  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.
