Dualising the Baryonic Branch: Dynamic SU(2) and confining backgrounds in IIA

In this paper we construct and examine new supersymmetric solutions of massive IIA supergravity that are obtained using non-Abelian T-duality applied to the Baryonic Branch of the Klebanov-Strassler background. The geometries display SU(2) structure which we show flows from static in the UV to dynamical in the IR. Confinement and symmetry breaking are given a geometrical interpretation by this change of structure. Various field theory observables are studied, suggesting possible ways to break conformality and flow in $N=1$ $T_N$ and related field theories.


Introduction And General Idea Of This Paper
The notion of duality is of course quite old, going back to well-known examples like the Maxwell equations in vacuum. The true power of the idea became clear around 1940 with the Kramers-Wannier [1] duality of the Ising model. In more recent times dualities have continued to be a driver of theoretrical progress with examples including Bosonisation [2], Montonen-Olive duality [3], S and T-dualities, Seiberg-Witten duality [4], Seiberg duality [5] and more general String dualities (U dualities). The duality conjectured by Maldacena [6], also called AdS/CFT or Gauge-Strings duality, is arguably the most powerful, widely applicable and conceptually deep duality of all known at present. All these dualities present common features: the degrees of freedom on both sides of the dual descriptions are in principle quite different; a strongly coupled (highly fluctuating) description of the system is characteristically mapped into a weakly coupled (semiclassical) one, in the same vein a phenomena that is 'local' in one set of variables becomes 'non-local' in the other (as exemplified by order-disorder operators and their typical 'uncertainty' relations), global symmetries are common to both dual descriptions, etc.
In this paper, we will mostly work with two dualities, the one conjectured by Maldacena and its extensions (see the papers [7] for a sample of representative work and reviews) together with what is called 'non-Abelian T-duality' [8]. We will use non-Abelian T-duality as a technique to generate new solutions to the equations of motion of Type II supergravity. Following the implementation of non-Abelian T-duality as a solution generating technique of RR backgrounds in [9], there have been a number of recent developments in the use of non-Abelian duality, see [10]- [19]. We will make use of many technical tools developed in these various papers.
We will consider backgrounds of Type II Supergravity that have a well understood (strongly coupled) field theory dual; we will then study the effect of this generating technique on the background. This will lead us to the construction of new solutions of ten-dimensional Supergravity and, as advocated in [14], we will use these new backgrounds to define new field theories at strong coupling. All of our backgrounds will be smooth and minimal supersymmetry in four dimensions (four supercharges) will be preserved. These new solutions will admit a description in terms of G-structures and we will explain how certain field theoretical phenonomena, like confinement and symmetry breaking are encoded in generic changes of the G-structure.
The system on which we will focus our study is the Baryonic Branch of the Klebanov-Strassler field theory [20], [21], [22]. This is perhaps, among the minimally SUSY examples known at the moment, the one that better passed test of the correspondence between geometry and (strongly coupled) field theoretical aspects. Besides, the Baryonic Branch field theory and geometry unifies the original Klebanov-Strassler system and the system of five branes wrapping a two cycle inside the resolved conifold [23]. Field theoretically, this unifi-cation can be thought as a Higgs-like mechanism and a particular limit where an accidental symmetry appears. See the papers in [24] for different geometric and physical aspects of this connection.
In this work, we will perform an SU(2) non-Abelian T-duality on the Baryonic Branch geometry. This is a geometry described by an SU(3)-structure. All features of the geometry are characterised by a couple of forms J 2 , Ω 3 that also encode many aspects of the strongly coupled dual field theory. Using non-abelian T-duality, we will obtain a new background in Massive Type IIA Supergravity. The G-structure will change to what is called SU(2)structure, characterised by forms j 2 , w 1 , v 1 , ω 2 . The SU(2)-structure will transition from being static in the large radius region of the geometry (corresponding to high energies in the dual field theory) to being dynamical once the small radius region of the geometry is considered. Hence, the phenomena of confinement and symmetry breaking are given a geometric description by the change in SU(2)-structure from static to dynamical.
The action of non-Abelian T-duality on the G-structures has been studied in many backgrounds which we take the opportunity to summarise in the table below. The contents of this paper are organised as follows. In Section 2 we will briefly summarise the original background and field theory corresponding to the Baryonic Branch of the Klebanov-Strassler field theory (the seed background/field theory pair on which we will apply our generating technique). In Section 3 we will present explictly the new solution.
In Section 4, we will organise all the previous information using the language of Gstructures. This will lead to a compact way of writing things, that can be very useful for other studies. We will study how the dynamical or static character of the G-structure depends on the field theoretic low energy dynamics captured by the original solution. In Section 5, we will discuss different aspects of the field theory dual to our new backgrounds. We close the paper with a list of possible future problems and conclusions. A number of technical and useful appendixes complement our presentation.

Generalities on the Baryonic Branch
The Klebanov-Strassler field theory is a two-group quiver with bifundamental matter, charged under a global symmetry of the form SU(2) × SU(2) × U(1) R × U(1) B . The ranks of the gauge groups are (N, N + M) and the bifundamental matter A 1 , A 2 , B 1 , B 2 self-interact via a superpotential of the form W ∼ ABAB. For a very clear explanation of many of the details of this quantum field theory, see [27], [28]. One detail that will be crucial to our present work is the fact that the so called 'duality cascade', a succesion of Seiberg dualities, ends in a situation where the quantum field theory may choose to develop VEV's for the Baryon and anti-Baryon operators.
In the last step of the duality cascade the gauge group is SU(M) × SU(2M). This theory has mesons M = (A a ) α i (B b ) i β and also baryonic operators [21] (2.1) and similar forB made out of (B i ) a l fields. One can see that both baryons and anti-baryons are neutral under SU(2) × SU(2) transformations.
The moduli space consists of two branches -the mesonic and the baryonic [28]. On the mesonic branch the baryons are zero (B =B = 0) and the mesons satisfy det M = Λ 4M . The non-perturbative contribution to the superpotential means that the associated moduli space can be identified with a symmetric product of the deformed conifold. On the baryonic branch the mesons are zero (M = 0) but the baryons acquire expectation values, where Λ is the strong coupling scale of the group SU(2M). Notice that both VEV's are equal only if ξ = 1. This corresponds to a Z 2 -symmetric point, represented by the exact solution in [20]. On this baryonic branch the U(1) B symmetry is spontaneously broken and the associated massless (pseudo-scalar) Goldstone mode corresponds to the phase of ξ. By supersymmetry this Goldstone lives in a chiral multiplet and comes along with scalar partner, the saxion, which corresponds to changing the modulus of ξ. As discussed in [28], the VEV of the operator, Let us focus on the situation where the field theory chooses to move to the purely baryonic branch. In this case, there is a smooth solution of the equations of motion of Type IIB su-pergravity, that describes the strong dynamics of this field theory, including the spontaneous breaking of the U(1) B symmetry [21], [22]. In the notation that we will adopt in this work, such background can be written compactly by introducing the (string frame) vielbein basis, Whereω i are the left invariant forms of SU (2). The metric, RR and NSNS fields are We have defined where κ is a constant that we will choose to be κ = e −Φ(∞) . The functions are, (2.8) The system has a radial coordinate ρ, on which (a, b, Φ, g, h, k) depend, and we have set α ′ g s = 1. The background is then determined by solving the equations of motion for the functions (a, b, Φ, g, h, k). A system of BPS equations is derived. These non-linear and coupled first-order equations can be arranged in a convenient form, by rewriting the functions of the background in terms of a combination of them, that decouples the equations (as explained in [30]- [31]). We will not go over these in the present paper. Enough will be for us to state that the whole dynamics of the string background is controlled by a single function P (ρ), subject to a second order non-linear and ordinary differential equation. This function P (ρ) can be determined numerically and has IR and UV behaviors There is only one independent parameter, c + > 0 (the constant h 1 is determined by c + ) and it is this parameter that can be identified with the Baryonic expectation value It is convenient to define a dimensionless quantity λ = 2 2/3 c + ǫ −4/3 where ǫ may be identified with the conifold deformation. See the paper [26] for a good account of the logic and technical details.

SU (3) structure of the Baryonic Branch
The Supergravity background above is characterised by what is called an SU(3) structure. That is, there exists a couple of formsĴ 2 andΩ 3 , in terms of which the BPS equations, the fluxes and various other quantities characterising the space can be written. The observation of [25], it that the formsĴ ,Ω, describing the full Baryonic Branch can be obtained from the simpler ones describing a set of D5 branes wrapping the two cycle of the resolved conifold. We will not repeat the details of the derivation here, but we quote the results to the extent that we will find useful.
In general, an SU(3) structure solution can be described by the following pure spinors in type-IIB [39], Where e 2A is the warp factor of the metric. Let us define where C 2 + S 2 = 1. It is possible to show that for zero axion field, that is F 1 = 0, SUSY requires the following equalities to hold (these are the BPS equations previously mentioned) The fluxes are determined as (2.14) The system of N c D5 branes wrapped on the resolved conifold is supported by just F 3 flux and is a solution to these equations when S = 0. The (string-frame) frame fields that describe this geometry can be obtained from those of eq.(2.5) by settingĥ = 1. In terms of these, the J 2 , Ω 3 ( denoted without hats to distinguish them from those of the Baryonic Branch) are given by J = e r3 + (cos αe ϕ + sin αe 2 ) ∧ e θ + (cos αe 2 − sin αe ϕ ) ∧ e 1 , Ω hol = e r + i e 3 ∧ (cos α e ϕ + sin α e 2 ) + i e θ ∧ (− sin α e ϕ + cos α e 2 ) + i e 1 , 4 Ω hol ∧Ω hol . The BPS equations for the functions h, g, k, a, b, Φ and the RR three-form flux, are Then the results of [25] show that theĴ,Ω of the full Baryonic Branch solution are obtained by introducing a non-zero phase or rotation parameter 2 ζ(r) in to (2.11) and defining: where e 2A is the warp factor of the Baryonic Branch solution. For further details on the geometry and physics implied by this 'scaling of forms', we refer the reader to the original papers [25] and [24].

A useful gauge transformation
Let us comment on a small subtlety that will be important in what follows. The above rotation argument makes it quite clear that by sending ζ → 0, the geometry becomes that of the wrapped D5 branes. On the other hand taking ζ → π 2 accompanied with λ → 0, the geometry becomes that given by Klebanov and Strassler i.e. the Z 2 point of the Baryonic Branch. Taking this limit is slightly delicate. One finds that sin ζ → 1 and cos ζ → 1 λ h KS where h KS is the Klebanov-Strassler warp factor. Expanding the functions (a, b, Φ, g, h, k) in the large λ limit and rescaling Minkowski coordinates x i → x i λ −1 one finds that leading term of the metric is independent of λ and reproduces the KS geometry. The limit applied on the NS two form is less trivial, in fact its expansion in inverse powers of λ is However the form of P 1 (the leading contribution of P (ρ) in this expansion) ensures that the pre-factor on the first term in this expression reduces to a constant and one recovers the Klebanov-Strassler NS two form modulo a pure gauge term.
In fact it is going to suit our purposes to perform a similar gauge transformation across the whole baryonic branch (2.6). We do this by defining In the KS limit this reduces to exactly the gauge transformation required in (2.18) and it has the effect of removing certain mixing between the angular directions and the radial direction in the NS two-form. 3 This will greatly simplify matters upon performing a duality transformation.

Non-Abelian duality on the Baryonic Branch
In this section, we will present the result for the non-Abelian T-duality when applied to one of the SU(2) isometries of the baryonic branch background in eq.(2.5)-(2.6). We extend the results of [14] in which the NS sector was established but full details of the RR sector were not provided. 4 We will perform the transformation described in [14] to the coordinates (θ,φ, ψ), present in the left-invariant forms of SU(2),ω i , i = 1, 2, 3 of eq.(2.5). We will choose a gauge where the new coordinates after the duality will be (v 2 , v 3 , ψ). We presents the results here and refer the reader to the Appendix C for details. We will start by specifying the vielbeins. The components do not change. The vielbeins in the (θ, ϕ) directions are also unchanged by the duality however we find it useful to introduce a rotation in (e θ , e ϕ ) such that the dual solution has no explicit ψ dependence.
3 This transformation leaves unchanged the gauge coupling defined through the integral of B 2 however it is non-vanishing at infinity and so one should exercise appropriate caution. 4 The results of [14] lead at first sight to a geometry that has a mixing between angular and radial directions. This is however a gauge artifact as will be made clear in Appendix C. By making the gauge transformation (2.19) to the seed geometry, as we do here, one removes this mixing. Alternatively one can perform the following coordinate transformation to the solution presented in [14] to obtain the solution presented here: v there where we have introduced left invariant SU(2) forms for the angles {θ, φ, ψ}. The vielbeins in the directions1,2,3 and NS 2-form potential can be compactly written in terms of the quantities defined as, The function Z was introduced as a gauge transformation to the seed solution already in (2.19). With these, we have (3.5) We will then have a metric that in terms of these vielbeins reads, ds 2 st = 10 i=1 (e i ) 2 . In terms of these vielbeins, the NS two-form B 2 reads, The dual dilaton is given by And the RR sector is given by, He k eθφ12 + v 2 e g eθφ23 .

(3.8)
Warning on potentially confusing nomenclature: The N c appearing in the above originated as the number of D5 branes wrapping the resolved conifold which was then rotated to give the Baryonic Branch and then T-dualised to this solution. Prior to T-duality, N c corresponds to the D5 charge which is also commonly denoted by M (which we will also use in section 5 when we specialised to the Klebanov-Tseytlin geometry). We hope the reader will not get overly confused by this point.

UV asymptotic behaviour
Using the semi analytic UV expansions that can be found, for example, in [26] it is possible to calculate the UV behaviour of the dual metric. The dual vielbeins at leading order in the UV are given by (3.9) Thus the dual 3-manifold shrinks as one flows towards the UV, in line with our expectations from abelian T-duality, where big circles are mapped to small circles.
One may worry that this vanishing manifold is a signal of a singularity in the UV, however, an explicit check shows that the curvature invariants: Ricci scalar, R µν R µν and R µνλκ R µνλκ are finite. In other words, both the g s and the α ′ expansions are under control and the background is trustable in the far UV. Notice that there is a one-cycle, labelled by the coordinate ψ in ω 3 , that shrinks to zero size in the large-ρ regime. This implies that strings wrapping this cycle will become light and will enter the spectrum of the dual QFT at high energies.
The dual dilaton is defined as e 2Φ = e 2Φ W where asymptotically, and so the dilaton is UV vanishing.

IR asymptotic behaviour
Let us now study the small radius regime of the metric, corresponding with the low energy regime of the dual QFT. Things are a bit less-simple. At leading order, terms in the metric depend explicitly of the original IR-parameters of the Baryonic Branch solution, but they also depend on the values of the v 2 , v 3 coordinates. The dual vielbeins in the IR tend to where we have defined for convenience. The function W tends to Here again, it happens that the dilaton is bounded and the Ricci scalar and Ricci and Riemann tensors squared are finite. This was expected, as we are performing a duality transformation on a space that in the small-ρ regime was of finite size (the S 3 in the deformed conifold). Dualities typically invert 'sizes' (or couplings). This example is not an exception. One may start with a background solution where Supergravity is a good approximation and obtain that in the far IR the new generated solution is still a trustable Supergravity background.
A point that we want to emphasize again is that in the far IR, the parameter that was labeling the different 'positions' on the Baryonic Branch (that is the different baryonic VEVs) still appears in the small-radius expansion above. There is a still a one-parameter family of solutions. Indeed, notice the dependence on the integration constants e Φ(0) and h 1 as defined in [25], both related to the number parametrising the Baryonic Branch.

SU (2) Structure of the background
We will now study the associated G-structure with this solution. Again, we will postpone details to the Appendix C. The geometry supports two pure spinors given by In the case at hand we find Here the framesẽ are obtained by a rotation, given by (B.19), of those in (3.5) and the parameters ∆, ζ i which enter into this rotation are specified by (C.15).
There are various immediate things to observe. If we move to the large radius region of the geometry, the functions sin α(ρ) ∼ a(ρ) ∼ b(ρ) → 0. The formulas simplify and we obtain, among other things that k → 0. This implies that, as happens in the paper [16], the two pure spinors are 'perpendicular' in the large radius regime of the solution and the SU(2)-structure is static. Similar behaviour was found in [18], where a dynamical SU(3)structure in 7-d becomes orthogonal in the UV. This changes as we evolve to the small radius regime of the background, the SU(2)-structure is said to become dynamical. In Section 5, we will discuss the physical effects that are associated with a change in the SU(2)-structure, from static in the far UV to dynamic in the IR.

Correspondence with Field Theory
In this section, we will connect our previous geometrical studies with aspects of the quantum field theory that our background is dual to. As it was anticipated in the paper [14], we believe that the field theory dual to our massive IIA background should be a non-conformal version of the Sicilian gauge theories presented in [32,33] or the linear quiver field theories studied in [34]. There are certain things that can be inferred immediately, like for example the confining character of the QFT. This follows from the fact that the calculation of the Wilson loop will proceed exactly as in the case of the Baryonic Branch field theory. Indeed, the R 1,3 × ρ part of the geometry is unchanged, hence, the Wilson loop will give the same result as before the non-Abelian T-duality. Nevertheless, many calculations done with the Klebanov-Strassler/Baryonic Branch background involved the 'internal' five dimensional space. The purpose of this section will be to learn how some of those calculations for field theory observables change (or not) for the new geometries in massive IIA.
The idea that will guide us is that for a given correlation function or related QFT observable, that in the original background was calculated in a way that is 'independent' of the SU(2) isometry used to perform the non-Abelian duality, will give the same result in the transformed background. We can think about those operators or correlators as 'uncharged' under the SU(2) symmetry in question. Ideas of this sort already worked in other solution generating techniques, like T-s-T dualities. Similar ideas also appeared in large N c (planar) equivalences between parent-daughter theories. The Physics of the common or 'uncharged' sector goes through to the new field theory. The rest of the paper deals with observables that are, in principle 'charged' under the SU(2) symmetry.
In the paper [14] it was shown that the cascade of Seiberg dualities-defined geometrically as a large gauge transformation of the NS two form and its effect on Page charges, persisted in the massive IIA background. In the paper [16], we started to geometrise some of the field theory effects corresponding to the Klebanov-Witten non-Abelian T-dual. In the rest of this section, we will focus our attention on the relation between the dynamical character of the SU(2)-structure and the field theoretical phenomena of confinement and discrete R-symmetry breaking. We will show how the presence of Domain Walls with an induced Chern-Simons dynamics on their world-volume follows as a consequence of the confinement and the dynamical character of the SU(2)-structure. Then, we will make clear that the symmetry associated with changes in the ψ-direction is related with an anomalous U(1) R R-symmetry in the field theory. We will define an instantonic object using an euclidean D0 brane; this will lead us to a possible definition for a Θ-angle and gauge coupling. We will find that this coupling has a non-conventional running in the far UV. We will then move into studying different aspects of the 'baryonic branch', also present in our new backgrounds. We will find that a given fluctuation of the RR background fields can be put in correspondence with a global continuous symmetry that the IR dynamics breaks spontaneously. We will find the associated Goldstone boson and an expression for the conformal dimension of such a baryonic operator.

Dynamic SU(2): A pathway to confinement
In this section, we will make more concrete the relation between the QFT phenomena of confinement and the dynamical character of the SU(2)-structure. The first observation is that the 'parallel projection' between both spinors, represented by k in eq.(C.18), is proportional to the quantity sin α. This quantity is related to the background functions as can be read from Appendix B of the paper [29], This is compatible with the expression in eq.(2.7) after following the algebra in Appendix B of the paper [29]. The presence of the functions a(ρ), b(ρ) in the Baryonic branch solution-see eqs.(2.5)-(2.8)-are responsible for the de-singularisation of the space (the appearance of a finite size S 3 ) and the IR minimization of the dilaton and warp factor. These have as a consequence the linear law, E QQ = σL QQ for large distance separations between the quark-antiquark pair. In other words, the functions a(ρ), b(ρ) and their effects on the warp factor and dilaton 'produce' confinement. In the same vein, at the level of the metric, the presence of a(ρ) implies the breaking of the symmetry ψ → ψ + ǫ into ψ → ψ + 2π. This is the remaining Z 2 symmetry after the spontaneous discrete R-symmetry breaking. So, we see clearly that confinement and spontaneous R-symmetry breaking go hand-in-hand with the function a(ρ). Hence, these phenomena in the dual QFT are closely related to the presence of k , which as we made clear is related to the dynamical character of the SU(2)-structure. In the papers [42,43], the point was made that the functions a(ρ), b(ρ) were directly related with the gaugino condensate. This suggests that in our massive IIA picture, there exists a relation of the form < λλ >∼ k . Similar ideas will be discussed in the paper [51].

A comment on domain walls
It was proposed in [14], that domain wall objects were realised in the Non-Abelian T-dual of the geometries we are considering, as D2 branes that extend on R 1,2 . Indeed, the induced metric, action and tension of a (2 + 1)-dimensional object are, If we also turn on a gauge field in the world-volume of this D2 brane, a Chern-Simons-Maxwell action will be induced, at leading order in α ′ on this D-brane, We have used that a new WZ-like term appears in Massive IIA as explained in [44]. The Chern-Simons term is quantised, being proportional to T D2 N c . 5 In the type IIB Baryonic Branch solution(s), domain walls were realised by D5-branes extended on R 1,2 and the three-sphereS 3 = [θ,φ, ψ]. Once a gauge field is turned on, a Chern-simons terms was induced, proportional to T D5 S3 F 3 . Naively, we can think that both objects are 'connected' by the non-abelian T-duality, under which the directions onS 3 disappear and we are left with a D2 brane as described above.
Supersymmetry gives support to this. Indeed, around eq.(6.19) of the paper [39], we are presented with the calibration form for a domain-wall like object, which is given by the real part of the pure spinor Ψ + . Using that |a| 2 = e A = e Φ/2ĥ−1/4 , we obtain that the BI action equals the calibration form. Notice also that this selects the k component of the pure spinor.
As it was shown in the paper [14], once the R-symmetry is broken in the Type IIB setup, the non-abelian T-duality maps these backgrounds to their partners in Massive IIA. In a minimally SUSY quantum field theory, the presence of domain-walls is tied up with confinement and the spontaneous breaking of the Z 2Nc -symmetry. As we emphasized, these phenomena are related to the 'dynamical' character of the SU(2)-structure, hence to the presence of the k part of the pure spinor.

The fate of the U (1) R anomaly
In the backgrounds presented in [14] and those of this paper it is somewhat natural to expect that the coordinate ψ is singled out as being related to an R-symmetry of any putative field theory dual. That this is true is by no means obvious, after all in the technical process of dualisation the fact that we retained the coordinate ψ was purely a result of a judicious gauge choice. Here we provide evidence that this is indeed the correct identification and furthermore that this U(1) is afflicted with an anomaly, breaking it down to a discrete subgroup.
A robust understanding of how ∂ ψ plays the role of the R-symmetry in the holographic dual was given in [40] with several important details of the supergravity solution clarified in [41]. The essential point of [40] is to introduce a bulk 5d gauge field that gauges this U(1) ψ by making the replacement dψ → χ = dψ − 2A in the metric. This must be supplemented with an appropriate ansatz for the fluxes. In the case of the Klebanov-Witten background one finds that the resultant gauge field is massless and is the dual fluctuation to the global U(1) R of the gauge theory. However, in the non-conformal cases, the correct ansatz for the fluxes actually yields a massive gauge field (the mass here comes from a Stückelberg rather than Brout-Englert-Higgs mechanism).
Let us begin our discussion with the non-abelian T-dual of the Klebanov-Witten backgound. The NS sector of the geometry is given by where σ3 = dψ + cos θdφ. This metric is supported by RR two and four form fluxes. The U(1) acting as ∂ ψ can be gauged by making the replacement σ3 →χ = σ3 − 2A in the NS sector above. The potentials corresponding to the correct modification of the RR forms that support this fluctuation are given by (cos θdφ + A) , where we introduce the volume form on the S 2 ,ω 2 = sin θdθdφ and ⋆ 5 is the Hodge dual in the AdS 5 directions. This solves the linearised equations of motions, linearised Einstein equations and Bianchi identities provided that the gauge field obeys the equation d ⋆ 5 dA. This, together with the fact that the Killing spinors of the geometry are charged under U(1) ψ identifies this as the dual to the R-symmetry. Upon substitution of this ansatz in to the action one finds all the gauge field dependance gives a field strength squared contribution, for some function f (v 2 , v 3 ) of the internal coordinates that will be integrated over in a reduction to a five-dimensional theory. Now we turn to the non-conformal geometry obtained by transformation of the Klebanov-Tseytlin geometry (since we are only interested in the UV behavior we will not need the full Klebanov-Strassler or baryonic branch). The NS sector, with the U(1) ψ gauged, is given by Here h(r) is the usual Klebanov-Tseytlin warp factor and . Without the gauging this is a solution of massive IIA with Romans' mass proportional to M. By examining how the non-abelian T-duality transformation acts on the ansatz given by Krasnitz in [41], we can determine a suitable ansatz for the fluxes: The remaining term in the three-form potential is given implicitly by 6 Here W is a gauge invariant 1-form that combines the gauge field A with a Stückelberg scalar scalar W = A − dλ though for practical purposes we follow [41] and chose a gauge in which W = A. This is a solution to the linearised flux equations and Bianchi identities provided the fields introduced obey the constraints on the ansatz required in [41]: The exterior derivative of right hand side of this expression vanishes on the equations (5.8).
Here ⋆ 5 is the Hodge dual with respect to the metric ds 2 3 . In [41] it was shown how these equations (5.8) can be diagonalised by defining The mode W 1 corresponds to a massive gauge field whose mass as a result of the spontaneous (anomalous) breaking of R-symmetry. The mass of this mode is given by [41]: The interpretation is identical here and we conclude therefor that the U(1) R symmetry is anomalously broken.

Dependance on ψ in the potentials and D0 brane instantons
To understand this breaking as an anomaly it is informative to look at the forms of the RR potentials. For the non-Abelian T-dual of the Klebanov-Witten we have following potentials For the dual of the Klebanov-Tseytlin (which has Romans mass proportional to M) we have Note how the dependence on ψ in C 1 is quite different in the potentials in the conformal and non-conformal cases.
Let us now consider D0 branes. These D0 branes will move in the v 3 direction, leaving all other coordinates fixed, in particular we will choose v 2 = 0. We can then calculate using (5.5) the induced metric for this D0 brane, relevant gauge potential and its BIWZ action, that will read We use now that We will equate the BIWZ action of this euclidean D0 brane with the gauge coupling and the Θ angle imposing that S BIW Z = 8π 2 g 2 + iΘ. In other words, we consider this D0 brane to be an instanton in the dual gauge theory.
Analysing the WZ term, we have that (like above, we choose g s = 1), Using that the theta angle should be periodic, we can impose that the allowed changes in the angle ψ get selected to be So, we see that there is a breaking of the global continuous symmetry into a discrete one. The residual discrete symmetry is determined by the domain of the coordinate v 3 . In the case in which we would like to impose this discrete symmetry to be the same as before the non-Abelian duality we should impose that L v 3 = 2. Indeed, one of the major challenges with understanding non-abelian T-duality is to identify the periodicities of the coordinates of the T-dual geometry. Here we see a direct link between a field theory property (the anomaly) and the global properties of the geometry. Let us look at the BI term. We have that the gauge coupling, associated is We can perform the integral explicitly, but it is perhaps more illuminating to look at the large radius limit of the expression above. After all, we are doing this calculation in the non-Abelian dual of the Klebanov-Tseytlin solution, we should only trust the result in the far UV. We have then, considering the leading term in the large-r expansion, this reproduces a result obtained by other means in [14].

The fate of U (1) B
The Klebanov-Witten SU(N) ×SU(N) conformal field theory coming from D3 branes at the tip of the conifold has a U(1) baryonic number symmetry acting as In the gravity dual this number current gives rise to a massless AdS 5 gauge field where ω 3 is the usual closed three form on T 1,1 . The non-abelian T-dual of the AdS 5 × T 1,1 geometry was obtained in [13]. In the T-dual geometry, this U(1) B mode translates into a perturbation, which solves the linearised supergravity equations of motion, given by The final two terms in δC 3 come from the a contribution from δC 6 under the T-duality transformation 7 . Although the two-form W 2 has a simple form it can not easily be written in terms of the invariant tensors that define the SU(2) structure of the geometry. The existence of this mode is suggestive that the field theory duals corresponding to the conformal geometries constructed in [13] have a global U(1) symmetry in addition to the preserved U(1) R . In fact, the geometry T-dual to the Klebanov Witten is closely related to those proposed in [33] as the gravity duals to N = 1 SCFT's formed by wrapping M5 branes on Riemann surfaces (which in this case is genus zero giving rise to many subtleties). These SCFT's do indeed have U(1) R × U(1) F Abelian global symmetries which are seen geometrically as isometries of the corresponding eleven-dimensional supergravity solution. Upon reduction to ten-dimension one of these U(1)'s gets degeometrized corresponding to the above gauge field δC 1 = A.
In this paper our main focus has been the cascading field theory where at the last step of the cascade when the gauge group is SU(M) × SU(2M) the baryons acquire expectation values, On this Baryonic branch the U(1) B symmetry is spontaneously broken. To see this from the gravity perspective it is sufficient to work with the Klebanov-Strassler geometry corresponding to the field theory at the Z 2 symmetric point of the Baryonic branch. As shown in [21], there is a massless glue ball corresponding to a Goldstone mode associated with changing the phase of ξ which is given by 8 The linearised supergravity equations are solved when the pseudo-scalar is a harmonic function in R 3,1 and the function f 2 (τ ) obeys a second order differential equation admitting a normalisable solution.
The non-abelian T-dual geometries considered also admits a similar mode, which can be obtained simply by performing a T-dualisation of the ansatz for the scalar modes in the seed IIB solutions. The T-dual of the Klebanov-Strassler geometry was obtained explicitly in [14]. Performing a dualisation of the ansatz (5.22) gives rise to a perturbation δF 2 and δF 4 . This perturbation solves the supergravity equations of motion when f 2 obeys the same differential equation as for the ansatz (5.22). The expressions for F 2 and F 4 are not particularly enlightening though for completeness let us provide a few details. Here we display the results in the UV regime where the geometry is given by (5.5). The corresponding deformations to the potentials are given by 23) The extra functions introduced above are completely determined by f 1 and f 2 according to where T (r) = 9 √ 2 M log r/r 0 and h(r) = 27 32r 4 (3M 2 + 8Nπ + 12M 2 log r/r 0 ). The existence of this mode suggests a spontaneously broken global U(1) in the field theories dual to the geometries obtained in section 3. In the conformal case, the unbroken U(1) becomes geometrized upon lifting to M-theory whereas these non-conformal backgrounds are solutions of massive IIA and so can not be lifted. This further underlines the expectation that a U(1) is broken.
In the same multiplet as the pseudo-scalar goldstone is a scalar perturbation corresponding to changing the magnitude of ξ. In the same vein as above, one could deduce the fate of this scalar perturbation under the T-duality transformation; it will give a similar, albeit complicated, perturbation in the dual IIA background. Since the full baryonic branch geometry found in [22] can be thought of as exponentiating such transformations to give arbitrary values of the Baryonic vev, implicitly in the geometries presented in section 3 we have already done just that.

The fate of the baryon condensate
In Klebanov-Witten theory the closest analogy to a baryon vertex -the object to which N external quarks can attach [36] -would be a D5 brane wrapping the T 1,1 space with world volume coordinates {x 0 , θ 1 , φ 1 , θ 2 , φ 2 , ψ} [37]. The primary reason for this identification follows the argument made in [36]; since we have the WZ term induces a charge to the world volume U(1) gauge field A via the coupling This introduces N units of charge which must be canceled by some other source to give zero net charge in a closed universe. This cancelation is achieved by N elementary strings stretching from the boundary to the brane whose end points are external quarks. A perhaps naive approach would be to suggest in the IIA geometry dual to the Klebanov-Witten theory a similar role could be played by a D2 brane wrapping the S 2 with world volume coordinates {x 0 , θ, φ}. Indeed, since in the case of T-dual to Klebanov-Witten we have C 1 ∝ cos θdφ the WZ coupling F ∧ C 1 produces a charge contribution for the gauge field that could be cancelled with external quarks just as in the Klebanov-Witten scenario. It would be of some interest to study the baryon vertex in the massive IIA backgrounds. 9 This baryon vertex should however be distinguished from the configuration representing the actual baryon condensate -which should be supersymmetric, gauge invariant and not require BIon spikes. The configuration that describes the baryon condensate is a Euclidean D5 brane wrapping the T 1,1 and the radial directions [37]. This D5 has D3 branes dissolved within [38] which are traded for a world volume gauge field. Following the logic applied to the baryon vertex one might anticipate that in the IIA geometries presented here, the role of the condensate is played by a wrapped Euclidean D2 brane on the S 2 × R with a world volume gauge field.
To determine the existence of such a configuration, rather than calculate the kappa symmetry projectors, we will harness the power of the G-structure and the calibration techniques of [39]. The condition for a supersymmetric Euclidean p brane on a cycle Σ is essentially the same as that of a Lorentzian p + 4 brane that is spacetime filling in the Minkowski directions. This condition is given by where the world volume field strength is F = B| Σ + 2πα ′ dA and the pure spinor entering the calibration form is given Φ = Ψ + for IIB and Φ = Ψ − for IIA. Before looking at this question in the context of the full baryonic branch let us address it in the conformal case -we would still anticipate a supersymmetric configuration to exist even. In the Klebanov-Witten theory the E5 configuration of a brane extended along Σ = {r, θ 1 , φ 1 , θ 2 , φ 2 , ψ} with a world volume gauge field A = 1 3 ζ(r) (dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 ) , (5.28) obeys the calibration condition (5.27) provided that which of course can be readily integrated.
In the IIA non-Abelian T-dual of the Klebanov-Witten geometry we find an E2 configuration extended along Σ = {r, θ, φ} at the point v 2 = 0 but with a non-trivial embedding v 3 = f (r). We search for a supersymmetric configuration solving the calibration condition (5.27) when supported by a gauge field From the calibration condition one finds firstly that the embedding f (r) and the gauge field should differ only by a constant c 0 . The gauge field should then obey an equation which can also be readily solved and one notices that when c 0 = 0 has the same form as the equation (5.29) governing the configuration in IIB.
Lets move up to the KT geometry working in the exact logarithmic solution 10 . First we recapitulate the calculation for the baryon condensate in the IIB background. Using the calibration technique one readily finds the E5 configuration is the same but with the gauge field equation of motion eq. (5.29) modified to be where T (r) = 9 √ 2 M log r/r 0 and h(r) = 27 32r 4 (3M 2 + 8Nπ + 12M 2 log r/r 0 ). This equation may be integrated to yield where c is a constant of integration which we now set to zero since its contributions are in any case sub-leading. Inserting this into to the DBI action one finds, changing variables to t = log r, In [37], e −S E5 was identified with the bulk field dual to the baryonic condensate. Using the standard asymptotic expansion the field theory scaling dimension can be extracted (at least in the large t regime) as reproducing exactly the result of [37] notable for the scaling dimension dependence on the energy scale of the baryons as anticipated from the field theory.
In the non-abelian T-dual the situation is already rather involved. We search for an E2 configuration extended along Σ = {r, θ, φ} at the point v 2 = 0 and now with v 3 = χ(r) and an ansatz for the gauge field We take the square of the calibration equation eq. (5.27) and first consider terms proportional to cos 2 θ. From these one finds a first equation relating the gauge field and the embedding in v 3 : α ′ (r) = χ ′ (r) . We let c 0 be the additive constant between α and χ. Then from the remaining terms in eq. (5.27) one finds a differential equation for the gauge field Changing variable to t = log r one can solve this equation on the exact logarithmic solution: (5.39) here c is an integration constant giving sub-leading contributions that we hence ignore.
Using the equation (5.38) we find that the DBI action is given by If we expand out asymptotically we find that where κ = T D2 vol(S 2 ) = 1 π . It would be interesting to pursue this line of reasoning further by extracting the value of the condensate across the baryonic branch. This is technically rather involved and we do not intend to do so in this report.

Conclusions and Future Directions
In this paper we have examined a new family of solutions of massive IIA supergravity. These new backgrounds were obtained by performing a non-abelian T-duality on the geometry that describes the non-perturbative Physics of the baryonic-branch of the Klebanov-Strassler field theory. We have explored the transition from SU(3) structure, characterising the 'seed' backgrounds to the dynamical SU(2)-structure that describes the resulting massive IIA solutions. We made clear-at least for the type of backgrounds studied here-that the dynamical character of the SU(2) structure is directly related to the phenomena of confinement and symmetry breaking. We believe that all these new features have not been discussed in previous literature, in a context as clear and unifying as the one presented here.
The new backgrounds discussed in this paper display a host of interesting non-perturbative phenomena that 'define' the dual field theory. Some of these are, • The non-conformality of the geometry is enabled by a non-zero Romans' mass.
• Whilst the UV geometries proposed in [14] are characterized by static SU(2) structure [16] the full IR complete geometry of this paper has dynamic SU(2) structure.
• The transition to dynamic SU(2) structure gives a geometric realization of confinement and permits supersymmetric D2 branes that act as domain walls in the IR. This realises geometrically the relation between confinement, the spontaneous breaking of a discrete R-symmetry and the presence of domain walls.
• The U(1) R symmetry is realized by the vector ∂ ψ and the corresponding fluctuation, which is a massless gauge field in the conformal case, acquires a mass indicating an anomalous breaking.
• Euclidean 'instantonic' branes reproduce this anomaly of the R-symmetry and at the same time suggest a non-conventional running for a suitably defined gauge coupling.
• A further U(1) (baryonic) symmetry is broken. In the conformal case of [14] this symmetry is unbroken and is realized geometrically by the M-theory circle. In our backgrounds, once conformality is broken by the addition of fractional branes, the symmetry is no longer geometrical as we are now in a massive IIA context. The U(1) B symmetry is spontaneously broken and we identified a corresponding massless glueball (the associated Goldstone boson).
• We give evidence that this U(1) B may be thought of as baryonic and that a baryonic condensate is given by a Euclidean D2 brane wrapping a two-cycle in the geometry.
Although we do not yet have a complete understanding of the field theory dual to this new geometry, the results of this paper together with those in [14] suggest that it may be a nonconformal and cascading version of the Sicilian theories of [32,33] or the linear quivers of [34]. We would like to close this paper on a forward looking note. We suggest that the features mentioned above may be prototypical of a wider class of holographic duals. The theories in [32,33] and also the IIA linear quivers of [34], present a wide new class of interesting examples of N = 1 SCFTs. We anticipate that by a modification of these theories (this paper suggests that the modification will involve adding D8 branes in IIA) one can obtain a variety of non-conformal gauge theories. Some of the non-perturbative features of these new field theories should be the ones we are describing in this paper.
Aside from this and on a more geometrical note, we believe the backgrounds presented in this paper may serve as a prototype for new dynamical SU(2) solutions of massive IIA supergravity that will be the corresponding string duals to the new field theories described above. This is, of course, in the same vein as the route from the conformal geometry of Klebanov-Witten to the non-conformal geometry of Klebanov-Strassler. In our view, these represent the most interesting avenues of further investigation.

Acknowledgments
Discussions with various colleagues helped to improve the contents and presentation of this paper. We wish to thank:

A.1 Supergravity
We work in string and the 10-d hodge dual is defined such that where F n are the RR fluxes of either type-IIA or type-IIB supergravity. The fluxes may be used to define a polyform F such that In terms of the polyform the Bianchi identities may be expressed as It is easy to show this is satisfied with the definition where C is a polyform constructed from the RR potentials in the same fashion as above and F 0 should be taken to be zero in type-IIB. The flux equations of motion are expressed as where the sum needs to me taken over the appropriate RR fluxes of type-IIA/IIB. The dilaton must obey the equation of motion while Einstein's equations are in type-IIA by with an equivalent equation holding in type-IIB.

A.2 Pure Spinors
Here we follow the conventions of [49] except for a difference in the self duality condition of the RR section which leads to a few sign differences. We work in string frame and consider solution with metrics that can be expressed as ds 2 = e 2A dx 2 3,1 + ds 2 6 (A. 8) and preserve N = 1 SUSY in 4-d with non trivial RR sector. This means that the internal space, with metric ds 2 6 , must support an SU(3) × SU(3)-structure [39]. We decompose the 10-d MW spinors into a 4 + 6 split as where in ǫ 2 the upper/lower signs should be taken in type-IIA/B, the ± indicates chirality of both 4-d and internal 6-d spinors and we choose a basis for the internal spinors such that (η + ) * = η − . It is possible to define two Clif f (6, 6) pure spinors on the internal space as which may be identified with polyforms under the Clifford map. The internal spinors are decomposed as where k 2 || + k 2 ⊥ = 1, η † + η + = χ † + χ + = 1 and χ † + η + = 0. The N = 1 SUSY conditions for such a SU(3) × SU(3)-structure solution are given by the differential conditions where λ(A n ) = (−1) A n andF is the internal part of RR polyform in type IIA/B where the RR forms are each decomposed such that As before upper/lower signs correspond to type IIA/B Clearly in general η 2 + is composed of a parts that is parallel and a part that is orthogonal to η 1 + . The SU(3) × SU(3)-structure can categorised into 3 distinct cases depending on the values of the coefficients k ⊥ and k || :

SU (3)-structure
When k ⊥ = 0 the internal spinors are parallel and the pure spinors define an SU(3)-structure in 6-d such that where J and Ω hol are the two and holomorphic three forms associated with SU(3), they are defined as in terms of the 6-d gamma matrices as Ω (hol) and satisfy 16) Orthogonal SU (2)-structure When k || = 0 the internal spinors are orthogonal and the pure spinors define an orthogonal SU(2)-structure in 6-d such that where the SU(2)-structure one forms v, w and two forms j, ω are defined as and obey the relations 19) Intermediate and Dynamical SU (2)-structure For intermediate SU (2)-structure k || and k ⊥ are non zero constants, this and the previous example are also referred to as static SU(2)-structure. For dynamical SU(2)-structure k || and k ⊥ are point dependent. For both these cases the pure spinors are given by where eq A.19 and eq A.18 still hold. In these conventions the SUSY conditions (here we consider type IIA, details of type IIB are given in appendix E) may be split up as follows: where the second of these gives a definition for H which can be combined with the first to give a definition of the NS potential, namely this is not the same as the NS potential generated by non-abelian T-duality but must match it up to an exact.
The rest of the SUSY conditions are (A.23) from which it is possible to define the higher forms of the RR sector as: where the RR potentials are given by: The calibration is given by where ± depends on our conventions in the WZ action. That S DBI + S W Z = 0 is trivial because in these convensions we have: This all works perfectly for the case θ − = 0 which is the dual of the wrapped D5 solution.
B Details of the non-Abelian T-duality on the D5 branes solution.
The purpose of this section is to give some details of the SU (2) isometry T-dual of Wrapped D5 branes on S 2 . This was first derived in [14], but in slightly different conventions and the G-structure was not found. This is the C = 1, S = 0 limit of the full baryonic branch dual solution, and as the procedure for find the the G-structure is the same in both case we hope that this more simple example will be instructive. Solution of wrapped D5 branes on S 2 [29] has string frame metric given by where the functions a, b, g, h, k and the dilaton Φ only depend on the holographic coordinate r. Theω i are SU(2) left invariant 1-forms which can be parametrised as ω 1 = cos ψdθ + sin ψ sinθdφ , ω 2 = − sin ψdθ + cos ψ sinθdφ , .

(B.2)
A convenient set of vielbeins is given by with respect to which the non trivial RR flux F 3 may be expressed as where the f i are given by eq 2.8. In these conventions the projections the 10-d Killing spinor ǫ obeys are Γ 12 ǫ = Γ θϕ ǫ, Γ r123 ǫ = (cos α + sin αΓ ϕ2 )ǫ, iǫ * = ǫ , (B.5) with respect to the 4 + 6 split we can define components of ǫ to be equal with positive chirality as where 2A = Φ. Once the usual decomposition of gamma matrices, is performed it is a simple matter to derive the SU(3)-structure forms of eq 2.15 using eq A.15, where we have chosen iγ rθϕ123 η + = η + . To do this it is helpful to perform a rotation in e ϕ , e 2 which will also be useful later e ϕ = cos αe ϕ + sin αe 2 e 2 = − sin αe ϕ + cos αe 2 e a = e a for a = ϕ, 2. (B.8) The rotated 6-d projections are then simplŷ γ ϕθ η + =γ r3 η + =γ 21 η + = i η + (B.9) and the SU(3)-structure becomes canonical. We want to T-dualise this wrapped D5-brane solution along the SU(2) isometry parametrised byω i . Section 2 and Appendix B of [14] give all the details of the algorithm one must follow to do this and so we direct the interested reader there for details of the NS sector. For the RR sector we only give details that will be relevant for later calculations.
The duality will drastically change the vielbeins that contain the SU(2) left invariant 1-forms e 1 , e 2 , e 3 and leave the others untouched. For the dual of the wrapped D5 brane solution gauge fixed such that the remaining dual coordinates are v 2 , v 3 and ψ, the cannonical vielbeins given by the procedure of [14] are with the remaining veilbeins still given by eq B.3, that is e a ′ = e a for a = 1, 2, 3. The ω i are defined as in eq B.2 but withθ → θ,φ → ϕ. It is possible to remove all the explicit angular dependence from the dual solution by performing a rotation in the θ, ϕ directions such that eθ = e h+Φ/2 ω 1 = cos ψe θ + sin ψe ϕ eφ = e h+Φ/2 ω 2 = − sin ψe θ + cos ψe ϕ , (B.11) and an additional rotation in 1',2',3' directions such that (B.12) Theses rotation make the expressions for the vielbeins and fluxes a lot more simple than they otherwise would be, they are given for the dual of the wrapped D5 solution as in section 3 but with S = 0, C = 1. However, it is the e a ′ vielbeins rather than the eâ ones that are more suited to calculating the G-structure of the dual solution.
It was shown explicitly in [16] that the 10-d MW Killing spinors transform under an SU(2) isometry T-duality asǫ where Ω is given by Ω = Γ (10) −Γ 123 + 3 a=1 ζ a Γ a 1 + ζ 2 (B.14) and for the wrapped D5 background we have Starting from eq B.10 we first rotate the veilbeins as in eq B.8 so that the projections are canonical. The Ω matrix then becomes where we have used γ 1ϕ3 η + = iη − . The new spinorǫ 2 is: It is clear here that, as long as sin α = 0, we are in the dynamical SU(2)-structure case, because α = α(r). In order to simplify the expressions we perform another transformation of the vielbein basis: We define a new basis: where the order is rθϕ123. In terms of this new basis, the spinor is: And the projections in this basis are still: Let us now express the forms of the geometric structure, following the conventions of appendix A.
which is a dynamical SU(2)-structure.

C Details of the non-Abelian T-duality on the Baryonic Branch solution
In this section we give some details of the SU(2) isometry T-dual of the Baryonic branch of Klebanov-Strassler. This was originally derived in [14] with gauge fixing such that v 1 = ϕ = θ = 0. The previous derivation indicated a departure in the T-dual from the log corrected AdS 5 asymptotics of the baryonic branch. Let as begin by giving some details of original calculation in our current convensions C.1 Dual of the baryonic branch without the shift in B 2 Once more we will start by specifying the dual vielbeins. The components do not change. The vielbeins in the θ, ϕ are also unchanged by the duality however we find it useful to introduce a rotation in e θ , e ϕ such that the dual solution has no explicit ψ dependence.
The vielbeins in the directions1,2,3 can be compactly written in terms of the quantities defined as, With these, we have where the rotation of eq B.12 has been performed 11 . We will then have a metric that in terms of these vielbeins reads, Notice that the quantity Λ in eq.(3.4) will, when squared to construct the metric with the vielbeins above, imply the existence of crossed terms g ρv 3 and also the change of the asymptotic behaviour of g ρρ away from log corrected AdS 5 .
In terms of these vielbeins, the NS two-form B 2 reads, The dual dilaton is given by And the RR sector is given by, 2C sin α N c b + a(e 2g cos 2 α − N c ) + e g+h sin 2α eθφ , (C.7) We will now proceed to show that the bad asymptotic behaviour and off diagonal ρ terms of the metric are actually a gauge artefact.

C.2 The dual of the baryonic branch with the shift in B 2
The NS 2-from of the original solution contains the term It is this term, when dualised, that gives rise to the undesirable behaviour as this will contribute to the dual metric in both g ρρ and g ρv3 via the dual vielbeins eˆi which will have legs in ρ. This happens because of the dρ ∧ω i term inB 2 which is not a spectator under the duality transformation 12 . However, one is always free to add an exact to the NS potential as this will not change the fluxes or metric of the original solution. Consider adding a closed form to the initial B 2 This precisely cancels the effect ofB 2 in the dual solution when Z ′ = − 1 2 Se 2k+Φ becausẽ As there is no longer a dρ ∧ω i term in the NS 2 form before dualisation, the dual vielbeins will have no legs in ρ and so there will no longer be a modification to g ρρ and g ρv 3 . The trade off is that the function Z will now enter into the dual solution.
We now once more follow the procedure of [14] with gauge fixing, as before, such that v 1 = ϕ = θ = 0. We are lead to the dual vielbeins 12 See section 2 of [14] for details of how the initial B 2 enters into the definition of the dual vielbeins.
which upon rotating according to eq B.12 give the vielbeins of eq 3.5.
A valid question at this point is whether there is a local diffeomorphism which maps us from the baryonic branch dual solution as defined in section C.1 to the solution defined as in section 3. The answer is yes, and it may be most easily found by comparing the dilaton as defined in eq 3.7 and eq C.6 . Examining these makes it clear that one needs to transform V 3 such that it is mapped to H. This may be achieved with a transformation in v 3 only and so vielbeins of eq C.4 are mapped to those of eq 3.5. The map on the RR sector also follows trivially whilst the NS 2-form of eq 3.6 is mapped to that of eq C.5 up to an exact. So it is clear that one may "cure" the bad asymptotics and g ρv 3 mixing of section C.1 either by a gauge transformation in the NS 2-from before dualisation, or by a local diffeomorpism on the dual coordinate v 3 after the duality procedure is performed.

C.3 Details of the Dual Baryonic Branch Structure
All that remains to compete the elucidation of the baryonic dual is to give supplementary details to section 4 on the dynamical SU(2) structure. Actually, the derivation of the structure is essentially the same as that of the dual of the wrapped D5 solution in section B, so we will only focus on the differences here. The 10-d MW Killing spinors of baryonic branch obey the same projection as the wrapped D5 spinors (see eq B.5). However, whilst the internal spinors are still parallel, they now differ by a point dependent phase e iζ(r) = C + iS ǫ 1 = e A (ξ + ⊗ (e iζ(r)/2 η + ) + ξ − ⊗ (e −iζ(r)/2 η − )), ǫ 2 = e A (ξ + ⊗ (e −iζ(r)/2 η + ) + ξ − ⊗ (e iζ(r)/2 η − )) (C.14) where the Minkowski warp factor is now e 2A = e Φ C . We now follow the steps illustrated between eqs B.7 and B.9 such that the SU(3)-structure of the baryonic branch takes canonical form.
The dual 10-d Killing spinors are given as in eqs B.13,B.14, however the ζ a entering into their definition are now given by The new spinorǫ 2 is:ǫ whereη 2 − is still given by eq D.26. The dynamic SU(2)-structure supported by the dual baryonic branch solution may be expressed as (C.17) The forms and functions entering into these expressions are given by with ζ a defined by C. 15. Specifically the vielbeinsẽ that the structure is expressed in terms of a rotation of those in eq C.11. First one preforms a rotation by α e ϕ = cos αe ϕ + sin αe 2 ′ e 2 = − sin αe ϕ + cos αe 2 ′ e a = e a for a = ϕ, 2 ′ , (C. 19) and then rotates these vielbeins to getẽ = Rê, where the matrix R is given by eq B.19 with ζ a by eq C.15.

D Details of the non-Abelian T-duality on D6 branes on S 3
In this appendix, we study another background, similar to the one described in the main part of this paper. We want to start with a solution of D6-branes wrapping a three-sphere in type IIA supergravity, that preserves N = 1 supersymmetry. We first describe such a solution, then we apply a non-Abelian T-duality to find a new type IIB supergravity solution. We study this transformation at the level of the geometric structure. We then take advantage of this example to make general statements on N = 1 type IIB supergravity solutions.

D.1 The type IIA solution
We are interested in finding a solution of D6-branes in type IIA supergravity. For that purpose, we start by considering eleven-dimensional supergravity. Because we only want D6branes, the M-theory solution is a background with no fluxes. Such a solution is described in [46] or [47] (we follow the notation of the latter). The metric of the solution is: where the seven-dimensional internal space has the metric with a, b, c, f, g, g 3 all functions of the radial coordinate r. Here the §, σ are left-invariant SU(2) forms: σ 1 = cos ψ 1 + sin ψ 1 sin θdϕ , § 1 = cos ψ 2 + sin ψ 2 sinθdφ , σ 2 = − sin ψ 1 + cos ψ 1 sin θdϕ , § 1 = − sin ψ 2 + cos ψ 2 sinθdφ , The BPS equations of this solution give [48] g = − af 2bc ,

(D.4)
To get a ten-dimensional solution, we reduce the solution above along a U(1) isometry. To accomplish our goal of getting D6-branes wrapping a three-sphere, we choose the isometry generated by the Killing vector ∂ ψ 1 + ∂ ψ 2 . After some algebra, we get the following type IIA solution in string frame: where the ω are defined as §, replacing ψ 2 with ψ = ψ 2 − ψ 1 .

D.2 Non-Abelian T-dual
Let us now take the solution from the previous section, and apply a non-Abelian T-duality on the SU(2) isometry parametrised by the ω. We follow Section 2 of [14] and fix the gauge asθ =φ = v 1 = 0. We obtain a type IIB supergravity solution. The metric, in string frame, is given by: and ω 1 = cos ψ dθ − sin ψ sin θ dϕ,ω 2 = − sin ψ dθ − cos ψ sin θ dϕ,ω 3 = dψ − cos θdϕ . (D.8) The dual dilatonΦ is defined through 9) and the two-form potential as (D.10) The RR sector has all possible fluxes turned on. F 1 and F 5 can be expressed as follows: (D.11) F 3 is considerably more complicated:
In terms of those forms, the pure spinors are defined as: Let us now look at the BPS equations of type IIB supergravity in the general case of SU(3)×SU(3)-structure, generalising the system of pure SU(3)-structure that exhibit a rotation.

E BPS equations for a solution of type IIB supergravity with a general SU(3) × SU(3)-structure
We again follow the conventions of Andriot's thesis in this section. We start with the following pure spinors: (E.1) For type IIB supergravity, the BPS equations are: .

(E.10)
Using all the equations we have so far, we can rewrite the three-form ones as: (E.11) Those equations have been written in such a way as to make the limits for θ + → 0, π/2 obvious, and to give the equations of the rotation present in [25] when taking k ⊥ → 0, k → 1 (limit of SU(3)-structure). The last equation, involving * 6 F 1 can be rewritten in the following way: 1 2 d(e −φ k sin θ + ) ∧ χ ∧ χ = * 6 F 1 + ( * 6 F 3 ) ∧ λ + ( * 6 F 5 ) ∧ λ ∧ λ . (E.13) Let us now look at the equations we get for Φ − . We first define: (E.14) We get for the BPS equations, after some simplifications: The equation we would get for the six-form is just the one for the four-form wedged with β, so it is not an additional independent equation. It is quite easy to check that, taking the pure SU(3) limit, that is k → 0, k ⊥ → 1, we recover the system we already knew from [25].
Finally, we want to explicitly specialise to the cases of θ + = 0 and θ + = π/2. First θ + = 0: (E.17) Those systems do not look much more complicated than the ones in the pure SU(3) case, but there does not seem to be an easy transformation starting from either θ + = 0 or θ + = π/2 and recovering the full system.