Conformal Bootstrap in Two Dimensions

Abstract

In this dissertation, we study bootstrap constraints on conformal field theories in two dimensions. The first half concerns two-dimensional (4,4) superconformal field theories of central charge c=6, corresponding to nonlinear sigma models on K3 surfaces. The superconformal bootstrap is made possible through a surprising relation between the BPS N=4 superconformal blocks with c=6 and bosonic Virasoro conformal blocks with c=28, and an exact moduli dependence of a certain integrated BPS four-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly the signature of a continuous spectrum above a gap at the singular moduli, and find numerically an upper bound on this gap that is saturated by the A1 N=4 cigar CFT. The second half concerns the semiclassical limit of two-dimensional CFTs, motivated by holography. In this limit, the conformal block decomposition of the four-point function is dominated a particular weight, and the crossing equation simplifies drastically. We find that if a certain "weakness" condition is satisfied, then the OPE coefficients follow a universal formula given by the semiclassical limit of the fusion kernel. This is matched with a bulk action evaluated on a geometry with three conical defects, analytically continued in the deficit angles beyond the range for which a metric with positive signature exists. The analytically continued geometry has a codimension-one coordinate singularity surrounding the heaviest conical defect. This singularity becomes a horizon after Wick-rotating to Lorentzian signature, suggesting a connection between universality and the existence of a horizon.