The Chern-Simons One-form and Gravity on a Fuzzy Space

These notes are a short review of the q-deformed fuzzy sphere S^2_{q,N}, which is a ``finite'' noncommutative 2-sphere covariant under the quantum group U_q(su(2)). We discuss its real structure, differential calculus and integration for both real q and q a phase, and show how actions for Yang-Mills and Chern- Simons-like gauge theories arise naturally. It is related to D-branes on the SU(2)_k WZW model for q = exp(\frac{i \pi}{k+2}).

We study a matrix model which is obtained by dimensional reduction of Chern-Simon theory on S^3 to zero dimension. We find that expanded around a particular background consisting of multiple fuzzy spheres, it reproduces the original theory on S^3 in the planar limit. This is viewed as a new type of the large N reduction generalized to curved space.

In continuum physics, there are important topological aspects like instantons, theta-terms and the axial anomaly. Conventional lattice discretizations often have difficulties in treating one or the other of these aspects. In this paper, we develop discrete quantum field theories on fuzzy manifolds using noncommutative geometry. Basing ourselves on previous treatments of instantons and chiral fermions (without fermion doubling) on fuzzy spaces and especially fuzzy spheres, we ...

Generalizing the previous works on evolving fuzzy two-sphere, I discuss evolving fuzzy CP^n by studying scalar field theory on it. The space-time geometry is obtained in continuum limit, and is shown to saturate locally the cosmic holographic principle. I also discuss evolving lattice n-simplex obtained by `compactifying' fuzzy CP^n. It is argued that an evolving lattice n-simplex does not approach a continuum space-time but decompactifies into an evolving fuzzy CP^n.

We consider a non-relativistic (NR) limit of $(2+1)$-dimensional Maxwell Chern-Simons (CS) gravity with gauge algebra [Maxwell] $\oplus \ u(1)\oplus u(1)$. We obtain a finite NR CS gravity with a degenerate invariant bilinear form. We find two ways out of this difficulty: To consider i) [Maxwell] $\oplus\ u(1)$, which does not contain Extended Bargmann gravity (EBG); or, ii) the NR limit of [Maxwell] $\oplus\ u(1)\oplus u(1)\oplus u(1)$, which is a Maxwellian generalization o...

In spacetime dimensions larger than 2, whenever a global symmetry G is spontaneously broken to a subgroup H, and G and H are Lie groups, there are Nambu-Goldstone modes described by fields with values in G/H. In two-dimensional spacetimes as well, models where fields take values in G/H are of considerable interest even though in that case there is no spontaneous breaking of continuous symmetries. We consider such models when the world sheet is a two-sphere and describe their ...

These lectures provide an elementary introduction to Chern Simons Gravity and Supergravity in $d=2n+1$ dimensions.

The content of this paper is completely contained in arXiv:1204.0418v2: "A Chern-Simons action for noncommutative spaces in general with the example SU_q(2)"

We formulate a model of noncommutative four-dimensional gravity on a covariant fuzzy space based on SO(1,4), that is the fuzzy version of the $\text{dS}_4$. The latter requires the employment of a wider symmetry group, the SO(1,5), for reasons of covariance. Addressing along the lines of formulating four-dimensional gravity as a gauge theory of the Poincar\'e group, spontaneously broken to the Lorentz, we attempt to construct a four-dimensional gravitational model on the fuzz...

This is a preliminary version, comments and inputs are welcome. Contents: 1. Introduction. 2. Fuzzy Spaces. 3. Star Products. 4. Scalar Fields on the Fuzzy Sphere. 5. Instantons, Monopoles and Projective Modules. 6. Fuzzy Nonlinear Sigma Models. 7. Fuzzy Gauge Theories. 8. The Dirac Operator and Axial Anomaly. 9. Fuzzy Supersymmetry. 10.Fuzzy Spaces as Hopf Algebras.