This dissertation concerns two topics. We first discuss the Yangian structure of deformed integral, four-dimensional N = 4 super Yang-Mills theories using Yangians. We use twist deformations in the Yangian coproducts, which are known to maintain the integrable structure. In a five-state subset of states we examine two explicit cases of deformation resulting in SU(2)xU(1)3 and SU(2∣1)xU(1) 2, which are subgroups of the N = 1 residual supersymmetry, PSU(2, 2∣1), in the full theory. While the full PSU(2, 2∣4)Yangian structure is manifest in the deformed theory, we show how the symmetry breaking to N = 1 is produced via twisted coproducts. For the second topic, we display the vertex operators for all states in the conformal supergravity sector of twistor string theory. Using canonical quantization of the open string, we compute N-point tree amplitudes for the supergraviton states. These include amplitudes involving the 'dipole' gravitons, which are not eigenstates of the translation generator. The conformal gravity amplitudes would be hard to access using conventional field theory methods.
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