Towards a covariant closed string theory

A very elegant construction of a covariant field theory for open strings and superstrings was given by Witten based on the pioneering use by Siegel of the 1st quantised BRST charge Q. The extension of the string-field product to closed strings has been attempted by Strominger. However the string product now has the defects of being non-associative, generating a background dependent kinetic energy, being non-local in the embedding space-time and giving a multiple cover of moduli space in the resulting multi-loop amplitudes. Other attempts using Φ have difficulties with string length. The approach proposed here will be based on the more geometrical attempt to build a general relatively of loop spaces. Difficulties were experienced in the introduction of a string field product. This is because the natural product in such an approach is [Φ(X)]n, where n = 2,3, ... and Φ is some functional of the string variables Xμ(σ), 0 ≤ σ ≥ 2π, μ = 1,...d. This was avoided when the discussion was restricted to open strings. It is the purpose here to indicate briefly how the approach of the authors may provide a more satisfactory method of resolving some of the difficulties (only strings are considered, though the extension to superstrings appears straightforward). (author)