We give a detailed derivation of a supersymmetric configuration of wrapped D5-branes on a two-cycle of a warped resolved conifold. Our analysis reveals that the resolved conifold should support a non-Kähler metric with an SU(3) structure. We use this as a starting point of the geometric transition in type IIB theory. A mirror, and a subsequent flop transition using an intermediate M-theory configuration with a G 2 structure, gives rise to the complete IR geometric transition in type IIA theory. A further mirror transformation gives the type IIB gravity dual of the IR gauge theory on the wrapped D5-branes. Expectedly non-Kähler deformations of the resolved and the deformed conifolds appear as the gravity duals of the confining gauge theories in type IIA and type IIB theories respectively, although in more generic cases these manifolds could also be non-geometric. In the local limit we reproduce precisely the scenarios presented in our earlier works. Our present work should therefore be viewed as providing a supergravity proof of geometric transitions in the full global scenarios in type II theories. B. Components of G-fluxes in type IIA mirror configuration 48 C. Metric components in type IIB after geometric transition 81where η is the Euclidean metric on C 4 . The prefactor of 2 here on the right-hand side of (2.22) is not present in [24] but is required by our normalization condition for K. From here, we get a 2-parameter family of SU(3) structures given by(2.23)Using (2.20), it is immediate to see that J and Ω satisfy the conditions (2.9) for an SU (3) structure.There is an extra phase parameter for Ω in [24], but we supress it here since the metric does not depend on this phase. In Appendix 1 we write down all the components of the metric. T-duality and Orientifold ProjectionThe blown-up conifold that we discussed above, has a product structure of P 1 × P 1 and we would like to discuss the T-duality to a IIA model. Our final aim is to see how orientifold projection effects the T-duality. To do so we first review the case of [31] for the resolution of a deformed A 2 singularity. This involves a natural way to introduce two P 1 cycles. which acts as follows on the affine coordinates ξ j of the two P 1 factors: 3.8) and projects to the following involution κ 0 of X 0 :(z, x, y, u) κ 0 −→ (−z, − y, − x, u) .(3.9)

A. Attractors in the Hamilton-Jacobi Formalism 33References 38 2 A non-minimal coupling can be written in this form after an appropriate Weyl transformation to Einstein frame [23,24].

Non-extremal solution with warped resolved-deformed conifold background is important to study the infrared limit of large N thermal QCD. Earlier works in this direction have not taken into account all the back-reactions on the geometry, namely from the branes, fluxes, and black-hole carefully. In the present work we make some progress in this direction by solving explicitly the supergravity equations of motions in the presence of the backreaction from the black-hole. The backreactions from the branes and the fluxes on the other hand and to the order that we study, are comparatively suppressed. Our analysis reveal, among other things, how the resolution parameter would depend on the horizon radius and how the RG flows of the coupling constants should be understood in these scenarios, including their effects on the background three-form fluxes. We also study the effect of switching on a chemical potential in the background and, in a particularly simplified scenario, compute the actual value of the chemical potential for our case.

We derive a novel deformation of the warped resolved conifold background with supersymmetry breaking ISD (1,2) fluxes by adding D7-branes to this type IIB theory. We find spontaneous supersymmetry breaking without generating a bulk cosmological constant. In the compactified form, our background will no longer be a Calabi-Yau manifold as it allows a non-vanishing first Chern class. In the presence of D7-branes the (1,2) fluxes can give rise to non-trivial D-terms. We study the Ouyang embedding of D7-branes in detail and find that in this case the D-terms are indeed non-zero. In the limit when we approach the singular conifold, the D-terms vanish for Ouyang's embedding, although supersymmetry appears to be broken.We also construct the F-theory lift of our background and demonstrate how these IIB (1,2) fluxes lift to non-primitive (2,2) flux on the fourfold. The seven branes correspond to normalisable harmonic forms. We briefly sketch a possible way to attain an inflaton potential in this background once extra D3-branes are introduced and point out some possibilities of restoring supersymmetry in our background that could in principle be used as the end point of the inflationary set-up. In a companion paper we will analyse in details the inflationary dynamics in this background.

Abstract:The question of what happens when the heterotic SO(32) instanton becomes small was answered sometime back by Witten. The heterotic theory develops an enhanced Sp(2k) gauge symmetry for k small instantons, besides the allowed SO(32) gauge symmetry. An interesting question now is to ask what happens when we take the large k limit. In this paper we argue that in some special cases, where Gauss' law allows the large k limit, the dynamics of the large k small instantons can be captured by a dual gravitational description. For the cases that we elaborate in this paper, the gravity duals are non-Kähler manifolds although in general they could be non-geometric. These small instantons are heterotic five-branes and the duality allows us to study the strongly coupled field theories on these five-branes. We review and elaborate on some of the recent observations pointing towards this duality, and argue that in certain cases the gauge/gravity duality may be understood as small instanton transitions under which the instantons smoothen out and consequently lose the Sp(2k) gauge symmetry. This may explain how branes disappear on the dual side and are replaced by fluxes. We analyse the torsion classes before and after the transitions, and discuss briefly how the ADHM sigma model and related vector bundles could be studied for these scenarios.