# Conformal bootstrap in two-dimensional conformal field theories with non-diagonal spectrums

## TL;DR

This paper extends the conformal bootstrap method to two-dimensional conformal field theories with non-diagonal spectra, providing explicit solutions, validating them numerically, and proposing a new family of non-rational theories.

## Contribution

It introduces an analytic bootstrap extension for non-diagonal spectra, constructs explicit structure constants, and proposes a new class of non-diagonal, non-rational CFTs for any central charge with validation.

## Key findings

- Explicit solutions for structure constants in non-diagonal theories.
- Numerical validation of crossing symmetry in minimal models.
- Proposal of a new family of non-diagonal CFTs extending Liouville theory.

## Abstract

In this thesis we study two-dimensional conformal field theories with Virasoro algebra symmetry, following the conformal bootstrap approach. Under the assumption that degenerate fields exist, we provide an extension of the analytic conformal bootstrap method to theories with non-diagonal spectrums. We write the equations that determine structure constants, and find explicit solutions in terms of special functions. We validate this results by numerically computing four-point functions in diagonal and non-diagonal minimal models, and verifying that crossing symmetry is satisfied. In addition, we build a proposal for a family of non-diagonal, non-rational conformal field theories for any central charges such that $\Re{c} < 13$. This proposal is motivated by taking limits of the spectrum of D-series minimal models. We perform numerical computations of four-point functions in these theories, and find that they satisfy crossing symmetry. These theories may be understood as non-diagonal extensions of Liouville theory.