Mathematical Aspects of Twistor Theory: Null Decomposition of Conformal Algebras and Self-dual Metrics on Intersections of Quadrics

<p>The conformal algebra of an n-dimensional affine space with a metric of arbitrary signature (p, q) with p + q = n is considered. The case of broken conformal invariance is studied, by considering the subalgebra of the enveloping algebra of the conformal algebra that commutes with the squared-mass operator. This algebra, denoted R, is generated by the generators of the Poincaré Lie algebra and an additional vector operator R which preserves the relevant information when the conformal invariance is broken. Due to the nonlinearity of the algebra, finding the Casimir invariants becomes extremely difficult. The R-algebra is constructed for arbitrary dimensions, but the Casimir invariants are only determined for n ≤ 5. The second part of this thesis describes the geometric properties of metrics on the twistor space on intersections of quadrics. Consider a generic pencil of quadrics in a complex projective space ℂℙ⁵. The base locus of this pencil is considered as a three-dimensional projective twistor space, such that each point of the associated space-time is a projective two-plane lying inside one quadric of the pencil. The time coordinate can be described as a hyperelliptic curve of genus two, over which the space time is fibered. The metrics arising on the associated twistor space of the completely null two-planes are studied. It emerges that for pencils generated by simultaneously diagonalizable quadrics, these metrics are always self-dual and, in certain cases, conformal to vacuum.