Tigran Hakobyan, Armen Nersessian
We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane). We also extend this picture to the N=2k superconformal mechanics described in terms of Lobachevsky superspace.
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The main purpose of the report is to provide some argumentation that three seemingly distinct approaches of 1. Giveon, Kutasov and Seiberg (hep-th/9806194); 2. Hemming, Keski-Vakkuri (hep-th/0110252); Maldacena, Ooguri (hep-th/0001053) and 3. I. Bars (hep-th/9503205) can be investigated by applying the mathematical methods of integral geometry on the Lobachevsky plane, developed previously by Gel'fand, Graev and Vilenkin. All these methods can be used for finding the transformations, leaving the Kac-Moody and Virasoro algebras invariant. The near-distance limit of the Conformal Field Theory of…
We propose Lobachevsky boundary conditions that lead to asymptotically H^2xR solutions. As an example we check their consistency in conformal Chern-Simons gravity. The canonical charges are quadratic in the fields, but nonetheless integrable, conserved and finite. The asymptotic symmetry algebra consists of one copy of the Virasoro algebra with central charge c=24k, where k is the Chern-Simons level, and an affine u(1). We find also regular non-perturbative states and show that none of them corresponds to black hole solutions. We attempt to calculate the one-loop partition function, find a rem…
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 show how the motion of a charged particle near the horizon of an extreme Reissner-Nordstrom black hole can lead to different forms of conformal mechanics, depending on the choice of the time coordinate.
We demonstrate that any scale-invariant mechanics of one variable exhibits not only 0+1 conformal symmetry, but also the symmetries of a full Virasoro algebra. We discuss the implications for the adS/CFT correspondence.
We derive the explicit form of the polar decomposition of the Wiener measure, and obtain the equation connecting functional integrals in conformal quantum mechanics to those in the Schwarzian theory. Using this connection we evaluate some nontrivial functional integrals in the Schwarzian theory and also find the fundamental solution of the Schroedinger equation in imaginary time in the model of conformal quantum mechanics.
It is found a relation between conformal quantum mechanics and Fick-Jacobs equation, which describes diffusion in channels. This relation is given between a family of channels and a family of conformal Hamiltonians. In addition, it is shown that a conformal Hamiltonian is associated with two channels with different geometry. Furthermore exact solutions for Fick-Jacobs equation are given for this family of channels.
In conformal quantum mechanics with the vacuum of a real scaling dimension and with a complete orthonormal set of energy eigenstates which is preferable under the unitary evolution, the dilatation expectation value between energy eigenstates monotonically decreases along the flow from the UV to the IR. In such conformal quantum mechanics there exist bounds on scaling dimensions of the physical states and the gauge operators.
Conformal Galilei algebra contains so(1,2) subalgebra which is the conformal algebra in one dimension. In this note we generalize methods previously developed for one-dimensional many-body systems and construct a unitary map relating a quantum mechanics invariant under the conformal Galilean transformations to a set of decoupled particles for which the same symmetry is realized in a nonlocal way. Possible applications of the map are discussed.
We suggest a method to compute the correlation functions in conformal quantum mechanics (CFT$_1$) for the fields that transform under a non-local representation of $\mathfrak{sl}(2)$ basing on the invariance properties. Explicit calculations of 2- and 3-point correlation functions are presented.
The quantum mechanics of one degree of freedom exhibiting the exact conformal SL(2,R) symmetry is presented. The starting point is the classification of the unitary irreducible representations of the SL(2,R) group (or, to some extent, its universal covering). The coordinate representation is defined as the basis diagonalizing the special conformal generator K. It is indicated how the resulting theory emerges from the canonical/geometric quantization of the Hamiltonian dynamics on the relevant coadjoint orbits.
Conformal many-body mechanics in Newton-Hooke spacetime is studied within the framework of the Lagrangian formalism. Global symmetries and Noether charges are given in a form convenient for analyzing the flat space limit. N=2 superconformal extension is built and a new class on N=2 models related to simple Lie algebras is presented. A decoupling similarity transformation on N=2 quantum mechanics in Newton-Hooke spacetime is discussed.
We construct a q-deformed version of the conformal quantum mechanics model of de Alfaro, Fubini and Furlan for which the deformation parameter is complex and the unitary time evolution of the system is preserved. We also study differential calculus on the q-deformed quantum phase space associated with such system.
In this paper a new supersymmetric extension of conformal mechanics is put forward. The beauty of this extension is that all variables have a clear geometrical meaning and the super-Hamiltonian turns out to be the Lie-derivative of the Hamiltonian flow of standard conformal mechanics. In this paper we also provide a supersymmetric extension of the other conformal generators of the theory and find their "square-roots". The whole superalgebra of these charges is then analyzed in details. We conclude the paper by showing that, using superfields, a constraint can be built which provides the exact …
A 0+1-dimensional candidate theory for the CFT$_1$ dual to AdS$_2$ is discussed. The quantum mechanical system does not have a ground state that is invariant under the three generators of the conformal group. Nevertheless, we show that there are operators in the theory that are not primary, but whose "non-primary character" conspires with the "non-invariance of the vacuum" to give precisely the correlation functions in a conformally invariant theory.
We discuss the possibility of realizing the infinite dimensional symmetries of conformal mechanics as time reparametrizations, generalizing the realization of the SL(2,R) symmetry of the de Alfaro, Fubini, Furlan model in terms of quasi--primary fields. We find that this is possible using an appropriate generalization of the transformation law for the quasi-primary fields.
We investigate the conformal and superconformal properties of a non-relativistic spinning particle propagating in a curved background coupled to a magnetic field and with a scalar potential. We derive the conditions on the couplings for a large class of such systems which are necessary in order their actions admit conformal and superconformal symmetry. We find that some of these conditions can be encoded in the conformal and holomorphic geometry of the background. Several new examples of conformal and superconformal models are also given.
We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a non-relativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan one-forms, examine various choices of the relevant coset spaces and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension D=d+1 (no central charges)…
We present the most general curvature obstruction to the deformed parabolic orthosymplectic symmetry subalgebra of the supersymmetric quantum mechanical models recently developed to describe Lichnerowicz wave operators acting on arbitrary tensors and spinors. For geometries possessing a hypersurface-orthogonal homothetic conformal Killing vector we show that the parabolic subalgebra is enhanced to a (curvature-obstructed) orthosymplectic algebra. The new symmetries correspond to time-dependent conformal symmetries of the underlying particle model. We also comment on generalizations germane to …
We find, at the Lagrangian off-shell level, the explicit equivalence transformation which relates the conformal mechanics of De Alfaro, Fubini and Furlan to the conformal mechanics describing the radial motion of the charged massive particle in the Bertotti-Robinson AdS$_2\times S^2$ background. Thus we demonstrate the classical equivalence of these two systems which are usually regarded as essentially different ``old'' and ``new'' conformal mechanics models. We also construct a similar transformation for N=2, $SU(1,1|1)$ superconformal mechanics in N=2 superfield formulation. Performing this …
Recently, there has been much interest in holographic computations of two-point non-equilibrium Green functions from AdS-Vaidya backgrounds. In the strongly coupled quantum field theory on the boundary, the dual interpretation of the background is an equilibration process called a holographic quench. The two dimensional AdS-Vaidya spacetime is a special case, dual to conformal quantum mechanics. We study how the quench is incorporated into a Hamiltonian $H + θ(t) ΔH$ and into correlation functions. With the help of recent work on correlation functions in conformal quantum mechanics, we first r…
We study N=4 supersymmetric quantum-mechanical many-body systems with M bosonic and 4M fermionic degrees of freedom. We also investigate the further restrictions of conformal and superconformal invariance. In particular, we construct conformal N=4 extensions of the A_{M-1} Calogero models, which for generic values of the coupling constant are not SU(1,1|2) superconformal. This class of models is also extended to arbitrary (even) N. We give both hamiltonian and (classical) lagrangean formulations. In the latter case we use both component and N=4 superfield formulations.
The consequences of coupling of the torsion (highest curvature) term to the Lagrangian of a massive spinless particle in four-dimensional space-time are studied. It is shown that the modified system remains spinless and possesses extended gauge invariance. Though the torsion term does not generate spin, it provides the system with a nontrivial mass spectrum, described by one-dimensional conformal mechanics. Under an appropriate choise of characteristic constants the system has solutions with a discrete mass spectrum.
We present a new infinite family of Type IIB backgrounds with an AdS$_2$ factor, preserving ${\cal N}=4$ SUSY. For each member of the family we propose a precise dual Super Conformal Quantum Mechanics (SCQM). We provide holographic expressions for the number of vacua (the "central charge"), Chern-Simons terms and other non-perturbative aspects of the SCQM. We relate the "central charge" of the one-dimensional system with a combination of electric and magnetic fluxes in Type IIB. The Ramond-Ramond fluxes are used to propose an extremisation principle for the central charge. Other physical and g…
We consider the quantum Lobachevsky space ${\bf L}_q^3$, which is defined as subalgebra of the Hopf algebra ${\cal A}_q(SL_2({\bf C}))$. The Iwasawa decomposition of ${\cal A}_q(SL_2({\bf C}))$ introduced by Podles and Woronowicz allows to consider the quantum analog of the horospheric coordinates on ${\bf L}_q^3$. The action of the Casimir element, which belongs to the dual to ${\cal A}_q$ quantum group $U_q(SL_2({\bf C}))$, on some subspace in ${\bf L}_q^3$ in these coordinates leads to a second order difference operator on the infinite one-dimensional lattice. In the continuos limit $q\righ…
In this paper we implement scale anisotropic transformations in the space-time in classical mechanics. The resulting system is consistent with the dispersion relation of gravity at a Lifshitz point recently considered in arXiv:0901.3775. Also, we show that our model is a generalization of the conformal mechanics of Alfaro, Fubini and Furlan. For arbitrary $z$ we construct the dynamical symmetries that correspond to the Schroedinger algebra. Furthermore, we obtain the Boltzman distribution for a gas of free particles compatible with anisotropic scaling transformations and compare our result wit…
Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e. the spectrum of one can be obtained from another by the q^2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra su_q(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of B…
Entanglement entropy and horizon temperature in conformal quantum mechanics
本文在共形量子力学中证明闵可夫斯基时空惯性真空对应热场双态，并由此重现菱形温度与Milne温度，同时计算得到点状边界下的紫外对数发散纠缠熵。
It is observed that some structures recently uncovered in the study of Calogero-Sutherland models and anyons are close analogs of well-known structures of boundary conformal field theory. These examples of ``boundary conformal quantum mechanics'', in spite of their apparent simplicity, have a rather reach structure, including some sort of T-duality, and could provide useful frameworks for testing general properties of boundary conformal theories. Of particular interest are the duality properties of anyons and Calogero-Sutherland particles in presence of boundary-violations of conformal invaria…
Conformal Quantum Mechanics, Holomorphic Factorisation, and Ultra-Spinning Black Holes
研究了相对论性CFT退化为共形量子力学的极限，发现其与引力对偶中的超旋转极限一致，提出超旋转黑洞对偶于有限维共形量子力学系综，并在全息因子化中扮演块的角色。
A model of an electron and a Dirac monopole interacting through an axially symmetric non-hermitian but \mathcal{PT}-symmetric potential is discussed in detail. The intriguing localization of the wave-packet as a result of the anomalous breaking of the scale symmetry is shown to provide a scale for the system. The symmetry algebra for the system, which is the conformal algebra SO(2,1), is discussed and is shown to belong to the enveloping algebra of the combined algebra, composed of the Virosoro algebra, \{L_n, n\in \mathbb{N}\} and an abelian algebra, \{P_n,n\in \mathbb{N}\}.
One of the simplest examples of a PT-symmetric quantum system is the scaling Yang-Lee model, a quantum field theory with cubic interaction and purely imaginary coupling. We give a historical review of some facts about this model in d <= 2 dimensions, from its original definition in connection with phase transitions in the Ising model and its relevance to polymer physics, to the role it has played in studies of integrable quantum field theory and of PT-symmetric quantum mechanics. We also discuss some more general results on PT-symmetric quantum mechanics and the ODE/IM correspondence, and ment…
N-particle quantum mechanics described by a sigma model with an N-dimensional target space with torsion is considered. It is shown that an SL(2,R) conformal symmetry exists if and only if the geometry admits a homothetic Killing vector $D^a$ whose associated one-form $D_a$ is closed. Further, the SL(2,R) can always be extended to Osp(1|2) superconformal symmetry, with a suitable choice of torsion, by the addition of N real fermions. Extension to SU(1,1|1) requires a complex structure I and a holomorphic U(1) isometry $D^a I_a{^b} \partial_b$. Conditions for extension to the superconformal grou…
Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional $N=2$ supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are $q$-isospectral, i.e. the spectrum of one can be obtained from another (with possible exception of the lowest level) by $q^2$-factor scaling. This construction allows easily to rederive a special self-similar potential found by Shabat and to show that for the latter a $q$-deformed harmonic oscillator algebra of Biedenharn a…
- We have suggested using the action-angle variables for the study of a (quasi)particle in quantum ring. We have presented the action-angle variables for three two-dimensional singular oscillator systems - We have suggested a procedure of constructing new integrable systems form the known ones, by adding a radial part to the angular Hamiltonian. Using this method we have constructed a class of integrable generalizations of oscillator and Coulomb systems on N-dimensional Euclidian space, sphere and hyperboloid. - We have developed the methods of study of conformal mechanics, based on separation…
We analyze the effects of noncommutativity in conformal quantum mechanics (CQM) using the $κ$-deformed space-time as a prototype. Upto the first order in the deformation parameter, the symmetry structure of the CQM algebra is preserved but the coupling in a canonical model of the CQM gets deformed. We show that the boundary conditions that ensure a unitary time evolution in the noncommutative CQM can break the scale invariance, leading to a quantum mechanical scaling anomaly. We calculate the scaling dimensions of the two and three point functions in the noncommutative CQM which are shown to b…
We discuss BPS solitons in gauged ${\cal N}=2$, D=4 supergravity. The solitons represent extremal black holes interpolating between different vacua of anti-de Sitter spaces. The isometry superalgebras are determined and the motion of a superparticle in the extremal black hole background is studied and confronted with superconformal mechanics. We show that the Virasoro symmetry of conformal mechanics, which describes the dynamics of the superparticle near the horizon of the extremal black hole under consideration, extends to a symmetry under the $w_{\infty}$ algebra of area-preserving diffeomor…
We study string theory in the background of a two-dimensional black hole which is described by an $SL(2, R)/U(1)$ coset conformal field theory. We determine the spectrum of this conformal field theory using supersymmetric quantum mechanics and give an explicit form of the vertex operators in terms of the Jacobi functions. We also discuss the applicability of SUSY quantum mechanics techniques to non-linear $σ$-models.
The geometry of gauged (super)conformal mechanics
本文系统研究了一维规范 sigma 模型的共形不变性，发现特定模型仅在规范后才完全共形不变，并解决了物理 Hilbert 空间上的量子排序歧义。
We show the natural relation between the Wigner Hamiltonian and the conformal Hamiltonian. It is presented a model in (super)conformal quantum mechanics with (super)conformal symmetry in the Wigner-Heisenberg algebra picture $ [x,p_{x}]= i(1+c{\bf P})$ (${\bf P}$ being the parity operator). In this context, the energy spectrum, the Casimir operator, creation and annihilation operators are defined. This superconformal Hamiltonian is similar to the super-Hamiltonian of the Calogero model and it is also an extension of the super-Hamiltonian for the Dirac Oscillator.
Conformally compactified phase space is conceived as an automorphism space for the global action of the extended conformal group. Space time and momentum space appear then as conformally dual, that is conjugate with respect to conformal reflections. If now the former, as generally agreed, is appropriate for the description of classical mechanics in euclidean geometrical form, then the latter results appropriate for the description of quantum mechanics in spinor geometrical form. In such description, fermion multiplets will naturally appear as consequence of higher symmetries and furthermore, t…
A simple conformal quantum mechanics model of a d-component variable is proposed, which exactly reproduces the retarded Green functions and conformal weights of conformally coupled scalar fields in de Sitter spacetime seen by a static patch observer. It is found that the action integral of this model is automatically expressed by a complex integral over the time variable t along a closed contour in a way which is typical to the Schwinger-Keldysh formalism of a thermofield theory. Hence this model is at finite temperature. The case of conformally coupled scalar fields in 3d Schwarzschild de Sit…
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 discuss in detail the parasupersymmetric quantum mechanics of arbitrary order where the parasupersymmetry is between the normal bosons and those corresponding to the truncated harmonic oscillator. We show that even though the parasusy algebra is different from that of the usual parasusy quantum mechanics, still the consequences of the two are identical. We further show that the parasupersymmetric quantum mechanics of arbitrary order p can also be rewritten in terms of p supercharges (i.e. all of which obey $Q_i^{2} = 0$). However, the Hamiltonian cannot be expressed in a simple form in term…
On conformal symmetry in large-$N$ quiver mechanics
研究AdS₂/CFT₁全息中极端超对称黑洞微观描述问题，通过局域化技术推导超共形箭图指标的不动点公式，以循环阿贝尔箭图为例，在大 N 极限下精确捕捉微观标度BPS指标贡献，为AdS₂/CFT₁对偶弦论实现助力。
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…
We revisit conformal quantum mechanics (CQM) from the perspective of sine-square deformation (SSD) and the entanglement Hamiltonian. The operators that correspond to SSD and the entanglement Hamiltonian are identified. Thus, the nature of SSD and entanglement can be discussed in a much simpler CQM setting than higher-dimensional field theories.
The equations of motion for $N$ non-relativistic particles attracting according to Newton's law are shown to correspond to the equations for null geodesics in a $(3N+2)$-dimensional Lorentzian, Ricci-flat, spacetime with a covariantly constant null vector. Such a spacetime admits a Bargmann structure and corresponds physically to a generalized pp-wave. Bargmann electromagnetism in five dimensions comprises the two Galilean electro-magnetic theories (Le Bellac and Lévy-Leblond). At the quantum level, the $N$-body Schrödinger equation retains the form of a massless wave equation. We exploit the …
Conformal mechanics related to the near horizon extreme Kerr-Newman-AdS-dS black hole is studied. A unique N=2 supersymmetric extension of the conformal mechanics is constructed.
We discuss quantum mechanics in the moduli space consisting of two maximally charged dilaton black holes. The quantum mechanics of the two black hole system is similar to the one of DFF model, and this system has the $SL(2,R)$ conformal symmetry. Also, we discuss the bound states in this system.
We study the generalized conformal quantum mechanics of the probe D0-brane in the near horizon background of the bound state of source D0-branes. We elaborate on the relationship of such model to the M theory in the light cone frame.
Higher-derivative analogues of multidimensional conformal particle and many-body conformal mechanics are constructed. Their Newton-Hooke counterparts are derived by applying appropriate coordinate transformations.
The lectures provide a pedagogical introduction to the methods of CFT as applied to two-dimensional critical behaviour.
In this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector-fields built over the symplectic space of the original system. Our supersymmetric Hamiltonian itself turns out to have a clear geometrical meaning being the Lie-derivative of the Hamiltonian flow of conformal mechanics. Using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original non-supersymmetric mode…
We study the relation between the (generalized) conformal quantum mechanics of 0-branes and the two-dimensional dilaton gravity. The two-dimensional actions obtained from the supergravity effective actions for the (dilatonic) 0-branes through the compactification on a sphere are related to known two-dimensional dilaton gravity models. The SL(2,R) symmetry of the (generalized) conformal quantum mechanics is realized within such two-dimensional models. The two-dimensional dilatonic gravity model derived from the non-dilatonic 0-brane action is related to the Liouville theory and therefore is con…
We investigate a one dimensional quantum mechanical model, which is invariant under translations and dilations but does not respect the conventional conformal invariance. We describe the possibility of modifying the conventional conformal transformation such that a scale invariant theory is also invariant under this new conformal transformation.
A canonical transformation which relates the model of a massive relativistic particle moving near the horizon of an extremal black hole in four dimensions and the conventional conformal mechanics is constructed in two different ways. The first approach makes use of the action-angle variables in the angular sector. The second scheme relies upon integrability of the system in the sense of Liouville.
Recently, Galajinsky, Lechtenfeld and Polovnikov proposed an elegant group-theoretical transformation of the generic conformal-invariant mechanics to the free one. Considering the classical counterpart of this transformation, we relate this transformation with the Weil model of Lobachewsky space.
Extending previous work on 2 -- and 3 -- point functions, we study the 4 -- point function and its conformal block structure in conformal quantum mechanics CFT$_1$, which realizes the SO(2,1) symmetry group. Conformal covariance is preserved even though the operators with which we work need not be primary and the states are not conformally invariant. We find that only one conformal block contributes to the four-point function. We describe some further properties of the states that we use and we construct dynamical evolution generated by the compact generator of SO(2.1).
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…
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