A scan for new Script N = 1 vacua on twisted tori

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.

Type IIA orientifold compactification on SU(2)-structure manifolds

“…The hyperkähler structure in turn is determined by the subspace Σ of the second cohomology class H 2 (K3, R) spanned by the two-forms J, ReΩ and ImΩ defined in (2.6), or equivalently, the space of self-dual harmonic two-forms on K3 [30]. The second cohomology class of K3 is a 22-dimensional vector space, equipped with a metric of signature (3,19) via the intersection product. One can express the intersection product in terms of the basis {ω α } of harmonic two-forms:…”

“…Since J and Ω span the subspace Σ of self-dual harmonic two-forms, H α β takes the form [25] H 19) where ξ xα = η αβ ξ x β . The next step is to determine the S-invariant subspace of M K3 or in other words determine the S-invariant deformations of the K3-metric.…”

“…Including orientifold planes this supersymmetry can be further reduced to N = 2 or N = 1 [14,19,11,20].…”

Matrix theory origins of non-geometric fluxes

“…Performing a T-duality along one direction of the torus the T-dual geometry is described by the three-dimensional nilmanifold, the twisted torus, whose non-trivial spin connection serves as a geometric flux. Such backgrounds lie in the heart of Scherk-Schwarz compactifications [29] and they were studied systematically in [30] and more recently in [28,[31][32][33][34]. A second T-duality along another direction of the torus takes the twisted torus background to a situation which is globally ill-defined.…”

New examples of flux vacua

“…However, a complete classification of the possible geometric backgrounds that can be found from this method has not been described. A scan of supersymmetric twisted tori vacua in the framework of generalized geometry can be found in [4].…”