# Kähler geometry for $su(1,N|M)$-superconformal mechanics

Erik Khastyan<EMAIL_ADDRESS>Yerevan Physics Institute, 2 Alikhanian
Brothers St., Yerevan 0036 Armenia  Sergey Krivonos<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Russia  Armen Nersessian<EMAIL_ADDRESS>Yerevan Physics
Institute, 2 Alikhanian Brothers St., Yerevan 0036 Armenia  Bogoliubov
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna, Russia  Institute of Radiophysics and Electronics, Ashtarak-2, 0203,
Armenia

###### Abstract

We suggest the $su(1,N|M)$-superconformal mechanics formulated in terms of
phase superspace given by the non-compact analogue of complex projective
superspace.We parameterized this phase space by the specific coordinates
allowing to interpret it as a higher-dimensional super-analogue of the
Lobachevsky plane parameterized by lower half-plane (Klein model). Then we
introduced the canonical coordinates corresponding to the known separation of
the ”radial” and ”angular” parts of (super)conformal mechanics. Relating the
”angular” coordinates with action-angle variables we demonstrated that
proposed scheme allows to construct the $su(1,N|M)$ supeconformal extensions
of wide class of superintegrable systems. We also proposed the superintegrable
oscillator- and Coulomb- like systems with a $su(1,N|M)$ dynamical
superalgebra, and found that oscillator-like systems admit deformed
$\mathcal{N}=2M$ Poincaré supersymmetry, in contrast with Coulomb-like ones.

##  I Introduction

Kähler manifolds are the Hermitian manifolds which possesses the symplectic
structure obeying the specific compatibility condition with the Riemann
(and/or complex) structure arnold . Being highly common objects in almost all
areas of theoretical physics, these manifolds usually appear as configuration
spaces of the particles and fields. Only in a limited number of physical
problems they appear as phase spaces, mostly for the description of various
generalizations of tops, Hall effect (including its higher-dimensional
generalizatons, see, e.g. nair  and refs therein), etc. Respectively, the
number of the known nontrivial (super)integrable systems with Kähler phase
spaces is very restricted, and their study does not attract much attention.
The widely known integrable model with Kähler phase space extensively studying
nowadays is compactified Ruijsenaars-Schneider model with excluded center of
mass, whose phase space is complex projective space RS .

On the other hand, there are some indications that Kähler phase spaces can be
useful for the study of conventional Hamiltonian systems, i.e. for the systems
formulated on cotangent bundle of Riemann manifolds. A very simple example of
such system is one-dimensional conformal mechanics formulated in terms of
Lobachevsky plane (“noncompact complex projective plane”) treated as a phase
space lobach . Such description, being quite elegant, allows immediate
construction of $\mathcal{N}=2M$ superconformal extension associated with
$su(1,1|M)$ superalgebra. Recently the similar formulation of some higher-
dimensional systems was given in terms of $su(1,N)$-symmetric Kähler phase
space treated as the non-compact version of complex projective space kns . In
such approach all symmetries of the generic superintegrable conformal-
mechanical systems acquire interpretation in terms of the powers of the
$su(1,N)$ isometry generators. The maximally superintegrable generalizations
of the Euclidean oscillator/Coulomb systems has also been considered, all the
symmetries of these superintegrable systems were expressed via $su(1,N)$
isometry generators as well. However, the supersymmetrization aspects of that
system was not considered there at all. In the present paper we construct the
$\mathcal{N}$-extended superconformal extensions of the systems considered in
kns , as it was done in lobach  for one-dimensional case. Namely, we consider
the systems with $su(1,N|M)$-symmetric $(N|M)_{\mathbf{C}}$-dimensional Kähler
phase superspace (in what follow we denote it by
$\widetilde{\mathbb{CP}}^{N|M}$) and relate their symmetries with the isometry
generators of the super-Kähler structure. We construct this superspace
reducing the $(N+1|M)_{\mathbf{C}}$-dimensional complex pseudo-Euclidean
superspace by the $U(1)$-group action and then identify the reduced phase
superspace with noncompact analogue of complex projective superspace
constructed in jmp . We parameterize this superspace by the complex bosonic
variable $w$, ${\rm Im}\;w<0$, by the $N-1$ complex bosonic variables
$z^{\alpha}\in[0,\infty),\;{\rm arg}\;z\in[0;2\pi)$, and by $M$ complex
fermionic coordinates $\eta^{A}$. Thus, it can be considered as the
$N$-dimensional extension of the Klein model of Lobachevsky plane dnf . This
allows us to connect the complex coordinate $w$ with the radial coordinate and
momentum of the conformal-mechanical system spanned by $su(1,1)$ subalgebra,
and separate the $su(1,1)$ generators interpreting them as Hamiltonian,
conformal boosts and dilatation operators. The rest bosonic generators
$z^{\alpha}$ parameterize the angular part of integrable conformal mechanics
with Euclidean configuration spaces111The convenience of the separation of the
radial coordinates from the angular one in the study of conformal mechanics
and in their supersymmetrization was demonstrated, e.g., in angular . Relating
the angular coordinates and momenta with the action-angle variables, we
describe all symmetries of the generic superintegrable conformal-mechanical
systems in terms of the powers of the $su(1,N)$ isometry generators. An
important aspect of proposed approach is the choice of canonical coordinates
where all fermionic degrees of freedom appear only in the angular part of the
Hamiltonian.

Furthermore, we construct the super-analogues of the maximally superintegrable
generalizations of the Euclidean oscillator/Coulomb systems considered in kns
as follows: we preserve the form of Hamiltonian expressed via generators of
$su(1,1)$ subalgebra but extend the phase space $\widetilde{\mathbb{CP}}^{N}$
to phase superspace $\widetilde{\mathbb{CP}}^{N|M}$. As a result, we find that
these superextensions preserve all symmetries of the initial bosonic
Hamiltonians and possess maximal set of functionally-independent fermionic
integrals, i.e. they remains superintegrable in the sense of super-Liouville
theorem. We also find, that the constructed oscillator-like systems (in
contrast with Coulomb-like ones) possess deformed $\mathcal{N}=2M,d=1$
Poincaré supersymmetry (see ivanovsidorov , and express all the symmetries of
these superintegrable systems via $su(1,N)$ isometry generators as well.

The paper organized as follows.
In Section 2 we present the basic facts on Kähler supermanifolds and
construct, by the Hamiltonian reduction, the non-compact complex projective
superspace $\widetilde{\mathbb{CP}}^{N|M}$ in the parametrization similar to
those of Klein model. In Section 3 we analyze the symmetry algebra of
$\widetilde{\mathbb{CP}}^{N|M}$ and extract from it the
$su(1,N|M)$-superconformal systems. In Section 4 we introduce the canonical
coordinates which naturally split radial and angular parts of the Hamiltonian
and relate the angular part with the systems formulating in terms of action-
angle variables. In the Section 5 we construct superintegrable
supergeneralizations of oscillator- and Coulomb-like systems. In Section 6 we
represent the Kähler structure of phase superspace in the Fubini-Study-like
form. We conclude the paper by the outlook and final remarks in Section 7.

##  II  Noncompact complex projective superspace

The (even) $(N|M)$-dimensional Kähler supermanifold can be defined as a
complex supermanifold with symplectic structure given by the expression

$\Omega=\imath(-1)^{p_{I}(p_{J}+1)}g_{I{\bar{J}}}dZ^{I}\wedge
d{\bar{Z}}^{J},\quad d\Omega=0,$ (1)

with $Z^{I}$ denoting $N$ complex bosonic coordinates and $M$ complex
fermionic ones. The $p_{I}:=p(Z^{I})$ is Grassmanian parity of coordinate: it
is equal to zero for bosonic coordinate and to one for the fermionic one.
Through the paper we will use the following conjugation rule:
$\overline{Z^{I}Z^{J}}=\overline{Z}^{I}\overline{Z}^{J}$,
$\overline{\overline{Z}^{I}Z^{J}}={Z}^{I}\overline{Z}^{J}$,$\overline{\overline{Z}^{I}\overline{Z}^{J}}={Z}^{I}{Z}^{J}$,
for both bosonic and fermionic variables.

The “metrics components” $g_{I{\bar{J}}}$ can then be locally represented in
the form

$g_{I{\bar{J}}}=\frac{\partial^{L}}{\partial
Z^{I}}\frac{\partial^{R}}{\partial{\bar{Z}}^{J}}K(Z,{\bar{Z}}),$ (2)

where $\partial^{L(R)}/\partial Z^{I}$ denotes left(right) derivatives.

The Poisson brackets associated with this Kähler structure looks as follows

$\\{f,g\\}=\imath\left(\frac{\partial^{R}f}{\partial\bar{Z}^{I}}g^{{\bar{I}}J}\frac{\partial^{L}g}{\partial
Z^{J}}-(-1)^{p_{I}p_{J}}\frac{\partial^{R}f}{\partial
Z^{I}}g^{\bar{J}I}\frac{\partial^{L}g}{\partial\bar{Z}^{J}}\right),\quad{\rm
where}\quad
g^{{\bar{I}}J}g_{J{\bar{K}}}=\delta^{\bar{I}}_{\bar{K}},\quad\overline{g^{\bar{I}J}}=(-1)^{p_{I}p_{J}}g^{\bar{J}I},$
(3)

As in the pure bosonic case, the isometries of Kähler manifolds are given by
the holomorphic Hamiltonian vector fields,

$\mathbf{V_{\mu}}:=\\{h_{\mu}(Z,\bar{Z}),\quad\\}=V^{I}(Z)\frac{\partial^{L}}{\partial
Z^{I}}+{\bar{V}}^{I}({\bar{Z}})\frac{\partial^{L}}{\partial{\bar{Z}}^{I}},$
(4)

where $h_{\mu}(Z,\bar{Z})$ are real functions called Killing potentials (see,
e.g. lnp ; jmp  for the details).

Our goal is to study the systems on the Kähler phase space with $su(1,N|M)$
isometry superalgebra. For the construction of such phase space it is
convenient, at first, to present the linear realization of $u(1,N|M)$
superconformal algebra on the complex pseudo-Euclidean superspace
$\mathbb{C}^{1,N|M}$ equipped with the canonical Kähler structure (and thus,
by the canonical supersymplectic structure)and then reduced it by the action
of $U(1)$ generator.

It is instructive to present this reduction in details. Let us equip, at
first, the $(N+1|M)$-dimensional complex superspace with the canonical
symplectic structure

$\Omega_{0}=\imath\sum_{a,b=0}^{N}\gamma_{a\bar{b}}dv^{a}\wedge
d{\bar{v}}^{b}+\sum_{A=1}^{M}d\eta^{A}\wedge d{\bar{\eta}}^{A},$ (5)

with $v^{a},\bar{v}^{a}$ being bosonic variables, and
$\eta^{A},{\bar{\eta}}^{A}$ being fermionic ones, and with the matrix
$\gamma_{a\bar{b}}$ chosen in the form

$\gamma_{a\bar{b}}=\left(\begin{array}[]{c|c}\begin{matrix}0&-i\\\
i&0\end{matrix}&\\\ \hline\cr&\begin{matrix}-1&\\\ &\ddots&\\\
&&-1\end{matrix}\end{array}\right),\qquad a,b=N,0,1,...,N-1.$ (6)

With this supersymplectic structure we can associate the Poisson brackets
given by the relations

$\\{v^{a},{\bar{v}}^{b}\\}=-\imath\gamma^{\bar{b}a},\quad\\{\eta^{A},\bar{\eta}^{B}\\}=\\{\bar{\eta}^{B},\eta^{A}\\}=\delta^{A\bar{B}},\qquad\gamma^{\bar{a}b}\gamma_{b\bar{c}}=\delta^{\bar{a}}_{\bar{c}}.$
(7)

Equivalently,

$\\{v^{0},\bar{v}^{N}\\}=1,\qquad\\{v^{N},\bar{v}^{0}\\}=-1,\qquad\\{v^{\alpha},\bar{v}^{\beta}\\}=\imath\delta^{\alpha\bar{\beta}},\qquad\\{\eta^{A},\bar{\eta}^{B}\\}=\\{\bar{\eta}^{B},\eta^{A}\\}=\delta^{A\bar{B}}$
(8)

Here we introduced the indices $\alpha,\beta=1,\ldots,N-1$.

On this superspace we can define the linear Hamiltonian action of
$u(1,N|M)=u(1)\times su(1,N|M)$ superalgebra

$\displaystyle\\{h_{a\bar{b}},h_{c\bar{d}}\\}=-\imath\left(h_{a\bar{d}}\gamma^{\bar{c}b}-h_{c\bar{b}}\gamma^{\bar{a}d}\right),\quad\\{\Theta_{A\bar{a}},{\bar{\Theta}}_{\bar{B}b}\\}=h_{b\bar{a}}\delta^{B\bar{A}}-R_{A\bar{B}}\gamma^{\bar{b}a},\quad\\{\Theta_{A\bar{a}},h_{b\bar{c}}\\}=-\imath\Theta_{A\bar{c}}\gamma^{\bar{b}a},$
(9)
$\displaystyle\\{R_{A\bar{B}},R_{C\bar{D}}\\}=\imath\left(R_{A\bar{D}}\delta^{B\bar{C}}-R_{C\bar{B}}\delta^{D\bar{A}}\right),\quad\\{\Theta_{A\bar{a}},R_{C\bar{D}}\\}=-\imath\Theta_{C\bar{a}}\delta^{D\bar{A}},$
(10)

where

$h_{a\bar{b}}={\bar{v}}^{a}v^{b},\qquad\Theta_{A\bar{a}}=\bar{\eta}^{A}v^{a},\qquad
R_{A\bar{B}}=\imath\bar{\eta}^{A}\eta^{B}.$ (11)

The $u(1)$ generator defining the center of $u(N.1|M)$ is given by the
expression

$J=\gamma_{a\bar{b}}v^{a}{\bar{v}}^{b}+\imath\eta^{A}\bar{\eta}^{A}\;:\\{J,h_{a\bar{b}}\\}=\\{J,\Theta_{A\bar{a}}\\}=\\{J,R_{A\bar{B}}\\}=0.$
(12)

Hence, reducing the system by the action of this generator we will get the
”non-compact” projective super-space $\widetilde{\mathbb{CP}}^{N|M}$ (i.e. the
supergeneralization of noncompact projective space
$\widetilde{\mathbb{CP}}^{N}$), which is $(2N|2M)$-(real)dimensional space.

For performing the reduction by the action of generator (12) we have to
choose, at first, the $2N$ real ($N$ complex) bosonic and $2M$ real ($M$
complex) fermionic functions commuting with $J$. Then, we have to calculate
their Poisson brackets and restrict the latters to the level surface

$J=g.$ (13)

As a result we will get the Poisson brackets on the reduced $(2N|2M)$-(real)
dimensional space, with that $U(1)$-invariant functions playing the role of
the latter’s coordinates.

The required functions could be easily found as

$w=\frac{v^{N}}{v^{0}},\quad
z^{\alpha}=\frac{v^{\alpha}}{v^{0}},\quad\theta^{A}=\frac{\eta^{A}}{v^{0}}\;:\qquad\\{w,J\\}=\\{z^{a},J\\}=\\{\theta^{A},J\\}=0,\qquad{\rm
and}\quad c.c..$ (14)

Calculating their Poisson brackets and having in mind the expression following
from (13),

$A:=\left.\frac{1}{v^{0}\bar{v}^{0}}\right\rvert_{J=g}=\frac{1}{g}\left(\imath(w-\bar{w})-\sum_{\gamma=1}^{N-1}z^{\gamma}\bar{z}^{\gamma}+\imath\sum_{C=1}^{M}\theta^{C}\bar{\theta}^{C}\right),$
(15)

we get the reduced Poisson brackets defined by the following non-zero
relations (and their complex conjugates)

$\\{w,\bar{w}\\}=-A(w-\bar{w}),\quad\\{z^{\alpha},\bar{z}^{\beta}\\}=\imath
A\delta^{\alpha\bar{\beta}},\quad\\{\theta^{A},\bar{\theta}^{B}\\}=A\delta^{A\bar{B}},\quad\\{w,\bar{z}^{\alpha}\\}=A\bar{z}^{\alpha},\quad\\{w,\bar{\theta}^{A}\\}=A\bar{\theta}^{A}.$
(16)

These Poisson brackets are associated with the supersymplectic structure

$\displaystyle\Omega=\frac{\imath}{g}$
$\displaystyle\left[\frac{1}{A^{2}}dw\wedge d\bar{w}-\frac{\imath
z^{\alpha}}{A^{2}}dw\wedge d\bar{z}^{\alpha}-\frac{\theta^{A}}{A^{2}}dw\wedge
d\bar{\theta}^{A}\right.$
$\displaystyle+\frac{\imath\bar{z}^{\alpha}}{A^{2}}dz^{\alpha}\wedge
d\bar{w}+\left(\frac{g\delta_{\alpha\bar{\beta}}}{A}+\frac{\bar{z}^{\alpha}z^{\beta}}{A^{2}}\right)dz^{\alpha}\wedge
d\bar{z}^{\beta}-\frac{\imath\bar{z}^{\alpha}\theta^{A}}{A^{2}}dz^{\alpha}\wedge
d\bar{\theta}^{A}$
$\displaystyle-\frac{\bar{\theta}^{A}}{A^{2}}d\theta^{A}\wedge
d\bar{w}+\frac{\imath\bar{\theta}^{A}z^{\alpha}}{A^{2}}d\theta^{A}\wedge
d\bar{z}^{\alpha}-\left.\left(\frac{\imath
g\delta_{A\bar{B}}}{A}+\frac{\bar{\theta}^{A}\theta^{B}}{A^{2}}\right)d\theta^{A}\wedge
d\bar{\theta}^{B}\right].$ (17)

It is defined by the Kähler potential

${{\mathcal{K}}}=-g\log(\imath({{w}}-{\bar{w}})-z^{\alpha}{\bar{z}}^{\alpha}+\imath\theta^{A}\bar{\theta}^{A}).$
(18)

In what follows we will call this space “ noncompact projective superspace
$\widetilde{\mathbb{CP}}^{N|M}$ ”. The isometry algebra of this space is
$su(1,N|M)$, which can be easily obtained by the restriction of the generators
(9),(10) to the level surface (13). It is defined by the following Killing
potentials

$\displaystyle H:=v^{N}\bar{v}^{N}|_{J=g}=\frac{w\bar{w}}{A},$ $\displaystyle
K:=v^{0}\bar{v}^{0}|_{J=g}=\frac{1}{A},$ $\displaystyle
D:=(v^{N}\bar{v}^{0}+v^{0}\bar{v}^{N})|_{J=g}=\frac{w+\bar{w}}{A},$ (19)
$\displaystyle
H_{\alpha}:=\bar{v}^{\alpha}v^{N}|_{J=g}=\frac{\bar{z}^{\alpha}w}{A},$
$\displaystyle
K_{\alpha}:=\bar{v}^{\alpha}v^{0}|_{J=g}=\frac{\bar{z}^{\alpha}}{A},$
$\displaystyle
h_{\alpha\bar{\beta}}:=\bar{v}^{\alpha}v^{\beta}|_{J=g}=\frac{\bar{z}^{\alpha}z^{\beta}}{A},$
(20) $\displaystyle
Q_{A}:=\bar{\eta}^{A}v^{N}|_{J=g}=\frac{\bar{\theta}^{A}w}{A},$ $\displaystyle
S_{A}:=\bar{\eta}^{A}v^{0}|_{J=g}=\frac{\bar{\theta}^{A}}{A},$
$\displaystyle\Theta_{A\bar{\alpha}}:=\bar{\eta}^{A}v^{\alpha}|_{J=g}=\frac{\bar{\theta}^{A}z^{\alpha}}{A},$
(21) $\displaystyle
R_{A\bar{B}}:=\imath\bar{\eta}^{A}\eta^{B}|_{J=g}=\imath\frac{\bar{\theta}^{A}\theta^{B}}{A}.$
(22)

Constructed super-Kähler structure can be viewed as a higher dimensional
analogue of the Klein model of Lobachevsky space, where the latter is
parameterized by the lower half-plane. One can choose, instead of non-diagonal
matrix (6), the diagonal one, $\gamma_{a\bar{b}}=diag(1,-1,\ldots,-1)$. In
that case the reduced Kähler structure will have the Fubini-Study-like form
(see Section VI). In the next Section we will analyze the isometry algebra
defined by these generators in details. Presented choice (6) is motivated by
its convenience for the analyzing superconformal mechanics. Indeed, in that
case the generators (19) define conformal subalgebra $su(1,1)$ and are
separated from the rest $su(N,1)$ generators. Thus they can be interpreted as
the Hamiltonian of conformal mechanics, the generator of conformal boosts and
the generator of dilatation.

In the next section we will analyze in details these superconformal mechanics
and their dynamical defined by the generators (19),(20),(21),(22).

##  III $su(1,N|M)$ superconformal algebra

The generators (Killing potentials) (19),(20),(21),(22) form $su(1,N|M)$
superalgebra given by (9),(10) with $\gamma_{a\bar{b}}$ defined in (6). Its
explicit expression with separated $su(1,1)$ subalgebra is represented below.
For the convenience it is divided into three sectors: ”bosonic”, ”fermionic”
and ”mixed” ones.

#### Bosonic sector: $su(1,N)\times u(M)$ algebra

The bosonic sector is the direct product of the $su(1,N)$ algebra defined by
the generators (19),(20), and the $u(M)$ algebra defined by the R-symmetry
generators (22). Explicitly, the $su(1,N)$ algebra is given by the relations

$\displaystyle\\{H,K\\}=-D,\quad\\{H,D\\}=-2H,\quad\\{K,D\\}=2K,$ (23)
$\displaystyle\\{H,K_{\alpha}\\}=-H_{\alpha},\quad\\{H,H_{\alpha}\\}=\\{H,h_{\alpha\bar{\beta}}\\}=0,$
(24)
$\displaystyle\\{K,H_{\alpha}\\}=K_{\alpha},\quad\\{K,K_{\alpha}\\}=\\{K,h_{\alpha\bar{\beta}}\\}=0,$
(25)
$\displaystyle\\{D,{K}_{\alpha}\\}=-K_{\alpha},\quad\\{D,H_{\alpha}\\}=H_{\alpha},\quad\\{D,h_{\alpha\bar{\beta}}\\}=0,$
(26)
$\displaystyle\\{K_{\alpha},K_{\beta}\\}=\\{H_{\alpha},H_{\beta}\\}=\\{K_{\alpha},H_{\beta}\\}=0,$
(27) $\displaystyle\\{K_{\alpha},{\overline{K}}_{\beta}\\}=-\imath
K\delta_{\alpha\bar{\beta}},\quad\\{H_{\alpha},{\overline{H}}_{\beta}\\}=-\imath
H\delta_{\alpha\bar{\beta}},\quad\\{h_{\alpha\bar{\beta}},h_{\gamma\bar{\delta}}\\}=\imath(h_{\alpha\bar{\delta}}\delta_{\gamma\bar{\beta}}-h_{\gamma\bar{\beta}}\delta_{\alpha\bar{\delta}}),$
(28) $\displaystyle\\{K_{\alpha},h_{\beta\bar{\gamma}}\\}=-\imath
K_{\beta}\delta_{\alpha\bar{\gamma}},\quad\\{H_{\alpha},h_{\beta\bar{\gamma}}\\}=-\imath
H_{\beta}\delta_{\alpha\bar{\gamma}},\qquad\\{K_{\alpha},{\overline{H}}_{\beta}\\}=h_{\alpha\bar{\beta}}+\frac{1}{2}\left(I-\imath
D\right)\delta_{\alpha\bar{\beta}},$ (29)

where

$I:=g+\sum_{\gamma=1}^{N-1}h_{\gamma\bar{\gamma}}+\sum_{C=1}^{M}R_{C\bar{C}}$
(30)

The R-symmetry generators form $u(M)$ algebra and commutes with all generators
of $su(1,N)$:

$\\{R_{A\bar{B}},R_{C\bar{D}}\\}=\imath(R_{A\bar{D}}\delta_{C\bar{B}}-R_{C\bar{B}}\delta_{A\bar{D}}),\quad\\{R_{A\bar{B}},(H;K;D;K_{\alpha};H_{\alpha};h_{\alpha\bar{\beta}})\\}=0.$
(31)

It is clear that the generators $H,D,K$ form conformal algebra $su(1,1)$, the
generators $h_{\alpha\bar{\beta}}$ form the algebra $u(N-1)$, and all together
- the $su(1,1)\times u(N-1)$ algebra. Notice, that the generator $I$ in (30)
defines the Casimir of conformal algebra $su(1,1)$:

$\mathcal{I}:=\frac{1}{2}I^{2}=\frac{1}{2}D^{2}-2HK.$ (32)

Hence, choosing $H$ as a Hamiltonian, we get that $H_{\alpha}$,
$h_{\alpha\bar{\beta}},R_{A\bar{B}}$ define its constant of motion. Similarly,
choosing the generator $K$ as a Hamiltonian, we get that it has constants of
motion $K_{\alpha},h_{\alpha\bar{\beta}},R_{A\bar{B}}$.

#### ”Fermionic” sector

The Poisson brackets between fermionic generators (21) have the form

$\displaystyle\\{S_{A},{\overline{S}}_{B}\\}=K\delta_{A\bar{B}},\quad\\{Q_{A},{\overline{Q}}_{B}\\}=H\delta_{A\bar{B}},\quad\\{S_{A},{\overline{Q}}_{B}\\}=-\imath
R_{A\bar{B}}+\frac{\imath}{2}\left(I-\imath D\right)\delta_{A\bar{B}},$ (33)
$\displaystyle\\{\Theta_{A\bar{\alpha}},{\overline{\Theta}}_{B\bar{\beta}}\\}=R_{A\bar{B}}\delta_{\beta\bar{\alpha}}+h_{\beta\bar{\alpha}}\delta_{A\bar{B}},\quad\\{S_{A},{\overline{\Theta}}_{B\bar{\alpha}}\\}=K_{\alpha}\delta_{A\bar{B}},\quad\\{Q_{A},{\overline{\Theta}}_{B\bar{\alpha}}\\}=H_{\alpha}\delta_{A\bar{B}},$
(34)
$\displaystyle\\{S_{A},S_{B}\\}=\\{Q_{A},Q_{B}\\}=\\{\Theta_{A\bar{\alpha}},\Theta_{B\bar{\beta}}\\}=\\{S_{A},Q_{B}\\}=\\{S_{A},\Theta_{B\bar{\alpha}}\\}=\\{Q_{A},\Theta_{B\bar{\alpha}}\\}=0.$
(35)

Hence, the functions $Q_{A}$ play the role of supercharges for the Hamiltonian
$H$, and the functions $S_{A}$ define the supercharges of the Hamiltonian
given by the generator of conformal boosts $K$.

#### ”Mixed” sector

The mixed sector is given by the relations

$\displaystyle\\{H,Q_{A}\\}=\\{H,\Theta_{A\bar{\alpha}}\\}=0,\quad\\{H,S_{A}\\}=-Q_{A},$
(36)
$\displaystyle\\{K,S_{A}\\}=\\{K,\Theta_{A\bar{\alpha}}\\}=0,\quad\\{K,Q_{A}\\}=S_{A},$
(37)
$\displaystyle\\{D,S_{A}\\}=-S_{A},\quad\\{D,Q_{A}\\}=Q_{A},\quad\\{D,\Theta_{A\bar{\alpha}}\\}=0$
(38)
$\displaystyle\\{Q_{A},{\overline{K}}_{\alpha}\\}=-\Theta_{A\bar{\alpha}},\quad\\{Q_{A},H_{\alpha}\\}=\\{Q_{A},{\bar{H}}_{\alpha}\\}=\\{Q_{A},{\bar{K}}_{\alpha}\\}=\\{Q_{A},h_{\alpha\bar{\beta}}\\}=0,$
(39)
$\displaystyle\\{S_{A},{\overline{H}}_{\alpha}\\}=\Theta_{A\bar{\alpha}},\quad\\{S_{A},{K}_{\alpha}\\}=\\{S_{A},{\bar{K}}_{\alpha}\\}=\\{S_{A},{H}_{\alpha}\\}=\\{S_{A},h_{\alpha\bar{\beta}}\\}=0,$
(40) $\displaystyle\\{\Theta_{A\bar{\alpha}},K_{\beta}\\}=\imath
S_{A}\delta_{\beta\bar{\alpha}},\quad\\{\Theta_{A\bar{\alpha}},H_{\beta}\\}=\imath
Q_{A}\delta_{\beta\bar{\alpha}},\quad\\{\Theta_{A\bar{\alpha}},{\bar{H}}_{\alpha}\\}=\\{\Theta_{A\bar{\alpha}},{\bar{K}}_{\alpha}\\}=0,\quad\\{\Theta_{A\bar{\alpha}},h_{\beta\bar{\gamma}}\\}=\imath\Theta_{A\bar{\gamma}}\delta_{\beta\bar{\alpha}},$
(41) $\displaystyle\\{S_{A},R_{B\bar{C}}\\}=-\imath
S_{B}\delta_{A\bar{C}},\quad\\{Q_{A},R_{B\bar{C}}\\}=-\imath
Q_{B}\delta_{A\bar{C}},\quad\\{\Theta_{A\bar{\alpha}},R_{B\bar{C}}\\}=-\imath\Theta_{B\bar{\alpha}}\delta_{A\bar{C}}.$
(42)

Looking to the all Poisson bracket relations together we conclude that

  * •

The bosonic functions $H_{\alpha}$, $h_{\alpha\bar{\beta}}$ , and the
fermionic functions $Q_{A}$, $\Theta_{A\bar{\alpha}}$ commute with the
Hamiltonian $H$ and thus, provide it by the superintegrability property 222In
accord with super-analogue of Liouville theorem shander  the system on
$(2N.M)$ phase superspace is integrable iff it possess $N$ commuting bosonic
integrals (with nonvanishing and functionally independent bosonic parts) and
$M$ fermionic ones ;

  * •

The bosonic functions $K_{\alpha}$, $h_{\alpha\bar{\beta}}$ and the fermionic
functions $S_{A}$,$\Theta_{A\bar{\alpha}}$ commute with the generator $K$.
Hence, the Hamiltonian $K$ defines the superintegrable system as well.

  * •

The triples $(H,H_{\alpha},Q_{A},)$ and $(K,K_{\alpha},S_{A},)$ transform into
each other under the discrete transformation

$(w,z^{\alpha},\theta^{A})\to(-\frac{1}{w},\frac{z^{\alpha}}{w},\frac{\theta^{A}}{w})\quad\Rightarrow
D\to-D,\quad\left\\{\begin{array}[]{ccc}(H,H_{\alpha},Q_{A},)&\to&(K,-K_{\alpha},-S_{A}),\\\
(K,K_{\alpha},S_{A})&\to&(H,H_{\alpha},Q_{A},)\end{array}\right..$ (43)

  * •

The functions $h_{\alpha\bar{\beta}},\Theta_{A\bar{\alpha}}$ are invariant
under discrete transformation (43). Moreover, they appear to be constants of
motion both for $H$ and $K$. Hence, they remain to be constants of motion for
any Hamiltonian being the functions of $H,K$. In particular, adding to the
Hamiltonian $H$ the appropriate function of $K$, we get the superintegrable
oscillator- and Coulomb-like systems with dynamical superconformal symmetry
(see Section V) .

  * •

The superalgebra $su(1,N|M)$ admits 5-graded decomposition 5gr ; grading

$su(1,N|M)=\mathfrak{f}_{-2}\oplus\mathfrak{f}_{-1}\oplus\mathfrak{f}_{0}\oplus\mathfrak{f}_{+1}\oplus\mathfrak{f}_{+2}\qquad\textrm{with}\qquad\left[\mathfrak{f}_{i},\mathfrak{f}_{j}\right]\subseteq\mathfrak{f}_{i+j}\quad\textrm{for}\
i,j\in\left\\{-2,-1,0,1,2\right\\},$ (44)

where $\mathfrak{f}_{i}=0$ for $|i|>2$ is understood. The subset
$\mathfrak{f}_{0}$ includes the generators
$D,h_{\alpha\bar{\beta}},\Theta_{A\bar{\alpha}},{\overline{\Theta}}{A\bar{\alpha}},R_{A\bar{B}}$,
the subsets $\mathfrak{f}_{-2}$ and $\mathfrak{f}_{2}$ contain only generators
$H$ and $K$, respectively, while the subsets $\mathfrak{f}_{-1}$ and
$\mathfrak{f}_{1}$ contain the generators
$H_{\alpha},{\bar{H}}_{\alpha},Q_{A},{\bar{Q}}_{A}$ and
$K_{\alpha},{\bar{K}}_{\alpha},S_{A},{\bar{S}}_{A}$.

Let us conclude this section by the following remark. It is easy to see, that
the generator (30) commutes the generators $H,D,K,S_{A},Q_{A},R_{A\bar{B}}$.
Hence, these generators form superconformal algebra $su(1,1|M)$ with central
charge $\sqrt{2\mathcal{I}}$ (32) (being the Casimir of $su(1,1|M))$ as well)

$\displaystyle\\{H,K\\}=-D,\quad\\{H,D\\}=-2H,\quad\\{K,D\\}=2K,\quad\\{S_{A},{\overline{S}}_{B}\\}=K\delta_{A\bar{B}},\quad\\{Q_{A},{\overline{Q}}_{B}\\}=H\delta_{A\bar{B}},$
$\displaystyle\\{S_{A},{\overline{Q}}_{B}\\}=-\imath
R_{A\bar{B}}+\frac{\imath}{2}\left(\sqrt{2\mathcal{I}}-\imath
D\right)\delta_{A\bar{B}},$
$\displaystyle\\{H,S_{A}\\}=-Q_{A},\quad\\{K,Q_{A}\\}=S_{A},\quad\\{H,Q_{A}\\}=\\{K,S_{A}\\}=0,\quad\\{D,S_{A}\\}=-S_{A},\quad\\{D,Q_{A}\\}=Q_{A},$
$\displaystyle\\{R_{A\bar{B}},R_{C\bar{D}}\\}=\imath(R_{A\bar{D}}\delta_{C\bar{B}}-R_{C\bar{B}}\delta_{A\bar{D}}),\quad\\{S_{A},R_{B\bar{C}}\\}=-\imath
S_{B}\delta_{A\bar{C}},\quad\\{Q_{A},R_{B\bar{C}}\\}=-\imath
Q_{B}\delta_{A\bar{C}}.$ (45)

In the next section we will express presented $su(1,N|M)$ generators in
appropriate canonical coordinates and in this way we will relate presented
formulae with the superextensions of conventional conformal mechanics.

##  IV Canonical coordinates and action-angle variables

To define the canonical coordinates we pass from the complex bosonic
coordinates $w$, $z^{\alpha}$

$w=x+\imath y,\quad z^{\alpha}=q_{\alpha}{\rm
e}^{\imath\varphi_{\alpha}},\quad{\rm where}\qquad y<0,\quad q_{\alpha}\geq
0,\quad\varphi_{\alpha}\in[0,2\pi),\quad
q^{2}:=\sum_{\alpha=1}^{N-1}q_{\alpha}^{2}<-2y.$ (46)

Then we re-define fermionic ones such that the new variables will have
canonical Poisson brackets.

For this purpose we write down the symplectic/Kähler one-form and identify it
with the canonical one

$\mathcal{A}=-\frac{g}{2}\frac{dw+d\bar{w}-\imath(z^{\alpha}d\bar{z}^{\alpha}-\bar{z}^{\alpha}dz^{\alpha})+\theta^{A}d\bar{\theta}^{A}+\bar{\theta}^{A}d\theta^{A}}{\imath(w-\bar{w})-z^{\gamma}\bar{z}^{\gamma}+\imath\theta^{C}\bar{\theta}^{C}}:=p_{x}dx+\pi_{\alpha}d\varphi_{\alpha}+\frac{1}{2}\chi^{A}d\bar{\chi}^{A}+\frac{1}{2}\bar{\chi}^{A}d\chi^{A}$
(47)

After some calculations and canonical transformation
$(p_{x},x)\rightarrow(-\frac{r^{2}}{2},\frac{p_{r}}{r})$, one can obtain

$w=\frac{p_{r}}{r}-\imath\frac{I}{r^{2}},\quad
z^{\alpha}=\frac{\sqrt{2\pi_{\alpha}}}{r}{\rm
e}^{\imath\varphi_{\alpha}},\quad\theta^{A}=\frac{\sqrt{2}}{r}\chi^{A},$ (48)

where $r,p_{r},\pi_{\alpha},\varphi_{\alpha},\chi^{A},\bar{\chi}^{A}$ are
canonical coordinates.

$\\{r,p_{r}\\}=1,\quad\\{\varphi_{\beta},\pi_{\alpha}\\}=\delta_{\alpha\beta},\quad\\{\chi^{A},\bar{\chi}^{B}\\}=\delta^{A\bar{B}},\qquad\pi_{a}\geq
0,\quad\varphi^{a}\in[0,2\pi),\quad r>0.$ (49)

They expresses via initial ones as follows

$p_{r}=\frac{w+\bar{w}}{2}\sqrt{\frac{2}{A}},\quad
r=\sqrt{\frac{2}{A}},\quad\pi_{\alpha}=\frac{z^{\alpha}\bar{z}^{\alpha}}{A},\quad\varphi_{\alpha}=\arg(z^{\alpha}),\quad\chi^{A}=\frac{\theta^{A}}{\sqrt{A}},\quad
c.c.,$ (50)

where

$I=g+\sum_{\alpha=1}^{N-1}\pi_{\alpha}+\sum_{A=1}^{M}\imath\bar{\chi}^{A}\chi^{A}\;,\qquad
A:=\frac{\imath(w-\bar{w})-z^{\gamma}\bar{z}^{\gamma}+\imath\theta^{C}\bar{\theta}^{C}}{g}=\frac{2}{r^{2}}.$
(51)

In these canonical coordinates the isometry generators read

$\displaystyle H=\frac{p_{r}^{2}}{2}+\frac{I^{2}}{2r^{2}},\quad
K=\frac{r^{2}}{2},\quad D=p_{r}r,$ (52) $\displaystyle
H_{\alpha}=\sqrt{\frac{\pi_{\alpha}}{2}}{\rm
e}^{-\imath\varphi_{\alpha}}\left({p_{r}}-\imath\frac{I}{r}\right),\quad
K_{\alpha}=r\sqrt{\frac{\pi_{\alpha}}{2}}{\rm
e}^{-\imath\varphi_{\alpha}},\quad
h_{\alpha\bar{\beta}}=\sqrt{\pi_{\alpha}\pi_{\beta}}{\rm
e}^{-\imath(\varphi_{\alpha}-\varphi_{\beta})},$ (53) $\displaystyle
Q_{A}=\frac{\bar{\chi}^{A}}{\sqrt{2}}\left(p_{r}-\imath\frac{\sqrt{2\mathcal{I}}}{r}\right),\quad
S_{A}=\frac{\bar{\chi}^{A}}{\sqrt{2}}r,\quad\Theta_{A\bar{\alpha}}=\bar{\chi}^{A}\sqrt{\pi_{\alpha}}{\rm
e}^{\imath\varphi_{\alpha}},\quad R_{A\bar{B}}=\imath\bar{\chi}^{A}\chi^{B}.$
(54)

Interpreting $r$ as a radial coordinate, and $p_{r}$ as radial momentum, we
get the superconformal mechanics with angular Hamiltonian given by

$\mathcal{I}=\frac{I^{2}}{2}:=\frac{1}{2}\left(I_{0}+(\bar{\chi}\chi)\right)^{2},\qquad{\rm
with}\qquad
I_{0}:=g+\sum_{\alpha=1}^{N-1}\pi_{\alpha},\quad(\bar{\chi}\chi):=\sum_{A=1}^{M}\imath\bar{\chi}^{A}\chi^{A}\;.$
(55)

So, the fermionic part of superconformal Hamiltonian is encoded in its angular
part.

The explicit dependence of the Hamiltonian $H$ and the supercharges $Q_{A}$ on
the fermions is as follows

$H=H_{0}+\frac{I_{0}(\bar{\chi}\chi)}{r^{2}}+\frac{(\bar{\chi}\chi)^{2}}{2r^{2}},\qquad
Q_{A}=-\frac{\bar{\chi}^{A}}{\sqrt{2}}\left(p_{r}-\imath\frac{I_{0}}{r}-\imath\frac{(\bar{\chi}\chi)}{r}\right),\qquad$
(56)

while the dependence of bosonic integrals $H_{\alpha}$ on fermions is given by
the expression

$H_{\alpha}=H^{0}_{\alpha}-\frac{K_{\alpha}(\bar{\chi}\chi)}{2K},$ (57)

where

$H_{0}:=\frac{p^{2}_{r}}{2}+\frac{I^{2}_{0}}{2r^{2}},\quad
H^{0}_{\alpha}=\sqrt{\frac{\pi_{\alpha}}{2}}{\rm
e}^{-\imath\varphi_{\alpha}}\left({p_{r}}-\imath\frac{I_{0}}{r}\right)\;:\quad\\{H^{0}_{\alpha},H^{0}\\}=0.$
(58)

So, proposed superconformal Hamiltonian $H$ inherits all symmetries of initial
Hamiltonian $H_{0}$ (given by $H^{0}_{\alpha},h_{\alpha\bar{\beta}}$).

Looking at the functional dependence of the angular Hamiltonian $\mathcal{I}$
from the angular variables $\varphi^{\alpha},\pi_{\alpha}$ one can expect that
the set of conformal mechanics admitting proposed $su(1,N|M)$ superconformal
extensions seems to be very restricted. However, it is not the case, since we
it is not necessary to interpret $\varphi^{\alpha}$ as a coordinate of the
configuration space, and $\pi_{\alpha}$ as its canonically conjugated
momentum. Instead, since $\pi_{\alpha}$ define a constant of motion of the
bosonic Hamiltonian $H_{0}$ (and of the respective angular Hamiltonian
$\mathcal{I}_{0}=H_{0}K/2-D^{2}$), we can interpret it as the action variable
$I_{\alpha}$, and consider $\varphi^{\alpha}$ as a respective angle variable
$\Phi_{\alpha}$ .

Furthermore, suppose that $\pi_{\alpha},\varphi_{\alpha}$ are related with the
action-angle variables $(I_{\alpha},\Phi_{\alpha})$ of some
$(N-1)$-dimensional angular mechanics by the relations

$\pi_{\alpha}=n_{\alpha}I_{\alpha},\quad\varphi_{\alpha}=\frac{\Phi_{\alpha}}{n_{\alpha}},\qquad{\rm
where}\quad
n_{\alpha}\in\mathbb{N},\qquad\\{\Phi_{\alpha},I_{\beta}\\}=\delta_{\alpha\beta},\qquad\Phi_{\alpha}\in[0,2\pi).$
(59)

Upon this identification the bosonic part of the angular Hamiltonian (55)
takes a form

$\widetilde{\mathcal{I}}_{0}=\frac{1}{2}\left(g+\sum_{\alpha=1}^{N-1}n_{\alpha}I_{\alpha}\right)^{2},\qquad{\rm
with}\quad n_{\alpha}\;\in\;\mathbb{N},$ (60)

but the bosonic generators $H_{\alpha},S_{\alpha},h_{\alpha\bar{\beta}}$,
become locally defined, $\varphi_{\alpha}\;\in\;[0,2\pi/n_{\alpha})$, and fail
to be constants of motion. To get the globally defined bosonic generators we
have to take their relevant powers,

${\widetilde{H}}_{\alpha}:=(H_{\alpha})^{n_{\alpha}},\quad{\widetilde{K}}_{\alpha}:=(K_{\alpha})^{n_{\alpha}},\quad{\widetilde{h}}_{\alpha\bar{\beta}}:=(h_{\alpha\bar{\beta}})^{n_{\alpha}n_{\beta}}.$
(61)

as well as replace the fermionic generator $\Theta_{A\bar{\alpha}}$ by the
following one

$\widetilde{\Theta}_{A\bar{\alpha}}=(H_{\alpha})^{n_{\alpha}-1}\Theta_{A\bar{\alpha}}.$
(62)

As a result, the dynamical (super)symmetry algebra becomes nonlinear
deformation of $su(1,N|M)$ .

The angular Hamiltonian (60) defines the class the superintegrable
generalizations of the conformal mechanics, and of the oscillator- and
Coulomb-like systems on the $N$-dimensional Euclidean spaces rapid . As a
particular case, this class of systems includes the ”charge-monopole” system
monopole , Smorodinsky-Winternitz system sw (for the explicit expressions of
the action-angle variables of these systems see, respectively,saghatel  and
galajinsky ), as well as the rational Calogero models 333To our best
knowledge, action-angle variables for the angular part of the rational
Calogero models are not yet constructed explicitly. However, we have at hand
the spectrum of the angular part of rational Calogero model flp . Taking its
(semi)classical limit we can conclude that it has the form (60), see,
e.g.rapid . Thus, proposed systems can be considered as their $2M$
superconformal extensions.

Since the generators $Q_{A},S_{A},R_{A\bar{B}}$ remain unchanged upon above
identification (as well as the expression of the angular Hamiltonian (32) via
generators $H,K,D$), we conclude that listed generators form superconformal
algebra $su(1,1|N)$ with central charge (45).

Finally, notice that in (60) the nonzero constant $g\neq 0$ appears, and the
range of validity of the action variables is fixed to be
$I_{\alpha}\in[0,\infty)$. As a result, standard free particle and conformal
mechanics cannot be included in the proposed description, since for these
systems we should choose $g=0,I_{\alpha}\in[0,\infty)$. To exclude this
constant we should replace the initial generators by the following ones

$\mathcal{H}:=H-\frac{g(g-2I)}{4K},\qquad\mathcal{H}_{\alpha}:={H}_{\alpha}+\imath
g\frac{K_{\alpha}}{2K},\qquad\mathcal{Q}_{A}:=Q_{A}-\imath g\frac{S_{A}}{2K}.$
(63)

This deformation will further “non-linearize” the dynamical supersymmetry
algebra $su(1,N|M)$.

##  V Oscillator- and Coulomb-like Systems

In the previous section we mentioned that the angular Hamiltonian (60) defines
the superintegrable deformations of $N$-dimensional oscillator and Coulomb
system rapid , while in kns  the examples of such systems on noncompact
projective space $\widetilde{\mathbb{CP}}^{N}$ playing the role of phase space
were constructed. So, one can expect that on the phase superspace
$\widetilde{\mathbb{CP}}^{N|M}$ one can construct the super-counterparts of
that systems, which presumably, possess (deformed) $\mathcal{N}=2M,d=1$
Poincaré supersymmetry. Below we examine this question and show that our claim
is corrects in some particular cases.

###  V.1 Oscillator-like systems

We define the supersymmetric oscillator-like system by the the phase space
$\widetilde{\mathbb{CP}}^{N|M}$ (equipped with the Poisson brackets (16)) by
the Hamiltonian

$H_{osc}=H+\omega^{2}K,$ (64)

where the generators $H,K$ are given by (19). In canonical coordinates (50)it
reads

$H_{osc}=\frac{p_{r}^{2}}{2}+\frac{(g+\sum_{\alpha=1}^{N-1}\pi_{\alpha}+\sum_{A=1}^{M}\imath\bar{\chi}^{A}\chi^{A})^{2}}{r^{2}}+\frac{\omega^{2}r^{2}}{2}.$
(65)

This system possesses the $u(N)$ symmetry given by the generators
$h_{\alpha\bar{\beta}}$ defined in (20)(among them $N-1$ constants of motion
$\pi_{\alpha}$ are functionally independent), the $U(M)$ R-symmetry given by
the generators $R_{A\bar{B}}$ (22) as well as $N-1$ hidden symmetries given by
the generators

$M_{\alpha\beta}=(H_{\alpha}+\imath\omega K_{\alpha})(H_{\beta}-\imath\omega
K_{\beta})=\frac{\bar{z}^{\alpha}\bar{z}^{\beta}}{A^{2}}(w^{2}+\omega^{2})\;:\quad\\{H_{osc},M_{\alpha\beta}\\}=0,$
(66)

The generators (66) and the $su(N)$ generators $h_{\alpha\bar{\beta}}$ form
the following symmetry algebra

$\\{h_{\alpha\bar{\beta}},M_{\gamma\delta}\\}=\imath\left(M_{\alpha\delta}\delta_{\gamma\bar{\beta}}+M_{\gamma\alpha}\delta_{\delta\bar{\beta}}\right),\quad\\{M_{\alpha\beta},M_{\gamma\delta}\\}=0,$
(67)
$\\{M_{\alpha\beta},{\overline{M}}_{\gamma\delta}\\}=\imath\left(4\omega^{2}Ih_{\alpha\bar{\delta}}h_{\beta\bar{\gamma}}-\frac{M_{\alpha\beta}\bar{M}_{\gamma\delta}}{h_{\alpha\bar{\gamma}}}\delta_{\alpha\bar{\gamma}}-\frac{M_{\alpha\beta}\bar{M}_{\gamma\delta}}{h_{\alpha\bar{\delta}}}\delta_{\alpha\bar{\delta}}-\frac{M_{\alpha\beta}\bar{M}_{\gamma\delta}}{h_{\beta\bar{\gamma}}}\delta_{\beta\bar{\gamma}}-\frac{M_{\alpha\beta}\bar{M}_{\gamma\delta}}{h_{\beta\bar{\delta}}}\delta_{\beta\bar{\delta}}\right),$
(68)

with $I$ given by (30) and summation over repeated indices is not assumed.

Besides, this system has a fermionic constants of motion
$\Theta_{A\bar{\alpha}}$ defined in (21).Hence, it is superintegrable system
in the sense of super-Liouville theorem, i.e. it has $2N-1$ bosonic and $2M$
fermionic, functionally independent, constants of motion shander . Further
generalization to the systems with angular Hamiltonian (60) is
straightforward.

Let us show, that for the even $M=2k$ this system possess the deformed
$\mathcal{N}=2k$ Poincaré supersymmetry, in the sense of papers ivanovsidorov
. For this purpose we choose the following Ansatz for supercharges

$\mathcal{Q}_{A}=Q_{A}+\omega C_{AB}{\bar{S}}_{B},$ (69)

with the constant matrix $C_{AB}$ obeying the conditions

$C_{AB}+C_{BA}=0,\qquad C_{AB}{\overline{C}}_{BD}=-\delta_{A\bar{D}}$ (70)

For sure, the condition (70) assumes that $M$ is an even number, $M=2k$.

Calculating Poisson brackets of the functions (69) we get

$\\{\mathcal{Q}_{A},\bar{\mathcal{Q}}_{B}\\}=H_{osc}\delta_{AB},\qquad\\{\mathcal{Q}_{A},\mathcal{Q}_{B}\\}=-\imath\omega\mathcal{G}_{AB},\qquad\\{\bar{\mathcal{Q}}_{A},\bar{\mathcal{Q}}_{B}\\}=\imath\omega\bar{\mathcal{G}}_{AB},$
(71)

where

$\mathcal{G}_{AB}:=C_{AC}R_{B\bar{C}}+C_{BC}R_{A\bar{C}},\qquad\mathcal{G}_{\bar{A}\bar{B}}:=\bar{\mathcal{G}}_{AB}=\bar{C}_{AC}R_{C\bar{B}}+\bar{C}_{BC}R_{C\bar{A}},\qquad\bar{\mathcal{G}}_{AB}=\bar{C}_{AC}\bar{C}_{DB}\mathcal{G}_{DC}.$
(72)

Then we get that the algebra of generators $\mathcal{Q}_{A}$,
$\mathcal{H}_{osc}$, $\mathcal{R}_{A}^{B}$ is closed indeed:

$\displaystyle\\{\mathcal{Q}_{A},H_{osc}\\}=\omega
C_{AB}\mathcal{Q}_{B},\qquad\\{\mathcal{G}_{AB},H_{osc}\\}=0,$ (73)
$\displaystyle\\{\mathcal{Q}_{A},\mathcal{G}_{BC}\\}=\imath(C_{AB}\mathcal{Q}_{C}+C_{AC}\mathcal{Q}_{B}),\qquad\\{\mathcal{Q}_{A},\bar{\mathcal{G}}_{BC}\\}=-\imath(\bar{C}_{BD}\mathcal{Q}_{D}\delta_{A\bar{C}}+\bar{C}_{CD}\mathcal{Q}_{D}\delta_{A\bar{B}}).$
(74)

Hence, for the $M=2k$ the above oscillator-like system (64) possesses deformed
$\mathcal{N}=4k$ supersymmetry. In the particular case $M=2$ the choice of the
matrix $C_{AB}$ is unique(up to unessential phase factor): $C_{AB}:={\rm
e}^{\kappa}\varepsilon_{AB}$. In that case the above relations define the
superalgebra $su(1|2)$-deformation of $\mathcal{N}=4$ Poincaré supersymmetric
mechanics studied in details in ivanovsidorov . For the $k\geq 2$ the choice
of matrices $C_{AB}$ is not unique, and we get the family of deformed
$\mathcal{N}=4k$ Poincaré supersymmetric mechanics.

Let us present other deformed $\mathcal{N}=2M$ Poincaré supersymmetric systems
whose bosonic part is different from those of (64) but nevertheless, has the
oscillator potential.

For this purpose we choose another Ansatz for supercharges (in contrast with
previous case $M$ is not restricted to be even number)

$\widetilde{\mathcal{Q}}_{A}=Q_{A}+\imath\omega S_{A}.$ (75)

These supercharges generates the $su(1|M)$ superalgebra, and thus generalizes
the systems considered in ivanovsidorov  to arbitrary $M$,

$\displaystyle\\{\widetilde{\mathcal{Q}}_{A},\bar{\widetilde{{\mathcal{Q}}}}_{B}\\}=\mathcal{H}_{osc}\delta_{AB}-\omega\mathcal{R}^{A}_{B},\qquad\\{\widetilde{\mathcal{Q}}_{A},\widetilde{\mathcal{Q}}_{B}\\}=0,\qquad\\{\mathcal{R}_{A}^{\;B},\mathcal{R}_{C}^{\;D}\\}=\imath(\mathcal{R}_{A}^{\;D}\delta^{B}_{C}-\mathcal{R}_{C}^{\;B}\delta^{D}_{A})$
(76)
$\displaystyle\\{\widetilde{\mathcal{Q}}_{A},\mathcal{R}^{C}_{B}\\}=\imath\left(\frac{1}{M}\widetilde{\mathcal{Q}}_{A}\delta_{B\bar{C}}+\widetilde{\mathcal{Q}}_{B}\delta_{A\bar{C}}\right),\qquad\\{\widetilde{\mathcal{Q}}_{A},\mathcal{H}_{osc}\\}=\imath\omega\frac{2M-1}{M}\widetilde{\mathcal{Q}}_{A},$
(77)

where

$\mathcal{H}_{osc}:=H_{osc}-\omega(I+\frac{1}{M}\sum_{C}R_{C\bar{C}}),\qquad\mathcal{R}_{A}^{\;B}:=R_{A\bar{B}}-\frac{1}{M}\delta_{A}^{B}\sum_{C}R_{C\bar{C}}$
(78)

with $I$ defined by (30). Hence, the Hamiltonian get the additional bosonic
term proportional to the Casimir of conformal group. In canonical coordinates
(50) it reads

$\mathcal{H}_{osc}=\frac{p_{r}^{2}}{2}+\frac{\mathcal{I}}{r^{2}}+\frac{\omega^{2}r^{2}}{2}-\omega\left(\sqrt{2\mathcal{I}}+\frac{1}{M}(\bar{\chi}\chi)\right).$
(79)

This Hamiltonian, seemingly, describes the oscillator-like systems specified
by the presence of external magnetic field.

So, choosing $\widetilde{\mathbb{CP}}^{N|M}$ as a phase superspace, we can
easily construct superintegrable oscillator-like systems which possess
deformed $\mathcal{N}=2M,d=1$ Poincaré supersymmetry.

###  V.2 Coulomb-like systems

Now, let us construct on the phase space $\widetilde{\mathbb{CP}}^{N|M}$ with
the Poisson bracket relations (16), the Coulomb-like system given by the
Hamiltonian

$H_{Coul}=H+\frac{\gamma}{\sqrt{2K}},$ (80)

where the generators $H,K$ are defined by (19).

The bosonic constants of motion of this system are given by the $u(N-1)$
symmetry generators $h_{\alpha\beta}$ , and by the $N-1$ additional constants
of motion

$R_{\alpha}=H_{\alpha}+\imath\gamma\frac{K_{\alpha}}{I\sqrt{2K}}\;:\quad\\{H_{Coul},R_{\alpha}\\}=\\{H_{Coul},h_{\alpha\bar{\beta}}\\}=0,$
(81)

where $H_{\alpha},K_{\alpha},\eta_{\alpha\bar{\beta}}$ are defined by (20).
These generators form the algebra

$\\{R_{\alpha},\bar{R}_{\bar{\beta}}\\}=-\imath\delta_{\alpha\bar{\beta}}\Bigg{(}H_{Coul}-\frac{\imath\gamma^{2}}{2I^{2}}\Bigg{)}+\frac{\imath\gamma^{2}h_{\alpha\bar{\beta}}}{2I^{3}},\quad\\{h_{\alpha\bar{\beta}},R_{\gamma}\\}=\imath\delta_{\gamma\bar{\beta}}R_{\alpha},\quad\\{R_{\alpha},R_{\beta}\\}=0.$
(82)

Besides, proposed system has $2M$ fermionic constants of motion given by
$\Theta_{A\bar{\alpha}}$, and $u(M)$ R-symmetry given by $R_{A\bar{B}}$.
Hence, it is superintegrable in the sense of super-Liouville theorem shander .
So, we constructed the maximally superintegrable Coulomb problem with
dynamical $SU(1,N|M)$ superconformal symmetry which inherits all symmetries of
initial bosonic system.

One can expect, that in analogy with oscillator-like system, our Coulomb-like
system would possess (deformed) $\mathcal{N}=2M$-super-Poincaré symmetry for
$M=2k$ and $\gamma>1$. However, it is not a case.

Indeed, let us choose the following Ansatz for supercharges

$\mathcal{Q}_{A}=Q_{A}+\sqrt{2\gamma}C_{AB}\frac{{\bar{S}}_{B}}{(2K)^{3/4}},$
(83)

with the constant matrix $C_{AB}$ obeying the conditions (70), $M=2k$ and
$\gamma>0$.

Calculating their Poisson brackets we find

$\displaystyle\\{\mathcal{Q}_{A},\bar{\mathcal{Q}}_{B}\\}=H_{Coul}\delta_{A\bar{B}}+\frac{3}{2}\frac{\sqrt{2\gamma}}{(2K)^{7/4}}\left(S_{A}\bar{C}_{BD}S_{D}+\bar{S}_{B}C_{AD}\bar{S}_{D}\right),$
(84)
$\displaystyle\\{\mathcal{Q}_{A},\mathcal{Q}_{B}\\}=-\frac{\imath\sqrt{2\gamma}}{2(2K)^{3/4}}(C_{BD}\mathcal{R}_{A}^{D}+C_{AC}\mathcal{R}_{B}^{D}),\quad\\{\mathcal{Q}_{A},\mathcal{R}_{B}^{\;C}\\}=-\imath\mathcal{Q}_{B}\delta_{A\bar{C}},$
(85)

where $\mathcal{R}^{A}_{B}$ is defined in (78).

Further calculating the Poisson brackets of $\mathcal{Q}_{A}$ with the
generators appearing in the r.h.s. of the above expressions we get that the
superalgebra is not closed. For example,

$\\{\mathcal{Q}_{A},H_{Coul}\\}=\frac{3\gamma}{(2K)^{3/2}}S_{A}+\frac{\sqrt{2\gamma}}{(2K)^{3/4}}C_{AB}\left(\mathcal{\bar{Q}}_{B}-\frac{3}{4K}\bar{S}_{B}D\right).$
(86)

Hence, proposed supercharges do not yield closed deformation of
$\mathcal{N}=2M$-super-Poincaré algebra.

Let us choose another Ansatz for supercharges (as above we assume that
$\gamma>0$)

$\widetilde{\mathcal{Q}}_{A}=Q_{A}+\imath\sqrt{2\gamma}e^{\imath\frac{\pi}{2}}\frac{S_{A}}{(2K)^{3/4}},$
(87)

which yields

$\\{\widetilde{\mathcal{Q}}_{A},\bar{\widetilde{\mathcal{Q}}}_{B}\\}=\mathcal{H}_{Coul}\delta_{A\bar{B}}+\frac{\sqrt{2\gamma}}{2(2K)^{3/4}}\mathcal{R}^{B}_{A},\qquad\\{\widetilde{\mathcal{Q}}_{A},\widetilde{\mathcal{Q}}_{B}\\}=0,\qquad\\{\widetilde{\mathcal{Q}}_{A},\mathcal{R}^{C}_{B}\\}=\imath\left(\frac{1}{M}\widetilde{\mathcal{Q}}_{A}\delta_{B\bar{C}}-\widetilde{\mathcal{Q}}_{B}\delta_{A\bar{C}}\right),$
(88)

where

$\mathcal{H}_{Coul}=H_{Coul}-\frac{\sqrt{2\gamma}}{(2K)^{3/4}}\left(I-\frac{1}{2M}\sum_{C}R_{C\bar{C}}\right),$
(89)

with $I$ and $\mathcal{R}^{A}_{B}$ are defined, respectively, in (55) and
(78). In canonical coordinates (50) this Hamiltonian reads

$\mathcal{H}_{Coul}=\frac{p_{r}}{2}+\frac{\mathcal{I}}{r^{2}}+\frac{\gamma}{r}-\frac{\sqrt{2\gamma}}{r^{3/2}}\left(g+\sum_{\alpha}\pi_{\alpha}+\frac{2M-1}{2M}(\bar{\chi}\chi)\right).$
(90)

However, one can easily check that proposed supercharges do not yield closed
deformation of Poincaré superalgebra as well, e.g.

$\\{\widetilde{\mathcal{Q}}_{A},\frac{\mathcal{R}^{C}_{B}}{(2K)^{3/4}}\\}=\frac{\imath}{(2K)^{3/4}}\left(\frac{1}{M}\widetilde{\mathcal{Q}}_{A}\delta_{B\bar{C}}-\widetilde{\mathcal{Q}}_{B}\delta_{A\bar{C}}\right)+\frac{3}{2}\frac{S_{A}}{(2K)^{7/4}}\mathcal{R}_{B}^{C}$
(91)

So, proposed superextensions of Coulomb-like systems, being well-defined from
the viewpoint of superintegrability, do not possess neither $\mathcal{N}=2M$
supersymmetry, no its deformation. The $su(1,N|M)$ superalgebra plays the role
of dynamical algebra of that systems.

##  VI  Fubini-Study-like Kähler structure

The above considered super-Kaḧler structure is obviously the higher-
dimensional super-analogue of the Klein model of Lobachevsky space. On the
other hand, Lobachevsky space has other common parametrization as well, which
is known as Poincaré disc dnf . The higher-dimensional generalization of
Poincaré disc parameterizing the noncompact complex projective space is quite
similar to the Fubini-Study structure for $\mathbb{CP}^{N}$. It is defined by
the Kähler potential

$\mathcal{K}=-g\log(1-\sum_{a=1}^{N}z^{a}\bar{z}^{a}).$ (92)

For the obtaining of the super-analogue of this potential from
$\mathbb{C}^{1,N|M}$, one should pass from the matrix (6) to the diagonal
matrix $\gamma_{a\bar{b}}=\operatorname{diag}(1,-1,\ldots,-1)$. This can be
done by the transformation

$v^{0}\to\frac{v^{0}+v^{N}}{\sqrt{2}},\qquad
v^{N}\to\frac{v^{0}-v^{N}}{\imath\sqrt{2}}.$ (93)

On the reduced phase space (93) corresponds to the transformation

$w\to\imath\frac{z^{N}-1}{z^{N}+1},\qquad
z^{\alpha}\to\sqrt{2}\frac{z^{\alpha}}{z^{N}+1},\qquad\theta^{A}\to\sqrt{2}\frac{\theta^{A}}{z^{N}+1}.$
(94)

Thus we will get the Fubini-Study-like Kähler potential

$\mathcal{K}=-g\log(1-z^{c}\bar{z}^{c}+\imath\theta^{C}\bar{\theta}^{C}),$
(95)

which defines the following Kähler structure

$\Omega=\frac{\imath}{g}\left[\left(\frac{g\delta_{a\bar{b}}}{\tilde{A}}+\frac{\bar{z}^{a}z^{b}}{\tilde{A}^{2}}\right)dz^{a}\wedge
d\bar{z}^{b}+\frac{\imath\bar{\theta}^{A}z^{a}}{\tilde{A}^{2}}d\theta^{A}\wedge
d\bar{z}^{a}-\frac{\imath\bar{z}^{a}\theta^{A}}{\tilde{A}^{2}}dz^{a}\wedge
d\bar{\theta}^{A}-\left(\frac{g\delta_{A\bar{B}}}{\tilde{A}}+\frac{\bar{\theta}^{A}\theta^{B}}{\tilde{A}^{2}}\right)d\theta^{A}\wedge
d\bar{\theta}^{B}\right],$ (96)

where we have used a similar notation as in (15)

$\tilde{A}:=\frac{1-z^{c}\bar{z}^{c}+\imath\theta^{C}\bar{\theta}^{C}}{g}.$
(97)

The respective Poisson brackets read:

$\\{z^{a},\bar{z}^{b}\\}=\imath{\tilde{A}}\left({\delta^{a\bar{b}}-z^{a}\bar{z}^{b}}\right),\qquad\\{z^{a},\bar{\theta}^{A}\\}=\imath{\tilde{A}}{z^{a}\bar{\theta}^{A}},\qquad\\{\theta^{A},\bar{\theta}^{B}\\}={\tilde{A}}\left({\delta^{A\bar{B}}+\theta^{A}\bar{\theta}^{B}}\right).$
(98)

Now let us introduce the canonical coordinates, but now taking the
symplectic/Kähler one form associated with the Kähler potential (95), i.e. the
one that define ”Fubini-Study”-like metric. Then, as before, one needs to
identify it with the canonical one, and this canonical coordinates will play
the role of ”Cartesian” coordinates instead of the ”spherical” ones discussed
above.

$\tilde{\mathcal{A}}=-\frac{g}{2}\frac{\imath(\bar{z}^{a}d{z}^{a}-{z}^{a}d\bar{z}^{a})+{\theta}^{A}d\bar{\theta}^{A}+\bar{\theta}^{A}d{\theta}^{A}}{1-{z}^{c}\bar{z}^{c}+\imath{\theta}^{C}\bar{\theta}^{C}}:=p_{a}d\varphi_{a}+\frac{1}{2}{\chi}^{A}d\bar{\chi}^{A}+\frac{1}{2}\bar{\chi}^{A}d{\chi}^{A}.$
(99)

It leads to the relations

${z}^{a}=\sqrt{\frac{p_{a}}{g+p-\imath{\chi}^{C}\bar{\chi}^{C}}}e^{\imath\varphi_{a}},\qquad{\theta}^{A}=\frac{\sqrt{2}}{r}{\chi}^{A},\qquad
p=\sum_{a}p_{a}\;,$ (100)

or

$p_{a}=\frac{z^{a}\bar{z}^{a}}{\tilde{A}},\qquad\varphi_{a}=\arg(z^{a}),\qquad\chi^{A}=\frac{\theta^{A}}{\sqrt{\tilde{A}}},$
(101)

where $\tilde{A}$ is defined by (97).

These coordinates are related with (50) as follows:

$p_{\alpha}=\pi_{\alpha},\quad
p_{N}=\frac{1}{4}\left(p^{2}_{r}+\left(r-\frac{\sqrt{2\mathcal{I}}}{r}\right)^{2}\right),\quad\varphi_{N}=\arctan\left(\frac{2xy}{(x-y)(x+y)}\right),$
(102)

where

$x=1-\frac{p_{r}^{2}}{r^{2}}-\frac{2\mathcal{I}}{r^{4}},\quad
y=\frac{p_{r}}{r},$ (103)

while ${\chi}^{A}$ and $\varphi_{\alpha}$ remains unchanged after transition
from one parameterization to the other.

Finally, let us draw readers attention to the complete similarity of the
bosonic part of (100) with the equations mapping parameterizing compactified
Ruijsenaars-Schneider model with excluded centre of mass to the complex
projective (phase) space $\mathbb{CP}^{N}$. This prompt us, at first, to
construct the conformal-invariant analogue of that model by replacing the
complex projective space by its noncompact analogue
$\widetilde{\mathbb{CP}}^{N}$. Then one can try to construct its
$su(1,N|M)$-supeconformal extension by further replacement of
$\widetilde{\mathbb{CP}}^{N}$ by $\widetilde{\mathbb{CP}}^{N|M}$.

##  VII Concluding remarks

In this paper we suggested to construct the $su(1,N|M)$-superconformal
mechanics formulating them on phase superspace given by the non-compact
analogue of complex projective superspace $\widetilde{\mathbb{CP}}^{N|M}$. The
$su(1,N|M)$ symmetry generators were defined there as a Killing potentials of
$\widetilde{\mathbb{CP}}^{N|M}$. We parameterized this phase space by the
specific coordinates allowing to interpret it as a higher-dimensional super-
analogue of the Lobachevsky plane parameterized by lower half-plane (Klein
model). Then we transited to the canonical coordinates corresponding to the
known separation of the ”radial” and ”angular” parts of (super)conformal
mechanics. Relating the ”angular” coordinates with action-angle variables we
demonstrated that proposed scheme allows to construct the $su(1,N|M)$
supeconformal extensions of wide class of superintegrable systems. We also
proposed the superintegrable oscillator- and Coulomb- like systems with a
$su(1,N|M)$ dynamical superalgebra, and found that oscillator-like systems
admit deformed $\mathcal{N}=2M$ Poincaré supersymmetry, in contrast with
Coulomb-like ones.

In fact, proposed scheme demonstrated the effectiveness of the
supersymmetrization via formulation of the initial systems in terms of Kähler
phase space and further superextension of the latters. In order to relate
considered systems with the conventional ones (with Euclidean configuration
spaces), we restricted ourself by the non-compact complex projective
superspace. So, we are sure that applying the same approach to the
conventional (compact) complex projective spaces we can find many new
integrable systems as well and construct their unexpected extended
supersymmetric extensions.

Proposed scheme could obviously be extended to the systems on complex
Grassmanians (and on their noncompact analogues). In particular, we expect to
find, in this way, the $\mathcal{N}$-supersymmetric extensions of compactified
spin-Ruijsenaars-Schneider models. Moreover, it seems to be straightforward
task to apply proposed approach to the systems with generic $U(N)$-invariant
Kähler phase spaces locally defined by the Kähler potential
$\mathcal{K}\left(z^{a}\bar{z}^{a}\right)$. We expect that it can be done in
terms of Kähler phase superspace locally defined by the potential

$\widetilde{\mathcal{K}}=\mathcal{K}\left(z^{a}\bar{z}^{a}+\imath\eta^{a}\bar{\eta}^{A}\right).$
(104)

In this way we expect to construct the $\mathcal{N}=2M$ supersymmetric
extensions of the systems with curved (Riemann) configuration space as well,
in particular, of the so-called $\kappa$-deformations (i.e.
spherical/hyperbolic generalizations) of conformal mechanics, oscillator and
Coulomb systemsranada ; shmavon .

Finally, notice that considered phase superspace is not associated with
external algebra of initial bosonic manifold, and thus, it is not related with
the superfield approach. Thus, it is interesting to consider the systems with
$(N|kM)$-dimensional Kähler phase superspaces defined by the potentials

$\widetilde{\mathcal{K}}=\mathcal{K}\left(z^{a}\bar{z}^{a}\right)+F\left(\imath
g_{a\bar{b}}\eta^{a}_{\alpha}\bar{\eta}^{b}_{\alpha}\right),\qquad
F^{\prime}(0)={\rm const},$ (105)

and construct, in this way, the $\mathcal{N}=kN$ supersymmetric mechanics.
Very preliminary attempt in this direction was done in npps  where the
$\mathcal{N}=2$ supersymmetric extensions of the systems with generic Kähler
phase space was considered. However, this promising direction was not further
developed since that time. We plan to consider listed problems elsewhere.

###### Acknowledgements.

This work was supported by the Russian Foundation of Basic Research grant
20-52-12003 (S.K., A.N.) and by the Armenian Science Committee projects
20RF-023, 21AG-1C062 (E.Kh., A.N.) and 21AA-1C001(E.Kh.). The work of E.Kh.
was completed within the Regional Doctoral Program on Theoretical and
Experimental Particle Physics Program sponsored by VolkswagenStiftung, and
within ICTP Network Program NT-04.

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