CFT Correlators and Analytics

Abstract

Analytic methods to solve conformal field theories (CFT) have yielded a lot of mileage in recent years. This dissertation builds up on these analytical techniques (lightcone methods and inversion formulas) and extends them to new avenues including defect CFTs and double-twist analytics. First, we use embedding formalism to construct correlators for $d$-dimensional CFT in the presence of $q$ co-dimensional defect. All possible invariants appearing in correlators of arbitrary representation of operators are constructed for the first time in a defect setting. This allows constraining the defect CFT by studying crossing relations of operators in arbitrary representations. Second, inversion formula is utilized to compute anomalous dimensions and three-point coefficient corrections for double-twist operators in arbitrary dimensions. We develop a new technique in Mellin space to compute closed form expression of these corrections which are valid at any finite value of conformal spin. Finally, a new connection is established between conformal correlator expansion and perturbative diagrammatic expansion in Wilson-Fisher theory in $4-\epsilon$ dimensions. To derive this connection we develop novel techniques for representing scalar and twist contributions to correlators using Mellin space. Our techniques generalize to other theories with $\epsilon$-expansion as well.