Issues in Complex Structure Moduli Inflation

TL;DR

The paper investigates whether complex structure moduli can drive or contribute to slow-roll inflation in Type IIB flux compactifications with moderately large volume, focusing on how Kähler-moduli stabilization via racetrack gaugino condensation (and KL tuning) constrains such dynamics. It finds that the racetrack barrier height is generically bounded by the KK scale, placing the maximal viable inflationary energy around ρ ≲ (M_KK)^4 and typically far below M_Pl^4, which undermines high-scale large-field scenarios. The analysis shows that even small deformations of complex structure during inflation distort volume stabilization through T–z mixing, yielding stringent limits on how z can evolve without destabilizing the vacuum. The authors discuss several potential workarounds (multi-field KYY-like constructions, topological inflation, or inflaton dynamics mediated by T–z mixing) but conclude that achieving robust complex-structure–driven inflation in this framework remains highly challenging and strongly model-dependent, motivating further exploration in refined string constructions.

Abstract

Supersymmetric compactification with moderately large radius (${\rm Re}< T > \sim {\cal O}(10)$ or more) not only accommodates supersymmetric unification, but also provides candidates for an inflaton in the form of geometric moduli; the value of ${\rm Re}< T > > 1$ may be used as a parameter that brings corrections to the inflaton potential under control. Motivated by a bottom-up idea "right-handed sneutrino inflation" scenario, we study whether complex structure moduli can play some role during the slow-roll inflation and/or reheating process in this moderately large radius regime. Even when we allow a tuning introduced by Kallosh and Linde, the barrier of volume stabilization potential from gaugino condensation racetrack superpotential can hardly be as high as $(10^{16} \; {\rm GeV})^4$ for generic choice of parameters in this regime. It is also found that even very small deformation of complex structure during inflation/reheating distorts the volume stabilization potential, so that the volume stabilization imposes tight constraints on large-field inflation scenario involving evolution of complex structure moduli. A few ideas of satisfying those constraints in string theory are also discussed.

Figures (6)

• Figure 1: This is a contour plot of the vacuum value $\left\langle {T} \right\rangle$ on the $(-r, N_1)$ plane; the curve above (solid) is the contour of $\left\langle {T} \right\rangle = 25$, and the one below (dashed) that of $\left\langle {T} \right\rangle =10$.
• Figure 2: The potential $V_{\rm racetrack}(T)$ for a couple of different choices of $r := \left\langle {a_2/a_1} \right\rangle$ and $N_1$ that lead to $\left\langle {T} \right\rangle = 25$. Panel (a): the curves shown in solid, long dashed, dashed and dotted lines are for parameters $(r,N_1)=(-1.05,68)$, $(-1.06,62)$, $(-1.07,57)$ and $(-1.08,53)$, respectively. The larger the value of $r$ is, the larger the height of the potential barrier becomes. Panel (b): the potential for the values of $(r,N_1) = (-1.05,68)$, $(-1.02,118)$, $(-1.007,238)$ and $(-1.002,626)$ is shown in a solid, long dashed, dashed and dotted line, respectively. The second (negative energy) minimum deepens for larger $r$.
• Figure 3: The potential barrier height $[V_{\rm racetrack}]^{\rm barrier}/M_{\rm Pl}^4$ changes by orders of magnitude for different choices of $(r, N_1)$. The data points are for $(r, N_1) \simeq (-1.1, 50)$, $(-1.67, 20)$, $(-6.1, 10)$, $(-9.9, 9)$, $(-18.7,8)$, and $(-48.5, 7)$.
• Figure 4: The potential $V_{\rm racetrack}(T)$ with $W^{({\rm tot})}$ in (\ref{['eq:Wtot-parametrize4racetrack']}) for various values of $\delta \widetilde{W}^{({\rm cpx})}_{\rm eff.}(z)/a_1$; the vacuum parameters in (\ref{['eq:vac-parameter']}) are used here, and the other deformation parameter $a_2/a_1$ is held fixed at its vacuum value in this figure. Panel (a): the four curves from top to bottom (solid, long dashed, dashed and dotted) correspond to the deformation parameters $\delta \widetilde{W}^{({\rm cpx})}_{\rm eff}/a_1 =0$, $-0.0005$, $-0.001$ and $-0.002$, respectively. Panel (b): the potential for the deformation $\delta \widetilde{W}^{({\rm cpx})}_{\rm eff}/a_1 = 0$, $0.002$, $0.004$ and $0.006$ are shown in the solid, long dashed, dashed and dotted curves, respectively.
• Figure 5: The potential $V_{\rm racetrack}(T)$ with $W^{({\rm tot})}$ in (\ref{['eq:Wtot-parametrize4racetrack']}) for various values of $a_2/a_1$; the value of $W^{({\rm cpx})}(z)$ is fixed at $W_0$ in this figure. Panel (a): the four curves (solid, long dashed, dashed and dotted) are for $a_2/a_1 \simeq - 1.0504$, $-1.0530$, $-1.0550$ and $-1.0600$, respectively. Panel (b): the parameter value is set at $a_2/a_1 = -1.0504$, $-1.0450$, $-1.0400$ and $-1.0350$ in the curves drawn in the solid, long dashed, dashed and dotted lines, respectively. The potential $V_{\rm racetrack}(T)$ drawn in the solid line in (a) and (b) are the same, the one at the vacuum $a_2/a_1 = \left\langle {a_2/a_1} \right\rangle$.