Conformal Theories, BRST Formalism and Representations of the Lie Superalgebras

The results in the theory of representations of the Lie superalgebras are discussed which were obtained in the framework of a variant of the quantum BRST formalism developed by the authors. The central object of the study in this variant is the complete algebra $\mathcal A$ of the BRST symmetry which coincides with the Lie superalgebra $l(1,1)$. The set of these results presents a nearly complete description of the theory of representations of $\mathcal A$. For infinite-dimensional representations, the criteria characterizing physical representations are established and a class of representations of $\mathcal A$ by unbounded operators in Krein spaces is constructed which is sufficiently large for all physical applications; all the problems concerning the operator domains are rigorously taken into account. For finite-dimensional representations, the complete solution of the decomposition problem is presented. All the series of irreducible and indecomposable representations of $\mathcal A$ are explicitly described and all cases which do not admit any decomposition over these series are singled out and reduced to definite unsolvable algebraic problems.