The motion of a particle near the Reissner-Nordstrom black hole horizon is described by conformal mechanics. In this paper we present an extended one-dimensional analysis of the N=4 superconformal mechanics coupled to n copies of N=8, d=1 vector supermultiplets. The constructed system possesses a special Kahler geometry in the scalar sector of the vector multiplets as well as an N=4 superconformal symmetry which is provided by a proper coupling to a dilaton superfield. The superconformal symmetry completely fixes the resulting action. We explicitly demonstrate that the electric and magnetic charges, presenting in the "effective black hole" action, appear as a result of resolving constraints on the auxiliary components of the vector supermultiplets. We present the component action, supercharges and Hamiltonian with all fermionic terms included. One of the possible ways to generalize the black hole potential is to consider a modified version of the N=4 superconformal multiplet where its auxiliary components acquire non-zero constant values. We explicitly write down the corresponding modified black hole potential.
We study quantum mechanical systems with $\mathfrak{osp}(4^{*}|4)$ superconformal symmetry. We classify unitary lowest-weight representations of this superconformal algebra and define an index which receives contributions from short and semi-short multiplets only. We consider the example of a quantum mechanical $σ$-model with hyper-Kähler target $\mathcal{M}$ equipped with a triholomorphic homothety. The superconformal index coincides with the Witten index of a novel form of supersymmetric quantum mechanics for a particle moving on $\mathcal{M}$ in a background magnetic field in which an unbro…
We consider supersymmetric quantum mechanics on a Kähler cone, regulated via a suitable resolution of the conical singularity. The unresolved space has a $\mathfrak{u}(1,1|2)$ superconformal symmetry and we propose the existence of an associated quantum mechanical theory with a discrete spectrum consisting of unitary, lowest weight representations of this algebra. We define a corresponding superconformal index and compute it for a wide range of examples.
We study the quantum mechanical sigma model arising in the discrete light-cone quantisation of N=4 supersymmetric Yang-Mills theory. The target space is a certain torus fibration over a scale-invariant special Kahler manifold. We show that the expected SU(1,1|4) light-cone superconformal invariance of the N=4 theory emerges in a limit where the volume of the fibre goes to zero and give an explicit construction of the generators. The construction given here yields a large new family of superconformal quantum mechanical models with SU(1,1|4) invariance.
We explicitly construct a family of ${\cal N}=4$ superconformal mechanics of dyonic particles, generalizing the work of Anninos et al.\,\cite{Anninos:2013nra} to an arbitrary number of particles. These mechanics are obtained from a scaling limit of the effective Coulomb branch description of ${\mathcal N}=4$ quiver quantum mechanics describing D-branes in type II Calabi-Yau compactifications. In the supergravity description of these D-branes this limit changes the asymptotics to AdS$_2\times $S$^2\times$CY$_3$. We exhibit the $D(1,2;0)$ superconformal symmetry and conserved charges of the mech…
A representation of the conformal Newton-Hooke algebra on a phase space of n particles in arbitrary dimension which interact with one another via a generic conformal potential and experience a universal cosmological repulsion or attraction is constructed. The minimal N=2 superconformal extension of the Newton-Hooke algebra and its dynamical realization in many-body mechanics are studied.
We report on the new approach to constructing superconformal extensions of the Calogero-type systems with an arbitrary number of involved particles. It is based upon the superfield gauging of non-abelian isometries of some supersymmetric matrix models. Among its applications, we focus on the new N=4 superconformal system yielding the U(2) spin Calogero model in the bosonic sector, and the one-particle case of this system, which is a new OSp(4|2) superconformal mechanics with non-dynamical U(2) spin variables. The characteristic feature of these models is that the strength of the conformal inve…
We present basics of the gauged superfield approach to constructing N-superconformal multi-particle Calogero-type systems developed in arXiv:0812.4276, arXiv:0905.4951 and arXiv:0912.3508. This approach is illustrated by the multi-particle systems possessing SU(1,1|1) and D(2,1;α) supersymmetries, as well as by the model of new N=4 superconformal quantum mechanics.
The issue of constructing an N=4 superconformal mechanics in one dimension is reconsidered with a special emphasis put on the realizations of the su(2)-subalgebra in the full su(1,1|2)-superalgebra. New dynamical realizations of su(1,1|2) are constructed which describe an interaction of the (0,4,4)-supermultiplet with either the (1,4,3)-, or (3,4,1)-, or (4,4,0)-supermultiplet. A relation of the N=4 superconformal mechanics with massive superparticles propagating near the black hole horizons is discussed. Background geometry associated with the model based on the (4,4,0)-supermultiplet is iden…
A new superconformal mechanics with OSp(4|2) symmetry is obtained by gauging the U(1) isometry of a superfield model. It is the one-particle case of the new N=4 super Calogero model recently proposed in arXiv:0812.4276 [hep-th]. Classical and quantum generators of the osp(4|2) superalgebra are constructed on physical states. As opposed to other realizations of N=4 superconformal algebras, all supertranslation generators are linear in the odd variables, similarly to the N=2 case. The bosonic sector of the component action is standard one-particle (dilatonic) conformal mechanics accompanied by a…
We propose Lagrangian formulations of a three-particles translation invariant $N=4$ superconformal mechanics based on the standard and twisted $(1,4,3)$ supermultiplets. We show that in the appropriate set of coordinates, in which the bosonic kinetic terms for each of the two cases take conformally-flat forms, the corresponding models produce just similar physical systems with the flipped angular part of bosonic potential $U(x)\rightarrow 1/U(x)$. This flipping looks so simple only being written in the "angle" variable, while in the standard variables it looks more complicated to preserve the …
The complex nonperturbative color-confining dynamics of QCD is well captured in a semiclassical effective theory based on superconformal quantum mechanics and its extension to the light-front. I describe here how this new approach to hadron physics incorporates confinement, the appearance of nearly massless pseudoscalar particles, and Regge spectroscopy consistent with experiment. It also gives remarkable connections between the meson and baryon spectrum across the light and heavy-light hadron spectrum. I also briefly discuss how higher spin states are consistently described in this framework …
In spacetime dimension two, pure Yang-Mills possesses no physical degrees of freedom, and consequently it admits a supersymmetric extension to couple to an arbitrary number, N say, of Majorana-Weyl gauginos. This results in (N,0) super Yang-Mills. Further, its dimensional reduction to mechanics doubles the number of supersymmetries, from N to N+N, to include conformal supercharges, and leads to a superconformal Yang-Mills quantum mechanics with symmetry group OSp(N|2,R). We comment on its connection to AdS_2 \times S^{N-1} and reduction to a supersymmetric Calogero model.
Proceeding from nonlinear realizations of (super)conformal symmetries, we explicitly demonstrate that adding the harmonic oscillator potential to the action of conformal mechanics does not break these symmetries but modifies the transformation properties of the (super)fields. We also analyze the possibility to introduce potentials in N=4 supersymmetric mechanics by coupling it with auxiliary fermionic superfields. The new coupling we considered does not introduce new fermionic degrees of freedom - all our additional fermions are purely auxiliary ones. The new bosonic components have a first or…
Bogoliubov Laboratory of Theoretical Physics, JINR（约飞联合核子研究所鲍里索布理论物理实验室）；Laboratory of Applied Mathematics and Theoretical Physics, TUSUR（托木斯克州立大学应用数学和理论物理实验室）
Superconformal mechanics from N-extended Euler-Calogero-Moser and Calogero models
Bogoliubov Laboratory of Theoretical Physics, JINR（鲍戈留博夫理论物理实验室，欧洲核子研究中心）；Tomsk State University of Control Systems and Radioelectronics（托姆斯克状态控制系统与电子大学）
We present a new multiparticle model of $\mathcal{N}{=}\,8$ mechanics with superconformal $F(4)$ symmetry. The system is constructed in terms of two matrix $\mathcal{N}{=}\,4$ multiplets. One of them is a bosonic matrix $({\bf 1, 4, 3})$ multiplet and another is a fermionic $({\bf 0, 4, 4})$ one. Off-diagonal bosonic components of the $({\bf 1, 4, 3})$ multiplet are chosen to take values in the flag manifold $\mathrm{U}(n)/[\mathrm{U}(1)]^n$ and they carry additional gauge symmetries. The explicit form of the $F(4)$ supersymmetry generators is found. We demonstrate that the $F(4)$ superalgebra…
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 $\mathbb{CP}^{N|M}$. 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 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 demo…
We present new explicit realizations of the most general N=4, d=1 superconformal symmetry D(2,1;α) in the models of N=4 superconformal mechanics based on the reducible multiplets (1,4,3)\oplus(0,4,4), (3,4,1)\oplus(0,4,4) and (4,4,0)\oplus(0,4,4). We start from the manifestly supersymmetric superfield actions for these systems and then descend to the relevant off- and on-shell component actions from which we derive the D(2,1;α) (super)charges by the Noether procedure. Some peculiarities of these realizations of D(2,1;α) are discussed. We also construct a new D(2,1;α) invariant system by joinin…
Superparticle models with $OSp(N|2)$ supersymmetry group are studied. We first consider the $N=4$ case and construct the models with $κ$-symmetry on the coset spaces of the $OSp(4|2)$ supergroup. In addition, within the canonical formalism we present an $OSp(4|2)$ superparticle model with semi-dynamical angular variables. For generic $N$ we construct a superparticle model on $AdS_2\times S^{N-1}$ with the reduced $κ$-symmetry. It is demonstrated that the Hamiltonian of this model has the same structure as the one for the $N=4$ case because additional fermions contribute to the second-class con…
We study superpaticle models with fermionic gauge symmetry on the coset spaces of the $SU(1,1|N)$ supergroup. We first construct $SU(1,1|N)$ supersymmetric extension of a particle on $AdS_2$ possessing the $κ$-symmetry. Including angular degrees of freedom and extending this model to a superparticle on the $AdS_2\times \mathbb{CP}^{N-1}$ background with two-form flux, one breaks the $κ$-symmetry down to a fermionic gauge symmetry with one parameter. A link of the background field configuration to the near horizon black hole geometries is discussed.
We study the large $N$ limit of the superconformal index of a large class of 5d $\mathcal{N}=1$ superconformal field theories and show it is given by the square of the partition function on the squashed five-sphere. We show this simple relation implies a Cardy formula in 5d, which is valid in an "extended" regime in which fugacities are finite and $N$ is large. For theories with weakly coupled gravity duals we conjecture this large $N$ Cardy formula universally accounts for the microscopic entropy of spinning black holes in AdS$_{6}$. We check this explicitly for known black hole solutions in …
We discuss the superconformal quantum mechanics arising from the M2-branes. We begin with a comprehensive review on the superconformal quantum mechanics and emphasize that conformal symmetry and supersymmetry in quantum mechanics contain a number of exotic and enlightening properties which do not occur in higher dimensional field theories. We see that superfield and superspace formalism is available for $\mathcal{N}\le 8$ superconformal mechanical models. We then discuss the M2-branes with a focus on the world-volume descriptions of the multiple M2-branes which are superconformal three-dimensi…
We survey the salient features and problems of conformal and superconformal mechanics and portray some of its developments over the past decade. Both classical and quantum issues of single- and multiparticle systems are covered.
We extend the differential form representation of N = (n,n) supersymmetric quantum mechanics to the superconformal case. We identify the superalgebras occurring for n = 1,2,4, give necessary and sufficient conditions for their existence, and give explicit geometric constructions of their generators and commutation relations. Quantum mechanics on the moduli space of instantons is considered as an example.
We formulate the equations which determine a potential function in an $\mathcal{N}=1$ higher derivative supersymmetric mechanics compatible with the $osp(2|1)\oplus so(d)$ symmetry and provide a few explicit examples.
Using the worldline SU(2|1) superfield approach, we construct N=4 superconformally invariant actions for the d=1 multiplets (1, 4, 3) and (2, 4, 2). The SU(2|1) superfield framework automatically implies the trigonometric realization of the superconformal symmetry and the harmonic oscillator term in the corresponding component actions. We deal with the general N=4 superconformal algebra D(2,1;$α$) and its central-extended $α$=0 and $α$=-1 psu(1,1|2)$\oplus$su(2) descendants. We capitalize on the observation that D(2,1;$α$) at $α\neq$0 can be treated as a closure of its two su(2|1) subalgebras,…
