Negative scalar potentials and the swampland: an Anti-Trans-Planckian Censorship Conjecture

In this paper, we derive a characterisation of negative scalar potentials, $V<0$, in $d$-dimensional effective theories of quantum gravity. This is achieved thanks to an Anti-Trans-Planckian Censorship Conjecture (ATCC), inspired by a refined version of the TCC. The ATCC relies on the fact that in a contracting universe, modes that become sub-Planckian in length violate the validity of the effective theory. In the asymptotics of field space, we deduce that $-V^{\mathrm{\prime }}\mathrm{/}V\ge c_0$ when $V^{\mathrm{\prime }}\ge 0$. The rate $c_0=2\mathrm{/}\sqrt{(d-1)(d-2)}$ is successfully tested in several string compactifications for $d\ge 4$. In addition, a new asymptotic condition, $V^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}V\ge c_0^2$, is derived. By extrapolation to anti-de Sitter solutions of radius $l$, we infer the existence of a scalar whose mass should obey $m^2{l}^2\lesssim -2$. This property is verified in many supersymmetric examples.