Matrix configurations for spherical 4-branes and non-commutative structures on S4 Original Research Article

We present a matrix theory action and matrix configurations for spherical 4-branes. The dimension of the representations is given by N=2(2j+1) (j=1/2,1,3/2,…). The algebra which defines these configurations is not invariant under SO(5) rotations but under SO(3)⊗SO(2). We also construct a non-commutative product ★ for field theories on S4 in terms of that on S2. An explicit formula of the non-commutative product which corresponds to the N=4 dim representation of the non-commutative S4 algebra is worked out. Because we use S2⊗S2 parametrization of S4, our S4 is doubled and the non-commutative product and functions on S4 are indeterminate on a great circle (S1) on S4. We will however, show that despite this mild singularity it is possible to write down a finite action integral of the non-commutative field theory on S4. NS–NS B field background on S4 which is associated with our matrix S4 configurations is also constructed.