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 (6)

• A Gravity Theory on Noncommutative Spaces
• Twist Symmetry and Gauge Invariance
• Twist to close
• Noncommutative Two Dimensional Gravities
• The Chern-Simons One-form and Gravity on a Fuzzy Space
• Non-anticommutative Supersymmetric Field Theory and Quantum Shift

Cited by in corpus (9)

• Emergent Gravity from Noncommutative Spacetime
• Lie algebraic Noncommutative Gravity
• Deformed Reissner--Nordstrom solutions in noncommutative gravity
• Newton equation for canonical, Lie-algebraic and quadratic deformation of classical space
• A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity
• Fuzzy Toric Geometries
• Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives
• Star Product and Invariant Integration for Lie type Noncommutative Spacetimes
• Domain wall solitons and Hopf algebraic translational symmetries in noncommutative field theories