By A. N. Schellekens

The booklet contains a collection of papers at the building of superstring theories, mostly written through the years 1984-1987. It covers ten-dimensional supersymmetric and non-supersymmetric strings, 4-dimensional heterotic strings and 4-dimensional type-II strings. An advent to more moderen advancements in conformal box thought with regards to string development is equipped

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In sect. 4 we define and study the spectrum of the closed string sector of the dual spinor model. In sect. 5 we establish the correspondence between supergravity the ories and the dual model of closed strings. 2. 1. 1) μν r, s half integers. 2) = v Smn [b , b^ ]+ = η δ^ μν η = (—+++), and — and + denote commutators and anticommutators, respectively. The states | ψ> obtained by applying creation operators on the vacuum |0) do not form a Hilbert space because of the indefinite metric. However, one can define a sector of states, called the physical sector, which has semi-definite positive metric.

Y ~ . 14) in D dimensions (one time, D — 1 space dimensions, D assumed even). Then one can show that ( ^ + ΐ ) 2 = (_ΐ)Φ/2-ΐ). 15) Left and right handed states can be defined for any even D. Just as in the Neveu-Schwarz model, physical states are ghost free for D< 10, and for D = 10 only transverse (positive norm) states are coupled. The difference does not affect the ground-state spinor, which is not a gauge field. Finally, under the U(N) group the fermions can belong to a different representa tion to that of the bosons.

Thus on-shell gauge invariance, together with the Lorentz condition proves that only the trans verse components of the photon are coupled in the dual model, at all orders in a. The number of physical degrees of freedom of the photon are then D — 2. All these statements hold for D< 10. For D= 10, the zero norm states are more numerous and one can show that not only is the photon transverse, but also all the states in the dual model. One can then reformulate the model in such a way that oscillators with only D — 2 = 8 Lorentz indices appear, so that the no-ghost theorem is trivial, but Lorentz invariance is less obvious (but true).