  We introduce a new numerical algorithm based on semidefinite programming to
efficiently compute bounds on operator dimensions, central charges, and OPE
coefficients in 4D conformal and N=1 superconformal field theories. Using our
algorithm, we dramatically improve previous bounds on a number of CFT
quantities, particularly for theories with global symmetries. In the case of
SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal
technicolor. In N=1 superconformal theories, we place strong bounds on
dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the
line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive
anomalous dimensions in this region. We also place novel upper and lower bounds
on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we
find examples of lower bounds on central charges and flavor current two-point
functions that scale with the size of global symmetry representations. In the
case of N=1 theories with an SU(N) flavor symmetry, our bounds on current
two-point functions lie within an O(1) factor of the values realized in
supersymmetric QCD in the conformal window.
