Theory HEP Seminar

A New Prescription for Conformal Blocks and Correlators

Valentina Prilepina

Université Laval

Conformal field theories (CFTs) are special quantum field theories (QFTs) invariant under a broad group of symmetries, the group SO(2, d). They represent fixed points in renormalization group flows, describe second order phase transitions in statistical physics systems, and shed light on the universal structure of the landscape of all QFTs. Moreover, they prescribe a nonperturbative approach for probing quantum gravity theories via the AdS/CFT correspondence. The desire for a deep understanding of these special theories cannot be overstated. The embedding space is a natural habitat where CFTs live. In this talk, I will describe a general CFT operator product expansion (OPE) that naturally arises in the embedding space, obtained using a new uplift for general quasi-primary operators. The uplift proposed here, which relies on standard projectors and operators carrying spinor indices only, enables a universal treatment of operators in arbitrary irreducible Lorentz representations.

I will introduce the most useful form of the OPE differential operator within the context of this formalism and establish its action on any embedding space quantity appearing in the computation of M-point correlation functions. I will then construct the most general form of the two-point, three-point, and four-point correlators, setting the stage for a general analysis of 4-point conformal blocks. I will lay out a convenient framework that renders the computation of conformal blocks in arbitrary representations essentially effortless, reducing the procedure to (1) determining the relevant group theoretic structures and (2) deriving and applying a simple set of conformal substitution rules. In this setup, the blocks may be compactly encoded as linear combinations of the Gegenbauer polynomials in a specific variable, upon which one subsequently applies a set of unique substitution rules.