Magnetic fields, branes and noncommutative geometry
Daniela Bigatti, Leonard Susskind
TL;DR
Bigatti and Susskind introduce a minimal physical model of a particle on the noncommutative plane using two opposite charges in a strong magnetic field connected by a spring, showing that in the large-B limit the system exhibits Moyal-bracket phases and links to open-string light-cone quantization on D-branes. They derive the light-cone string vertex and demonstrate that planar diagrams in noncommutative Yang–Mills theories reproduce the commutative results up to trivial external phases, implying equivalence in the 't Hooft large-N limit, while nonplanar diagrams are more convergent due to the noncommutative phases. The work provides a concrete physical mechanism for NC phase factors, connecting quantum mechanics in magnetic fields with string theory in B-field backgrounds within a perturbative framework. Overall, it clarifies how noncommutative geometry manifests in scattering vertices and perturbation theory, highlighting planar-versus-nonplanar behavior and potential implications for AdS/CFT and matrix-model perspectives.
Abstract
We construct a simple physical model of a particle moving on the infinite noncommutative 2-plane. The model consists of a pair of opposite charges moving in a strong magnetic field. In addition, the charges are connected by a spring. In the limit of large magnetic field, the charges are frozen into the lowest Landau level. Interaction of such particles include Moyal bracket phases characteristics of field theory on noncommutative space. The simple system arises in lightcone quantization of open strings attached to D-branes in a.s. tensor background. We use the model to work out the general form of lightcone vertices from string splitting. We then consider Feynman diagrams in uncompactified NC YM theories and find that for all planar diagrams the comm. and noncomm. theories are the same. This means large N theories are equivalent in the 't Hooft limit. Non planar diagrams convergence is improved.