AdS-phobia, the WGC, the Standard Model and Supersymmetry

It has been recently argued that an embedding of the SM into a consistent theory of quantum gravity may imply important constraints on the mass of the lightest neutrino and the cosmological constant $\Lambda_{4}$. The constraints come from imposing the absence of any non-SUSY AdS stable vacua obtained from any consistent compactification of the SM to 3 or 2 dimensions. This condition comes as a corollary of a recent extension of the Weak Gravity Conjecture (WGC) by Ooguri and Vafa. In this paper we study $T^2/Z_N$ compactifications of the SM to two dimensions in which SM Wilson lines are projected out, leading to a considerable simplification. We analyze in detail a $T^ 2/Z_4$ compactification of the SM in which both complex structure and Wilson line scalars are fixed and the potential is only a function of the area of the torus $a^2$. We find that the SM is not robust against the appearance of AdS vacua in 2D and hence would be by itself inconsistent with quantum gravity. On the contrary, if the SM is embedded at some scale $M_{SS}$ into a SUSY version like the MSSM, the AdS vacua present in the non-SUSY case disappear or become unstable. This means that WGC arguments favor a SUSY version of the SM, independently of the usual hierarchy problem arguments. In a $T^2/Z_4$ compactification in which the orbifold action is embedded into the $B-L$ symmetry the bounds on neutrino masses and the cosmological constant are recovered. This suggests that the MSSM should be extended with a $U(1)_{B-L}$ gauge group. In other families of vacua the spectrum of SUSY particles is further constrained in order to avoid the appearance of new AdS vacua or instabilities. We discuss a possible understanding of the little hierarchy problem in this context.