Covariant Methods for Superconformal Field Theorie

In this thesis, we develop manifestly covariant methods for 4 dimensional, N = 1 superconformal field theories. First, we generalize the embedding formalism in conformal field theories (CFTs) to N = 1 superconformal field theories (SCFTs). As applications we construct manifestly superconformally covariant expressions for land 3-point correlation functions involving the supercurrent multiplet or the global symmetry current superfield. Next, we combine this superembedding formalism with the shadow formalism in CFTs into a new method for computing superconformal blocks appearing in 4-point functions of SCFTs. This new method, called the supershadow formalism, expresses a superconformal block as manifestly covariant integrations over a product of 3-point functions. The supershadow formalism is much more efficient computationally than the brute force methods used previously in the literature. We obtain the superconformal blocks appearing in the 4-point functions of general scalar operator and then specialize to the 4-point functions involving chiral and global symmetry current multiplets. The results in the chiral case can be further generalized to N = 2 SCFTs. Finally, we present a systematic algorithm to extract the correlation functions of conformal primary component operators from the superfield correlation functions. We implemented this algorithm in Mathematica and applied it to the superfield 2-point function between general operators, from which we obtain all the component 2-point functions and all possible shortening conditions for a N = 1 superconformal multiplet. We also discuss a few potential directions for future researches.