Mirror quintic vacua: hierarchies and inflation

TL;DR

This work advances flux-compactification phenomenology by computing the full, convergent Picard-Fuchs periods for the IIB mirror-quintic setup across the entire complex-structure moduli space, enabling an exact assessment of vacua and inflationary regions. It shows that higher-order period corrections modify previously known hierarchies and can invalidate fake vacua found at leading order, while confirming generic 4D–6D scale hierarchies near the conifold. The study finds no viable single-field axion monodromy inflation; instead, slow-roll regions arise in multi-field directions and persist along conifold-to-orbifold paths under suitable fluxes, with some configurations admitting small ε but often requiring multi-field dynamics. Overall, the results underscore the critical role of exact period data for accurate vacuum structure and inflationary prospects in no-scale flux compactifications, and point to future work with Kähler moduli stabilization and extended monodromy analyses.

Abstract

We study the moduli space of type IIB string theory flux compactifications on the mirror of the CY quintic 3-fold in P4. We focus on the dynamics of the four dimensional moduli space, defined by the axio-dilaton τ and the complex structure modulus z. The z-plane has critical points, the conifold, the orbifold and the large complex structure with non trivial monodromies. We find the solutions to the Picard-Fuchs equations obeyed by the periods of the CY in the full z-plane as a series expansion in z around the critical points to arbitrary order. This allows us to discard fake vacua, which appear as a result of keeping only the leading order term in the series expansions. Due to monodromies vacua are located at a given sheet in the z-plane. A dS vacuum appears for a set of fluxes. We revisit vacua with hierarchies among the 4D and 6D physical scales close to the conifold point and compare them with those found at leading order in [1, 2]. We explore slow-roll inflationary directions of the scalar potential by looking at regions where the multi-field slow-roll parameters ε and η are smaller than one. The value of ε depends strongly on the approximation of the periods and to achieve a stable value, several orders in the expansion are needed. We do not find realisations of single field axion monodromy inflation. Instead, we find that inflationary regions appear along linear combinations of the four real field directions and for certain configurations of fluxes.

Figures (13)

• Figure 1: The figure represents the three different critical points of the CS moduli space of the mirror of the quintic CY on $\mathbb{P}^4$ on the complex $z_M$ plane. As well we represent the regular point where we also constructed the solution to the PF equation in order to improve convergence, $z_M=(1-e^{i p})/5^5$ with $-\pi/3<p<\pi/3$. The yellow dashed line represents the branch cut chosen for the conifold periods. Recall that $z_C=1-5^5 z_M$ and $z_O=1/z_M$. The LCS, conifold and orbifold series convergence regions are coloured in green, pink and blue respectively.
• Figure 2: The figure represents three paths leading to monodromies around the critical points of the CS moduli space of the mirror of the quintic CY on $\mathbb{P}^4$ on the complex $z_M$-plane. The paths around the LCS, conifold and orbifold points leading to monodromies $\mu_M$, $\mu_C$ and $\mu_O$ are green, pink and blue respectively.
• Figure 3: The plots on the first row show $t_1$ and $g_s$ vacua true values (dots) for the set of fluxes $F_1=80$, $H_4=1$ and variable $H_3$, with the red line representing the hierarchical solution of Giddings:2001yu. First plot on the second row represents the absolute value of $|z_{old}|$ (\ref{['zH']}) for the solution of Giddings:2001yu (dashed line) compared with the true vacua solutions $|z_0|$ (dots). Second plot on that row represents $|z_{new}|$, the corrected equation (\ref{['zHok']}) (yellow line) compared with the true vacuum $|z_0|$(dots).
• Figure 4: The first plot shows the exact $|z_0|$ for vacua solutions vs. $H_3$ (blue dots) together with approximation (\ref{['aproxG']}) (red line). The second plot is a zoom of the first, showing more clearly the difference between the exact solution and the approximation. The third and fourth plots show the exact vacua solutions (dots) and the approximation (\ref{['aproxG']}) vs. $F_3$, and its zoom. Turning on $H_3$ and $F_3$, leaves $g_s$ fixed and leads to vacua close to the conifold. This implies a hierarchy between the four and six dimensional scales.
• Figure 5: Vacuum solutions vs. the order in the period expansion for two sets of non-zero fluxes. The vacuum solution on the left does not converge inside the conifold convergence region after order 200th, and thus does not correspond to a true vacuum. Instead, the solution on the right converges very quickly to a stable value.