Non commutative gravity from the ADS/CFT correspondence

Abstract

The exclusion principle of Maldacena and Strominger is seen to follow from deformed Heisenberg algebras associated with the chiral rings of S_N orbifold CFTs. These deformed algebras are related to quantum groups at roots of unity, and are interpreted as algebras of space-time field creation and annihilation operators. We also propose, as space-time origin of the stringy exclusion principle, that the $ADS_3 \times S^3$ space-time of the associated six-dimensional supergravity theory acquires, when quantum effects are taken into account, a non-commutative structure given by $SU_q(1,1) \times SU_q (2)$. Both remarks imply that finite N effects are captured by quantum groups $SL_q(2)$ with $q= e^{i π\over {N + 1}}$. This implies that a proper framework for the theories in question is given by gravity on a non-commutative spacetime with a q-deformation of field oscillators. An interesting consequence of this framework is a holographic interpretation for a product structure in the space of all unitary representations of the non-compact quantum group $SU_q(1,1)$ at roots of unity.