Non-commutative Field Theory on S^4

TL;DR

This paper develops a matrix-based framework for non-commutative S^4, enabling the expansion of matrices and construction of scalar field theories on this space, which resembles known structures on non-commutative S^2.

Contribution

It introduces a matrix configuration for non-commutative S^4 and constructs a scalar field theory, extending previous work on non-commutative geometries.

Findings

Matrices can be expanded in terms of the NC4S configuration.

A scalar field theory on NC4S is successfully constructed.

The matrix configuration describes two S^4's joined at a circle.

Abstract

In the previous paper (hep-th/0402010) we proposed a matrix configuration for a non-commutative S^4 (NC4S) and constructed a non-commutative (star) product for field theories on NC4S. In the present paper we will show that any matrix can be expanded in terms of the matrix configuration representing NC4S just like any matrix can be expanded into symmetrized products of the matrix configuration for non-commutative S^2. Then a scalar field theory on NC4S is constructed. Our matrix configuration describes two S^4's joined at the circle and the Matrix theory action contains a projection matrix inside the trace to restrict the space of matrices to that for one S^4.