Ghost cohomologies and new discrete states in supersymmetric c=1 model - by Omar Khaled El Deeb

Abstract

As of today, string theory appears to be one of the most promising physical models unifying the fundamental interactions in nature, such as electromagnetic (gauge) interactions and the gravity. It is also one of the best candidates for constructing the consistent theory of quantum gravity. While the perturbative theory of strings appears to be well explored by now, we still lack an adequate formulation of string-theoretic formalism in the non-perturbative, or strongly coupled regime. One of the approaches, allowing us to explore the non-perturbative dynamics of strings (as well as of other physical theories with gauge degrees of freedom) is the formalism of ghost cohomologies, studied in this thesis. This approach is based on the fact that virtually all the crucial information on non- perturbative physics of gauge theories (including superstring theory with its reparametrizational gauge symmetry and local worldsheet supersymmetry) is carried by physical operators, which are physical (BRST invariant and non-trivial), but not manifestly gauge-invariant. Typically, these operators belong to a very special sector of the Hilbert space of gauge theories, where the matter and the ghost degrees of freedom are mixed. These physical operators are defined as elements of ghost cohomologies, studied in this thesis. In this work, we explore the formalism of ghost cohomologies on the example of supersymmetric c = 1 model (or, e quivalently, two-dimensional supergravity) which is one of the simplist models of superstrings, with elegant and transparent structure of the spectrum of physical states (vertex operators). We show how the presence of the ghost cohomologies enlarges the spectrum of states and leads to new intriguing symmetries of the theory and points to possible nontrivial relations of two-dimensional supergravity to physical theories in higher dimensions. We also develop general prescription for constructing BRST -invariant and nontrivial vertex operators.