We discuss the construction of $κ$-Poincaré invariant actions for gauge theories on $κ$-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the $κ$-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for gauge theories. We show that within a natural differential calculus based on a distinguished set of twisted derivations, d=5 is the unique value for the classical dimension at which the gauge action supports both the gauge invariance and the $κ$-Poincaré invariance. Within standard (untwisted) differential calculi, we show that the full gauge invariance cannot be achieved, although an invariance under a group of transformations constrained by the modular (Tomita) operator stemming from the $κ$-Poincaré invariance still holds.
Algebraic properties of the BRST symmetry associated to the twisted gauge symmetry occurring in the $κ$-Poincaré invariant gauge theories on the $κ$-Minkowski space are investigated. We find that the BRST operation associated to the gauge invariance of the action functional can be continuously deformed together with its corresponding Leibniz rule, into a nilpotent twisted BRST operation, leading to a twisted BRST symmetry algebra which may be viewed as a noncommutative analog of the usual Yang-Mills BRST algebra.
Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by a Jordanian twist built up from dilatation and momenta generators. The second is the light-like $κ$-deformation of the Poincare algebra extended to the conformal algebra, obtained by a twist corresponding to the extended Jordanian r-matrix. The $κ$-Minkowski spacetime is covariant quantum space under both of these deformations. The extension of the conformal algebra by the noncommutative coordinates is presented in two cases. The differential realizations for $κ$-Minkows…
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…
Two one-parameter families of twists providing kappa-Minkowski * -product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfeld's twisting tensor technique. The other one relies on an appropriate extension of "deformed realizations" of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us t…
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 present the star-product algebra of the kappa-deformation of Minkowski space and the formulation of Poincare covariant differential calculus. We use these tools to construct a twisted K-cycle over the algebra and a twisted cyclic cocycle.
$κ$-deformed commutation relation between quantum operators is constructed via abelian twist deformation in $κ$-Minkowski spacetime. The commutation relation is written in terms of universal $R$-matrix satisfying braided statistics. The equal-time commutator function turns out to vanish in this framework.
We study four dimensional $κ$-Minkowski spacetime constructed by the twist deformation of $U(igl(4,R))$. We demonstrate that the differential structure of such twist-deformed $κ$-Minkowski spacetime is closed in four dimensions contrary to the construction of $κ$-Poincaré bicovariant calculus which needs an extra fifth dimension. Our construction holds in arbitrary dimensional spacetimes.
We review the application of twist deformation formalism and the construction of noncommutative gauge theory on $κ$-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider ${U}(1)$ gauge theory. Different methods of obtaining a gauge invariant action and related problems are thoroughly discussed.
The differential structure on the kappa-Minkowski spacetime from Jordanian twist of Weyl algebra is constructed, and it is shown to be closed in 4-dimensions in contrast to the conventional formulation. Based on this differential structure, we have formulated a scalar field theory in this kappa-Minkowski spacetime.
In this paper, we investigate the twisted algebra of the fermionic oscillators associated with Dirac field defined in $κ$-Minkowski space-time. Starting from $κ$-deformed Dirac theory, which is invariant under the undeformed $κ$-Poincare algebra, using the twisted flip operator, we derive the deformed algebra of the creation and annihilation operators corresponding to the Dirac field quanta in $κ$-Minkowski space-time. In the limit $a\rightarrow 0$, the deformed algebra reduces to the commutative result.
We consider two new classes of twisted D=4 quantum Poincaré symmetries described as the dual pairs of noncocommutative Hopf algebras. Firstly we investigate a two-parameter class of twisted Poincaré algebras which provide the examples of Lie-algebraic noncommutativity of the translations. The corresponding associative star-products and new deformed Lie-algebraic Minkowski spaces are introduced. We discuss further the twist deformations of Poincaré symmetries generated by the twist with its carrier in Lorentz algebra. We describe corresponding deformed Poincaré group which provides the quadrati…
An algebraic characterization of vacuum states in Minkowski space is given which relies on recently proposed conditions of geometric modular action and modular stability for algebras of observables associated with wedge-shaped regions. In contrast to previous work, continuity properties of these algebras are not assumed but derived from their inclusion structure. Moreover, a unique continuous unitary representation of spacetime translations is constructed from these data. Thus the dynamics of relativistic quantum systems in Minkowski space is encoded in the observables and state and requires n…
Using the twist deformation of $U(igl(4,R))$, the linear part of the diffeomorphism, we define a scalar function and construct a free scalar field theory in four-dimensional $κ$-Minkowski spacetime. The action in momentum space turns out to differ only in integration measure from the commutative theory.
We present a spectral triple for $κ$-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the $κ$-Poincaré algebra. The weight satisfies a KMS condition and its associated modular operator plays an important role in the construction. This forces us to introduce two ingredients which have a modular flavor: the first is a twisted commutator, used to obtain a boundedness condition for the Dirac operator, the second is a weight replacing the usual operator trace, used to measure the growth o…
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…
Kappa-Minkowski space-time is an example of noncommutative space-time with potentially interesting phenomenological consequences. However, the construction of field theories on this space, although operationally well-defined, is plagued with ambiguities. A part of ambiguities can be resolved by clarifying the geometrical picture of gauge transformations on the kappa-Minkowski space-time. To this end we use the twist approach to construct the noncommutative U(1) gauge theory coupled to fermions. However, in this approach we cannot maintain the kappa-Poincaré symmetry; the corresponding symmetry…
We consider the issue of statistics for identical particles or fields in kappa-deformed spaces, where the system admits a symmetry group G. We obtain the twisted flip operator compatible with the action of the symmetry group, which is relevant for describing particle statistics in presence of the noncommutativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the ordering prescription.
Kappa-Minkowski space-time is an example of noncommutative space-time with potentially interesting phenomenology. However, the construction of field theories on this space is plagued with ambiguities. We propose to resolve certain ambiguities by clarifying the geometrical picture of gauge transformations on the kappa-Minkowski space-time in the twist approach. We construct the action for the noncommutative U(1) gauge fields in a geometric way, as an integral of a maximal form. The effective action with the first order corrections in the deformation parameter is obtained using the Seiberg-Witte…
Recently the anomalous dimension of twist two operators in N=4 SYM theory was computed by Gubser, Klebanov and Polyakov in the limit of large 't Hooft coupling using semi-classical rotating strings in AdS_5. Here we reproduce their results for large angular momentum by using the cusp anomaly of Wilson loops in Minkowski signature also computed within the AdS/CFT correspondence. In this case the anomalous dimension is related to an Euclidean worldsheet whose properties are completely determined by the symmetries of the problem. This gives support to the proposed identification of rotating strin…
Within the context of the twisted Poincaré algebra, there exists no noncommutative analogue of the Minkowski space interpreted as the homogeneous space of the Poincaré group quotiented by the Lorentz group. The usual definition of commutative classical fields as sections of associated vector bundles on the homogeneous space does not generalise to the noncommutative setting, and the twisted Poincaré algebra does not act on noncommutative fields in a canonical way. We make a tentative proposal for the definition of noncommutative classical fields of any spin over the Moyal space, which has the d…
Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski x two-dimensional black hole. This model may be thought of as an SL(2,R)/U(1) version of Gepner models and corresponds to a conifold point on the moduli space of Calabi-Yau compactifications. We directly deal with N=2, c=9 unitary superconformal characters. Modular invariance is achieved by requiring the string to have a momentum along an extra noncompact direction, in agreement with the picture of singular CFTs advocated by …
A bicovariant calculus on the twisted inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type, and on the corresponding quantum planes, is found by means of a projection from the bicovariant calculus on $B_{n+1}$, $C_{n+1}$, $D_{n+1}$. In particular we obtain the bicovariant calculus on a dilatation-free $q$-Poincaré group $ISO_q (3,1)$, and on the corresponding quantum Minkowski space. The classical limit of the $B_n,C_n,D_n$ bicovariant calculus is discussed in detail.
Freed-Hopkins-Teleman expressed the Verlinde algebra as twisted equivariant K-theory. We study how to recover the full system (fusion algebra of defect lines), nimrep (cylindrical partition function), etc of modular invariant partition functions of conformal field theories associated to loop groups. We work out several examples corresponding to conformal embeddings and orbifolds. We identify a new aspect of the A-D-E pattern of SU(2) modular invariants.
Classification of differential forms on $κ$-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the $κ$-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differe…
We compute explicitly a star product on the Minkowski space whose Poisson bracket is quadratic. This star product corresponds to a deformation of the conformal spacetime, whose big cell is the Minkowski spacetime. The description of Minkowski space is made in the twistor formalism and the quantization follows by substituting the classical conformal group by a quantum group.
We show that the realizations of noncommutative coordinates that are linear in the Lorentz generators form a closed Lie algebra under certain conditions. The star product and the coproduct for the momentum generators are obtained for these Lie algebras and the corresponding twist satisfies the cocycle and normalization conditions. We also obtain the twisted flip operator and the $\mathcal R$-matrix that define the statistics of particles or quantum fields propagating in these noncommutative spacetimes. The Lie algebra obtained in this work contains a special case which has been used in the lit…
We employ the modification of the basic Penrose formula in twistor theory, which allows to introduce commuting composite space-time coordinates. It appears that in the course of such modification the internal symmetry SU(2) of two-twistor system is broken to U(1). We consider the symplectic form on two-twistor space, permitting to interpret its 16 real components as a phase-space. After a suitable change of variables such a two-twistor phase space is split into three mutually commuting parts, describing respectively the standard relativistic phase space (8 degrees of freedom), the spin sector …
Unified graded differential algebra, generated by $κ$-Minkowski noncommutative (NC) coordinates, Lorentz generators and anticommuting one-forms, is constructed. It is compatible with $κ$-Poincaré-Hopf algebra. For time- and space-like deformations, the super-Jacobi identities are not satisfied. By introducing additional generator, interpreted as exterior derivative, we find a new unique algebra that satisfies all super-Jacobi identities. It is universal and valid for all type of deformations (time-, space-, and light-like). For time-like deformations this algebra coincides with the one in \cit…
$κ$-Poincaré invariant gauge theories on $κ$-Minkowski space-time, which are noncommutative analogs of the usual $U(1)$ gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the $κ$-Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermitic…
We provide an explicit twistorial construction of quaternion-Kahler manifolds obtained by deformation of c-map spaces and carrying an isometric action of the modular group SL(2,Z). The deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing NS5-brane instanton effects in string compactifications with N=2 supersymmetry. In this context the modular invariant parametrization of twistor lines found in this work yields the complete non-perturbative mirror map between type IIA and type IIB physical fields.
We investigate the implications for fermion mass models in heterotic orbifolds of the modular symmetry mixing twisted states localized at different fixed points. We show that, unlike in the case of continuous gauge symmetries, the mass eigenstates do not mix under the symmetry; thus coupling constants in the low-energy theory are invariant functions of the moduli.
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…
In this paper we study the deformed statistics and oscillator algebras of quantum fields defined in $κ$-Minkowski spacetime. The twisted flip operator obtained from the twist associated with the star product requires an enlargement of the Poincaré algebra to include the dilatation generators. Here we propose a novel notion of a fully covariant flip operator and show that to the first order in the deformation parameter it can be expressed completely in terms of the Poincaré generators alone. The $R$-matrices corresponding to the twisted and the covariant flip operators are compared up to first …
We demonstrate how to find modular discrete symmetry groups for $Z_N$ orbifolds. The $Z_7$ orbifold is treated in detail as a non-trivial example of a $(2,2)$ orbifold model. We give the generators of the modular group for this case which, surprisingly, does not contain $\sltz^3$ as had been speculated. The treatment models with discrete Wilson lines is also discussed. We consider examples which demonstrate that discrete Wilson lines affect the modular group in a non-trivial manner. In particular, we show that it is possible for a Wilson line to break $SL(2,{\bf Z})$.
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…
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…
We consider a class of non-unitary Toda theories based on the Lie superalgebras $A^{(1)}(n,n)$ in two-dimensional Minkowski spacetime, which can be twisted into a topological sector. In particular we study the simplest example with $n=1$ and real fields, and show how this theory can be treated as an integrable perturbation of the $A(1,0)$ superconformal model. We construct the conserved currents and the vertex operators which are chiral primary fields in the conformal theory. We then define the chiral ring of the $A^{(1)}(1,1)$ Toda theory and compute topological correlation functions in the t…
It is shown that deformations of twistor space compatible with the Moyal deformation of Minkowski space-time must take the form recently suggested by Kapustin, Kuznetsov and Orlov.
We obtain the action for a curved superconformal abelian M5 brane with the background R-symmetry gauge field turned on. We then restrict ourselves to superconformal M5 brane on a sphere times flat Minkowski space. We choose R-symmetry SO(1,4) instead of SO(5), which enables us to partially twist on Minkowski space and replace it by some curved Lorentzian manifold. We obtain M5 brane actions on $M_{1,1} \times S^4$ and $M_{1,2} \times S^3$ where actions and all fields, including the background gauge field, are real. Dimensional reduction along time gives real 5d SYM actions with nonabelian gene…
Massless particle dynamics in $D=10$ Minkowski space is given an $E_7$-covariant formulation, including both space-time and twistor variables. $E_7$ contains the conformal algebra as a subalgebra. Analogous constructions apply to $D=3,4$ and 6.
In the 5-dimensional twisted $U(1)$-like gauge theory on $κ$-Minkowski, the one-loop one-point (tadpole) function was computed in arXiv:2107.14462. This article summarizes the construction of such a gauge theory and discusses the non-vanishing of the tadpole.
We show that rotating p-brane solutions admit an analytical continuation to become twisted Sp-branes. Although a rotating p-brane has a naked singularity for large angular momenta, the corresponding S-brane configuration is regular everywhere and exhibits a smooth bounce between two phases of Minkowski spacetime. If the foliating hyperbolic space of the transverse space is of even dimension, such as for the twisted SM5-brane, then for an appropriate choice of parameters the solution smoothly flows from a warped product of two-dimensional de Sitter spacetime, five-dimensional Euclidean space an…
It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An explicit algebraic characterization of the kernel is given. It is also shown that the conductor, i.e. the order of the Dehn-twist is bounded by a function of the number of primary fields, allowing for a systematic enumeration of the modular representations coming from RCFTs. Restrictions on the spectrum of the Dehn-twist and arithmetic properties of modular matrix elements are presented.
Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N=4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M_{24}, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the …
We present a deformation of the Minkowski space as embedded into the conformal space (in the formalism of twistors) based in the quantum versions of the corresponding kinematic groups. We compute explicitly the star product, whose Poisson bracket is quadratic. We show that the star product although defined on the polynomials can be extended differentiably. Finally we compute the Eucliden and Minkowskian real forms of the deformation.
The most basic structure of chiral conformal field theory (CFT) is the Verlinde ring. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFT's associated to loop groups, as twisted equivariant K-theory. We build on their work to express K-theoretically the structures of full CFT. In particular, the modular invariant partition functions (which essentially parametrise the possible full CFTs) have a rich interpretation within von Neumann algebras (subfactors), which has led to the developments of structures of full CFT such as the full system (fusion ring of defect lines), nimrep (cyl…
In this work we study the generalization of twisted homology to geometric and non-geometric backgrounds. In the process we describe the necessary conditions to wrap a network of D-branes on twisted cycles. If the cycle is localized in time, we show how by an instantonic brane mediation, some D-branes transform into fluxes on different backgrounds including non-geometric fluxes. As a consequence, we show that in the case of a IIB six-dimensional torus compactification on a simple orientifold, the flux superpotential is not invariant by this brane-flux transition, allowing the connection among d…
The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data DHg fits into a family $D^ωHg_{2n+1}$, where $n\ge 0$ and $ω\in \bbZ_{2n+1}$. We show $D^0 Hg_{2n+1}$ is related to the subfactors Izumi hypothetically associates to the cyclic groups $Z_{2n+1}$. Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of I…
We consider the quantum state seen by an observer in the diamond-shaped region, which is a globally hyperbolic open submanifold of the Minkowski space-time. It is known from the operator-algebraic argument that the vacuum state of the quantum field transforming covariantly under the conformal group looks like a thermal state on the von Neumann algebra generated by the field operators on the diamond-shaped region of the Minkowski space-time. Here, we find, in the case of the free massless Hermitian scalar field in the 2-dimensional Minkowski space-time, that such a state can in fact be identifi…
We construct interpolating functions fully compatible with S-duality. We then consider the problem of resumming perturbative expansions for anomalous dimensions of low twist non-protected operators in N=4 super Yang-Mills theory. When the rank of the gauge group is small, the interpolations suggest that anomalous dimensions of leading twist operators take their maximum value at the point $τ=\exp(iπ/3)$. For fixed spin and large enough rank, there is a level-crossing region, where the anomalous dimension of the leading twist operator reaches its maximum and then bounces back.
We describe the twisted space-time symmetries which imply the quantum Poincaré covariance of noncommutative Minkowski spaces, with constant, Lie algebraic and quadratic commutators. Further we present the relativistic and nonrelativistic particle models invariant respectively under twisted relativistic and twisted Galilean symmetries.
We construct quantum deformation of Poincaré group using as a starting point $SU(2,2)$ conformal group and twistor-like definition of the Minkowski space. We obtain quantum deformation of $SU(2,2)$ as a real form of multiparametric $GL(4,C)_{q_{ij},r}$. It is shown that Poincaré subgroup exists for special nonstandard one-parametric deformation only, the deformation parameter $r$ being equal to unity. This leads to commuting affine structure of the corresponding Minkowski space and simple structure of the corresponding Lie algebra, the deformation of the group being non-trivial.
