The effective action of warped M-theory reductions with higher derivative terms — part I

Abstract

M-theory accessed via eleven-dimensional supergravity admits globally consistent warped solutions with eight-dimensional compact spaces if background fluxes and higher derivative terms are considered. The internal background is conformally Kähler with vanishing first Chern class. We perturb these solutions including a finite number of Kähler deformations of the metric and vector deformations of the M-theory three-form. Special emphasis is given to the field-dependence of the warp-factor and the higher-derivative terms. We show that the three-dimensional two-derivative effective action takes a surprisingly simple form in terms of a single higher-curvature building block due to numerous non-trivial cancellations. Both the ansatz and the effective action admit a moduli dependent scaling symmetry of the internal metric. Furthermore, we find that the required departure from Ricci-flatness and harmonicity of the zero-mode eigenforms does not alter the effective theory.

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

“…To keep the presentation simple we have moved most technical details to the four appendices. 1 The observations of the three previous paragraphs might be relevant in understanding the nontrivial cancellations observed in [12] among some of the terms in the 3d N = 2 effective action at order l 6 P . 2 The precise statement is that the integral of the four-form over any divisor of the base B of the elliptically-fibered CY, and the integral of the four-form over the elliptic fibration of any holomorphic curve in B must vanish.…”

“…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.…”

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.…”

“…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. One of the main conclusions of [46] is that even when one consistently takes into account l P -corrections, the internal manifold of the compactification is still topologically a Calabi-Yau four-fold. This strongly suggests that the conclusion of reference [10] is solid after properly taking into account l P -corrections.…”