Drinfeld Twist and General Relativity with Fuzzy Spaces

Abstract

We give a simplified formula for the star product on CP^n_L, which enables us to define a twist element suited for discussing a Drinfeld twist like structure on fuzzy complex projective spaces. The existence of such a twist will have several consequences for field theories on fuzzy spaces, some of which we discuss in the present paper. As expected, we find that the twist of the coproduct is trivial for the generators of isometries on CP^n_L. Furthermore, the twist allows us to define a covariant tensor calculus on CP^n_L from the perspective of the standard embedding of CP^n in flat Euclidean space. That is, we find a representation of a truncated subgroup of the diffeomorphisms on CP^n on the algebra of functions on CP^n_L. Using this calculus, we eventually write down an Einstein-Hilbert action on the fuzzy sphere, which is invariant under twisted diffeomorphisms.

24 pages; uses youngtab.sty; v2: minor changes; version to appear in CQG

References in corpus (35)

• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• A Gravity Theory on Noncommutative Spaces
• Noncommutative Geometry and Gravity
• New concept of relativistic invariance in NC space-time: twisted Poincaré symmetry and its implications
• Spin and Statistics on the Groenewold-Moyal Plane: Pauli-Forbidden Levels and Transitions
• Monopoles and Solitons in Fuzzy Physics
• Topologically nontrivial field configurations in noncommutative geometry
• Untwisting Noncommutative R^d and the Equivalence of Quantum Field Theories
• Fuzzy Complex Projective Spaces and their Star-products
• Spectrum of Schroedinger field in a noncommutative magnetic monopole
• Field Theory on a Supersymmetric Lattice
• The Origin of Chiral Anomaly and the Noncommutative Geometry
• Finite Gauge Theory on Fuzzy CP^2
• The Fuzzy Ginsparg-Wilson Algebra: A Solution of the Fermion Doubling Problem
• Field Theory on the q-deformed Fuzzy Sphere I
• Matrix ${\varphi }^4$ Models on the Fuzzy Sphere and their Continuum Limits
• Twist Symmetry and Gauge Invariance
• Twist to close
• Drinfeld-Twisted Supersymmetry and Non-Anticommutative Superspace
• Gravitational fields on a noncommutative space
• Non-Commutative Geometry and Twisted Conformal Symmetry
• The Fuzzy Supersphere
• Twist-Deformed Supersymmetries in Non-Anticommutative Superspaces
• Noncommutative Two Dimensional Gravities
• The Star Product on the Fuzzy Supersphere
• Noncommutative gravity: fuzzy sphere and others
• Twisted Conformal Symmetry in Noncommutative Two-Dimensional Quantum Field Theory
• Lorentz invariant and supersymmetric interpretation of noncommutative quantum field theory
• An invariant approach to dynamical fuzzy spaces with a three-index variable
• The Chern-Simons One-form and Gravity on a Fuzzy Space
• Non-Linear Sigma Model on the Fuzzy Supersphere
• Gravity on a fuzzy sphere
• Fuzzy Nambu-Goldstone Physics
• SUSY anomalies break N=2 to N=1: The supersphere and the fuzzy supersphere
• Non-anticommutative Supersymmetric Field Theory and Quantum Shift

Cited by in corpus (21)

• Emergent Gravity from Noncommutative Gauge Theory
• Emergent Gravity from Noncommutative Spacetime
• Lie algebraic Noncommutative Gravity
• Deformed Reissner--Nordstrom solutions in noncommutative gravity
• Fuzzy Scalar Field Theory as a Multitrace Matrix Model
• Geodesic equation in $k$-Minkowski spacetime
• Newton equation for canonical, Lie-algebraic and quadratic deformation of classical space
• Gauge theories on quantum spaces
• Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives
• A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity
• Fuzzy Toric Geometries
• $\kappa$-Minkowski star product in any dimension from symplectic realization
• Field Theory on Curved Noncommutative Spacetimes
• Graded Hopf Maps and Fuzzy Superspheres
• Star Product and Invariant Integration for Lie type Noncommutative Spacetimes
• Uniformly accelerated detector in the $\kappa$-deformed Dirac vacuum
• Tensor calculus on noncommutative spaces
• Contravariant Gravity on Poisson Manifolds and Einstein Gravity
• Domain wall solitons and Hopf algebraic translational symmetries in noncommutative field theories
• Dequantization via quantum channels
• Gauge theory models on $\kappa$-Minkowski space: Results and prospects