This thesis is devoted to the study of the deformation and rigidity of infinite dimensional Lie algebras which are not subject to the Hochschild-Serre factorization theorem. In particular, we consider $bms_{3}$, Virasoro-Kac-Moody and $bms_{4}$ algebras and their central extensions which are respectively obtained as asymptotic and/or near horizon symmetry algebras for Einstein gravity on $3d$ flat, AdS$_{3}$ and $4d$ flat spacetimes. We also explore possible deformations of the Maxwell-BMS algebra, which is obtained as asymptotic symmetry algebra of the Chern-Simons gravity theory invariant under the $2+1$ dimensional Maxwell algebra. We find that these algebras are not rigid and can be deformed into new non isomorphic infinite dimensional (family of) algebras. We study these deformations by direct computations and also by cohomological analysis. We then classify all the algebras obtained through deformation of these algebras as well as all possible central extensions thereof. We propose/conjecture an extension of the Hochschild-Serre factorization theorem for infinite dimensional algebras as well as introducing a new notion of rigidity for family algebras obtained through deformation. We also explore physical realizations and significance of the family of algebras we obtain through the deformation procedure.
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed S…
We present the minimal realization of the $\ell$-conformal Galilei group in 2+1 dimensions on a single complex field. The simplest Lagrangians yield the complex Pais-Uhlenbeck oscillator equations. We introduce a minimal deformation of the $\ell$=1/2 conformal Galilei (a.k.a. Schrödinger) algebra and construct the corresponding invariant actions. Based on a new realization of the d=1 conformal group, we find a massive extension of the near-horizon Kerr-dS/AdS metric.
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed Snyder to Snyder space is constructed. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and $κ$-Minkowski in Maggiore-type realizations are discussed.
We study all possible deformations of the Maxwell algebra. In D=d+1\neq 3 dimensions there is only one-parameter deformation. The deformed algebra is isomorphic to so(d+1,1)\oplus so(d,1) or to so(d,2)\oplus so(d,1) depending on the signs of the deformation parameter. We construct in the dS(AdS) space a model of massive particle interacting with Abelian vector field via non-local Lorentz force. In D=2+1 the deformations depend on two parameters b and k. We construct a phase diagram, with two parts of the (b,k) plane with so(3,1)\oplus so(2,1) and so(2,2)\oplus so(2,1) algebras separated by a c…
Realizations of $κ$-Minkowski space linear in momenta are studied for time-, space- and light-like deformations. We construct and classify all such linear realizations and express them in terms of $\mathfrak{gl}(n)$ generators. There are three one-parameter families of linear realizations for time-like and space-like deformations, while for light-like deformations, there are only four linear realizations. The relation between deformed Heisenberg algebra, star product, coproduct of momenta and twist operator is presented. It is proved that for each linear realization there exists Drinfeld twist…
Covariance of a quantum space with respect to a quantum enveloping algebra ties the deformation of the multiplication of the space algebra to the deformation of the coproduct of the enveloping algebra. Since the deformation of the coproduct is governed by a Drinfeld twist, the same twist naturally defines a covariant star product on the commutative space. However, this product is in general not associative and does not yield the quantum space. It is shown that there are certain Drinfeld twists which realize the associative product of the quantum plane, quantum Euclidean 4-space, and quantum Mi…
Covariance ties the noncommutative deformation of a space into a quantum space closely to the deformation of the symmetry into a quantum symmetry. Quantum deformations of enveloping algebras are governed by Drinfeld twists, inner automorphisms which relate the deformed to the undeformed coproduct. While Drinfeld twists naturally define a covariant star product on the space algebra, this product is in general not associative and does not yield a quantum space. It is reported that, nevertheless, there are certain Drinfeld twists which realize the quantum plane, quantum Euclidean 4-space, and qua…
We give a realization of quantum affine Lie algebra $U_q(\hat A_{N-1})$ in terms of anyons defined on a two-dimensional lattice, the deformation parameter $q$ being related to the statistical parameter $ν$ of the anyons by $q = e^{iπν}$. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel fermionic construction of undeformed affine Lie algebra.
We give a realization of the quantum affine Lie algebras $\uqa$ and $\uqc$ in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter $q$ being related to the statistical parameter $ν$ of the anyons by $q = e^{iπν}$. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel fermionic construction of undeformed affine Lie algebras.
The quasiconformal method provides us with a unified approach to the construction of minimal unitary representations (minrep) of noncompact groups, their deformations as well as their supersymmetric extensions. We review the quasiconformal construction of the minrep of SO(d,2), its deformations and their applications to unitary realizations of AdS_{(d+1)}/CFT_d higher spin algebras and their deformations for arbitrary d and supersymmetric extensions for dimensions d less than seven. AdS_{(d+1)}/CFT_d higher spin algebras, their deformations and supersymmetric extensions are given by the envelo…
We study the asymptotic properties of kinks in connection with the deformation procedure. We show that, upon deformation of the field-theoretic model, the asymptotics of kinks can change or remain unchanged, depending on the properties of the deforming function. The cases of both explicit and implicit kinks are considered. In addition, we show that the the deformation procedure can be applied to the important case of implicit kinks. We also prove that for any kink with a power-law tail, the stability potential decreases as the inverse square of the coordinate. The physical consequences of the …
Explicit boundary conditions are given at spatial infinity for four-dimensional supergravity, which provide a realization of the super-BMS algebra of Awada, Gibbons and Shaw. The results are then generalized to the $N$- extended super-BMS algebras.
We discuss quantum deformation of the affine transformation algebra. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators.
We study all possible deformations of the Maxwell algebra. In D=d+1\neq 3 dimensions there is only one-parameter deformation. The deformed algebra is isomorphic to so(d+1,1)\oplus so(d,1) or to so(d,2)\oplus so(d,1) depending on the signs of the deformation parameter. We construct in the dS (AdS) space a model of massive particle interacting with Abelian vector field via non-local Lorentz force. In D=2+1 the deformations depend on two parameters b and k. We construct a phase diagram, with two parts of the (b,k) plane with so(3,1)\oplus so(2,1) and so(2,2)\oplus so(2,1) algebras separated by a …
A realization of the Yangian double with center $DY_\hbar(sl_2)_k$ of level $k(\not=0,-2)$ in terms of free boson fields is constructed. The screening currents are also presented, which commute with $DY_\hbar(sl_2)$ modulo total difference. In the $\hbar\to 0$ limit, the currents of Yangian double $DY_\hbar(sl_2)_k$ becomes the Feigin-Fuchs realization of affine Lie $sl(2)_k$, while the screening currents of Yangian double $DY_\hbar(sl_2)_k$ becomes the screening currents of the affine Lie algebra $sl(2)_k$.
We study another realization of the Kerr/CFT correspondence. By imposing new asymptotic conditions for the near horizon geometry of Kerr black hole, an asymptotic symmetry which contains all of the exact isometries can be obtained. In particular, the Virasoro algebra can be realized as an enhancement of SL(2,R) symmetry of the AdS geometry. By using this asymptotic symmetry, we discuss finite temperature effects and show the correspondence concretely.
We consider the change in the asymptotic behavior of solutions of the type of flat domain walls (i.e., kink solutions) in field-theoretic models with a real scalar field. We show that when the model is deformed by a bounded deforming function, the exponential asymptotics of the corresponding kink solutions remain exponential, while the power-law ones remain power-law. However, the parameters of these asymptotics, which are related to the wall thickness, can change.
Differential realizations in coordinate space for deformed Lie algebras with three generators are obtained using bosonic creation and annihilation operators satisfying Heisenberg commutation relations. The unified treatment presented here contains as special cases all previously given coordinate realizations of $so(2,1),so(3)$ and their deformations. Applications to physical problems involving eigenvalue determination in nonrelativistic quantum mechanics are discussed.
A general method based on the polynomial deformations of the Lie algebra sl(2,R) is proposed in order to exhibit the quasi-exactly solvability of specific Hamiltonians implied by quantum physical models. This method using the finite-dimensional representations and differential realizations of such deformations is illustrated on the sextic oscillator as well as on the second harmonic generation.
We investigate N-point string scattering amplitudes in AdS_3 space. Based on recent observations on the solutions of KZ and BPZ-type differential equations, we discuss how to describe the string theory in AdS_3 as a marginal deformation of a (flat) linear dilaton background. This representation resembles the called "discrete light-cone Liouville" realization as well as the FZZ dual description in terms of the sine-Liouville field theory. Consequently, the connection and differences between those and this realization are discussed. The free field representation presented here permits to underst…
A 3-parametric two-sided deformation of Heisenberg algebra (HA), with p,q-deformed commutator in the l.h.s. of basic defining relation and certain deformation of its r.h.s., is introduced and studied. The third deformation parameter μappears in an extra term in the r.h.s. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p,q,μ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor μ in the considered deformed HA inevitably depends explicitly on part…
We investigate the algebra of vector fields on the sphere. First, we find that linear deformations of this algebra are obstructed under reasonable conditions. In particular, we show that $hs[λ]$, the one-parameter deformation of the algebra of area-preserving vector fields, does not extend to the entire algebra. Next, we study some non-central extensions through the embedding of $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(S^2)$ into $\mathfrak{v}\mathfrak{e}\mathfrak{c}\mathfrak{t}(\mathbb{C}^*)$. For the latter, we discuss a three parameter family of non-central extensions which contain…
We investigate the deformations and rigidity of boundary Heisenberg-like algebras. In particular, we focus on the Heisenberg and $\text{Heisenberg}\oplus\mathfrak{witt}$ algebras which arise as symmetry algebras in three-dimensional gravity theories. As a result of the deformation procedure we find a large class of algebras. While some of these algebras are new, some of them have already been obtained as asymptotic and boundary symmetry algebras, supporting the idea that symmetry algebras associated to diverse boundary conditions and spacetime loci are algebraically interconnected through defo…
We study a Lie algebra type $κ$-deformed space with undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. Space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is …
A method to construct free field realizations for the form factors of diagonal factorized scattering theories is described. Form factors are constructed from linear functionals over an associative `form factor algebra', which in particular generate solutions of the deformed Knizhnik-Zamolodchikov equations with parameter $2π$. We show that there exists a unique deformation off the (`Rindler') value $2π$ that preserves the original $S$-matrix and which allows one to realize form factors as vector functionals over an algebra of generalized vertex operators.
We study realizations of polynomial deformations of the sl(2,R)- Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite- and infinite-dimensional representations. The linear, quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are studied in the context of these ``nonlinear'' Lie algebras and some examples dealing with quantum optics are pointed out.
We review the W_N algebra and its quantum deformation, based on free field realizations. The (quantum deformed) W_N algebra is defined through the (quantum deformed) Miura transformation, and its singular vectors realize the Jack (Macdonald) polynomials. (Talk at the Nankai-CRM joint meeting on ``Extended and Quantum Algebras and their Applications to Physics'', Tianjin, China, August 19-24, 1996.)
Geometric realization of stress-tensor deformed field theory
Center for Theoretical Physics and College of Physics, Jilin University（吉林大学理论物理中心及物理学院）；Institute of Fundamental Physics and Quantum Technology & School of Physical Science and Technology, Ningbo University（宁波大学基础物理与量子技术研究所及物理科学与技术学院）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））
We present Lie-algebraic deformations of Minkowski space with undeformed Poincare algebra. These deformations interpolate between Snyder and kappa-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and kappa-Minkowski in Maggiore-type realizations are discussed. Our construction leads to a new class of deformed special relativity theories.
By considering a set of $N$ anyonic oscillators ( non-local, intrinsic two-dimensional objects interpolating between fermionic and bosonic oscillators) on a two-dimensional lattice, we realize the $SU_q(N)$ quantum algebra by means of a generalized Schwinger construction. We find that the deformation parameter $q$ of the algebra is related to the anyonic statistical parameter $ν$ by $q=exp({\rm i}πν)$.
A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.
Asymptotically locally conical (ALC) metric of exceptional holonomy has an asymptotic circle bundle structure that accommodates the M theory circle in type IIA reduction. Taking Spin(7) metrics of cohomogeneity one as explicit examples, we investigate deformations of ALC metrics, in particular that change the asymptotic S^1 radius related to the type IIA string coupling constant. When the canonical four form of Spin(7) holonomy is taken to be anti-self-dual, the deformations of Spin(7) metric are related to the harmonic self-dual four forms, which are given by solutions to a system of first or…
The leading classical asymptotics of Virasoro conformal blocks on the Riemann sphere with n generic and n-3 "heavy" degenerate field insertions can be described in terms of the geometry of Garnier system describing the monodromy preserving deformations of second order Fuchsian differential equations on an n-punctured sphere. This allows us to characterise the leading classical asymptotics of Virasoro conformal blocks completely, and to clarify in which sense conformal field theory represents a quantisation of the isomonodromic deformation problem.
We recast the joint $J\bar{T}$, $T\bar{J}$ and $T\bar{T}$ deformations as coupling the original theory to a mixture of topological gravity and gauge theory. This geometrizes the general flow triggered by irrelevant deformations built out of conserved currents and the stress-energy tensor, by means of a path integral kernel. The partition function of the deformed theory satisfies a diffusion-like flow equation similar to that found in the pure $T\bar{T}$ case. Our proposal passes two stringent tests. Firstly, we recover the classical deformed actions from the kernel, reproducing the known expre…
A method for obtaining complex analytic realizations for a class of deformed algebras based on their respective deformation mappings and their ordinary coherent states is introduced. Explicit results of such realizations are provided for the cases of the $q$-oscillators ($q$-Weyl-Heisenberg algebra) and for the $su_{q}(2)$ and $su_{q}(1,1)$ algebras and their co-products. They are given in terms of a series in powers of ordinary derivative operators which act on the Bargmann-Hilbert space of functions endowed with the usual integration measure. In the $q\rightarrow 1$ limit these realizations …
Motivated by the representation of the super Virasoro constraints as generalized Dirac-K{ä}hler constraints $(d \pm d^\dagger)|ψ> = 0$ on loop space, examples of the most general continuous deformations $d \to e^{-W} d e^W$ are considered which preserve the superconformal algebra at the level of Poisson brackets. The deformations which induce the massless NS and NS-NS backgrounds are exhibited. Hints for a manifest realization of S-duality in terms of an algebra isomorphism are discussed. It is shown how the first order theory of 'canonical deformations' is reproduced and how the deformation o…
The quantum Euclidean space R_{q}^{N} is a kind of noncommutative space which is obtained from ordinary Euclidean space R^{N} by deformation with parameter q. When N is odd, the structure of this space is similar to R_{q}^{3}. Motivated by realization of R_{q}^{3} by differential operators in R^{3}, we give such realization for R_{q}^{5} and R_{q}^{7} cases and generalize our results to R_{q}^{N} (N odd) in this paper, that is, we show that the algebra of R_{q}^{N} can be realized by differential operators acting on C^{infinite} functions on undeformed space R^{N}.
Boundary flow and geometric realization in holographic $T\bar T$-deformed BCFT
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
We study the behaviour of a scalar field coupled to a noncommutative black hole which is described by a $κ$-cylinder Hopf algebra. We introduce a new class of realizations of this algebra which has a smooth limit as the deformation parameter vanishes. The twisted flip operator is independent of the choice of realization within this class. We demonstrate that the $R$-matrix is quasi-triangular up to the first order in the deformation parameter. Our results indicate how a scalar field might behave in the vicinity of a black hole at the Planck scale.
Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous spectrum. The general scheme of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schroedinger Hamiltonian shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation. An algorithm for the mapping of energy levels is described.
For all three--dimensional Lie algebras the construction of generators in terms of functions on 4-dimensional real phase space is given with a realization of the Lie product in terms of Poisson brackets. This is the classical Jordan--Schwinger map which is also given for the deformed algebras ${\cal {SL}}_{q}(2,\R)$, ${\cal E}_{q} (2)$ and ${\cal H}_q(1)$. The ${\cal U}_{q}(n)$ algebra is discussed in the same context.
We determine the one-loop deformation of the conformal symmetry of a general N}=2 superconformally invariant Yang-Mills theory. The deformation is computed for several explicit examples which have a realization as world-volume theories on a stack of D3 branes. These include (i) N=4 SYM with gauge groups SU(N), USp(2N) and SO(N); (ii) USp(2N) gauge theory with one hypermultiplet in the traceless antisymmetric representation and four hypermultiplets in the fundamental; (iii) quiver gauge theory with gauge group SU(N)xSU(N) and two hypermultiplets in the bifundamental representations (N,\bar N) a…
Finsler geometry is a well known generalization of Riemannian geometry which allows to account for a possibly non trivial structure of the space of configurations of relativistic particles. We here establish a link between Finsler geometry and the sort of models with curved momentum space and DSR-relativistic symmetries which have been recently of interest in the quantum-gravity literature. We use as case study the much-studied scenario which is inspired by the $κ$-Poincaré quantum group, and show that the relevant deformation of relativistic symmetries can be implemented within a Finsler geom…
We generalize some results concerning affine algebras at the critical level to the corresponding quantum algebras. In particular, we show that the Wakimoto realization provides a homomorphism of Poisson algebras from the center of a quantum affine algebra to a Heisenberg-Poisson algebra. This homomorphism is a q-deformation of the Miura transformation. It is given by the same formulas as the spectra of transfer-matrices of the corresponding quantum integrable models. The image of the center in the Heisenberg-Poisson algebra is a Poisson subalgebra of the latter, which is a q-deformation of the…
Deformed $\W$--algebra $\W_{q,t}(\g)$ associated to an arbitrary simple Lie algebra $\g$ is defined together with its free field realizations and the screening operators. Explicit formulas are given for generators of $\W_{q,t}(\g)$ when $\g$ is of classical type. These formulas exhibit a deep connection between $\W_{q,t}(\g)$ and the analytic Bethe Ansatz in integrable models associated to quantum affine algebras $U_q(\G)$ and $U_t(\GL)$. The scaling limit of $\W_{q,t}(\g)$ is closely related to affine Toda field theories.
We show that the CFT with symmetry group $G_{k_1}\times G_{k_2}\times \cdots \times G_{k_n}$ consisting of WZW models based on the same group $G$, but at arbitrary integer levels, admits an integrable deformation depending on $2(n-1)$ continuous parameters. We derive the all-loop effective action of the deformed theory and prove integrability. We also calculate the exact in the deformation parameters RG flow equations which can be put in a particularly simple compact form. This allows a full determination and classification of the fixed points of the RG flow, in particular those that are IR st…
We conjecture that Vasiliev's theory of higher spin gravity in four-dimensional de Sitter space (dS) is holographically dual to a three-dimensional conformal field theory (CFT) living on the spacelike boundary of dS at future timelike infinity. The CFT is the Euclidean Sp(N) vector model with anticommuting scalars. The free CFT flows under a double-trace deformation to an interacting CFT in the IR. We argue that both CFTs are dual to Vasiliev dS gravity but with different future boundary conditions on the bulk scalar field. Our analysis rests heavily on analytic continuations of bulk and bound…
Superinsulators are dual superconductors, dissipationless magnetic monopole condensates with infinite resistance. The long-distance field theory of such states of matter is QED with dynamical matter coupled via a compact BF topological interaction. We will quantize the 2D model in the functional Schrödinger picture and show how strong entanglement of charges leads to a phase which is a single-color, asymptotically free version of QCD in which the infinite resistance is caused by the linear confinement of charges. This phase has been experimentally detected in TiN, NbTiN and InO thin films, inc…
We discuss a marginal deformation of the SL(2,R) x SU(2) x U(1)^4 WZW model, which describes string theory on AdS_3 x S^3 x T^4, that corresponds to warping the S^3 factor. This deformation breaks part of the N=(4,4) supersymmetry of the undeformed dual CFT to N=(0,4) supersymmetry. In the spirit of work by Giveon, Kutasov, and Seiberg, we construct the asymptotic spacetime symmetry algebra from worldsheet operators and find a restoration of (4,4) supersymmetry at discrete values of the deformation parameter. We explain this result from various perspectives: the worldsheet, supergravity, and f…
We show that some factors of the universal R-matrix generate a family of twistings for the standard Hopf structure of any quantized contragredient Lie (super)algebra of finite growth. As an application we prove that any two isomorphic superalgebras with different Cartan matrices have isomorphic q-deformations (as associative superalgebras) and their standard comultiplications are connected by such twisting. We present also an explicit relation between the generators of the second Drinfeld's realization and Cartan-Weyl generators of quantized affine nontwisted Kac-Moody algebras. Further develo…
