Noncommutative gravity in three dimensions coupled to point-like sources

Noncommutative gravity in three dimensions with no cosmological constant is reviewed. We find a solution which describes the presence of a torsional source. 1 What does ‘noncommutative’ mean? Please see [1] for a review of noncommutative field theory, if you want to study more. Consider noncommutative coordinates, for example [x, y ] = i θ , (1) where θ is a real constant. Then the ‘uncertainty’ lies between x and y, namely, ∆x∆y ≥ θ , (2) (where a numerical factor has omitted). This means the existence of the minimal length scale. If complex combinations of the coordinates, z = x+ iy and z = x− iy, are introduced, they satisfy [ z, z̄ ] = 2θ . (3) There are different representations to describe the noncommutativity; the commutative coordinate formalism with the star product, the Fock space (operator) formalism, etc. (see [1] for details). In this talk, we simply use them identically, unless the identification leads to confusions (thus, we do not use ⋆ in this talk). For example, we denote the equalities 1 = ∞