Supersymmetric backgrounds from generalized Calabi‐Yau manifolds

Abstract

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kähler form e iJ and the holomorphic form Ω. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: e iJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. Modulo a different action of the B-field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA.February 1, 2008

Supersymmetric completion of M-theory 4D-gauge algebra from twisted tori and fluxes

“…In the above formulas φ I a are the metric moduli of the internal manifold and span the coset manifold GL(7, R)/SO (7). the scalar φ is related to the volume V 7 of the torus:…”

“…Recently superstring and M-theory compactifications on manifolds admitting globally defined spinors [1,2,3,4,5,6,7,8] (for a recent review on string/M-theory flux-compactifications see also [9]), but with broken supersymmetry, have renewed considerable attention in as much as they offer examples of theories with a low-energy effective Lagrangian exhibiting spontaneously broken local supersymmetry as well as Higgs phases of certain gauge symmetries.…”

“…The resulting gauged supergravity [21] is defined by an embedding tensor transforming in the 78 +3 [35] with respect to the E 6(6) × O(1, 1) subgroup of E 7 (7) . In the basis of the 56 in which the 28 electric vector fields are A Λ µ = {A u µ , A 0 µ }, where A u µ , u = 1, .…”

Type IIB flows with 𝒩 = 1 supersymmetry

“…We then explore the backgrounds for possible types of flows in relation to the components of the fluxes. This is useful in scanning the backgrounds for new supergravity solutions and dual gauge/gravity theories, in studying flux compactifications with hierarchy of scales [6] and with stabilized moduli [7], in constructing suitable cosmological models [8,9] which might allow to probe stringy signatures left over from the early universe (see [10] for a review for instance), in studying mirror symmetry in flux compactifications (see [11,12,1] for instance), and in looking for stable flux vacua which break supersymmetry. The equations we present could be used to study specific flows on type IIB backgrounds with N = 1 supersymmetry with appropriate metric and flux ansatz.…”

“…In the last five years, group structures have been used to deal with supergravity backgrounds with torsion which preserve N = 1 and N = 2 supersymmetries. See [15,12,16,17,1,2,18,19] for instance. When the extra six dimensional space of type II strings is compactified over a generalized Calabi-Yau with group Gstructures, the components of the torsion and the fluxes fall in representations of G. The supersymmetry conditions translate into constraints on the balance among the components of the fluxes, the torsion, the running of the dilaton and the warp factor decomposed and organized representation by representation such that the appropriate number of supersymmetries is preserved.…”