Seed conformal blocks in 4D CFT

We compute in closed analytical form the minimal set of \u201cseed\u201d conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation (\u2113, \u2113) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0, |\u2113 12 \u2113|) and one (|\u2113 12 \u2113|, 0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any (\u2113, \u2113), by using an educated ansatz and reducing the problem to an algebraic linear system. Various details on the form of the ansatz have been deduced by using the so called shadow formalism. The complexity of the conformal blocks depends on the value of p = |\u2113 12 \u2113| and grows with p, in analogy to what happens to scalar conformal blocks in d even space-time dimensions as d increases. These results open the way to bootstrap 4-point functions involving arbitrary spinor/tensor operators in four dimensional conformal field theories

The effective bootstrap

We study the numerical bounds obtained using a conformal-bootstrap method - advocated in ref. [1] but never implemented so far - where different points in the plane of conformal cross ratios z and z¯ are sampled. In contrast to the most used method based on derivatives evaluated at the symmetric point z=z¯=1/2, we can consistently "integrate out" higher-dimensional operators and get a reduced simpler, and faster to solve, set of bootstrap equations. We test this "effective" bootstrap by studying the 3D Ising and O(n) vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature. We also determine the scaling dimensions of certain scalar operators in the O(n) vector models, with n=2,3,4, which have not yet been computed using bootstrap techniques. ArXI

Bounds on OPE coefficients in 4D Conformal Field Theories

We numerically study the crossing symmetry constraints in 4D CFTs, using previously introduced algorithms based on semidefinite programming. We study bounds on OPE coefficients of tensor operators as a function of their scaling dimension and extend previous studies of bounds on OPE coefficients of conserved vector currents to the product groups SO(N) 7SO(M). We also analyze the bounds on the OPE coefficients of the conserved vector currents associated with the groups SO(N), SU(N) and SO(N) 7SO(M) under the assumption that in the singlet channel no scalar operator has dimension less than four, namely that the CFT has no relevant deformations. This is motivated by applications in the context of composite Higgs models, where the strongly coupled sector is assumed to be a spontaneously broken CFT with a global symmetry. \ua9 The Authors

Deconstructing Conformal Blocks in 4D CFT

We show how conformal partial waves (or conformal blocks) of spinor/tensor correlators can be related to each other by means of differential operators in four dimensional conformal field theories. We explicitly construct such differential operators for all possible conformal partial waves associated to four-point functions of arbitrary traceless symmetric operators. Our method allows any conformal partial wave to be extracted from a few \u201cseed\u201d correlators, simplifying dramatically the computation needed to bootstrap tensor correlators. \ua9 2015, The Author(s)

CFTs and the Bootstrap

This thesis deals with investigations in the field of higher dimensional CFTs. The first part is focused on the technology neccessary for the calculation of general conformal blocks in 4D CFTs. These special functions are neccessary for general boostrap analysis in 4D CFTs. We show how to reduce the calculation of arbitrary conformal blocks to the calculation of a minimal set of "seed" conformal blocks through the use of differential operators. We explicitly write the set of operators necessary and show a general basis for the case of external traceless symmetric operators. We then compute in closed analytical form this set of seeds. We write in a compact form the set of quadratic Casimir equations and proceed to solve them in closed form with the use of an educated Ansatz. Various details on the form of the ansatz are deduced with the use of the so called shadow formalism. The second part of this thesis deals with numerical investigations of the bootstrap equation for external scalar operators. We compute bounds on the OPE coefficients in 4D CFTs for theories with and without global symmetries, and write the bootstrap equations for theories with SO(N ) 7 SO(M ) and SU (N ) 7 SO(M ) symmetries. The last part of the thesis presents the Multipoint bootstrap, a conformal-bootstrap method advocated in ref. [25]. In contrast to the most used method based on derivatives evaluated at the symmetric point z = z = 1/2, \u304 we can consistently \u201cintegrate out" higher-dimensional operators and get a reduced, simpler, and faster to solve, set of bootstrap equations. We test this \u201ceffective" bootstrap by studying the 3D Ising and O(n) vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature. We also determine the scaling dimensions of certain scalar operators in the O(n) vector models, with n = 2, 3, 4, which have not yet been computed using bootstrap techniques