You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Conformal mechanics related to the near horizon extreme Kerr–Newman-AdS-dS black hole is studied. A unique $\mathcal{N}=2$ supersymmetric extension of the conformal mechanics is constructed.'
---

LMP-TPU–3/11\

[**$\mathcal{N}=2$ superparticle near horizon of extreme**]{}\
0.5cm [**Kerr–Newman-AdS-dS black hole**]{}\
$
\textrm{\Large Anton Galajinsky and Kirill Orekhov\ }
$ 0.7cm [*Laboratory of Mathematical Physics, Tomsk Polytechnic University,\
634050 Tomsk, Lenin Ave. 30, Russian Federation*]{}\
[Emails: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>]{}

PACS: 04.70.Bw; 11.30.-j; 11.30.Pb\
Keywords: extreme Kerr–Newman-AdS-dS black hole, conformal mechanics, $\mathcal{N}=2$ supersymmetry

0

[**1. Introduction**]{}\
A large class of extreme black holes in four and five dimensions exhibits $SO(2,1)$ symmetry in the near horizon limit [@klr; @cl]. The most interesting example of such a kind is the Kerr black hole in four dimensions [@bh]. The idea in [@str] to consider excitations around the near horizon extreme Kerr geometry and to extend the corresponding $SO(2,1) \times U(1)$ symmetry group to the Virasoro group of asymptotic symmetries has paved the way for extensive investigation of the Kerr/CFT correspondence[^1].

A useful means of studying geometry of vacuum solutions of the Einstein equations is provided by particle mechanics on a curved background. Apart from the issue of geodesic completeness, the knowledge of conserved charges of a particle helps uncover symmetries of the background. For example, the discovery of a quadratic first integral for a massive particle in the Kerr space–time [@car] preceded the construction of the second rank Killing tensor for the Kerr geometry [@wp]. Because $SO(2,1)$ is the conformal group in one dimension, particle models derived from the near horizon geometries with the $SO(2,1)$ isometry will automatically be conformal invariant.

Conformal mechanics associated with the near horizon extreme Kerr black hole in four dimensions has been studied recently in [@ag]. It was shown that in the near horizon limit the Killing tensor degenerates into a quadratic combination of Killing vectors corresponding to the $SO(2,1) \times U(1)$ isometry group. Furthermore, because $SO(2,1) \times U(1)$ is the bosonic subgroup of $SU(1,1|1)$, an $\mathcal{N}=2$ superconformal extension is feasible. While a relation between first integrals of a particle propagating on a curved background and Killing vectors characterizing a background geometry is well known, a link between supersymmetry charges and Killing spinors is poorly understood. A possibility to construct $\mathcal{N}=2$ superparticle moving near the horizon of the extreme Kerr black hole thus raises the question about geometric interpretation of the supercharges.

The purpose of this work is to extend the analysis in [@ag] to the case of the near horizon extreme Kerr–Newman-AdS-dS black hole [@strominger]. There are two motivations for this study. First, the analysis in [@kp; @ck] indicates that the extreme Kerr–Newman-AdS black hole is supersymmetric. Then one can hope to relate supersymmetry generators of an $\mathcal{N}=2$ superparticle to Killing spinors and to explore a limit in which the cosmological constant tends to zero and the electric and magnetic charges of the black hole vanish. This might give an answer to the question raised above. Second, in [@ag] an $\mathcal{N}=2$ superparticle was constructed in an [*ad hoc*]{} manner following the recipe which was earlier applied to an $\mathcal{N}=4$ superparticle in Bertotti–Robinson space [@bgik; @gala2]. Below we give a proof that the $\mathcal{N}=2$ model is essentially unique and is constructed in a purely group theoretical way.

The organization of this paper is as follows. In the next section background fields are briefly discussed. In Section 3 conformal mechanics associated with the near horizon extreme Kerr–Newman-AdS-dS black hole is studied within the framework of the Hamiltonian formalism. First integrals corresponding to the $SO(2,1) \times U(1)$ isometry group are identified. Section 4 is devoted to $\mathcal{N}=2$ supersymmetric extension of the conformal mechanics which is done within the canonical formalism. Our analysis is fully general and shows that the $\mathcal{N}=2$ model is unique. We summarize our results and discuss possible further developments in Section 5.

[**2. Background fields**]{}\
The Kerr-Newman–AdS–dS black hole is a solution of the Einstein-Maxwell equations with a non–vanishing cosmological constant [@carter]. In Boyer–Lindquist–type coordinates it reads[^2] \[kna\] && ds\^2= (d t- \^2 d )\^2 - d r\^2 - d\^2 -\^2(a d t- d )\^2,\
&& A=-( dt - d ) -( a d t - d) , where && \_r=(r\^2+a\^2)(1+)-2Mr+q\^2, \_=1-\^2,\
&& \^2=r\^2+a\^2\^2,=1-,q\^2=q\_e\^2+q\_m\^2. The parameters $M$, $a$, $q_e$ and $q_m$ are linked to the mass, angular momentum, electric and magnetic charges of the black hole, respectively (for explicit relations see e.g. [@strominger]). $l^2$ is taken to be positive for AdS and negative for dS and is related to the cosmological constant via $\Lambda=-3/l^2$.

It is evident that (\[kna\]) is invariant under the time translation and rotation around $z$–axis \[tp\] t’=t+, ’=+. A less obvious fact is that this solution admits the second rank Killing tensor [@car; @wp; @carter1] which obeys \[KT\] K\_[mn]{}=K\_[nm]{}, \_[(n]{} K\_[mp)]{}=0, \^m K\_[mp]{}+[F\_[p]{}]{}\^m K\_[mn]{}=0. In Boyer–Lindquist–type coordinates it reads ($x^m=(t,r,\t,\phi)$) \[kt\] K\_[mn]{}=Q\_[mn]{}+r\^2 g\_[mn]{}, where Q\_[mn]{}=(

[cccccc]{} -\_r & 0 & 0 & \^2\
\
0 & & 0 & 0\
\
0 & 0 & 0 & 0\
\
\^2 & 0 & 0 & - \^4\

). Generally, a Killing tensor is a key ingredient in establishing complete integrability of the geodesic equations. It also helps to separate variables for some important field equations in a given gravitational background, including Klein–Gordon and Dirac equations (for a review and further references see e.g. [@fk]). It should also be mentioned that, when considering superparticle models on a curved background which admits Killing–Yano tensors, extra supersymmetry charges can be constructed [@grh] whose Poisson bracket yields Killing tensors[^3].

The extreme solution is characterized by the condition that the inner and outer horizons of the black hole coalesce. Denoting the corresponding value of the radial coordinate by $r_+$, from $\Delta_r (r_+)=0$, $\Delta_r' (r_+)=0$ one finds $$a^2 = \frac{r_+^2(1+3r_+^2/ l^2)-q^2}{1-r_+^2/ l^2},\qquad
M = \frac{r_+[(1+r_+^2/ l^2)^2-q^2/ l^2]}{1-r_+^2/ l^2},$$ and \_r(r)=[(r-r\_+)]{}\^2\[[(r+r\_+)]{}\^2+2 r\_+\^2+l\^2+a\^2\]/l\^2.

In order to describe the near horizon geometry, one redefines the coordinates [@strominger] \[red\] r r\_+ + r\_0 r, t t r\_0/, + , where $r_0$ is a constant to be fixed below and then takes the limit $\epsilon \rightarrow 0$. The first prescription in (\[red\]) followed by $\epsilon \rightarrow 0$ provides a natural definition of the near horizon region. The last two prescriptions are designed so as to make $\lim_{\epsilon \rightarrow 0 } {d s}^2$ nonsingular[^4] \[nhm\] && ds\^2=()- ()(d+krdt)\^2, A=f()(d+ k r dt), where && ()=, ()=, ()=, && \_+\^2=r\_+\^2+a\^2\^2,r\_0\^2= ,k = ,\
&& f()= . This is a solution of the Einstein-Maxwell equations [@strominger] which generalizes the near horizon extreme Kerr metric constructed in [@bh]. It reduces to the latter when $q_e=q_m=0$ and $l^2\rightarrow\infty$.

Near the throat the symmetry group is enhanced. In addition to (\[tp\]) it includes the dilatation \[tp1\] t’=t+t, r’=r-r, and the special conformal transformation \[sp\] t’=t+(t\^2+) , r’=r-2 tr, ’=- . Altogether they form $SO(2,1) \times U(1)$ group [@strominger].

Concluding this section we derive the second rank Killing tensor in the near horizon region. Implementing the same limit as above one finds that the second term in (\[kt\]) reduces to the metric which gives a trivial contribution to the Killing tensor and can be discarded. The first terms yields \[kt1\] K\_[nm]{} d x\^n d x\^m=()\^2 . Given the background fields (\[nhm\]), it is straightforward to verify that (\[kt1\]) meets all the requirements formulated in (\[KT\]). Note that in a four–dimensional space–time with four isometries the Killing tensor can not be irreducible. Reducibility of the Killing tensor is a priori guaranteed by the enhanced isometry. However, the explicit formula for the reduction of the Killing tensor into Killing vectors may be useful and is given below in Section 3. Note that up to a conformal factor (\[kt1\]) coincides with the $AdS_2$ metric in Poincaré coordinates.

[**3. Conformal mechanics**]{}\
First it is worth briefly reminding a relation between symmetries of a background and conserved charges of a particle propagating in it. The conventional action functional and equations of motion read \[AC\] S=-(m ds+e A ) m (+\^n\_[mp]{} )=e g\^[nm]{} F\_[mp]{} , where $m$ is the mass and $e$ is the electric charge of the particle. In general, the coordinate transformation $x'^n=x^n+\xi^n(x)$ generated by a Killing vector $\xi^n(x)$ leaves the background vector potential invariant if \[sc\] \^m F\_[mn]{}+\_n (\^m A\_m)=0 holds. Then from (\[AC\]) and (\[sc\]) one finds that each Killing vector $\xi^n (x)$ gives rise to the integral of motion \[ekv\] \^n (x) (m g\_[nm]{} +e A\_n), which is linear in $\frac{d x^m}{d s}$. Likewise, each Killing tensor $K_{mn}$ yields the integral of motion \[cqk\] K\_[nm]{} , which is quadratic in $\frac{d x^m}{d s}$. That (\[cqk\]) is conserved is verified with the use of the rightmost constraint in (\[KT\]).

The action functional for a massive charged particle propagating near the horizon of the extreme Kerr-Newman–AdS–dS black hole is \[start\] S= -d t  ( m +e f() ), where the overdot denotes the derivative with respect to $t$. We choose the Hamiltonian formalism to analyze the model. Introducing momenta $(p_r,p_\t,p_\phi)$ canonically conjugate to the configuration space variables $(r,\t,\phi)$ \[momenta\] && p\_r=,\
&& p\_=,\
&& p\_=-e f(), one readily finds the identity \[iden\] && =\
&& = . Then (\[momenta\]) and (\[iden\]) can be used to derive conserved charges, including the Hamiltonian $H$, the generator of special conformal transformations $K$, the generator of dilatations $D$ and the generator $P$ of rotations around $z$–axis, directly from the Killing vector fields (\[tp\]), (\[tp1\]), (\[sp\]) and the general formula (\[ekv\]) \[hamc\] && H=r ( -k p\_),\
&& K= ( +k p\_)+\
&& +t\^2 H+2trp\_r, D=t H+r p\_r, P=p\_. Under the Poisson bracket they form $so(2,1)\oplus u(1)$ algebra \[confalg\] {H,D }=H, {H,K }=2D, {D,K } =K.

The Killing tensor (\[kt1\]) yields the integral of motion quadratic in momenta \[L\] L=m\^2 ()+p\_\^2 / ()+() [\[p\_+e f()\]]{}\^2 / (). This can be used to derive the explicit formula for the reduction of the Killing tensor into Killing vectors. Computing the Casimir element of $so(2,1)$ one finds \[rela\] H K-D\^2=L-k\^2 P\^2. Translating this back to the Lagrangian formalism one can verify that contributions linear in the vector potential are canceled as well as those quadratic in the vector potential. The rest yields K\_[nm]{}=12 (\^[(1)]{}\_n \^[(3)]{}\_m+\^[(3)]{}\_n \^[(1)]{}\_m )-\^[(2)]{}\_n \^[(2)]{}\_m+k\^2 \^[(4)]{}\_n \^[(4)]{}\_m. Here $(\xi^{(1)}_n,\xi^{(2)}_n,\xi^{(3)}_n,\xi^{(4)}_n)$ denote the Killing vectors corresponding to the time translation, dilatation, special conformal transformation and rotation around $z$–axis, respectively. As usual, the index is lowered with the use of the metric. In view of the enhanced isometry similar formulae hold for the extremal Kerr throat geometry [@ag], for the near-horizon geometry of the extremal Kerr-NUT-AdS-dS black hole [@jr], and for the weakly charged extremal Kerr throat geometry [@fr].

Finally, we note that Hamiltonian equations of motion can be treated by analogy with the extreme Kerr throat case [@ag]. The dynamics of the radial pair $(r,p_r)$ is fixed from the conserved charges $H$ and $D$. The momentum $p_\phi$ is a constant of motion while $p_\t$ is expressed algebraically via other variables with the use of $L$. The remaining equations of motion for $\t$ and $\phi$ can be integrated by quadratures only.

[**4. $\mathcal{N}=2$ superconformal mechanics**]{}\
In this section we construct $\mathcal{N}=2$ supersymmetric extension of the particle propagating near the horizon of the extreme Kerr-Newman–AdS–dS black hole. In addition to the $so(2,1)\oplus u(1)$ generators discussed in the previous section, $su(1,1|1)$ superalgebra involves supersymmetry generators $Q$, $\bar Q$, superconformal generators $S$, $\bar S$, and $u(1)$ $R$–symmetry generator $J$. Here and in what follows the bar denotes complex conjugation. The structure relations read[^5] $$\begin{aligned}
\label{sa}
&
\{Q,\bar Q \}=-2i H, && \{K,Q \}=S, && \{Q,\bar S \}=2i (D+i J),
\nonumber\\[2pt]
&
\{D,Q \}=-\frac 12 Q, && \{H,S \}=-Q, && \{D,S \}=\frac 12 S,
\nonumber\\[2pt]
&
\{S,\bar S \}=-2iK, && \{J,Q \}=-\frac i2 Q, && \{J,S \}=-\frac i2 S,
\nonumber\\[2pt]
&
\{H,D \}=H, && \{H,K \}=2D, && \{D,K \} =K,\end{aligned}$$ plus complex conjugate relations.

A conventional way to accommodate $\mathcal{N}=2$ supersymmetry in the bosonic model under consideration is to introduce two fermonic degrees of freedom $\p$ and $\bar\p$ and impose the canonical bracket[^6] \[br\] {,|}=-i. The most general form of the supersymmetry charges is Q=A e\^[i B]{} , |Q=A e\^[-i B]{} |, where $A$ and $B$ are real functions on the full phase space $(r,\t,\phi,p_r,p_\t,p_\phi)$ to be fixed from the structure relations of the superalgebra and the requirement that in the bosonic limit a supersymmetric Hamiltonian reduces to the first line in (\[hamc\]). From $\{Q,\bar Q \}=-2i H$ one finds \[fullH\] H=12 A\^2+12 { A\^2,B } |, while the bosonic limit gives \[A\] H\_0=12 A\^2=r ( -k p\_). For a supersymmetric model the generators of dilatations and special conformal transformations can be chosen as in the preceding section \[DK\] D=t H+r p\_r, K=K\_0+t\^2 H +2 t r p\_r, where we denoted \[K0\] K\_0= ( +k p\_), and $H$ is now the full supersymmetric Hamiltonian (\[fullH\]).

Given $Q$ and $K$, superconformal generator $S$ and $R$–symmetry generator $J$ are uniquely fixed by the structure relations of the superalgebra (\[sa\]). Thus it remains to compute all the brackets and derive restrictions on $B$.

The bracket of $D$ and $Q$ yields[^7] \[r1\] {r p\_r,B }=0, which means that $B$ is a function of $(r p_r)$. Computing the bracket of $H$ and $K$ one gets {K\_0,{H\_0,B} }=0 {H\_0,{K\_0,B } }=0, which implies that $\{K_0,B \}$ is the integral of motion of the original bosonic theory. In the last line the Jacobi identity was used.

The bracket of $K$ and $Q$ determines $S$ S=(-t+i{K\_0,B}- ) Q, while $\{S,\bar S \}=-2iK$ gives \[r2\] {K\_0,B }\^2==, with $L$ from (\[L\]). Note that the numerator coincides with the Casimir element of $so(2,1)$ realized in the bosonic theory. The full fraction commutes with $H_0$ in agreement with what was found above.

Finally, computing the bracket of $Q$ and $\bar S$ one derives the $R$–symmetry generator J=H\_0 { K\_0,B} +(1+ ) |. It turns out that the rest of the algebra does not impose further restrictions on the form of $B$. Thus, in order to construct an $\mathcal{N}=2$ superparticle moving near the horizon of the extreme Kerr-Newman–AdS–dS black hole, it is sufficient to solve (\[r1\]) and (\[r2\]) for $B$.

As was mentioned above, (\[r1\]) means that $B$ depends on $r$ and $p_r$ in the form of the single argument $(r p_r)$. Taking into account the definition of $K_0$ from (\[r2\]) one derives a linear inhomogeneous first order partial differential equation for $B$. Its general solution is the sum of a particular solution to the inhomogeneous equation and the general solution to the homogeneous equation \[B\] B=-+B\_[hom]{}. The latter is an arbitrary function of the integrals of the associated system of ordinary differential equations (see e.g. [@smirnov]). It turns out that they all depend on $\phi$. Because rotation around $z$–axis is the symmetry of the bosonic model, the requirement that the momentum $p_\phi$ be conserved in the full supersymmetric theory rules out $B_{hom}$.

Thus, $\mathcal{N}=2$ supersymmetric extension is essentially unique. Substituting (\[B\]) in the generators above one gets $$\begin{aligned}
&
H=H_0- \left( {K_0}^{-1} \sqrt{L-k^2 p_\phi^2}  \right) \p\bar\p, && J=\frac 12 \p\bar\p+\sqrt{L-k^2 p_\phi^2},
\nonumber\\[2pt]
&
D=t H+r p_r, && K=K_0+t^2 H +2 t r p_r,
\nonumber\\[2pt]
&
Q=-i {(K_0/2)}^{-1/2} \left(r p_r+i \sqrt{L-k^2 p_\phi^2} \right) \p, && S=-t Q+i {(2K_0)}^{1/2} \p.\end{aligned}$$ Taking into account the fact that $H_0$, $K_0$, $r p_r$ obey the structure relations of $so(2,1)$ and $H_0 K_0-{(r p_r)}^2=L-k^2 p_\phi^2$ is the Casimir element, one concludes that the $\mathcal{N}=2$ model is constructed in a purely group theoretical way.

A few comments are in order. First, the second term in the definition of $J$ might seem odd. It is the square root of the Casimir element of $so(2,1)$ which commutes with all the generators of $su(1,1|1)$. If desirable, instead of including it in $J$, one could keep this term as the central charge in $\mathcal{N}=2$ superalgebra. Only the bracket of $Q$ and $\bar S$ would be altered. In fact, it is this way how the $R$–symmetry generator is usually treated in the context of non–relativistic $\mathcal{N}=2$ many–body mechanics (see e.g. [@gl; @ag1]). Second, by construction $p_\phi$ commutes with all the generators of $su(1,1|1)$. The full symmetry group of the superparticle is thus $SU(1,1|1) \times U(1)$. Third, $\mathcal{N}=2$ supersymmetry does not impose any restriction on the particle parameters. This is to be contrasted with the $\mathcal{N}=4$ superparticle moving near the horizon of the extreme Reissner–Nordström black hole where the higher symmetry leads to a kind of $BPS$ condition [@gala2]. Finally, our consideration implies that the $\mathcal{N}=2$ superparticle associated with the near horizon extreme Kerr geometry [@ag] is unique as well.

[**5. Conclusion**]{}

To summarize, in this work we studied the conformal mechanics resulting from the near horizon geometry of the extreme Kerr-Newman–AdS–dS black hole. Conserved charges of the particle were identified. A unique $\mathcal{N}=2$ supersymmetric extension of the conformal mechanics was constructed within the framework of the Hamiltonian formalism.

Proceeding to open questions, the first to mention is a link of the supersymmetry charges and the superconformal charges constructed in this work to Killing spinors characterizing the near horizon extreme Kerr-Newman–AdS–dS geometry. A related issue is a Lagrangian formulation for the $\mathcal{N}=2$ superparticle both in components and in superfields. Then it is worth constructing a canonical transformation to conventional conformal mechanics in the spirit of [@gala2]. The resulting angular sector might be interesting in the context of recent studies in [@hln] (see also references therein). It is also intriguing to study a possible link of the model in this work to the matrix model for $AdS_2$ proposed in [@as]. Finally, one could try to construct $\mathcal{N}>2$ supersymmetric extensions of the conformal mechanics, albeit it is not clear what they might correspond to in geometric terms.

[**Acknowledgements**]{}\
This work was supported by the Dynasty Foundation and in part by grants RFBR 09-02-00078, LSS 3558.2010.2.

[nn]{} H.K. Kunduri, J. Lucietti, H.S. Reall, Class. Quant. Grav. [**24**]{} (2007) 4169, arXiv:0705.4214. H. K. Kunduri, J. Lucietti, J. Math. Phys. [**50**]{} (2009) 082502, arXiv:0806.2051. J.M. Bardeen, G.T. Horowitz, Phys. Rev. D [**60**]{} (1999) 104030, hep-th/9905099. M. Guica, T. Hartman, W. Song, A. Strominger, Phys. Rev. D [**80**]{} (2009) 124008, arXiv:0809.4266. T. Azeyanagi, N. Ogawa, S. Terashima, [*On non–chiral extension of Kerr/CFT*]{}, arXiv:1102.3423. B. Carter, Phys. Rev. [**174**]{} (1968) 1559. M. Walker, R. Penrose, Commun. Math. Phys. [**18**]{} (1970) 265. A. Galajinsky, JHEP [**1011**]{} (2010) 126, arXiv:1009.2341. T. Hartman, K. Murata, T. Nishioka, A. Strominger, JHEP [**0904**]{} (2009) 019, arXiv:0811.4393. V.A. Kostelecký, M.J. Perry, Phys. Lett. B [**371**]{} (1996) 191, hep-th/9512222. M.M. Caldarelli, D. Klemm, Nucl. Phys. B [**545**]{} (1999) 434, hep-th/9808097. S. Bellucci, A. Galajinsky, E. Ivanov, S. Krivonos, Phys. Lett. B [**555**]{} (2003) 99, hep-th/0212204. A. Galajinsky, Phys. Rev. D [**78**]{} (2008) 044014, arXiv:0806.1629. B. Carter, Commun. Math. Phys. [**10**]{} (1968) 280. B. Carter, Phys. Rev. D [**16**]{} (1977) 3395. V. Frolov, D. Kubiznak, Class. Quant. Grav. [**25**]{} (2008) 154005, arXiv:0802.0322. G.W. Gibbons, R.H. Rietdijk, J.W. van Holten, Nucl. Phys. B [**404**]{} (1993) 42, hep-th/9303112. J.P. Ngome, P.A. Horváthy, J.W. van Holten, J. Phys. A [**43**]{} (2010) 285401, arXiv:1003.0137. J. Rasmussen, J. Geom. Phys. [**61**]{} (2011) 922, arXiv:1009.4388. A. Al Zahrani, V. Frolov, A. Shoom, [*Particle dynamics in weakly charged extreme Kerr throat*]{}, arXiv:1010.1570. A. Galajinsky, O. Lechtenfeld, Phys. Rev. D [**80**]{} (2009) 065012, arXiv:0907.2242. A. Galajinsky, Nucl. Phys. B [**832**]{} (2010) 586, arXiv:1002.2290. V.I. Smirnov, [*A course of higher mathematics*]{}, Vol. 2, Pergamon Press, 1964.

T. Hakobyan, O. Lechtenfeld, A. Nersessian, A. Saghatelian, J. Phys. A [**44**]{} (2011) 055205, arXiv:1008.2912. A. Strominger, JHEP [**0403**]{} (2004) 066, hep-th/0312194.

[^1]: By now there is an extensive literature on the subject. For a more complete list of references see e.g. a recent work [@aot].

[^2]: We use metric with mostly minus signature and put $c=1$, $G=1$. In these conventions the Einstein–Maxwell equations read $R_{nm}-\frac 12 g_{nm} (R+2 \Lambda)=-2(F_{ns} {F_m}^s-\frac 14 g_{nm} F^2)$, $\partial_n (\sqrt{-g} F^{nm})=0$.

[^3]: For some recent developments see [@hh].

[^4]: As to the vector potential, usually one calculates the near horizon field strength and derives the vector potential from there.

[^5]: For a discussion of this algebra in the context of non–relativistic many–body mechanics see e.g. [@gl; @ag1] and references therein.

[^6]: When constructing a supersymmetric generalization, the action functional (\[start\]) is extended by the fermionic kinetic term $\int dt \Big(\frac{i}{2} \bar\p \dot \p-\frac{i}{2} \dot{\bar\p} \p \Big)$ as well as other contributions describing boson–fermion couplings. Within the framework of the Hamiltonian formalism there appear fermionic second class constraints, (\[br\]) being the Dirac bracket associated to them.

[^7]: Here and in what follows it proves helpful to use the brackets $\{r p_r, H_0 \}=-H_0$, $\{r p_r, K_0 \}=K_0$, and $\{K_0,H_0 \}=-2 r p_r$ which reproduce the structure relations of $so(2,1)$.