These lectures provide an introduction to noncommutative geometry and its origins in quantum mechanics and to the construction of solitons in noncommutative field theory. These ideas are applied to the construction of D-branes as solitons of the tachyon field in noncommutative open string theory. A brief discussion is given of the K-theory classification of D-brane charge in terms of the K-theo...

In this letter we show that supersymmetry like geometry can be approximated using finite dimensional matrix models and fuzzy manifolds. In particular we propose a nonperturbative regularization of N = 2 supersymmetric U(n) gauge action in 4D. In some planar large N limits we recover exact SUSY together with the smooth geometry of Rθ. Noncommutative geometry [3] is the only known modification of...

The “noncommutative geometry” of complex algebraic curves is studied. As first step, we clarify a morphism between elliptic curves, or complex tori, and C-algebras Tθ = {u, v | vu = e2πiθuv}, or noncommutative tori. The main result says that under the morphism isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra...

We introduce the notion of a semi-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of semiRiemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative semi-Riemannian geometry are not Hilbert spaces any more but Krein spaces, and Dirac operators are Kreinselfadjoint. We show that the noncommuta...

We give a comment on the possible rôle of the sliver state in the generic boundary conformal field theory. We argue that for each Cardy state, there exists at least one projector in the string field theory. ∗E-mail address: [email protected] One of the long standing problem in the string theory is the description of the geometry. Generically we do not need the space-time at all to def...