The spectra of type IIB flux compactifications at large complex structure

Abstract

Abstract:We compute the spectra of the Hessian matrix, H, and the matrix M that governs the critical point equation of the low-energy effective supergravity, as a function of the complex structure and axio-dilaton moduli space in type IIB flux compactifications at large complex structure. We find both spectra analytically in an h 1,2 − + 3 real-dimensional subspace of the moduli space, and show that they exhibit a universal structure with highly degenerate eigenvalues, independently of the choice of flux, the details of the compactification geometry, and the number of complex structure moduli. In this subspace, the spectrum of the Hessian matrix contains no tachyons, but there are also no critical points. We show numerically that the spectra of H and M remain highly peaked over a large fraction of the sampled moduli space of explicit Calabi-Yau compactifications with 2 to 5 complex structure moduli. In these models, the scale of the supersymmetric contribution to the scalar masses is strongly linearly correlated with the value of the superpotential over almost the entire moduli space, with particularly strong correlations arising for g s < 1. We contrast these results with the expectations from the much-used continuous flux approximation, and comment on the applicability of Random Matrix Theory to the statistical modelling of the string theory landscape.

Universal properties of type IIB and F-theory flux compactifications at large complex structure

“…In this section we extend the results of [1] by showing that the regions of the moduli space where the low-energy theory is well-described by equation (2.12) or equation (2.15) exhibit certain universal properties, such as simple expressions for the gradient of the scalar potential and the spectrum of the Hessian matrix. Our method is very similar to that of [17] and is quite straight-forward: for a given point p in the moduli space, we make a coordinate transformation φ i → ϕ a to canonically normalise the Kähler metric.…”

“…These results are in stark contrast to the expectations derived from the continuous flux approximation and the proposed RMT models (which predict a smooth spectrum), and thus serve to indicate a limit of applicability of these statistical techniques and approximations. Moreover, it was shown in [1] that the scalar potential in this region of the moduli space is too steep to support inflation, thus presenting a significant obstacle for realising inflation at large complex structure. In this paper, we extend the results of [1] to the entire 2(h 1,2 + 1) dimensional moduli space for flux compactifications of type IIB string theory at large complex structure.…”

“…Moreover, it was shown in [1] that the scalar potential in this region of the moduli space is too steep to support inflation, thus presenting a significant obstacle for realising inflation at large complex structure. In this paper, we extend the results of [1] to the entire 2(h 1,2 + 1) dimensional moduli space for flux compactifications of type IIB string theory at large complex structure. We show that as long as the superpotential is dominated by terms cubic in the complex structure moduli, the spectrum of the Hessian is given by equation (1.2).…”

“…, h 1,1 + (M 3 ). 1 We are interested in compactifications in which integrally quantised RR (F 3 ) and NS-NS (H 3 ) fluxes wrap some non-trivial three-cycles of M 3 ,…”

“…The remaining components can be determined from the identity, 17) as reviewed in [1]. For canonically normalised fields this expression reads, 18) and the tensor Z AB can be expressed in matrix representation as,…”