On M-theory fourfold vacua with higher curvature terms

Abstract

We study solutions to the eleven-dimensional supergravity action, including terms quartic and cubic in the Riemann curvature, that admit an eight-dimensional compact space. The internal background is found to be a conformally Kahler manifold with vanishing first Chern class. The metric solution, however, is non-Ricci-flat even when allowing for a conformal rescaling including the warp factor. This deviation is due to the possible non-harmonicity of the third Chern-form in the leading order Ricci-flat metric. We present a systematic derivation of the background solution by solving the Killing spinor conditions including higher curvature terms. These are translated into first-order differential equations for a globally defined real two-form and complex four-form on the fourfold. We comment on the supersymmetry properties of the described solutions.Comment: 14 page

3d N = 1 $$ \mathcal{N}=1 $$ effective supergravity and F-theory from M-theory on fourfolds

“…More precisely this means that to any finite order in l P the metric of the fourfold can be corrected so that N = 2 supersymmetry is preserved to that order. This has recently been partially verified in [5] where it was shown that the metric gets corrected at order l 6 P away from the Ricci-flat CY metric while still retaining its Kähler property.…”

“…The expansions for the light modes will generally get modified at order l 6 P , where the fourfold geometry gets corrected away from Ricci-flatness [5,12]. However these modifications are subleading, as far as the mass spectrum of the light modes is concerned, and will not be necessary for our analysis.…”

“…They were shown to solve the eleven-dimensional equations of motion of M-theory to order l 3 P in [4], as recently reviewed in [5] in a scheme where the 11d equations of motion are solved perturbatively in an expansion in powers of l P . The four-form flux in these vacua is "small" as it is quantized in units of l 3 P .…”

On four-derivative terms in IIB Calabi-Yau orientifold reductions

“…However, a complete discussion of the reduction results of the R 4 -sector on a Calabi-Yau threefold involves a proper treatment of the ten-dimensional background solution at this order, with a successive dimensional reduction on this modified background. The analog scenario of the R 4 -sector in a dimensional reduction on a CalabiYau fourfold with Kähler deformations was discussed in the context of M-theory/F-theory in [34][35][36][37][38], where a complete treatment involving the corrected background was provided. Furthermore, the dimensional reduction of the R 4 -sector was discussed in related set-ups in [39,40].…”

M-theory on non-Kähler eight-manifolds

“…Of course, this problem is well-known to experts on the field, but unfortunately, as long as the elevendimensional Supergravity l P -corrected Killing spinor equation is not known, it seems not possible to solve it in a completely rigorous way. Important steps in this direction have been made in references [45][46][47], where a thoroughly and consistent analysis of M-theory compactifications in the presence of l P -corrections has been made, and even an educated guess for the l P -corrected Killing spinor equation has been proposed. Remarkably enough, the integrability condition of the l P -corrected proposal for the Killing spinor equation is compatible with the l P -corrected equations of motion, which definitely suggests that if the educated guess is not already the correct l P -corrected Killing spinor equation, it cannot be far from being it.…”