W-Infinity Algebras from Noncommutative Chern-Simons Theory

We examine Chern-Simons theory written on a noncommutative plane with a `hole', and show that the algebra of observables is a nonlinear deformation of the $w_\infty$ algebra. The deformation depends on the level (the coefficient in the Chern-Simons action), and the noncommutativity parameter, which were identified, respectively, with the inverse filling fraction and the inverse density in a recent description of the fractional quantum Hall effect. We remark on the quantization of our algebra. The results are sensitive to the choice of ordering in the Gauss law.Comment: 9 page

Non-commutative AdS2/CFT1AdS_2/CFT_1 duality: the case of massless scalar fields

We show how to construct correlators for the $CFT_1$ which is dual to non-commutative $AdS_2$ ($ncAdS_2$). We do it explicitly for the example of the massless scalar field on Euclidean $ncAdS_2$. $ncAdS_2$ is the quantization of $AdS_2$ that preserves all the isometries. It is described in terms of the unitary irreducible representations, more specifically discrete series representations, of $so(2,1)$. We write down symmetric differential representations for the discrete series, and then map them to functions on the Moyal-Weyl plane. The Moyal-Weyl plane has a large distance limit which can be identified with the boundary of $ncAdS_2$. Killing vectors can be constructed on $ncAdS_2$ which reduce to the $AdS_2$ Killing vectors near the boundary. We therefore conclude that $ncAdS_2$ is asymptotically $AdS_2$, and so the $AdS/CFT$ correspondence should apply. For the example of the massless scalar field on Euclidean $ncAdS_2$, the on-shell action, and resulting two-point function for the boundary theory, are computed to leading order in the noncommutativity parameter. The results agree with those of the commutative scalar field theory, up to a field redefinition.Comment: 25 page

On geodesics in space-times with a foliation structure: A spectral geometry approach

Motivated by the Horava-Lifshitz type theories, we study the physical motion of matter coupled to a foliated geometry in non-diffeomorphism invariant way. We use the concept of a spectral action as a guiding principle in writing down the matter action. Based on the deformed Dirac operator compatible with the reduced symmetry - foliation preserving diffeomorphisms, this approach provides a natural generalization of the minimal coupling. Focusing on the IR version of the Dirac operator, we derive the physical motion of a test particle and discuss in what sense it still can be considered as a geodesic motion for some modified geometry. We show that the apparatus of non-commutative geometry could be very efficient in the study of matter coupled to the Horava-Lifshitz gravity.Comment: 17 page

On spectral geometry approach to Horava-Lifshitz gravity: Spectral dimension

We initiate the study of Horava-Lifshitz models of gravity in the framework of spectral geometry. As the first step, we calculate the dimension of space-time. It is shown, that for the natural choice of a Dirac operator (or rather corresponding generalized Laplacian), which respects both the foliation structure and anisotropic scaling, the result of Horava on a spectral dimension is reproduced for an arbitrary, non-flat space-time. The advantage and further applications of our approach are discussed.Comment: References, a figure and minor clarifications added. To match the version to be published in CQ