k ) .
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 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 supermultiplet.

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 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 2) 3) The action (5.1) possesses D(2, 1; α) superconformal invariance. Using the Nöther procedure, we find the D(2, 1; α) generators. Quantum counterpart of them are 9) The symbol ... denotes Weyl ordering.
It can be directly checked that the generators (5.4)-(5.9) form the D(2, 1; α) superalgebra due to the quantum brackets (5.14) In (5.10)-(5.13) we use the notation To find the quantum spectrum, we make use of the realization for the bosonic operators where v + i is a commuting complex SU(2) spinor, as well as the following realization of the odd operators where ψ i are complex Grassmann variables. The full wave function Φ = A 1 + ψ i B i + ψ i ψ i A 2 is subjected to the constraints 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 + i of the degree c: On the physical states (5.17), (5.18) Casimir operator takes the value On the same states, the Casimir operators of the bosonic subgroups SU(1, 1), SU(2) R and SU(2) L , T 2 = r 0 (r 0 − 1) , J 2 = j(j + 1) , take the values listed in the Table The fields B ′ i and B ′′ i form doublets of SU(2) R generated by J ik , whereas the component fields A i ′ = (A 1 , A 2 ) form a doublet of SU(2) L generated by I i ′ k ′ .
Each of A i ′ , B ′ i , B ′′ i carries a representation of the SU(1,1) group. Basis functions of these representations are eigenvectors of the generator R = 1 2 (a −1 K + aH) , where a is a constant of the length dimension. These eigenvalues are r = r 0 + n, n ∈ N.

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 (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 findingout a role of the contribution of the center of mass in the case of D(2, 1; α), α =0, invariant systems.
• Obtaining other superextensions of the Calogero model distinct from the A n−1 type (related to the root system of SU(n) group), by applying the gauging procedure to other gauge groups.