Scalar potential bounds on one-parameter Calabi-Yau threefolds with fluxes

Abstract

We study the refined de Sitter conjecture (RdSC), the Trans-Planckian Censorship Conjecture (TCC) and the species scale ${\Lambda }_s$ in the multifield flux potential of type IIB string theory compactified on one-parameter Calabi--Yau threefolds. The scalar potential depends on the axio-dilaton and on the single complex-structure modulus, and is no-scale along the Kähler directions. Restricting to $h^{2,1}=1$ allows us to work with the exact periods obtained from the Picard--Fuchs equations rather than with their large-complex-structure expansion, and therefore to probe the conifold region. For the mirror quintic, $(h^{2,1},h^{1,1})=(1,101)$, we scan 200 flux configurations and find unstable extrema in $96\%$ of the cases, with $c^{\prime }=0.18$. Once the Kähler runaway is taken into account, the no-scale structure fixes the logarithmic slope of the potential at every critical point to the universal value $\sqrt{6}$, so that the first RdSC condition is satisfied throughout. Statistically, the complex structure is the heavier sector and the axio-dilaton dominates the unstable directions. For the TCC we obtain analytic results valid for the whole class of one-parameter models: the slope approaches the universal value $2$ at large complex structure, diverges at the conifold, and equals exactly $\sqrt{2}$ in the axio-dilaton directions, where the leading potential takes a single $SL(2)$-covariant form, as the multifield evolutions confirm numerically. The TCC further translates into a lower bound on the compactification volume, $v>v_0\sim e^N$, controlled by the number of e-folds. Finally, the field displacements along our trajectories are compatible with the species-scale field-range bound, and restoring the volume dependence of ${\Lambda }_s$ shows that $V\lesssim 3{\Lambda }_s^2$ is implied by the TCC volume bound, which is thus the genuine constraint on these models.