Li-Yuan ChiangTzu-Chen HuangYu-tin HuangWei LiLaurentiu RodinaHe-Chen Weng2025-01-142025-01-142024-02-26https://scholars.lib.ntu.edu.tw/handle/123456789/724807We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on R+⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above ∆gap∗, where c−112<Δgap∗[removed]AdS-CFT CorrespondenceConformal and W SymmetryField Theories in Lower Dimensionsjournal article10.1007/jhep02(2024)209