Week 19.02.2023 – 25.02.2023

These lectures aim to provide a self-contained introduction to the modern conformal bootstrap method. The study of conformal field theory (CFT) will first be motivated and the “old” way of studying CFTs as endpoints of RG flows will be explained. The set of ideas necessary to understand the conformal bootstrap method will then be introduced, and both analytic and numerical implementations of the conformal bootstrap method will be discussed.

Higher-spin gravity is a curious beast of mathematical physics: a cousin of supergravity and string theory that seems comfortable with 4 spacetime dimensions and positive cosmological constant. On the other hand, general arguments show that this theory must be pathologically non-local at quartic order. In this talk, I claim that the non-locality arguments rely on Lorentzian boundary signature. For Euclidean boundary, explicit calculation shows that the feared non-locality is absent. This implies that the theory is healthy in de Sitter space, but not in (Lorentzian) AdS. The surprising possibility of such signature-dependent locality has long been implicit in the CFT/holography literature. Higher-spin gravity provides the first explicit example.

Chern-Simons (CS) theory provides an attractive framework for quantizing 3d gravity, at least around a fixed saddle-point. But how do we describe matter in CS gravity while retaining its useful features? In this talk I will focus on the CS description of Euclidean de Sitter space about its three-sphere saddle. I will introduce a "Wilson spool," which can be interpreted as a collection of Wilson loops winding arbitrarily many times around the three-sphere and which provides an effective description of massive one-loop determinants. Constructing and subsequently evaluating the spool will require us to revisit starting assumptions about unitarity of the representations appearing in the Wilson loops as well as the library of "exact methods" available to CS theories on the three-sphere. The result will be an object that reduces to the scalar one-loop determinant on the three-sphere in the limit that Newton's constant vanishes yet can be evaluated at in any order in G_N perturbation theory. Time remaining, I will either discuss potential further applications of the Wilson spool (either to spinning fields or to contexts outside of de Sitter) or (unresolved) implications of CS gravity for the dS/CFT dictionary.

Local Schur operators in 4d N=2 SCFTs form a protected class of operators giving rise to a 2d vertex operator algebra. Following the local operator picture, we introduce classes of conformal extended operators (lines, surfaces) and study these in twisted Schur cohomology. We show how these operators support a vertex algebra structure, extending the VOA picture of local Schur operators.

Local Schur operators in 4d N=2 SCFTs form a protected class of operators giving rise to a 2d vertex operator algebra. Following the local operator picture, we introduce classes of conformal extended operators (lines, surfaces) and study these in twisted Schur cohomology. We show how these operators support a more general algebraic structure compared to the local operators, giving rise to an extension of the vertex algebra known for local Schur operators.