We construct a duality cycle which provides a complete supergravity description of geometric transitions in type II theories via a flop in M-theory. This cycle connects the different supergravity descriptions before and after the geometric transitions. Our construction reproduces many of the known phenomena studied earlier in the literature and allows us to describe some new and interesting aspects in a simple and elegant fashion. A precise supergravity description of new torsional manifolds that appear on the type IIA side with branes and fluxes and the corresponding geometric transition are obtained. A local description of new G 2 manifolds that are circle fibrations over non-Kähler manifolds is presented.Contents 1 Notice that this term also plays a crucial role in the S-duality conjecture of [5].2 The manifold that we will eventually get in type IIA side later in this paper, will however be more general than the half-flat manifold in the sense that both dΩ ± = 0.3 The outcome of these T-dualities is different from the ones of [21],[22] and [23] where one T-duality takes a type IIB picture to a type IIA brane configuration. 4 There have been previous attempts to relate the resolved conifold and the deformed conifold by starting with the deformed conifold [24]. As the deformed conifold does not admit a T 3fibration using this manifold as a starting point may seem more problematic, although, we have been informed that there are some papers that overcome this problem [25].

We complete the duality cycle by constructing the geometric transition duals in the type IIB, type I and heterotic theories. We show that in the type IIB theory the background on the closed string side is a Kähler deformed conifold, as expected, even though the mirror type IIA backgrounds are non-Kähler (both before and after the transition). On the other hand, the Type I and heterotic backgrounds are non-Kähler. Therefore, on the heterotic side these backgrounds give rise to new torsional manifolds that have not been studied before. We show the consistency of these backgrounds by verifying the torsional equation.

We continue our study of geometric transitions in type II and heterotic theories. In type IIB theory we discuss an F-theory setup which clarifies many of our earlier assumptions and allows us to study gravity duals of N = 1 gauge theories with arbitrary global symmetry group G. We also point out the subtle differences between global and local metrics, and show that in many cases the global descriptions are far more complicated than discussed earlier. We determine the full global description in type I/heterotic theory.In type IIA, our analysis gives rise to a local non-Kähler metric whose global description involves a particular orientifold action with gauge fluxes localised on branes. We are also able to identify the three form fields that allow for a smooth flop in the M-theory lift. We briefly discuss the issues of generalised complex structures in type IIB theory and possible half-twisted models in the heterotic duals of our type II models. In a companion paper we will present details on the topological aspects of these models.the Klebanov-Strassler solution an effective theory. The only difference is that in their case the solution of [1] is supersymmetric, whereas our starting point, the solution of [21] is not. This can be understood from the fact that [1] uses integer and fractional D3 branes, the latter being wrapped D5 on vanishing cycles. This is a susy configuration. When the vanishing cycle becomes finite (the conical singularity is resolved), the fractional D3 branes become genuine D5 wrapped on 2-cycles which 4 , together with the integer D3 branes, are in general a non-susy system 5 .From the type IIB solution one can infer a mirror IIA solution [18]. The result is a family of non-Kähler manifolds, the non-Kählerity coming from the presence of type IIB NS flux. This means that the geometric transitions of [3] can actually be generalized to non-Kähler manifolds, as presented in [18].The non-Kählerity in type IIA theory is proportional to the NS flux so the non-Kähler geometry can be considered as a function of the type IIB NS flux. However, in type IIB we know from [9] that the NS flux is proportional to the coupling constant of the dual gauge 4 We will also refer to these branes as fractional D3 branes.5 As we will discuss in sec. 2.1, it doesn't really matter if there might exist such a solution that preserves supersymmetry (i.e allows primitive three form fluxes). Of course, to discuss the geometric transition of N = 1 SU (N ) theory with fundamental flavor transforming under a group G, we have to introduce D7/O7 branes along with the fractional and whole D3 branes [26] (see also [27] where somewhat equivalent construction, but only with D7 branes, are made to study Klebanov-Strassler model [1] with fundamental flavors). Our local metric studied earlier in [18], [19] is much more robust and it only depends on the topology of the resolved conifold.