Supersymmetric $SU(5) \times U(1)_\chi$ and the Weak Gravity Conjecture

Understand

The gauge symmetry $SU(5) \times U(1)_\chi$ is the unique maximal subgroup of SO(10) which retains manifest unification at $M_{GUT}$ of the Standard Model gauge couplings, especially if low scale supersymmetry is present.

• The spontaneous breaking of $U(1)_\chi$ at some intermediate scale leaves unbroken a $Z_2$ symmetry which is precisely `matter' parity.
• This yields a stable supersymmetric dark matter particle as well as topologically stable cosmic strings.
• Motivated by the weak gravity conjecture we impose unification of $SU(5)$ and $U(1)_\chi$ at an ultraviolet cutoff $\Lambda \sim \alpha_\Lambda ^{1/2} M_{P} \approx 5 \times 10^{17}$ GeV, where $\alpha_\Lambda$ denotes the $SU(5)$ gauge coupling at $\Lambda$ and $M_P \approx 2.4 \times 10^{18}$ GeV is the reduced Planck Scale.