Geometric transitions, flops and non-Kähler manifolds: II

Abstract

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.

Dipole-deformed bound states and heterotic Kodaira surfaces

“…One can now see that for our type IIB D5/D7 bound state, the dilaton is given by 16) which reduces to (2.10) and is therefore not globally defined. As we observed above, this…”

“…Our conclusion then was that the backreactions would be small, and therefore the local geometry would be no different from the ones we examined earlier in [17], [21], and [16]. In this section, we aim to compute precisely the backreaction and see how far we are from the predicted local geometry of [17], [21], [16], which was determined for a separated system of D5 and D7 branes in a non-trivial geometry 3 . We will show that at this point in the moduli space, the local metric is surprisingly simple to determine even after we take into account all possible backreactions.…”

“…The geometries are conformally Kähler on both sides, but full global metrics are not known. However, the precise local geometries on any given patch are determined in [17], [21], [16] and [18] up to possible subtleties mentioned therein.…”

Lumps in the throat

“…The original Klebanov-Strassler model [21] has a similar structure at the far IR of the geometry, but differs from our case by the absence of three and seven branes. The geometry that is closest to our set-up is the one that was studied in [38] (for short reviews see [39]). We will soon provide a more rigorous way to analyse these coefficients (which was not attempted before).…”

“…Then semi-local defects can form on the D3-brane(s) worldvolume with a global symmetry G, provided the energetics also allow for this. The generic breaking pattern of the groups in this case will be: 39) where the subscript g and l refer to global and local symmetries respectively. The corresponding coset manifold M G , for which every point p corresponds to a semilocal defect that cannot be deformed into another one with finite energy, is given by…”