We present evidence for the following conjecture: when quantized, the magnetic monopole soliton solutions constructed by 't Hooft and Polyakov, as modified by Prasad, Sommerfield and Bogomolny, form a gauge triplet with the photon, corresponding to a Lagrangian similar to the original Georgi-Glashow one, but with magnetic replacing electric charge.

We study the connection between the dual spinor model and supersymmetry. We show that in the low-energy region, the dual spinor model yields a supersymmetric Yang-Mills theory with O(4) internal symmetry and a supergravity theory with O(4) internal symmetry.

We find that the spinor dual model is locally supersymmetric not only in the two-dimensional surface spanned by the string, but also with respect to the embedding space-time.

If the exact gauge symmetry of nature consists of the U(1)EM generated by the electric charge operator Q and the colour group K, with Q a colour singlet, then, if g is a possible magnetic charge, exp(4πigQ) must equal an element of the colour group. For colour singlet particles this reduces to Dirac's condition eg = 12n. In general, possible monopoles correspond to points of intersection of the colour and electromagnetic groups. If the colour is semi-simple and compact, there can at most be a finite number p of such points (p = N if K = SU(N)). The existence of non-trivial (not equal to unity) solutions to our condition means that there must be fractionally charged (with p the fraction) coloured particles and magnetic monopoles emanating colour magnetic flux as well as electromagnetic flux.

A group theoretical calculation is given of all the functions, defined in terms of infinite matrices, which occur in the dual theory of fermions and off-mass shell states when treated in the operator formalism.

The single planar dual-loop amplitudes are recalculated in the critical dimension of space time paying particular attention to the unitary property by ensuring that the only states propagating internally in the loop are those needed to factorize tree diagram residues in a positive definite way. The two new technical features which make this possible are 1.(i) our newly discovered physical state projection operator valid if the space-time dimension takes the critical value;2.(ii) the use of Feynman's tree theorem whereby it is sufficient to have the above projection operator on the mass shell.

We find the modified Lorentz invariant propagator which, when used on Neveu-Schwarz meson lines joining fermion lines, guarantees the coupling of just the ghost-free transverse spectrum of meson states. The modification is necessary to take account of the new form of reflected Ward identity valid for such lines. Thus we can write down dual amplitudes for processes involving four external fermions which possess a high degree of inner consistency, no-ghost theorems and Lorentz invariance, but we have not yet been able to evaluate them.

We present the physical state projection operator for the simple dual resonance model in twenty-six dimensions of space-time and unit intercept in two forms-(which are proved to be equivalent). One form obviously projects onto the so-called transverse states and the other obviously gives unity when inserted into any residue of a pole in a dual amplitude. Thus we explicitly verify the completeness of transverse states in 26 dimensions and have a weapon for constructing genuine unitary dual loop amplitudes (as shown in a second paper). We also present the corresponding results for the Neveu-Schwarz model.

Study of the non-planar orientable single dual loop diagrams in 26 space-time dimensions has revealed an infinite positive-definite spectrum of “pomeron” intermediate states which couple to reggeons via a bilinear pomeron-reggeon vertex operator. General algebraic techniques are developed to derive the behaviour of this vertex with respect to the Virasoro gauge operators. A reflection and transmission behaviour is found, reminiscent of the behaviour of a wave incident at the interface between two different media (in this case reggeonic and pomeronic). These gauge properties are such as to guarantee the desired “good properties”, namely completeness of the transverse reggeon states when coupled between physical reggeon states on one side, and on the other side, either physical pomeron states or else physical reggeon states created via an intermediate pomeron. This is yet another example of the amazing and gratifying self-consistency of the dual model with respect to duality, transversality and unitarity.

We construct the general dual vertex describing the transition of a Ramond fermion into a Neveu-Schwarz meson by the emission of a general excited fermion state. Our vertex thus generalizes the previous results of Thorn and Schwarz and is now put into a relatively compact and manageable form which enables us to check that certain of the general gauge conditions are correctly converted between the meson and fermion legs by the vertex. The particular construction of the vertex can be applied to the other older dual theories and leads to some new insights into them.

Using the operatorial expression for the gauge conditions, we derive an explicit form for the twisting operator and the semi-twisting operator which takes theP states intoV states. The expression for the last operator facilitates the proof of theV states factorization previously found by us. We show that the twisting operator proposed in the literature is inconsistent with multiple factorization,i.e. factorization of amplitudes with extermal spinning particles. The correct twisting operator depends on the integration variables of the twisted line, and automatically satisfies double-twist invariance.

We exhibit a transformation of variables which allows the simultaneous factorization of a multiparticle Veneziano term and its twisted counterpart (obtained by reversing the order of the intial or final particles). The states which factorize the amplitude are linear combinations of those previously obtained which were not invariant under twisting. The behaviour of the couplings under twisting allows the definition of a signature factor. The invariance under double twist is automatically satisfied. It is shown that the properties under the aforementioned transformation lead to relations between the couplings of the states to scalar particles (some of them are the ones called «Ward identities» in the literature).