Non commutative geometry and super Yang-Mills theory

TL;DR

This paper explores the connection between noncommutative geometry and super Yang-Mills theory, providing a framework to relate parameters of rational foliations on a torus with the gauge group U(N), and discusses the transition to irrational parameters.

Contribution

It introduces a prescription linking noncommutative geometry with super Yang-Mills theory via rational parameters and boundary conditions, clarifying the limiting process to irrational parameters.

Findings

A relation between rational foliation parameters and U(N) gauge theory.

A method to study noncommutative geometry with rational parameters.

Insights into the limit process to irrational parameters.

Abstract

We aim to connect the non commutative geometry ``quotient space'' viewpoint with the standard super Yang Mills theory approach in the spirit of Connes-Douglas-Schwartz and Douglas-Hull description of application of noncommutative geometry to matrix theory. This will result in a relation between the parameters of a rational foliation of the torus and the dimension of the group U(N). Namely, we will be provided with a prescription which allows to study a noncommutative geometry with rational parameter p/N by means of a U(N) gauge theory on a torus of size \Sigma / N with the boundary conditions given by a system with p units of magnetic flux. The transition to irrational parameter can be obtained by letting N and p tend to infinity with fixed ratio. The precise meaning of the limiting process will presumably allow better clarification.