Conformal field theories (CFTs) play central roles in modern theoretical physics. Many CFTs are strongly coupled and cannot be studied using perturbative method. Conformal bootstrap provides a non-perturbative approach to study CFTs, which only employs the crossing symmetry and unitarity while not depending on classical Lagrangian description. This method has been successfully applied to solve the 3D Ising model and O(N) vector model. In this thesis, the conformal bootstrap approach will be applied to study the 5D O(N) vector model and 4D N = 1 supersymmetric conformal field theories (SCFT). For the 5D O(N) vector model, we bootstrap the mixed four-point correlators of the leading O(N) vector ?i and the O(N) singlet ?. By imposing mild gaps in the spectra, we are able to isolate the scaling dimensions (??, ??) in a small island for large N = 500, which is highly consistent with the results obtained from large N expansion. For smaller N <= 100, the islands disappear after increasing ? which suggests a lower bound on the critical value Nc > 100, below which the interacting O(N) CFTs turn into nonunitary. To bootstrap SCFTs, it needs analytical expression of the superconformal block function, which is the summation of several conformal block functions with coefficients determined by supersymmetry. We have calculated the most general 4D N = 1 superconformal block function of scalar operators based on the supershadow approach and superembedding formalism. In superembedding space the 4D N = 1 superconformal transformation T ? SU(2, 2|1) is realized linearly and the superconformal covariant three-point correlator functions can be constructed directly. Based on these results, the minimal SCFT with lowest c central charge among all the known 4D N = 1 SCFTs has been studied through bootstrapping the mixed correlators of the chiral operator ? and the operator X ~ ??+. The scaling dimensions (??, ?x) have been isolated into a small island! The results further confirm the existence of the promising minimal SCFT and reveal several interesting properties of this theory.
Conformal field theories (CFTs) play central roles in modern theoretical physics. Many CFTs are strongly coupled and cannot be studied using perturbative method. Conformal bootstrap provides a non-perturbative approach to study CFTs, which only employs the crossing symmetry and unitarity while not depending on classical Lagrangian description. This method has been successfully applied to solve the 3D Ising model and O(N) vector model. In this thesis, the conformal bootstrap approach will be applied to study the 5D O(N) vector model and 4D N = 1 supersymmetric conformal field theories (SCFT).
For the 5D O(N) vector model, we bootstrap the mixed four-point correlators of the leading O(N) vector ?i and the O(N) singlet ?. By imposing mild gaps in the spectra, we are able to isolate the scaling dimensions (??, ??) in a small island for large N = 500, which is highly consistent with the results obtained from large N expansion. For smaller N <= 100, the islands disappear after increasing ? which suggests a lower bound on the critical value Nc > 100, below which the interacting O(N) CFTs turn into nonunitary.
To bootstrap SCFTs, it needs analytical expression of the superconformal block function, which is the summation of several conformal block functions with coefficients determined by supersymmetry. We have calculated the most general 4D N = 1 superconformal block function of scalar operators based on the supershadow approach and superembedding formalism. In superembedding space the 4D N = 1 superconformal transformation T ? SU(2, 2|1) is realized linearly and the superconformal covariant three-point correlator functions can be constructed directly. Based on these results, the minimal SCFT with lowest c central charge among all the known 4D N = 1 SCFTs has been studied through bootstrapping the mixed correlators of the chiral operator ? and the operator X ~ ??+. The scaling dimensions (??, ?x) have been isolated into a small island! The results further confirm the existence of the promising minimal SCFT and reveal several interesting properties of this theory.