Cohomological Problems arising in Quantum Field Theory and String Theory

This dissertation is devoted to the solution of some cohomological problems arising in supersymmetric field theories and in superstring theory.;It is clear now that the notion of cohomology can be applied very successfully in quantum field theory and string theory.;The cohomological approach has turned out to be especially useful to the study quantization of systems with a complicated space of fields, for instance constrained systems (BRST quantization). It is also useful in many other situations, for example, in the pure spinor formalism of supersymmetric gauge theories, superstrings, and supergravity.;We present a complete calculation of the homology and cohomology groups of the super Lie algebra of supersymmetries and of the super Poincare Lie algebra for (non-extended) SUSY in various dimensions less than or equal to eleven. For dimensions D = 10, 11, this study also describes the cohomology of reduction of a supersymmetric Lie algebra to lower dimensions. The method can also be applied to extended supersymmetric Lie algebras.;We solve some homological problems that were formulated in the paper by Berkovits and Howe [1] devoted to the study of integrals invariant with respect to supersymmetry. These problems appear also in other situations, in particular, in the pure spinor formalism of supergravity. One of these problems is equivalent to the calculation of cohomology groups of the super Lie algebra of supersymmetries, the second one is closely related to this calculation.;In the study of Type II superstrings, it is useful to consider the BRST complex associated to the sum of two pure spinors (it is related to the massless vertex operators). The corresponding cohomology classes describe the infinitesimal deformations of the given SUGRA solution. The cohomology of this complex is an infinite-dimensional vector space. It is also a finite-dimensional algebra over the algebra of functions of a single pure spinor. This dissertation determines the zero mode BRST cohomology, as well as its multiplicative structure, i.e. the cohomology ring.