In this talk I will introduce new constraints on couplings of operators in
effective field theory derived from Unitarity and Lorentz invariance in the
UV. These constraints come in the form of positive geometries in the space
of couplings. The origin of this positivity originates from the remarkable
positive properties of Gegenbauer polynomials, which are the basis
polynomials dictated by Lorentz invariance. Unitarity then dictates that
the discontinuity of the S-matrix has a positive expansion on these
polynomials. This double positivity give rises to the mathematical object
that we named the “EFThedron”, for which all consistent QFT must reside in.
Adding that the underlying theory has a worldsheet description, allows us
to begin to carve out the string landscape in an on-shell fashion.