Universal class of type-IIB flux vacua with analytic mass spectrum

We report on a new class of flux vacua generically present in Calabi-Yau compactifications of type-IIB string theory. At these vacua, the mass spectrum of the complete axiodilaton/complex structure sector is given, to leading order in ${\alpha }^{\prime }$ and $g_s$, by a simple analytic formula independent of the choice of Calabi-Yau. We provide a method to find these vacua and construct an ensemble of 17,054 solutions for the Calabi-Yau hypersurface ${\mathbb{WP}}_{[1,1,1,6,9]}^4$, where the masses of the axiodilaton and the 272 complex structure fields can be explicitly computed.

Figure 1

Numerically generated distribution of flux vacua on the fundamental domain of the reduced field space, with $\mathrm{Re}\tau \in [-1/2,1/2)$, $\mathrm{Im}\tau >1$ and $\mathrm{Re}\widehat{z}\in [-1/2,1/2)$ for the ${\mathbb{WP}}_{[1,1,1,6,9]}^4$ model. The plot represents a total of 28,683 vacua obtained by reducing the EFT along the monodromy direction $v^i=(1,1)$. We indicated in red vacua with large ($>5\%$) instanton corrections to the Kähler metric and $m_{3/2}$, and in blue (17,054 solutions) those with corrections $<5\%$.

Figure 2

Density of flux vacua in the reduced complex structure space in terms of the parameter $\xi$. The plot shows the numerical distribution obtained directly form the ensemble of 206,479 flux vacua. We have indicated in (dark and light) blue the 95,626 vacua with small ($<5\%$) instanton corrections and in orange those where the instanton contribution is large ($>5\%$). The dashed line represents the analytic distribution (b7) normalized in the range $\xi \in [0.001,0.12]$, where most vacua are in the LCS regime and the continuous flux approximation holds (dark blue).

Figure 3

Distribution of the string coupling $g_s$, in terms of $g_s^{-1}=\mathrm{Im}\tau$. The histogram represents the normalized distribution of the data points lying in the interval $\xi \in [0.001,0.12]$, while the dashed curve is the expected result from the continuous flux approximation.

Figure 4

Numerical distributions for the normalized scalar masses ${\tilde{\mu }}_{\pm \lambda }^2={\mu }_{\pm \lambda }^2/m_{3/2}^2$ of the $\mathcal{G}$-invariant modes, $\lambda =\{0,1,2\}$ in the ensemble of 95,626 vacua at LCS. In each figure, the dashed line represents the analytic formula (b11) for each value of $\lambda$, normalized in the same range $\xi \in [0.001,0.12]$. The darker regions represent the solutions for which the continuous flux approximation applies.

Figure 5

Complete distribution of scalar masses ${\tilde{\mu }}_{\lambda }^2$, $\lambda =\{0,1,2\}$ in the ensemble of 95,626 vacua at LCS. In each histogram, solutions with (a) $\xi \in [0.001,0.02]$ and (b) $\xi \in [0.13,0.5)$ have been highlighted. The insets in these plots show the mass distribution near ${\tilde{\mu }}^2\approx 0$, illustrating the presence of an asymptotically massless mode at vacua near the LCS point (a), and the absence of light modes in the spectrum when solutions near the LCS point are excluded (b).