A scan for new N=1 vacua on twisted tori

TL;DR

This work directly searches for $\mathcal{N}=1$ Minkowski vacua in type II string theory on compact six-dimensional nil- and solvmanifolds (twisted tori) by reformulating supersymmetry in terms of generalized complex geometry and pure spinors. It identifies three solution classes: T-duals of IIB on a conformal $\mathrm{T}^6$ with self-dual flux (fully localizable via warp factor), genuinely new multi-source Minkowski vacua not dual to Calabi–Yau backgrounds, and AdS$_4$ possibilities on flat solvmanifolds. The authors provide explicit models (including several for which fluxes and tadpoles are balanced by orientifolds and D-branes) and show that many new vacua are not Calabi–Yau nor connected to CY by dualities, highlighting a rich phase space of flux backgrounds beyond CY geometries. They also develop a detailed treatment of T-duality, mirror symmetry, and potential non-geometric extensions within the generalized complex framework. The results advance understanding of flux compactifications on twisted tori, offering concrete global and local solutions and pointing to further AdS and non-geometric directions for exploration.

Abstract

We perform a systematic search for N=1 Minkowski vacua of type II string theories on compact six-dimensional parallelizable nil- and solvmanifolds (quotients of six-dimensional nilpotent and solvable groups, respectively). Some of these manifolds have appeared in the construction of string backgrounds and are typically called twisted tori. We look for vacua directly in ten dimensions, using the a reformulation of the supersymmetry condition in the framework of generalized complex geometry. Certain algebraic criteria to establish compactness of the manifolds involved are also needed. Although the conditions for preserved N=1 supersymmetry fit nicely in the framework of generalized complex geometry, they are notoriously hard to solve when coupled to the Bianchi identities. We find solutions in a large-volume, constant-dilaton limit. Among these, we identify those that are T-dual to backgrounds of IIB on a conformal T^6 with self-dual three-form flux, and hence conceptually not new. For all backgrounds of this type fully localized solutions can be obtained. The other new solutions need multiple intersecting sources (either orientifold planes or combinations of O-planes and D-branes) to satisfy the Bianchi identities; the full list of such new solution is given. These are so far only smeared solutions, and their localization is yet unknown. Although valid in a large-volume limit, they are the first examples of Minkowski vacua in supergravity which are not connected by any duality to a Calabi-Yau. Finally, we discuss a class of flat solvmanifolds that may lead to AdS_4 vacua of type IIA strings.

Figures (1)

• Figure 1: Integrable structures on the 34 six--dimensional nilpotent Lie groups, from CG.