Gauged spinning models with deformed supersymmetry

New models of the SU(2|1) supersymmetric mechanics based on gauging the systems with dynamical (1, 4, 3) and semi-dynamical (4, 4, 0) supermultiplets are presented. We propose a new version of SU(2|1) harmonic superspace approach which makes it possible to construct the Wess-Zumino term for interacting (4,4,0) multiplets. A new N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} extension of d = 1 Calogero-Moser multiparticle system is obtained by gauging the U(n) isometry of matrix SU(2|1) harmonic superfield model.


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The actions of the same type describe semi-dynamical degrees of freedom [14,15], the use of which proved to be of pivotal importance for constructing the many-particle supersymmetric d = 1 systems [13] (see also the review [16]). Additional important technical ingredients of the N = 4 model-building which essentially exploit the WZ type d = 1 actions are the pure gauge "topological" multiplet and the superfield gauging procedure relating diverse models [17,18]. In this paper we construct new models of the N = 4 deformed supersymmetric mechanics that make use of a few different types of SU(2|1) supermultiplets: dynamical, semi-dynamical and pure gauge supermultiplets. The outcome are new SU(2|1)-invariant one-particle model with spinning degrees of freedom, as well as new SU(2|1) superextension of the Calogero-Moser multi-particle system.
The harmonic superspace (1.8) is not directly applicable for tackling these tasks. The main problem roots in the algebra of the covariant constraints to be imposed on the relevant harmonic superfields Ψ for singling out various irreducible SU(2|1) multiplets. The Grassmann analyticity conditions in the harmonic superspace (1.8) (specifically, D + Ψ = 0, D + Ψ = 0) necessarily entail the harmonic condition (specifically, D ++ Ψ = 0). However, such harmonic constraints turn out to be too strong if we wish to describe some supermultiplets in the harmonic approach, e.g. the "topological" gauge multiplet which is the main object of the d = 1 gauging [17,18] efficiently exploited in refs. [13][14][15][16]. As we will see, the only way around is to pass to an extended SU(2|1) harmonic superspace involving two sets of harmonic variables: those associated with the group SU(2) int and those parametrizing the external automorphism group SU(2) ext .
In section 2 we introduce new harmonic superspace with two sets of harmonic variables, including the standard (unitary) harmonics on SU(2) ext . As a result, we gain an opportunity to perform a gauging procedure and define interacting dynamical and semi-dynamical multiplets. In section 3 we construct the system of dynamical (1, 4, 3) multiplet interacting with a semi-dynamical (4, 4, 0) multiplet. This coupling is used to define the WZ term for the (4, 4, 0) multiplet, which, as was noticed in [3], is impossible in the framework of the harmonic superspace (1.8). The gauging procedure relevant to this SU(2|1) invariant system is described in section 5. In section 6 we present a matrix generalization of the SU(2|1) invariant model with dynamical, semi-dynamical and pure gauge supermultiplets. When reduced on shell, it describes SU(2|1) supersymmetrization of the Calogero-Moser multi-particle system [19][20][21][22], with the mass specified by the deformation parameter m of the su(2|1) algebra. Section A contains the concluding remarks. In appendix we present the "master" SU(2|1) harmonic formalism which yields the settings developed in [3] and in section 2 of the present paper upon two different reductions with respect to the extra harmonic variables.

SU(2|1) harmonic superspace revisited
As opposed to the "minimal" harmonic coset (1.8), we will use now the coset
The relation with the standard SU(2|1) superspace coordinates is given by where w ± i are the non-unitary harmonics which define the "minimal" complex harmonic coset (1.8) and are related to the harmonics (2.2) as [24,25] The relations (2.4) imply [3] The fermionic SU(2|1) transformations induced by the left shifts of the coset representative (2.3) are written as where It follows from the transformations (2.7) that the SU(2|1) harmonic superspace contains an analytic harmonic subspace parametrized by the reduced coordinate set, The transformations (2.7) rewritten through harmonics w ± i defined in (2.5) take just the form given in [3] δt A = 2i η −θ+ −η − θ + , The extra coordinate z ++ transforms in this basis as Applying the routine coset techniques to the coset (2.1) (see, for example, [1]) we derive the following expressions for the covariant derivatives (2.14) The partial harmonic derivatives in these expressions are defined as 20) andF ,Ĩ 0 ,Ĩ ++ are matrix parts of the generators F , I 0 , I ++ properly acting on the matrix indices of the superfields and the operators. In particular, note the U(1) assignments

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which will be used below. Note the non-zero commutation relation Also, the notable property is The covariant derivatives act on the harmonic superfields Ψ (q) (t A , θ ± ,θ ± , u ± , z ++ ) = Ψ (q) (ζ H ) which are assumed to transform under SU(2|1) supersymmetry in accord with the general rules of the (super)coset realizations As usual, these superfields are eigenfunctions of the harmonic U(1) charge operator D 0 : (2.25) We treat the dependence of Ψ (q) (t A , θ ± ,θ ± , u ± , z ++ ) on two sorts of harmonic variables in the same way as in [25]. Namely, we assume the polynomial dependence on z ++ and the standard harmonic expansion in u ± [23].
In what follows we will mainly limit our study to the harmonic superfields subjected to some additional covariant conditions as well as the constraint The constraint (2.29) effectively eliminates the dependence of the harmonic superfields on the variable z ++ where Φ (q) satisfies the condition as a consequence of (2.25) and has the standard expansion in u ± . The superfield solution (2.30) can be rewritten as where w ± i and ζ H were defined in (2.5) and (1.8).

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The constraint (2.28) is the self-consistency condition for the covariant definition of the analytic SU(2|1) superfields which live on the analytic subspace (2.9). This definition amounts to the Grassmann-analyticity constraints which, due to the relation following from (2.15), necessarily imply (2.28). Similar to (2.32), the analytic harmonic superfields are expressed as (2.35) As opposed to the approach of ref. [3], the constraints (2.33) and (2.34) by no means require the condition D ++ Ψ (q) = 0. Of course the latter can be imposed as an independent additional constraint, but it is not necessitated now by the Grassmann analyticity conditions (2.33). The relationship between two alternative SU(2|1) harmonic approaches is explained in appendix.
The constraint (2.27) leads to some simplification of the expressions for other covariant derivatives. For example, on harmonic superfields obeying the constraints (2.25)-(2.29) the covariant derivative D ++ (2.18) takes the form The general transformation law (2.24) for the superfields subjected to the constraints (2.25)-(2.29) is simplified to the form One more comment concerns the possibility to use, along with the harmonic basis (u ± i , z ++ ), the basis (w ± i , z ++ ) with the non-unitary harmonics. Due to the relation (2.5), these two bases are equivalent to each other, while many formulas and constraints are simplified in the second basis. The dictionary between these bases is as follows For instance, in the (w ± i , z ++ ) basis the constraint (2.29) becomes just the condition of (2.40)

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Its SU(2|1) covariance immediately follows from the property δ ∂ ∂z ++ = 0 . Also, it is instructive to present the (w ± i , z ++ ) form of the pure harmonic part of the covariant derivative D ++ (2.18): In construction of the superfield particle actions we will need the expressions for the invariant integration measures over the full harmonic and the harmonic analytic superspaces [3]: and The multiplet (1, 4, 3) is described by the Grassmann-even real superfield X subjected to the conditions (2.25)-(2.29), and additional constraints Here, , because of (2.14) and (2.21).

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After passing to the central basis coordinates by (2.4), we observe that the θ expansion of the superfield (3.4) in the central basis takes the form [1] where the component fields The fermionic SU(2|1) transformations of component fields are the following (3.10) The free X-action reads Integrating in it over the θ-variables and harmonics, 2 we obtain the component action [1] Another description of the multiplet (1, 4, 3) is through an analytic real prepotential V(ζ A ) (D + V =D + V = 0). Its pregauge freedom can be exploited to show that V(ζ A ) describes just the multiplet (1, 4, 3) (by choosing the appropriate WZ gauge). The superfield V(ζ A ) is related to the superfield X in the central basis by the harmonic integral transform where the vertical bar means that the expressions ) should be substituted into the integrand. Then, from (3.14) we can identify the fields appearing in the WZ gauge for V with the fields in (3.4) The representation (3.14) generalizes the analogous transform in the "flat" non-deformed N =4 supersymmetric mechanics [14,15,17,18].
The fermionic SU(2|1) transformation of Z + is a particular case of the general transformation law (2.38), δZ + = m ǫ −θ+ +ǭ − θ + Z + . This property is harmless because V is not subject to any extra harmonic constraints. One can formally define D ++ V, and it is a covariant SU(2|1) analytic superfield living on the superspaceζA (2.9) and having a linear dependence on z ++ (in the (w ± i , z ++ ) basis).
The corresponding component action S WZ = dt L WZ with the component Lagrangian is invariant under the SU(2|1) transformations (3.10), (3.25).

Total action
Now we consider a system with the action given by the sum S X + S WZ . Making use of the component form of these actions defined in (3.12) and (3.27), eliminating the auxiliary fields φ,φ, ϕ,φ, N ik from this sum by their algebraic equations of motion ) and, finally, redefining z k → z k / √ x, we obtain

(3.29)
In contrast to the analogical model of the N = 4 supersymmetric mechanics [14,15], the action (3.29) contains mass term (oscillator term) for the component field x. But the spinning variables z i prove to be not restricted by any constraint besides the second class constraints produced by the first order kinetic term for these variables. As a result, the quantum spectrum of this composite model involves an infinite number of the states, like in its "flat" prototype. For getting the finite number of physical states it is necessary to impose an additional constraint which amounts to the gauging procedure described in the next section.

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4 Gauging of coupled dynamical multiplet (1,4,3) and semi-dynamical multiplet (4, 4, 0) The WZ action (3.26) and the total action S X + S WZ are invariant with respect to the global U(1) transformations Now we require local invariance of this action, with the parameter in (4.1) being promoted to an analytic superfield λ = λ(ζ A ) satisfying the conditions To secure this local symmetry in the considered system we introduce the Grassmanneven analytic gauge superfield V ++ , which satisfies the conditions and is defined up to the gauge transformations The gauge superfield V ++ covariantizes the derivative D ++ . As a result, the complex analytic superfield Z + ,Z + , instead of the constraints (3.21), gets subjected to the covariantized harmonic constraints We can also add to the total action the gauge-invariant Fayet-Iliopoulos (FI) term So, we will consider the action S = S X + S WZ + S FI . (4.8) Using the U(1) gauge freedom (4.5), (4.1) we can choose the WZ gauge  The solution of the constraint (4.6) in the WZ gauge (4.9) is

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where Plugging the expressions (4.11) and (3.15) into the action (3.26) and integrating there over θs and harmonics, we obtain the component form of the WZ action (4.13) The fermionic fields φ, ϕ are auxiliary. The action is invariant under the residual local U(1) transformations (and similar phase transformations of the fermionic fields). The total action (4.8) in the WZ gauge takes the following on-shell form (like in (3.29), we make the redefinition z k → z k / √ x) The last term in the bosonic action (4.16) produces first class constraintz k z k − c ≈ 0 restricting the quantum spectrum to a single supermultiplet.

Matrix model
Now we are going to generalize the model of the previous section to the U(n), d=1 gauge theory following the papers [13,16]. The matrix model to be constructed involves the following U(n) entities: • n 2 commuting superfields X a b = ( X b a ), a, b = 1, . . . , n, forming the hermitian n×nmatrix superfield X = (X b a ) in adjoint representation of U(n); • n commuting complex superfields Z + a forming the U(n) spinor Z + = (Z + a ),Z + = (Z +a ); • n 2 non-propagating "gauge superfields" The local U(n) transformations are given by where λ b a (ζ A ) ∈ u(n) is the "hermitian" analytic matrix parameter, λ = λ. The SU(2|1) supersymmetric matrix model with U(n) gauge symmetry is described by the action S matrix = S X + S WZ + S FI .

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The first term in (5.2), is the gauged action of the (1,4,3) multiplets. Now the superfields X = (X b a ) are subjected to the constraints (3.1) and whereas the second term, is a WZ action describing coupling of n commuting analytic superfields Z + a and the singlet U(1) part X 0 ≡ Tr (X). The real analytic superfield V 0 (ζ, w) is defined by the integral transform (3.14) for the trace part: The n multiplets (4, 4, 0) are described by the superfields Z + a defined by the constraints (3.19)- (3.21) in which the constraint D ++ Z + = 0 is gauge-covariantized: Using the gauge freedom (5.1) we can choose the WZ gauge where now A(t A ) is an n×n matrix field. In this gauge we have The solution to the constraints (3.1) and the constraints (5.4), (5.5) for matrix field X is similar to (5.5) and it is as follows:

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Here, 14) ) The quantities The solution of the constraints (3.19)-(3.21) with the covariantization (5.9) for U(n) spinor superfield Z + is similar to (4.11): where are covariant derivatives of U(n) spinor d=1 fields Z i a ,Z a i = (Z i a ). Inserting the expressions (5.12), (5.18) in the action (5.2) and eliminating the fields N ik , φ,φ, ϕ,φ by their equations of motion we obtain, in the WZ gauge, and (Z i Z k ) ≡Z a i Z ka , (∇Z k Z k ) ≡ ∇Z a k Z k a . Let us consider the bosonic limit of S matrix , i.e. the action (5.21). Using the residual gauge invariance of the action (5.21), X ′ = e iλ X e −iλ , Z ′k = e iλ Z k , A ′ = e iλ A e −iλ − i e iλ (∂ t e −iλ ), where λ b a (t) ∈ u(n) are ordinary d=1 gauge parameters, we can impose the gauge

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i.e. X b a = X a δ b a and X 0 = n a=1 X a . As a result of this, and after eliminating A b a , a = b, by the equations of motion, the action (5.21) takes the following form (instead of Z i a we introduce the new fields Z ′i a = (X 0 ) 1/2 Z i a and omit the primes on these fields), where we used the following notation: (5.25) and no sum over the repeated index a in (5.24) is assumed. The terms a A a a Z a k Z k a − c in (5.23) produce n constraints (for each index a) for the fields Z k a . The constraints (5.26) generate abelian gauge [U(1)] n symmetry, Z k a → e iϕa Z k a , with local parameters ϕ a (t). Due to the constraints (5.26), the fields Z k a describe n sets of the target harmonics. After quantization, these variables become purely internal (U(2)-spin) degrees of freedom. So, in the Hamiltonian approach, the kinetic WZ term for Z in (5.23) gives rise to the following Dirac brackets: With respect to these brackets the quantities (5.24) for each index a form u(2) algebras [(S a ) i j , (S b ) k l ] D = iδ ab δ l i (S a ) k j − δ j k (S a ) i l . (5.28) As a result, after quantization the variables Z k a describe n sets of fuzzy spheres. The action (5.23) contains a potential in the center-of-mass sector with the coordinate X 0 (last term in (5.23)). Modulo this extra potential, the bosonic limit of the system constructed is none other than the U(2)-spin Calogero-Moser model which is a massive generalization of the U(2)-spin Calogero model [26,27] in the formulation of [28][29][30].

Concluding remarks and outlook
In this paper, we proposed new models of SU(2|1) supersymmetric quantum mechanics as a deformation of the corresponding "flat" N = 4, d = 1 supersymmetric models. The characteristic features of these models is the use of different types of supermultiplets: dynamical, semi-dynamical and pure gauge ones. In considered models, dynamical multiplets JHEP11(2016)103 are the (1, 4, 3) ones. The prepotential superfield description of them has provided an opportunity to build the WZ action for the (4, 4, 0) multiplets and thereby to use the latter for describing semi-dynamical degrees of freedom. The SU(2|1) version of the superfield gauging procedure of refs. [17,18] involving the appropriate gauge multiplets allowed us to gauge away some of the dynamical and semi-dynamical fields on shell.
We have studied these new SU(2|1) supersymmetric mechanics models both in the one-particle case and in the multi-particle one. In the latter case the system is described off shell by the matrix theory with U(n) gauging. After elimination of auxiliary and pure gauge fields this matrix theory yields new N = 4 superextensions of the A n−1 Calogero-Moser model. The mass (frequency) of the physical states is defined by the deformation parameter of the SU(2|1) supersymmetry.
The N = 4 superextensions of the Calogero-Moser model play a crucial role in applying the multiparticle integrable Calogero-type systems to the black hole physics. As was argued in [31], N = 4 supersymmetric extension of the conformal Calogero model can provide a microscopic description of the extreme Reissner-Nordström black hole in the near-horizon limit. At the same time, the corresponding physical states are identified with the eigenstates of the Calogero-Moser Hamiltonian. The deformed N = 4 supersymmetric generalization of the Calogero-Moser system found here can shed more light on these issues. One can expect, e.g., that this new multiparticle SU(2|1) model exhibits a trigonometric realization of the d = 1 superconformal group D(2, 1; α) along the lines of refs. [32][33][34].
Finally, it is worth pointing out that we have obtained N = 4 supersymmetric extension of the A n−1 Calogero-Moser system by dealing with the matrix model with the U(n) gauging. Superextensions of the Calogero-Moser models corresponding to other root systems could presumably be obtained by choosing other gauge groups and/or representations for the matrix and WZ superfields.