We present the TCFH of 11-dimensional supergravity and so demonstrate that the form bilinears of supersymmetric solutions satisfy a generalisation of the conformal Killing-Yano equation with resepct to the TCFH connection. We also compute the Killing-Stäckel, Killing-Yano and closed conformal Killing-Yano tensors of all spherically symmetric M-branes that include the M2-brane, M5-brane, KK-monopole and pp-wave and demonstrate that their geodesic flows are completely integrable by giving all independent conserved charges in involution. We then find that all form bilinears of pp-wave and KK-monopole solutions generate (hidden) symmetries for spinning particle probes propagating on these backgrounds. Moreover, there are Killing spinors such that some of the 1-, 2- and 3-form bilinears of the M2-brane solution also generate symmetries for spinning particle probes. We also explore the question on whether the form bilinears are sufficient to prove the integrability of particle probe dynamics on 11-dimensional supersymmetric backgrounds.
We find that spinning particles with suitable couplings propagating in certain supersymmetric backgrounds of $D=4$, $N=2$ and $D=5$, $N=1$ minimal supergravities are invariant under symmetries generated by the twisted covariant form hierarchies of these theories. We also compare our results with the symmetries of spinning particles generated by Killing-Yano forms which are responsible for the separability properties of some gravitational backgrounds.
We present the twisted covariant form hierarchies (TCFH) of type IIA and IIB 10-dimensional supergravities and show that all form bilinears of supersymmetric backgrounds satisfy the conformal Killing-Yano equation with respect to a TCFH connection. We also compute the Killing-Stäckel, Killing-Yano and closed conformal Killing-Yano tensors of all spherically symmetric type II brane backgrounds and demonstrate that the geodesic flow on these solutions is completely integrable by giving all independent charges in involution. We then identify all form bilinears of common sector and D-brane backgro…
These are introductory notes on the study of the Dirac equation in curved spacetime and its relation to hidden symmetries of the dynamics. We present general results on the relation between special spacetime tensors and hidden symmetries, both for the full Dirac equation and for its semi-classical limit, the spinning particle. A concrete application of the general results is provided by the case of rotating higher dimensional black holes with cosmological constant, which we discuss. For these metrics the Dirac equation is separable and the relation between this and hidden symmetries is explain…
In this paper, we discuss hidden symmetries in rotating black hole spacetimes. We start with an extended introduction which mainly summarizes results on hidden symmetries in four dimensions and introduces Killing and Killing-Yano tensors, objects responsible for hidden symmetries. We also demonstrate how starting with a principal CKY tensor (that is a closed non-degenerate conformal Killing-Yano 2-form) in 4D flat spacetime one can "generate" 4D Kerr-NUT-(A)dS solution and its hidden symmetries. After this we consider higher-dimensional Kerr-NUT-(A)dS metrics and demonstrate that they possess …
We demonstrate that the rotating black holes in an arbitrary number of dimensions and without any restrictions on their rotation parameters possess the same `hidden' symmetry as the 4-dimensional Kerr metric. Namely, besides the spacetime symmetries generated by the Killing vectors they also admit the (antisymmetric) Killing-Yano and symmetric Killing tensors.
We propose a different approach to the analysis of symmetries in the near-horizon region of black holes. The idea is presented here for spherically symmetric black holes, for which we have shown that the generators of hidden symmetries can be extracted by solving the conformal Killing equation under certain assumptions. Our result is in agreement with that obtained in literature by the analysis of the wave equation in the near-horizon region.
A framework associating quantum cosmological boundary conditions to minisuperspace hidden symmetries has been introduced in \cite{7}. The scope of the application was, notwithstanding the novelty, restrictive because it lacked a discussion involving realistic matter fields. Therefore, in the herein letter, we extend the framework scope to encompass elements from a scalar-tensor theory in the presence of a cosmological constant. More precisely, it is shown that hidden minisuperspace symmetries present in a pre-big bang model suggest a process from which boundary conditions can be selected.
In this dissertation we study hidden symmetries within the framework of string theory. There are two kinds of hidden symmetries investigated in this work: the first type is associated with dynamics of quantum fields and the second type is related to integrability of strings on various backgrounds. The goals of this thesis are twofold: extension of hidden symmetries known in general relativity to stringy backgrounds in higher dimensions and construction of new integrable string theories. In the context of the first goal we study hidden symmetries of stringy backgrounds, with and without supersy…
The Eisenhart-Duval lift allows embedding non-relativistic theories into a Lorentzian geometrical setting. In this paper we study the lift from the point of view of the Dirac equation and its hidden symmetries. We show that dimensional reduction of the Dirac equation for the Eisenhart-Duval metric in general gives rise to the non-relativistic Levy-Leblond equation in lower dimension. We study in detail in which specific cases the lower dimensional limit is given by the Dirac equation, with scalar and vector flux, and the relation between lift, reduction and the hidden symmetries of the Dirac e…
It is known that odd-dimensional de Sitter space acts as a transparent potential for free fields. Previous studies have explained this phenomena by relating de Sitter free field equations of motion to the time-independent Schrodinger equation with known transparent potentials. In this work we show that de Sitter's transparency is a consequence of an infinite set of "hidden" symmetries. These symmetries arise from an accidental symmetry for the zero-mode of matter fields, as well as the boost isometry of de Sitter space. For simplicity, we consider the case of massive Klein-Gordon theory. We sh…
We study hidden symmetries, the symmetries associated with the Killing tensors, of the near horizon geometry of odd-dimensional Kerr-AdS-NUT black hole in two limits: generic extremal and extremal vanishing horizon (EVH) limits. Starting from Kerr-AdS-NUT black hole in ellipsoidal coordinates which admits integrable geodesic equations, we obtain the near horizon extremal/EVH geometries and their principal and Killing tensors by taking the near horizon limit. We explicitly demonstrate that geodesic equations are separable and integrable on theses near horizon geometries. We also compute the con…
Important classes of null geodesics and hidden symmetries in the Sen black hole are investigated. First, we obtain the principal null geodesics and circular photon orbits. Then, an irreducible rank-two Killing tensor and a conformal Killing tensor are derived, which represent the hidden symmetries. Analyzing the properties of Killing tensors, we clarify why the Hamilton-Jacobi and wave equations are separable in this spacetime. We also investigate the gravitational capture of photons by the Sen black hole and compare the result with those by the various charged/rotating black holes and naked s…
It is well known that the Kerr-NUT-AdS-dS black hole admits two linearly independent Killing vectors and possesses a hidden symmetry generated by a rank-2 Killing tensor. The near-horizon geometry of an extremal Kerr-NUT-AdS-dS black hole admits four linearly independent Killing vectors, and we show how the hidden symmetry of the black hole itself is carried over by means of a modified Killing-Yano potential which is given explicitly. We demonstrate that the corresponding Killing tensor of the near-horizon geometry is reducible as it can be expressed in terms of the Casimir operators formed by…
This is a short pedagogical introduction to the subject of Killing-Stackel and Killing-Yano tensors and their role in the integrability of various types of equations that are of physical interest in curved spacetime, the main application being higher dimensional rotating black holes with cosmological constant.
We consider a general charged, rotating black hole in five-dimensional STU supergravity, and show that its six-dimensional Kaluza-Klein lift admits a Killing-Yano 3-form with torsion. This underlies its known Killing tensors in five dimensions, and is related to the separability of torsion-modified Dirac equations. In the generalization to gauged supergravity, we present a five-dimensional Killing-Yano 3-form with torsion when two of the gauge fields are equal, and a Killing-Stackel tensor in the general 3-charge Wu solution.
We prove that the most general solution of the Einstein equations with the cosmological constant which admits a principal conformal Killing-Yano tensor is the Kerr-NUT-(A)dS metric. Even when the Einstein equations are not imposed, any spacetime admitting such hidden symmetry can be written in a canonical form which guarantees the following properties: it is of the Petrov type D, it allows the separation of variables for the Hamilton-Jacobi, Klein-Gordon, and Dirac equations, the geodesic motion in such a spacetime is completely integrable. These results naturally generalize the results obtain…
We investigate the hidden symmetries and existence of Killing spinors of D=5 minimal gauged supergravity solutions which admit a Killing-Maxwell system in the sense of Carter, such as the Chong-Cveti$\check{c}$-Lü-Pope (CCLP) Kerr-dS black hole spacetime and a 5-dimensional minimal gauged supergravity solution endowed with a Sasaki structure deformed by torsion, a limiting case of the generalization of the 5-dimensional toric Sasaki-Einstein $L^{a,b,c}$ spacetime. We note that when an electromagnetic tensor is present and an associated Killing-Maxwell system can be constructed in the sense of …
In this thesis we study higher-dimensional rotating black holes. Such black holes are widely discussed in string theory and brane-world models at present. We demonstrate that even the most general known Kerr-NUT-(A)dS spacetime, describing the general rotating higher-dimensional asymptotically (anti) de Sitter black hole with NUT parameters, is in many aspects similar to its four-dimensional counterpart. Namely, we show that it admits a fundamental hidden symmetry associated with the principal conformal Killing-Yano tensor. Such a tensor generates towers of hidden and explicit symmetries. The …
Invited talk at the conference "Fundamental Interactions: From Symmetries to Black Holes" in honor of Francois Englert, 24 - 27 March, Universite Libre de Bruxelles, Belgium
Field theories with p-form gauge potentials can possess ``hidden'' symmetries leaving the field strengths invariant on-shell without being gauge symmetries on-shell. The relevance of such symmetries to supersymmetric models is discussed. They provide central charges of supersymmetry algebras, play a particular role in duality relations, and lead to peculiar interactions. A multiplet of N=2 supersymmetry in four dimensions with two hidden central charges is presented.
We review briefly a stream of ideas concerning the role of quantum groups as hidden symmetries in conformal field theories, paying particular attention to the field theoretical representations of quantum groups based on Coulomb gas methods. An extensive bibliography is also included.
We study the hidden symmetries arising in the dimensional reduction of d=5, N=2 supergravity to three dimensions. Extending previous partial results for the bosonic part, we give a derivation that includes fermionic terms, shedding light on the appearance of the local hidden symmetry SO(4) in the reduction.
We discuss hidden symmetries of three-dimensional field configurations revealed at the one-particle level by the use of pseudoclassical particle models. We argue that at the quantum field theory level, these can be naturally explained in terms of manifest symmetries of the reduced phase space Hamiltonian of the corresponding field systems.
We observe and study new non-linear global space-time symmetries of the full ghost+matter action of RNS superstring theory. We show that these surprising new symmetries are generated by the special worldsheet currents (vertex operators) of RNS superstring theory, violating the equivalence of superconformal ghost pictures. We review the questions of BRST invariance and non-triviality of picture-dependent vertex operators and show their relation to hidden space-time symmetries and hidden space-time dimensions. In particular, we relate the space-time transformations, induced by the picture-depend…
We determine the most general time-independent Noether symmetries of two-field cosmological models with rotationally-invariant scalar manifold metrics. In particular, we show that such models can have hidden symmetries, which arise if and only if the scalar manifold metric has Gaussian curvature $-3/8$, i.e. when the model is of elementary $α$-attractor type with a fixed value of the parameter $α$. In this case, we find explicitly all scalar potentials compatible with hidden Noether symmetries, thus classifying all models of this type. We also discuss some implications of the corresponding con…
We examine the precise structure of the loop algebra of `dressing' symmetries of the Principal Chiral Model, and discuss a new infinite set of abelian symmetries of the field equations which preserve a symplectic form on the space of solutions.
Hidden symmetries, described by higher order in momenta integrals of motion that generate nonlinear algebras, are explored at the level of classical and quantum mechanics in a variety of physical systems related to conformal and superconformal invariance.
A worldvolume action for membrane is considered to study the target space local symmetries. We introduce a set of generators of canonical transformations to exhibit the target space symmetries such as the general coordinate transformation and the gauge transformation of antisymmetric tensor field. Similar results are derived for type IIB string with manifestly S-duality-invariant worldsheet action.
We investigate hidden symmetries of P,T-invariant system of topologically massive U(1) gauge fields. For this purpose, we propose a pseudoclassical model giving rise to this field system at the quantum level. The model contains a parameter, which displays a quantization property at the classical and the quantum levels and demonstrates a nontrivial relationship between continuous and discrete symmetries. Analyzing the integrals of motion of the pseudoclassical model, we identify U(1,1) symmetry and S(2,1) supersymmetry as hidden symmetries of the corresponding quantum system. Representing the h…
In this review we discuss hidden symmetries of toroidal compactifications of eleven-dimensional supergravity. We recall alternative versions of this theory which exhibit traces of the hidden symmetries when still retaining the massive Kaluza-Klein states. We reconsider them in the broader perspective of M-theory which incorporates a more extended variety of BPS states. We also argue for a new geometry that may underly these theories. All our arguments point towards an extension of the number of space-time coordinates beyond eleven.
We study the presence of abelian discrete symmetries in globally consistent orientifold compactifications based on rational conformal field theory. We extend previous work [1] by allowing the discrete symmetries to be a linear combination of U(1) gauge factors of the visible as well as the hidden sector. This more general ansatz significantly increases the probability of finding a discrete symmetry in the low energy effective action. Applied to globally consistent MSSM-like Gepner constructions we find multiple models that allow for matter parity or Baryon triality.
Hidden symmetries in a covariant Hamiltonian formulation are investigated involving gauge covariant equations of motion. The special role of the Stackel-Killing tensors is pointed out. A reduction procedure is used to reduce the original phase space to another one in which the symmetries are divided out. The reverse of the reduction procedure is done by stages performing the unfolding of the gauge transformation followed by the Eisenhart lift in connection with scalar potentials.
In the present work, using the recently introduced framework of local geometric deformations, special types of vector fields - so-called hidden Killing vector fields - are constructed, which solve the Killing equation not globally, but only locally, i.e. in local subregions of spacetime. Taking advantage of the fact that the vector fields coincide locally with Killing fields and therefore allow the consideration of integral laws that convert into exact physical conservation laws on local scales, balance laws in dynamical systems without global Killing symmetries are derived that mimic as close…
Representations of a group $G$ in vector spaces over a field $K$ form a category. One can reconstruct the given group $G$ from its representations to vector spaces as the full group of monoidal automorphisms of the underlying functor. This is a special example of Tannaka-Krein theory. This theory was used in recent years to reconstruct quantum groups (quasitriangular Hopf algebras) in the study of algebraic quantum field theory and other applications. We show that a similar study of representations in spaces with additional structure (super vector spaces, graded vector spaces, comodules, braid…
The study of hidden symmetries within Dirac's formalism does not possess a systematic procedure due to the lack of first-class constraints to act as symmetry generators. On the other hand, in the Faddeev-Jackiw approach, gauge and reparametrization symmetries are generated by the null eigenvectors of the sympletic matrix and not by constraints, suggesting the possibility of dealing systematically with hidden symmetries throughout this formalism. It is shown in this paper that indeed hidden symmetries of noninvariant or gauge fixed systems are equally well described by null eigenvectors of the …
Many two-dimensional classical field theories have hidden symmetries that form an infinite-dimensional algebra. For those examples that correspond to effective descriptions of compactified superstring theories, the duality group is expected to be a large discrete subgroup of the hidden symmetry group. With this motivation, we explore the hidden symmetries of principal chiral models and symmetric space models.
We study particle dynamics in a space-time invariant under the $DISIM_b(2)$ group - the deformation of the $ISIM(2)$ symmetry group of very special relativity. We find that the Lorentz violation leads to the creation of higher order (hidden) symmetries which are connected to those broken at the space-time level. Through the perspective of the conserved quantities of the special relativistic case the Lorentz violation is linked to specific non-commutative relations in phase-space.
We organize the eight variables of the four-dimensional bosonic string ({\dot X}^μ, X'^μ) into a 2 x 2 x 2 hypermatrix a_{AA'A''} and show that in signature (2,2) the Nambu-Goto Lagrangian is given by \sqrt{Det a} where Det is Cayley's hyperdeterminant. This is invariant not only under [SL(2,R)]^{3} but also under interchange of the indices A, A' and A''. This triality reveals hitherto hidden discrete symmetries of the Nambu-Goto action.
The ten dimensional heterotic string effective action with graviton, dilaton and antisymmetric tensor fields is dimensionally reduced to two spacetime dimensions. The resulting theory, with some constraints on backgrounds, admits infinite sequence of conserved nonlocal currents. It is shown that generators of the infinitesimal transformations associated with these currents satisfy Kac-Moody algebra.
It is known for ten years that self-dual Yang-Mills theory is the effective field theory of the open N=2 string in 2+2 dimensional spacetime. We uncover an infinite set of abelian rigid string symmetries, corresponding to the symmetries and integrable hierarchy of the self-dual Yang-Mills equations. The twistor description of the latter naturally connects with the BRST approach to string quantization, providing an interpretation of the picture phenomenon in terms of the moduli space of string backgrounds.
Apart from a few remarks on lattice systems with global or gauge symmetries, most of this talk is devoted to some interesting ancient examples of symmetries and their breakdowns in elasticity theory and hydrodynamics. Since Galois Theory is in many ways the origin of group theory as a tool to analyse (hidden) symmetries, a brief review of this profound mathematical theory is also given.
We discuss hidden symmetries of M-theory, its feedback on the construction of the M-theory effective action, and a response of the effective action when locality is preserved. In particular, the locality of special symmetries of the duality-symmetric linearized gravity constraints the index structure of the dual to graviton field in the same manner as it is required to separate the levels 0 and 1 generators subalgebra from the infinite-dimensional hidden symmetry algebra of gravitational theory. This conclusion fails once matter fields are taken into account and we give arguments for that. We …
We study the role of Chern--Simons couplings for the appearance of enhanced symmetries of Cremmer--Julia type in various theories. It is shown explicitly that for generic values of the Chern--Simons coupling there is only a parabolic Lie subgroup of symmetries after reduction to three space-time dimensions but that this parabolic Lie group gets enhanced to the full and larger Cremmer--Julia Lie group of hidden symmetries if the coupling takes a specific value. This is heralded by an enhanced isotropy group of the metric on the scalar manifold. Examples of this phenomenon are discussed as well …
We study the hidden symmetries of the fermionic sector of D=11 supergravity, and the role of K(E10) as a generalised `R-symmetry'. We find a consistent model of a massless spinning particle on an E10/K(E10) coset manifold whose dynamics can be mapped onto the fermionic and bosonic dynamics of D=11 supergravity in the near space-like singularity limit. This E10-invariant superparticle dynamics might provide the basis of a new definition of M-theory, and might describe the `de-emergence' of space-time near a cosmological singularity.
This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description of physical systems as varied as non-relativistic, relativistic, with or without gravity, classical or quantum, and are related to the existence of conserved quantities of the dynamics and integrability. In recent years their study has grown intensively, due to the discovery of non-trivial examples that apply to different types of theories and different numb…
We construct the Lax connection of the Green-Schwarz superstring in AdS_5 x S^5 within the Hamiltonian formalism and obtain precisely that used in 0810.4136. It differs in a crucial way from the Bena-Polchinski-Roiban connection by terms proportional to the Hamiltonian constraints. These extra terms ensure firstly that the integrals of motion are all first class and secondly that the Lax connection is flat in the strong sense.
We report on recent developments regarding the supersymmetric Free Differential Algebra describing the vacuum structure of $D=11$ supergravity. We focus on the emergence of a hidden superalgebra underlying the theory, explaining the group-theoretical role played by the nilpotent fermionic generator naturally appearing for consistency of the construction. We also discuss the relation between this hidden superalgebra and other superalgebras of particular relevance in the context of supergravity and superstring, involving a fermionic generator with 32 components.
We study the action of picture-changing and spectral flow operators on a ground ring of ghost number zero operators in the chiral BRST cohomology of the closed N=2 string and describe an infinite set of symmetry charges acting on physical states. The transformations of physical string states are compared with symmetries of self-dual gravity which is the effective field theory of the closed N=2 string. We derive all infinitesimal symmetries of the self-dual gravity equations in 2+2 dimensional spacetime and introduce an infinite hierarchy of commuting flows on the moduli space of self-dual metr…
