The web of swampland conjectures and the TCC bound

We consider the swampland distance and de Sitter conjectures, of respective order one parameters λ and c. Inspired by the recent Trans-Planckian Censorship conjecture (TCC), we propose a generalization of the distance conjecture, which bounds λ to be a half of the TCC bound for c, i.e. $$ \lambda \ge \frac{1}{2}\sqrt{\frac{2}{3}} $$ in 4d. In addition, we propose a correspondence between the two conjectures, relating the tower mass m on the one side to the scalar 1 potential V on the other side schematically as $$ m\sim {\left|V\right|}^{\frac{1}{2}} $$ , in the large distance limit. These proposals suggest a generalization of the scalar weak gravity conjecture, and are supported by a variety of examples. The lower bound on λ is verified explicitly in many cases in the literature. The TCC bound on c is checked as well on ten different no-go theorems, which are worked-out in detail, and V is analysed in the asymptotic limit. In particular, new results on 4d scalar potentials from type II compactifications are obtained.