Multiparticle $\mathcal{N}{=}\,8$ mechanics with $F(4)$ superconformal symmetry

We present a new multiparticle model of $\mathcal{N}{=}\,8$ mechanics with superconformal $F(4)$ symmetry. The system is constructed in terms of two matrix $\mathcal{N}{=}\,4$ multiplets. One of them is a bosonic matrix $({\bf 1, 4, 3})$ multiplet and another is a fermionic $({\bf 0, 4, 4})$ one. Off-diagonal bosonic components of the $({\bf 1, 4, 3})$ multiplet are chosen to take values in the flag manifold $\mathrm{U}(n)/[\mathrm{U}(1)]^n$ and they carry additional gauge symmetries. The explicit form of the $F(4)$ supersymmetry generators is found. We demonstrate that the $F(4)$ superalgebra constructed contains as subalgebras two different $D(2,1;\alpha\,{=}{-}1/3)$ superalgebras intersecting over the common $sl(2,\mathbb{R})\oplus su(2)$ subalgebra.


Introduction
The models of superconformal mechanics occupy a notable place in the study of the AdS/CFT correspondence in supersymmetric gauge theories. This is basically due to the fact that the one-dimensional conformal SL(2, R) symmetry naturally emerges as a symmetry of the near horizon geometries of the black-hole solutions of the appropriate supergravities.
The superconformal mechanics systems, pioneered in eighties in the papers [1,2,3], have been so far worked out mainly up to the case of N = 4, d = 1 extended supersymmetry (see, e.g., refs. [4,5,6,7,8,9] and the review [10]). The models with N = 8 superconformal symmetry were studied to much less extent [11,12,13,14]. 1 But it is just N =8 superconformal mechanics which is most important from the standpoint of the AdS/CFT correspondence (see a recent review [17] and references therein). Moreover, the important role in this context is played by the exceptional N =8 superconformal symmetry F (4) (see, for example, recent papers [18,19]). 2 The first example of N = 8 superconformal mechanics with F (4) supersymmetry was presented in [12]. The one-particle system considered there was underlain by an interaction of two N = 8 multiplets: the dynamical (1,4,3) and the semi-dynamical (0, 4, 4) ones. In the present paper we consider a matrix generalization of this system and, as an outcome, obtain a new model of the multiparticle N = 8 superconformal mechanics.
The matrix models are an efficient tool of constructing conformally invariant systems [22,23,24]. In particular, it was found in [25,26,27] that the matrix one-dimensional superfield models yield Calogero-like systems with N = 4 supersymmetries after exploiting the appropriate gauging procedure [28]. The physical bosonic degrees of freedom were described by the diagonal elements of dynamical bosonic matrix of the (1,4,3) multiplets. The off-diagonal entries of this matrix proved to represent the purely gauge degrees of freedom.
As opposed to the gauging approach of refs. [25,26,27], in this paper all bosonic variables including the off-diagonal components of the (1, 4, 3) matrix multiplet are treated as dynamical. These off-diagonal fields parametrize the target space of flags U(n) U 1 (1) ⊗ . . . ⊗ U n (1) . So, they can be interpreted as a kind of the target harmonics, while the corresponding part of the worldline action as that of supersymmetric d = 1 sigma model on such a manifold. 3 The plan of the paper is as follows. In Section 2 we present N = 4 harmonic superfield description of the matrix multiplets (1,4,3) and (0, 4, 4). Also we introduce N = 8 superconformally invariant interaction of these multiplet and find, by Noether procedure, the supercharges generating the relevant N = 8 conformal superalgebra. In Section 3 we split the matrix n 2 bosonic fields into the sets of n diagonal (radial) and n 2 −n non-diagonal (angular) ones. The latter fields are identified with the target U(n) U 1 (1) ⊗ . . . ⊗ U n (1) harmonics on which some additional gauge symmetries are realized. In Section 4 we eliminate auxiliary fields and pass to the physical variables. Then we fulfill the Hamiltonian analysis of the system. We find the corresponding Hamiltonian and the relevant set of constraints. With respect to 2 Superconformal coupling of the matrix multiplets (1,4,3) and (0, 4, 4) The powerful approach to constructing N = 4, d = 1 supersymmetric models and finding interrelations between them is N = 4, d = 1 harmonic formalism which was proposed in [6]. As compared to the description in the usual superspace with the coordinates z = (t, θ i ,θ i ), (θ i ) * =θ i and covariant derivatives the harmonic description involves additional commuting harmonic variables In the harmonic analytic basis half of the N =4 covariant spinor derivatives D ± = u ± i D i ,D ± = u ± iD i becomes short: This implies the existence of the harmonic analytic superfields defined on the analytic subspace of the full harmonic superspace: It is closed under both N = 4 supersymmetry and N = 4 superconformal symmetry. The integration measure in the harmonic analytic subspace is defined as dudζ (−2) = dudt A dθ + dθ + . An important tool of the formalism is the harmonic derivatives: (2.6) The harmonic derivative D ++ is distinguished in that it commutes with the spinor derivatives (2.4) and so preserves the analyticity.
Here we presented only the definitions of the basic notions which will be used below. The full description of the harmonic superspace approach to d = 1 models is given in ref. [6] (details of the harmonic formulation of the multiplets which will be considered in this paper can be found in [10,35]).

Bosonic matrix multiplet (1, 4, 3)
The off-shell n 2 multiplets (1, 4, These constraints are solved by ik) or, in the more detailed notation, The same constraints (2.10), being rewritten in the harmonic superspace, read The extra harmonic constraint guarantees the harmonic independence of M in the central basis.

Superconformal coupling
Proceeding from the description of the multiplet (1, 4, 3) through the analytic prepotential V, it is easy to construct its superconformal coupling to Ψ +A [28] S (M,Ψ) This action is superconformal at any α = 0 and it also respects the gauge invariance (2.13) as a consequence of the constraint (2.7). An analysis based on dimensionality and the Grassmann character of the superfields Ψ +A , Ψ −A = D −− Ψ +A shows that the coupling (2.22) is the only possible coupling of this fermionic multiplet to the multiplet (1,4,3), such that it preserves the canonical number of time derivatives in the component action (no more than two for bosons and no more than one for fermions).
It is easy to find the component-field representation of (2.22) The total action is the sum of the superconformal ( (2.24) The variation of the total component action (2.24) with respect to the transformations (2.8), (2.17) can be represented as the integral Using this property, the Noether N = 4 supercharges generating the ε-transformations are computed to be where P := {M,Ṁ} stands for the matrix momenta. Note that Tr The Noether charges associated with the odd η-transformations are It is worth noting that the supercharges (2.27) and (2.28) involve only physical fields and their momenta. The corresponding transformations of auxiliary fields can be found using their algebraic equations of motion.

Implicit N = 4, d = 1 supersymmetry
As was shown in [12], in the one-particle case (n = 1) the total action (2.24) is invariant with respect to extra implicit N = 4 supersymmetry transformations. The multiparticle (matrix) generalization of these transformations [37] reads where ξ iA are fermionic parameters. The superfield transformations (2.29) amount to the following ones for the component fields The Noether charges of this hidden supersymmetry are then easily computed to be In the next sections we will prove that the closure of the supersymmetry transformations (2.8), (2.17), (2.30) generated by the supercharges (2.27), (2.28), (2.33) is just N = 8 conformal superalgebra F (4) [38,39].

Harmonic variables in the Hermitian matrix model
The kinetic terms of bosonic and fermionic fields in the component actions (2.20) and (2.22) are not fully flat. In this section we extract, from the complete set of bosonic fields, n fields having flat kinetic terms. The residual n(n − 1) bosonic variables are described by a non-trivial d = 1 non-linear sigma models and admit a suggestive interpretation as the target harmonics. After the appropriate redefinitions, the kinetic terms of all fermionic fields will acquire the flat form.
The basic step in proving these assertions will be the spectral decomposition of the matrix M. The Hermitian matrix M = M † is unitarily diagonalizable and its eigen valuedecomposition takes the form (see, for example, [40]) In terms of the entries of U, with the real eigenvalues y α = (y α ) * . Thus, the components of the matrix M defined in (3.1) are expressed as In this paper we will consider the option with unequal eigenvalues of the matrix M (3.1), i.e. with y α = y β for all α, β.
Taking into account the diagonal form of the matrix Y and the decomposition (3.1), we observe that the components of the unitary matrix U are defined up to local [U (1) where ϑ α (t) (α = 1, . . . , n) are local real parameters. Thus, the matrix U is defined up to the right local transformations U → Uh , (3.8) where the matrix h is diagonal with the components e iϑα , α = 1, . . . , n. In a fixed gauge with respect to these right local shifts, the matrices U involve n 2 − n essential parameters and so parametrize the cosets U(n)/H with the abelian stability subgroups H = U 1 (1)⊗. . .⊗U n (1).
Then the variables u β a andū a β can be interpreted as the Similar harmonics were considered in [31] for the case n = 3 and in [41,42] for arbitrary n.
In the n = 2 case we face just the target space analogs of the standard SU(2) U(1) harmonics defined in (2.2) [43,44].
In accord with what has been said above, the U(n) transformations act on the indices a, b whereas the indices α, β are subject to the U 1 (1) ⊗ . . . ⊗ U n (1) transformations. The harmonics u β a ,ū a β play the role of the bridges connecting the quantities with different types of symmetry, U(n) and [U(1)] n .
Let us rewrite the total action (2.24) (the sum of the component actions (2.21) and (2.23)) in terms of the variables y α , u β a ,ū a β . The crucial role will be played by the relation [24] (3.10) The first term in the Lagrangian of the action (2.21) takes the form The remaining terms in the actions (2.21) and (2.23) can be simplified after introducing new fermionic matrix variables, and new bosonic matrix variables, The definitions (3.12), (3.13) imply the relations (3.14) Using them, the total action (2.24) can be rewritten in terms of the new variables (3.5), (3.3), (3.12), (3.13) as The fields A ∼ ik , F ∼ A ,F ∼ A are auxiliary. In the next section, before performing the Hamiltonian analysis, we will eliminate them and diagonalize the kinetic terms for the bosonic y-variables and for the fermionic ones.
produces the 4-fermionic terms. As the result, we obtain the total on-shell superconformal action in the form Redefining field variables as 5 we cast the action (4.2) in the more convenient form Here and the 4-fermionic term L (4−f ) reads The Lagrangian (4.5) contains flat kinetic terms for x-variables and fermions. Harmonics are dynamical in this model: their second order kinetic term is proportional to K 2 which is just the relevant target space sigma-model metric. In the one-particle case (n = 1), the action (4.4) is reduced to the on-shell action from ref. [12] and, at α = −1/3, to the action from ref. [35].

Hamiltonian and harmonic constraints
The Lagrangian (4.5) yields the following explicit expressions for the momenta: (4.11) The canonical Hamiltonian takes the form and Ω β α was defined in (4.7). We point out that only the components K α β at α = β were used in the calculation of the Hamiltonian (4.13). These quantities are expressed as

(4.15)
Remind that we consider the case with y α = y β for any α = β and so all ∆ − αβ in (4.15) are non-vanishing.
The expressions (4.10) produce the standard second class constraints for odd variables. After introducing Dirac brackets for them, odd momenta Π kα β ,Π k α β , Π kAα β are removed from the phase space. The non-vanishing canonical Dirac brackets for the residual variables (at equal times) are where ǫ 12 = ǫ 21 = 1. Taking into account (3.4), we note that in the expressions for the harmonic momenta It should be pointed out that the quantities D α appearing in (4.20) coincide with D α β from (4.14) at α=β: D α =D α α . The quantities D α β defined in (4.14) form u(n) algebra with respect to Dirac brackets (4.17), 23) and commute, in a weak sense, with the quantities G α β , defined in (4.21), The non-vanishing Dirac brackets of the constraints (4.21) and (4.22) are (4.26) Therefore, n constraints D α (4.20) are first class, whereas the constraints G α β (4.21) and g α β (4.22) form n 2 pairs of second class constraints. We take account of the second class constraints (4.21), (4.22) by introducing Dirac brackets for them: The new Dirac brackets for the u(n) generators (4.14) also retain the old form: In what follows, this property will be crucial for analyzing the superconformal symmetry.
where p α =ẋ α as in (4.9) and ∆ ± αβ were defined in (4.6). The generators of the superconformal boosts (2.28) are With taking into account the relations (3.1), (3.9), (3.12) and (4.3), the generators of the second N = 4 supersymmetry (2.33) acquire the following form in the new variables

N = 8 superalgebras in the n = 1 case
In the one-particle case the indices α, β take only one value and the harmonic variables are absent. The Hamiltonian (4.13) and the supercharges (5.1), (5.2), (5.3) in this case read x , (5.5) Using the n = 1 case form of (4.28), we arrive at the following Dirac brackets for the fermionic generators (5.5), (5.6), (5.7) 3 .

(5.10)
Here, the bosonic generators H (defined in (4.13)) and the generators Finally, defining the quantities Q µii ′ , T µν , I we find the closed superalgebra of the full set of generators: This is none other than the standard form of the superalgebra D(2, 1; α=−1/3).
3 . The quantities defined similarly to (5.19), are combined into the generators The crossing Dirac brackets among the supercharges of the two N = 4 supersymmetries are vanishing: The only non-vanishing Dirac brackets are where Using (5.29) and introducing the second two-rank spinor ψ kk ′ as Also, in terms of the quantities (5.47) the su(2) generators (5.12), (5.19) become The bosonic generators T µν form sl(2, R) algebra. The generators N (ij)k ′ A refer to the SO(7) SU(2) ⊗ SU(2) ⊗ SU (2) coset and, together with the su(2) ⊕ su(2) ⊕ su(2) generators J ij , AB , form just so(7) R-symmetry algebra 6 so (7) : These bosonic generators and T µν , together with the odd generators Q µii ′ , Q µiA , constitute F (4) superalgebra. We observe that the F (4) superalgebra obtained in this way has the following notable structure It includes two D(2, 1; α =−1/3) superalgebras with the common sl(2, R) and su(2) generators T µν and J ij and so can be treated as a closure of these two superalgebras.