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This thesis presents recent developments within the pure spinor formalism, which has simplified amplitude computations in perturbative string theory, especially when spacetime fermions are involved. Firstly the worldsheet action of both the minimal and the non-minimal pure spinor formalism is derived from first principles, i.e. from an action with two dimensional diffeomorphism and Weyl invariance. Secondly the decoupling of unphysical states in the minimal pure spinor formalism is proved.
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Infinite-dimensional Clifford algebras and their Fock representations originated in the quantum mechanical study of electrons. In this book the authors give a definitive account of the various Clifford algebras over a real Hilbert space and of their Fock representations. A careful consideration of the latter's transformation properties under Bogoliubov automorphisms leads to the restricted orthogonal group. From there, a study of inner Bogoliubov automorphisms enables the authors to construct infinite-dimensional spin groups. Apart from assuming a basic background in functional analysis and operator algebras, the presentation is self-contained with complete proofs, many of which offer a fresh perspective on the subject. The book will therefore appeal to a wide audience of graduate students and researchers in mathematics and mathematical physics.
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