Toward an Effective Theory of the Volume Modulus

TL;DR

This paper provides a 10D analysis of the warped volume modulus stabilized by gaugino condensation in KKLT-like warped flux backgrounds. By identifying the correct 10D fluctuation as a shift of the warp factor (with compensator and Weyl factor) rather than a simple scaling of the internal metric, it derives a master equation for the warp-factor fluctuation and computes the 4D modulus mass across several simplified geometries. Across large-volume limits, strongly warped throats, and throat+bulk setups, the results consistently show that the modulus mass is set by the 4D AdS curvature scale, m^2 ~ O(1) |\hat{R}_{AdS}|, with only O(1) corrections from boundary terms and IASD flux; warping does not generically suppress the mass. These findings support, but also nuance, AdS scale separation expectations and have important implications for the KKLT EFT and warped-model building, underscoring the need for a careful 10D treatment of moduli in warped backgrounds.

Abstract

We investigate the 4-dimensional effective theory of the warped volume modulus in the presence of stabilizing effects from gaugino condensation by analyzing the linearized 10-dimensional supergravity equations of motion. Warping is generally expected to scale down the masses of bulk modes to the IR scale at the tip of a throat. We find that the mass of the warped volume modulus evades expectations and is largely insensitive to the effects of warping, even in strongly warped backgrounds. Instead, the mass is parametrically tied to the 4-dimensional AdS curvature scale $m^2 \sim {\mathcal O}(1) |\hat R_{\rm AdS}|$, presenting a challenge for scale separation in these backgrounds. We trace this effect to a universal contribution arising from the 10-dimensional equations of motion, and comment on the importance of a 10-dimensional treatment of the warped volume modulus for effective field theories and model building.

Figures (3)

• Figure 1: $\delta\alpha_1(r)/\epsilon$ vs $r/\ell$ for two different values of $r_c$ (as labeled in subfigures) and three values $\delta a_1=0.1\epsilon$ (red dotted), $0.3\epsilon$ (black dashed), $0.5\epsilon$ (blue solid). The range is $r_0\leq r\leq r_m$ with $r_0=\ell/1000$ and $r_m=10\ell$.
• Figure 2: $\delta\alpha_2(r)/\epsilon^2$ vs $r/\ell$ for two different values of $r_c$ (as labeled in subfigures) and three values $\delta a_2=0.1\epsilon^2$ (red dotted), $0.3\epsilon^2$ (black dashed), $0.5\epsilon^2$ (blue solid). The flux parameters are $g_1=3\epsilon \ell^{6-\delta}$ and $\delta=3/2$. The range is $r_0\leq r\leq r_m$ with $r_0=\ell/1000$ and $r_m=10\ell$.
• Figure 3: $\delta\alpha_2(r)/\epsilon^2$ vs $r/\ell$ for two different values of $r_c$ (as labeled in subfigures) and three values $g_1=3\epsilon\ell^6$ (red dotted), $0.3\epsilon^2$ (black dashed), $0.5\epsilon^2$ (blue solid). The flux parameters are $g_1=3\epsilon\ell^{6-\delta}$ (red dotted), $3\epsilon\ell^{6-\delta}$ (black dashed), and $7\epsilon\ell^{6-\delta}$ (blue solid) and $\delta=3/2$ in all cases. The range is $r_0\leq r\leq r_m$ with $r_0=\ell/1000$ and $r_m=10\ell$, and the boundary condition is $\delta a_2=0.3\epsilon^2$.