Noncommutative-geometry model for closed bosonic strings

It is shown how Witten's (1986) noncommutative geometry may be extended to describe the closed bosonic string. For closed strings, an explicit representation is provided of the integral operator needed to construct an action and of an associative product on string fields. The proper choice of the action of the integral operator and the associative product in order to give rise to a reasonable theory is explained, and the consequences of such a choice are discussed. It is shown that the ghost numbers of the operator and associative product can be chosen arbitrarily for both open and closed strings, and that this construct can be used as an action for interacting closed bosonic strings.