The Leigh-Strassler family of N=1 marginal deformations of the N=4 SYM theory admits a Hopf algebra symmetry which is a quantum group deformation of the SU(3) part of the R-symmetry of the Ncal=4 theory. We investigate how this quantum symmetry might be expressed on the gravity side of the AdS/CFT correspondence. First, we discuss the twist leading to the Hopf algebra structure for the well-known beta-deformation as well as a unitarily equivalent theory that we call the w-deformation. We then show how this Hopf twist can be used to define a star product between the three scalar superfields of these theories which encodes the deformed global symmetry. Turning to the gravity side, we adapt this star product to deform the pure spinors of six-dimensional flat space in its generalised geometry description. This leads to an N=2 NS-NS solution of IIB supergravity. Starting from this precursor solution, adding D3-branes and taking the near-horizon limit reproduces the dual gravitational solution to the above theories, first derived by Lunin and Maldacena using TsT techniques. This indicates that the Hopf algebra symmetry can play a useful role in constructing the supergravity duals of the general Leigh-Strassler deformations.
We unify k-Poincare algebra and k-Minkowski spacetime by embeding them into quantum phase space. The quantum phase space has Hopf algebroid structure to which we apply the twist in order to get k- deformed Hopf algebroid structure and k-deformed phase space. We explicitly construct k-Poincare-Hopf algebra and k-Minkowski spacetime from twist. It is outlined how this construction can be extended to k-deformed super algebra including exterior derivative and forms. Our results are relevant for constructing physical theories on noncommutative spacetime by twisting Hopf algebroid phase space struct…
We construct five new quantum Newton-Hooke Hopf algebras with the use of Abelian twist procedure. Further we demonstrate that the corresponding deformed space-times with quantum space and classical time are periodic or expanding in time.
We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $\mathbb{H}$ and quantum Poincare Hopf group $\widehat{\mathbb{G}}$. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one $% \mathcal{H}^{(10,10)}$ describing dynamics on quantum group algebra $% \widehat{\mathbb{G}}$ provided by the Heisenberg double algebra $\mathcal{HD=% }\mathbb{H}\rtimes \widehat{\mathbb{G}}$, and second, denoted by $\mathcal{% …
Lead by examples we introduce the notions of Hopf algebra and quantum group. We study their geometry and in particular their Lie algebra (of left invariant vectorfields). The examples of the quantum sl(2) Lie algebra and of the quantum (twisted) Poincare Lie algebra iso_θ(3,1) are presented.
The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The $κ$-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are infinitely many such realizations related by similarity transformations. For a given realization we construct corresponding coproducts of commutative coordinates and momenta (bialgebroid structure). The $κ$-deformed phase space has twisted Hopf algebroid structure. General method for the construction of twist operator (satisfying cocycle and normalization condition) c…
We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT.
In this article we discus the energy-momentum conservation principle for two-particle system in the case of canonically and Lie-algebraically twist-deformed Galilei Hopf algebra. Particularly, we provide consistent with the coproducts energy and momentum addition law as well as its symmetric with respect the exchange of particles counterpart. Besides, we show that the vanishing of total fourmomentum for two Lie-algebraically deformed kinematical models leads to the discret values of energies and momenta only in the case of the symmetrized addition rules.
We twist the Hopf algebra of igl(n,R) to obtain the kappa-deformed spacetime coordinates. Coproducts of the twisted Hopf algebras are explicitly given. The kappa-deformed spacetime obtained this way satisfies the same commutation relation as that of the conventional kappa-Minkowski spacetime, but its Hopf algebra structure is different from the well known kappa-deformed Poincare algebra in that it has larger symmetry algebra than the kappa-Minkowski case. There are some physical models which consider this symmetry. Incidentally, we obtain the canonical (theta-deformed) non-commutative spacetim…
We consider two twist operators that lead to kappa-Poincare Hopf algebra, the first being an Abelian one and the second corresponding to a light-like kappa-deformation of Poincare algebra. The advantage of the second one is that it is expressed solely in terms of Poincare generators. In contrast to this, the Abelian twist goes out of the boundaries of Poincare algebra and runs into envelope of the general linear algebra. Some of the physical applications of these two different twist operators are considered. In particular, we use the Abelian twist to construct the statistics flip operator comp…
I discuss motivations for introducing Hopf algebra symmetries in noncommutative field theories and briefly describe twisting of main symmetry transformations. New results include an extended list of twisted gauge invariants (which may help to overcome the problem of inconsistency of equations of motion) and a gauge-covariant twist operator (leading to a gauge-covariant star product).
We describe the twist quantization of string worldsheet theory, which unifies the description of quantization and the target space symmetry, based on the twisting of Hopf and module algebras. We formulate a method of decomposing a twist into successive twists to analyze the twisted Hopf and module algebra structure, and apply it to several examples, including finite twisted diffeomorphism and extra treatment for zero modes.
We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry …
Ten Abelian twist deformations of acceleration-enlarged Newton-Hooke Hopf algebra are considered. The corresponding quantum space-times are derived as well. It is demonstrate that their contraction limit $τ\to \infty$ leads to the new twisted acceleration-enlarged Galilei spaces.
The three new deformed Poincare Hopf algebras are constructed with use of twist procedure. The corresponding relativistic space-times providing the sum of canonical and Lie-algebraic type of noncommutativity are proposed. Finally, the nonrelativistic contraction limits to the corresponding Galilei Hopf algebras are performed.
Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion of twist is analyzed for $κ$-deformed phase space in Hopf algebroid setting. It is outlined how the twist in the Hopf algebroid setting reproduces the full Hopf algebra structure of $κ$-Poincaré algebra. Several examples of realizations are worked out in details.
In quantum groups coproducts of Lie-algebras are twisted in terms of generators of the corresponding universal enveloping algebra. If representations are considered, twists also serve as starproducts that accordingly quantize representation spaces. In physics, requirements turn out to be the other way around. Physics comes up with noncommutative spaces in terms of starproducts that miss a suiting quantum symmetry. In general the classical limit is known, i.e. there exists a representation of the Lie-algebra on a corresponding finitely generated commutative space. In this setup quantization can…
We provide fifteen twist-deformed doubly enlarged Newton-Hooke quantum space-times. In $τ$ approaching infinity limit the twisted doubly enlarged Galilei spaces are obtained as well.
We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the …
We examine nonlinear sigma models, in particular the Skyrme model with a twist (the twisted Skyrmion string), which comprises a vortex solution with an added dependence on a twist term $mkz$, where $z$ is the vertical coordinate. The topological and Hopf charges of a twisted Skyrmion string are calculated and are shown to be equivalent to the winding number $n$ of the vortex and proportional to $nmk$, respectively. The principle of conservation of the Hopf charge, which is a standard property of the baby Skyrmion model, is generalised to cover the non-compact twisting solutions studied here.
We investigate the Hopf algebra structure in string worldsheet theory and give a unified formulation of the quantization of string and the space-time symmetry. We reformulate the path integral quantization of string as a Drinfeld twist at the worldsheet level. The coboundary relation shows that the Drinfeld twist defines a module algebra which is equivalent to operators with normal ordering. Upon applying the twist, the space-time diffeomorphism is deformed into a twisted Hopf algebra, while the Poincaré symmetry is unchanged. This suggests a characterization of the symmetry: unbroken symmetri…
This is an introduction to work on the generalisation to quantum groups of Mackey's approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces $\CH^2(H,A)$ which classify them. They form a new kind of topological quantum number in physics which is visible only in the quantum world. These same cross product quantisations can also be viewed as trivial quantum principal bundles in quantum group gauge theory. We also relate this…
By correctly identifying the role of central extension in the centrally extended Heisenberg algebra h, we show that it is indeed possible to construct a Hopf algebraic structure on the corresponding enveloping algebra U(h) and eventually deform it through Drinfeld twist. This Hopf algebraic structure and its deformed version U^F(h) are shown to be induced from a more fundamental Hopf algebra obtained from the Schroedinger field/oscillator algebra and its deformed version, provided that the fields/oscillators are regarded as odd-elements of the super-algebra osp(1|2n). We also discuss the possi…
We investigate the quasi-Hopf twist of the quantum toroidal algebra of $\mathfrak{gl}_1$ as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter $p$ is identified as the elliptic modulus. Computing the quasi-Hopf twist of the $R$ matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on $\mathbb{R}^4 \times T^2$. The same $R$ matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum…
Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making possible the quantization are solved. The spectrum of the single-particle Hamiltonians is computed. The multi-particle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d=2 dimensions the rotational invariance is preserved, while in d=3 the so(3) rotational invariance is broken down to an so(2) invariance.
In this article we will use the Drinfeld twist leading to light-like $κ$-deformations of Poincaré algebra. We shall apply the standard Hopf algebra methods in order to define the star-product, which shall be used to formulate a scalar field theory compatible with $κ$-Poincaré-Hopf algebra. Using this approach we show that there is no problem with formulating integration on $κ$-Minkowski space and no need for introducing a new measure. We have shown that the $\star$-product obtained from this twist enables us to define a free scalar field theory on $κ$-Minkowski space that is equivalent to a co…
A generalisation of discrete torsion is introduced in which different discrete torsion phases are considered for the different fixed points or twist fields of a twisted sector. The constraints that arise from modular invariance are analysed carefully. As an application we show how all the different resolutions of the T^7/Z_2^3 orbifold of Joyce have an interpretation in terms of such generalised discrete torsion orbifolds. Furthermore, we show that these manifolds are pairwise identified under G_2 mirror symmetry. From a conformal field theory point of view, this mirror symmetry arises from an…
We show that the crossed modules and bicovariant different calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisation of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra $\C[x]\bicross_{\hbar,\grav}\C[p]$. This is a quantum group which has an $\hbar\to 0$ limit as the functions on a classical but non-Abelian group and a $\grav\to 0$ limit as flat spac…
Using normal coordinates in a Poincaré-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.
In this talk I discuss a recently developed "Unfolded Quantization Framework". It allows to introduce a Hamiltonian Second Quantization based on a Hopf algebra endowed with a coproduct satisfying, for the Hamiltonian, the physical requirement of being a primitive element. The scheme can be applied to theories deformed via a Drinfeld twist. I discuss in particular two cases: the abelian twist deformation of a rotationally invariant nonrelativistic Quantum Mechanics (the twist induces a standard noncommutativity) and the Jordanian twist of the harmonic oscillator. In the latter case the twist in…
Introducing a non-compact version of the Hopf map, we demonstrate remarkable close relations between quantum Hall effect and twistor theory. We first construct quantum Hall effect on a hyperboloid based on the noncompact 2nd Hopf map of split-quaternions. We analyze a hyperbolic one-particle mechanics, and explore many-body problem, where a many-body groundstate wavefunction and membrane-like excitations are derived explicitly. In the lowest Landau level, the symmetry is enhanced from $SO(3, 2)$ to the $SU(2, 2)$ conformal symmetry. We point out that the quantum Hall effect naturally realizes …
In a previous paper, we investigated the Hopf algebra structure in string theory and gave a unified formulation of the quantization of the string and the space-time symmetry. In this paper, this formulation is applied to the case with a nonzero B-field background, and the twist of the Poincare symmetry is studied. The Drinfeld twist accompanied by the B-field background gives an alternative quantization scheme, which requires a new normal ordering. In order to obtain a physical interpretation of this twisted Hopf algebra structure, we propose a method to decompose the twist into two successive…
In previous work, starting from the Moyal plane, we formulated interacting theories of matter and gauge fields with only the former fields twisted. In this approach, gauge theories, including the standard model, can be formulated without new gauge degrees of freedom. We show their underlying symmetry algebra to be Poincaré quasi-Hopf . The associated spacetime algebra is hence non-associative.
The properties of Hopf star operations and twisted Hopf stars operations on quantum groups are discussed in relation with the theory of representations (star representations). Invariant Hermitian sesquilinear forms (scalar products) on modules or module-algebras are then defined and analyzed. Particular attention is paid to scalar products that can be associated with the Killing form (when it exists) or with the left (or right) invariant integrals on the quantum group. Our results are systematically illustrated in the case of a family of non semi-simple and finite dimensional quantum groups th…
We show that an $L_\infty$-algebra can be extended to a graded Hopf algebra with a codifferential. Then we twist this extended $L_\infty$-algebra with a Drinfel'd twist, simultaneously twisting its modules. Taking the $L_\infty$-algebra as its own (Hopf) module, we obtain the recently proposed braided $L_\infty$-algebra. The Hopf algebra morphisms are identified with the strict $L_\infty$-morphisms, while the braided $L_\infty$-morphisms define a more general $L_\infty$-action of twisted $L_\infty$-algebras.
We compare two ways of force terms generating in the model of nonrelativistic particle moving in the presence of constant field force $\vec{F}$. First of them uses the twist-deformed acceleration-enlarged Newton-Hooke quantum space-times while the second one incorporates the doubly enlarged Newton-Hooke transformations of classical space. Particulary, we find the conditions for which the both treatments provide the same force terms.
In this paper, we study the perturbative aspects of a "B-twisted" two-dimensional $(0,2)$ heterotic sigma model on a holomorphic gauge bundle $\mathcal E$ over a complex, hermitian manifold $X$. We show that the model can be naturally described in terms of the mathematical theory of ``Chiral Differential Operators". In particular, the physical anomalies of the sigma model can be reinterpreted as an obstruction to a global definition of the associated sheaf of vertex superalgebras derived from the free conformal field theory describing the model locally on $X$. In addition, one can also obtain …
The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al., Felder). Fronsdal made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebra U_q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universal R matrix of U_q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. …
We introduce the quasi-Hopf superalgebras which are $Z_2$ graded versions of Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is the face type $B_{q,λ}(G)$ and the vertex type $A_{q,p}[\hat{sl(n|n)}]$ (and $A_{q,p}[\hat{gl(n|n)}]$), where $G$ is any Kac-…
We show how to construct, starting from a quasi-Hopf (super)algebra, central elements or Casimir invariants. We show that these central elements are invariant under quasi-Hopf twistings. As a consequence, the elliptic quantum (super)groups, which arise from twisting the normal quantum (super)groups, have the same Casimir invariants as the corresponding quantum (super)groups.
We provide the all-loop structure of gauge-variant operators required for the renormalisation of Green's functions with insertions of twist-two operators in Yang-Mills theory. Using this structure we work out an explicit basis valid up to 4-loop order for an arbitrary compact simple gauge group. To achieve this we employ a generalised gauge symmetry, originally proposed by Dixon and Taylor, which arises after adding to the Yang-Mills Lagrangian also operators proportional to its equation of motion. Promoting this symmetry to a generalised BRST symmetry allows to generate the ghost operator fro…
This is the second in a series of papers discussing in the framework of gerbe theory canonical and geometric aspects of the 2d nonlinear sigma model in the presence of conformal defects in the worldsheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of rigid symmetries of the sigma model is carried out with emphasis on algebraic structures on generalised tangent bundles over the target space of the theory and over its state space that give rise to a realisation of the symmetry algebra on states. The analysis leads to a proposal…
We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting nonassociativity is characterised by the associator 3-cocycle which cont…
Given a Hopf algebra H, we study modules and bimodules over an algebra A that carry an H-action, as well as their morphisms and connections. Bimodules naturally arise when considering noncommutative analogues of tensor bundles. For quasitriangular Hopf algebras and bimodules with an extra quasi-commutativity property we induce connections on the tensor product over A of two bimodules from connections on the individual bimodules. This construction applies to arbitrary connections, i.e. not necessarily H-equivariant ones, and further extends to the tensor algebra generated by a bimodule and its …
We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed $D=4$ quantum inhomegeneous conformal Hopf algebras $\mathcal{U}_{θ}(su(2,2)\ltimes T^{4}$) and $\mathcal{U}_{\barθ}(su(2,2)\ltimes\bar{T}^{4}$), where $T^{4}$ describe complex twistor coordinatesand $\bar{T}^{4}$ the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently we introduce the quantum deformations of $D=4$ Heisenberg-conformal algebra (H…
We analyze bicovariant differential calculus on $κ$-Minkowski spacetime. It is shown that corresponding Lorentz generators and noncommutative coordinates compatible with bicovariant calculus cannot be realized in terms of commutative coordinates and momenta. Furthermore, $κ$-Minkowski space and NC forms are constructed by twist related to a bicrossproduct basis. It is pointed out that the consistency condition is not satisfied. We present the construction of $κ$-deformed coordinates and forms (super-Heisenberg algebra) using extended twist. It is compatible with bicovariant differential calcul…
Transformation of operator algebras under Hopf algebra twist is studied. It is shown that that adjoint module algebras are stable under the twist. Applications to vector fields on non-commutative space-time are considered.
We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twis…
Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as underlying module algebras (=quantum spaces). Smash product construction combines these two into the new algebra which, in fact, does not depend on the twist. However, we can turn it into bialgebroid in the twist dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both techniques indicated in the title: twisting of a bialgebroid or …
