Supersymmetric configurations, geometric transitions and new non-Kähler manifolds

Abstract

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)

Fermions on the antibrane: Higher order interactions and spontaneously broken supersymmetry

“…The IIB background now automatically has the three-form fluxes, as well as a five-form flux. confirming the low-energy effective D3-brane behavior, and the following background for the threeand the five-form fluxes: 25) with the type IIB dilaton e φ B = e −φ . One may verify that (2.24) and (2.25) together solve the type IIB EOMs.…”

“…However there exists another way, using a series of duality transformations, to study the background in the presence of the three-form fluxes. The steps have been elaborated in [24,25,22]. The solutions we will study here are specific realizations of the general solutions found and analyzed in [22], where supersymmetry of the final 'dualized' solution was explicitly confirmed 12 .…”

“…13 There is some subtlety in interpreting the final background with fluxes or with sources. This has been discussed in [25] which the readers may refer to for details. 14 One may generalize this to make the warp factors Fi functions of all coordinates except (θ1, φ1), i.e the directions of the wrapped brane.…”

“…The powerful thing about the fermionic completion is that the T-duality rules follow exactly the formula laid out for the bosonic fields, except now all the fields are replaced by their fermionic completions. This can be illustrated as 25 :…”

“…Using our T-duality ideas, and using the Gamma matrix algebras (4.21), it is easy to see that the two-form (4.22) appears naturally with an overall Γ m Γ n matrix provided we impose: 25) without loss of generalities as transformations with even number of Gamma matrices will not change any results. The puzzle however is if (4.24) takes the following form: 26) where (σ 1 , I 2 ) are the first Pauli matrix and 2 dimensional identity matrix respectively; and C 16×16 is another 16 dimensional matrix constructed out of Gamma matrices, IIB fields and covariant derivatives.…”