Generalized structures of Script N = 1 vacua

Abstract

We characterize N = 1 vacua of type II theories in terms of generalized complex structure on the internal manifold M . The structure group of T (M )⊕T * (M ) being SU(3)×SU(3) implies the existence of two pure spinors Φ1 and Φ2. The conditions for preserving N = 1 supersymmetry turn out to be simple generalizations of equations that have appeared in the context of N = 2 and topological strings. They are (d+H∧)Φ1 = 0 and (d+H∧)Φ2 = FRR. The equation for the first pure spinor implies that the internal space is a twisted generalized Calabi-Yau manifold of a hybrid complex-symplectic type, while the RR-fields serve as an integrability defect for the second. September 11, 2018 been noted already in [7] for cases in which T has SU(3) structure. The SU(3) structure vacua are however just special cases of the more general SU(3)×SU(3) on T ⊕ T * considered here. The SU(3) structure vacua correspond to either complex (and with vanishing c 1 ) or symplectic manifolds, which are the two particular cases which inspired the definition of generalized Calabi-Yau. In the generic SU(3)×SU(3) case, vacua can be a complex-symplectic hybrid, namely a manifold that is locally a product of a complex k-fold times an 6 − 2k symplectic manifold.Physically, the generalized Calabi-Yau condition has also been argued to imply the existence of a topological model [8,9], not necessarily coming from the twisting of a (2, 2) model, which generalizes the A and B models. In other words, we find that all N = 1 Minkowski vacua have an underlying topological model. When there is a (2, 2) model, both pure spinors are closed under (d + H∧) [2], which reflects the fact that two topological models can be defined. This condition unifies the Calabi-Yau case and the (2, 2) models of [10]. It corresponds to an unbroken N = 2 in the target space, and it has recently been found from supergravity in [3]. Although the N = 2 requirement of having two twisted closed pure spinors looks like our N = 1 equations (1.1) for F = A = 0, we stress again that (1.1) applies only when the RR fluxes are non zero. Therefore we cannot obtain from there the N = 1 equations for pure NS flux, which correspond to a (2, 1) model.Another feature of (1.1) is that they are essentially identical for IIA and IIB. This suggests there must exist some form of mirror symmetry for these compactifications exchanging the even and odd pure spinors [11][12][13]. As far as we know, mirror symmetry could even be present when supersymmetry is spontaneously broken [11,13,14]. For the case at hand of unbroken N = 1, however, this is made particularly concrete by the remark above that all vacua have an underlying topological model; mirror symmetry has long been viewed [15] as an exchange of topological models, without necessarily involving Calabi-Yau's.Presumably there are connections to more recent lines of thought relating Hitchin functionals [16] to topological theories [17][18][19]. Particularly promising seems the results in [20] about the quantization of the functional, which relate directly to ...

Enhanced supersymmetry from vanishing Euler number

“…The Ricci tensor (in flat indices) is defined by contraction with the dual vielbein, 15) and the Ricci scalar as its trace…”

“…We need to work a little harder to describe the orthogonal complement T * 4 Y explicitly, as it is non-trivially fibered over Y . We begin therefore by considering the quotientỸ = T 6 /Z τ 2 , where we can specify a global basis for T * 4Ỹ , spanned by 15) and dx 1 and dx 2 . This shows thatỸ is a twisted six-torus, with T * Ỹ spanned by dx i , u i and v i , i = 1, 2.…”

“…Our analysis shows that before SU(2) doublets are included, this prepotential is also cubic, consistent with mirror symmetry. 15 The SU(2) doublet deformations will fiber over these modes, giving the full prepotential. Therefore, the Picard-Fuchs equations should drastically simplify in a parametrization of the holomorphic three-form compatible with the SU(2) structure.…”

“…We will see that θ drops out of all four-dimensional expressions. All other components are given by 15) where the covariant derivatives are defined as We see that if θ were torsion-free, the Riemann tensor would obey the Bianchi identity and the first term would be zero, thereby imposing that the connection be Ricci-flat. In The failure to satisfy the Bianchi identity is measured by R θ ∧ e = ∇T = dT + θT (as can be seen from the definition of the torsion tensor T ), therefore …”

Complex/Symplectic Mirrors

“…More recently the general conditions for preserving N = 1 supersymmetry in supergravity have been reduced to geometrical conditions [6]; in particular, the manifold has to be generalized complex [7]. The most prominent examples of generalized complex manifolds are complex and symplectic manifolds, neither necessarily Kähler.…”