On Exact Symmetries and Massless Vectors in Holographic Flows and other Flux Vacua

We analyze the isometries of Type IIB flux vacua based on the Papadopolous-Tseytlin ansatz and identify the related massless bulk vector fields. To this end we devise a general ansatz, valid in any flux compactification, for the fluctuations of the metric and p-forms that diagonalizes the coupled equations. We then illustrate the procedure in the simple case of holographic flows driven by the RR 3-form flux only. Specifically we study the fate of the isometries of the Maldacena-Nunez solution associated to wrapped D5-branes.

Due to the presence of fluxes, fields belonging to different sectors tend to mix with one another, compatibly with the residual (super)symmetry. Resolving the mixing and finding the spectrum of excitations is extremely laborious [19,20,21] as witnessed by the enormous effort needed to accomplish the task for the metric and active 1 scalar modes in holographic flows described by the Papadopoulos-Tseytlin (PT) ansatz [22,23,24,25]. Our aim is to extend this kind of analysis to the vector sector [19,20,21,26,27]. To this end, we will start by studying the fate of bulk symmetries of the Type IIB supergravity solutions. Although we will mostly adopt a 10-d perspective, we will also present the 5-d viewpoint, that has a more direct applicability in Holographic Renormalization [28,20,21,29,30,31,32,33,34] and Holographic QCD [35,36,37,38,39].

Symmetries can be divided into three classes
• Exact Symmetries: not only the metric admits Killing vectors but also fluxes are invariant [19,20,21,26,27].
• Partially Broken Symmetries: Metric invariant, some fluxes are not • Broken Symmetries: Metric and fluxes only asymptotically invariant [22,23,40,41] The PT ansatz [5] enjoys SU(2) × SU(2) isometry for arbitrary choices of the 'radial' functions. On the contrary, the U(1) R , associated to shifts of the coordinate ψ, is broken except for very special cases. The breaking is spontaneous from the bulk viewpoint, i.e. the would-be massless vector field becomes massive after 'eating' a Goldstone boson. The Stückelberg formalism for the gauging of axionic shift symmetries is particularly convenient in this respect [19,20,21,26,27,22,23,40,41]. Except for some very general remarks, we will neither have much further to say about broken symmetries and massive vectors nor discuss at all massless vectors related to harmonic forms [7] and to probe branes [35]. The latter give rise to chiral 'flavor' symmetries (breaking) and mesons. The former to baryonic symmetries.
The plan of the paper is as follows. In section 2 we describe the 5-d Lagrangian governing the dynamics of gauge fields and their mixing with would-be axions. Next, in Section 3, we check that the PT ansatz indeed admits full SU(2) × SU (2) symmetry, in that not only the metric but also the other background field (strengths) are invariant. In Section 4, we identify the bulk vector fields that remain massless by means of an ansatz for the fluctuations of the metric and p-forms, that diagonalizes the coupled equations. Finally, in Section 5 we illustrate the procedure in the simple case of holographic flows and other flux vacua with F 3 only, which are invariant under Ω. Specifically we study the MN solution [4] associated to wrapped D5-branes, i.e. 'fractional' D3-branes. Our conclusions and summary are contained in Section 6.

Vector fields in Holographic Renormalization
The 5-d Lagrangian describing vector fields and their possible mixing with inert (pseudo)scalars e.g. axions reads [19,20,21,26,27] L The gauge kinetic function K ij and the axion metric h AB may depend on the (active) scalars, while, in the present parametrization M A i are constant mass parameters 2 . The Lagrangian is invariant under gauge transformations of the form The square mass matrix is semi-positive definite. Zero eigenvalues correspond to exactly massless vectors, associated to isometries or to harmonic forms present in the background solution. Non-zero eigenvalues correspond to massive vectors and broken symmetries. Introducing gauge invariant combinations for the latter and denoting by A io µ the former yields 2 In general, after gauging some isometry of the scalar metric G ab (φ), covariant derivatives are given Here we focus on the gauging of axionic shifts After diagonalization, one finds a collection of decoupled vector bosons 3 each described by where K(φ) is the resulting gauge kinetic function and M(φ) is the possibly vanishing mass.

Field equations and PT ansatz
To set the stage for our analysis, let us now briefly recall Papadopoulos and Tseytlin (PT) ansatz for flux vacua in Type IIB supergravity and its symmetries. The main motivation behind PT ansatz is to identify a subset of fields that form a consistent truncation of Type IIB supergravity and allow to study flux vacua with reduced or no supersymmetry at all. The reader familiar with Type IIB supergravity and the PT ansatz can skip the following part and go directly to Section 3.2.
In the Einstein frame, Type IIB supergravity equations read ∇ M (e φF M N P ) = whereF with

PT ansatz
The consistent truncation of 10-d Type IIB supergravity found by Papadopoulos and Tseytlin is based on the following ansatz.
The remaining scalar fields {p, x, g, a, b, φ, h 1 , h 2 } are governed by a 5-d effective Lagrangian with (almost) diagonal metric G ab (only h 1 and h 2 mix with each other) and a complicated potential that play no role in our analysis.

Killing vectors
For arbitrary choices of the functions x, g, p, a, φ, b, h 1 , h 2 , (h 3 , K) of the radial coordinate u, the metric and p-forms are invariant under SU(2) × SU(2) isometry generated by the six Killing vectors ξ a Notice that ξ a have only components in the internal directions, i.e. ξ M = δ M i ξ i with M = 1, ..., 10 and i = 6, ..., 10, and the contra-variant components displayed above only depend on the internal 'angular' variables, i.e. ∂ µ ξ M = 0 with µ = 1, ..., 5. Although the metric does not mix the angular variables with the non-compact variables, after lowering the indices the components of the Killing vector acquire a u dependence due to warping. Clearly PT preserves Poincarè symmetry in the 'boundary' space-time directions, too.
It is easy to check that also the following two-forms (31) as well as the one-formω As a consequence, all background field-strengths are invariant i.e.
While admitting SU(2) × SU(2) isometry, the PT ansatz generically 'breaks' the abelian isometry associated to the vector fieldξ M ∂ M = ∂/∂ ψ . The latter may be identified 4 Lie derivatives act according to with the 'anomalous' U(1) R-symmetry of the dual N = 1 SYM theory on the boundary. In the bulk it is broken to Z 2N by the background 3-form and 5-form and then broken to Z 2 by non-perturbative effects (string or D-brane instantons, depending on the choice of wrapped branes). As discussed in Section 2, the bulk counterpart of the anomalous divergence of the R-symmetry current is a Higgs or rather Stückelberg mechanism [40], whereby a would-be massless vector field eats an axion and becomes massive. This effect has been studied in some details in [22,23] in the case of the KT background (a singular 'relative' of KS solution), confirming the expected value for the 'mass' predicted by [19].
For the case of MN solution, some considerations about the required axion can be found in [4,41].

Discrete symmetries and closed subsectors
There are two Z 2 symmetries and their product that allow to truncate Type IIB field equations in D = 10 to closed sets of fields mixing only with one another. The first is world-sheet parity Ω. The second is fermion parity in the L-moving sector Later on we will focus on the subsector invariant under Ω. For the PT ansatz, this means MN solution for wrapped D5-branes [4] belongs to this class, i.e. it is invariant under Ω. Its dual wrapped NS5-brane solution belongs to the class invariant under (−) F L . Standard AdS 5 × S 5 , i.e. near-horizon D3-branes, is invariant under (−) F L Ω. Finally KS and KT solutions (related to the conifold) do not preserve any of the above discrete symmetries and are thus more involved to study [24,25].

Exact symmetries and Massless vectors
In this Section, we would like to discuss the fate of the SU(2) × SU(2) isometry, that should correspond to the global 'flavor' symmetry of the boundary theory, possibly acting trivially on the lowest states relevant in the deep IR. Even if from the vantage point of the holographic duality, the presence of this isometry might be annoying, the analysis is quite general and applies to any isometry in any flux compactification.
We will find that the Killing vectors generating SU(2) are associated to truly massless vectors in the bulk that correspond to an exact global 'flavor' symmetry of any solution based on PT ansatz. For the first SU(2) factor the situation is subtler, at least in the case of MN solution [4].
First of all notice that invariance of the metric under isometry generated by a Killing vector ξ M reads as well as and Invariance under diffeomorphisms suggests the existence of a trivial massless zeromode for the metric fluctuations Taking It suggests an ansatz for the metric fluctuations of the form  Let us then consider the general case of an n-form X n , whose background (n + 1)-form field strength Y n+1 = dX n is invariant under some isometry generated by a Killing vector ξ Thanks to Bianchi identity dY n+1 = 0 one has (locally) where Z ξ n−1 is a (n − 1)-form defined up to an exact form δZ ξ n−1 = dW n−2 . Under a diffeomorphism generated by The last term can be cancelled by a gauge transformation of the n-form X n . This suggest that the correct ansatz for the 'massless' vector A 1 associated to the coupled fluctuations of the metric and n-form X n along ξ be of the form In this way gauge invariance under δA 1 = dα would not only be a consequence of general covariance but also of the n-form gauge invariance. For the fluctuations of the (n+1)-form field-strength Y n+1 one then finds In principle the procedure applies to any background n-form in PT or even more general flux vacua. The analysis can be performed in quite general terms but it drastically simplifies in backgrounds where F 5 = 0, H 3 = 0, F 1 = 0, thanks to invariance under worldsheet parity Ω, or else where F 5 = 0, F 3 = 0, F 1 = 0, thanks to invariance under (−) F L . In both cases mixing between C 2 and B 2 are excluded, and one can safely set A 4 = 0 and even δφ = 0, as we will see.

Dilaton equation (Consistency check)
Let us first check that it be consistent to set δφ = 0. Using the 3-form ansatz, one finds For a solution of the PT kind one has F ujk F i jk = 0, i = 6, ..., 10. Also one obtains = 0 for each one of the six Killing vectors ξ a . Therefore δF 2 = 0, consistently with the ansatz δφ = 0.

3-form equation
Let us focus on the 3-form equation Decomposing into space-time (ν = 1, ..., 5) and internal indices (j = 6, ..., 10) one has Keeping only non-vanishing components yields Plugging the anstaze one eventually finds √ĝ denote the dependence of √ g on the internal coordinates. Finally, using i ξ F 3 = dµ ξ one arrives at the following constraint for µ ξ One can check that this is satisfied for all µ ξ in any background of the PT kind. Anyway, we expect that it should always be possible to satisfy the constraint by adding to µ ξ an exact form dη ξ , where η ξ is an appropriate function.
• Equations for N = ν, P = l Keeping only non-zero components yields Plugging in the ansatz for the fluctuations yields After various cancellations, one finally arrives at the Dynamical Equation for the vector fields ∂ µ √ ge φ g µρ g νλ f ρλ µ l ξ = 0 (69) When µ i ξ = e −φ/2 ξ i , this further simplifies into which neatly displays the correspondence between bulk 5-d massless vector fields and exact Killing vectors.

Einstein equations
It is straightforward but very laborious to show that Einstein equations lead to the same results, i.e. the very same dynamical equation for A µ . For simplicity we will restrict our attention on the case in which µ ξ

Let us start with the source term
Defining the first order fluctuation of the Ricci tensor reads Moreover one finds and also Simplifying terms in the left and right hand side yields and In particular for M = µ and N = i, the remaining terms in the source fluctuations are while the remaining terms in the Ricci fluctuations read and finally After a number of cancellation one eventually finds when the 3-form equations are satisfied, showing that the fluctuations ansatz then satisfies Einstein equations, too.
It is easy to check that the other components (i, j) and (µ, ν) are satisfied as well. For brevity, we refrain to present the details here.
For MN background [4] a compact form for µ ξ in terms of the (rescaled) KV obtains
From the above scalar products one can read the gauge kinetic functions for the massless vectors of SU(2) × SU(2) in 5-d. For SU(2) one finds where the first term is a contribution from the Einstein-Hilbert term, while the second comes from the F 2 3 in the Type IIB action 6 . For SU(2), µ = e −φ/2 ξ and one simply finds The internal volume factor V − 1 3 arises from the Weyl scaling of the 10-d metric with pure space-time components so as to have canonical E-H term in 5-d, i.e. g (10) µν .

Massless Vectors in MN background
We will now explicitly apply the above analysis to the case of MN solution for wrapped D5-branes [4]. For simplicity we will focus on the SU(2) factor, for which µ ξ = e −φ/2 ξ.

MN Solution
In MN solution for wrapped D5-branes [4] one has h 1 = h 2 = 0 (no D3-branes F 5 = 0, no NS5-branes H 3 = 0, χ = 0) and b = a. Denoting the radial variable by u, the metric reads where The RR 3-form flux is given by The asymptotic behavior of the radial functions in the UV (u → 0) and IR (u → ∞) are found to be For later use, notice that e 5φ e 4g sin 2 θ sin 2θ

Spectrum of massless vector harmonics
As we have seen, in order to find the spectrum of bound-states that are holographically dual to the massless bulk vectors associated to the three Killing vectors of SU (2), one should solve There are two cases to consider ν =ν and ν = u.
For ν = u one simply gets that allows to express A u in terms of the longitudinal component of Aμ.
For ν =ν one gets ∂μfμν + 1 Setting Aμ = aμ(u)e ip·x and A u = b(u)e ip·x one can solve for aμ(u) and b(u). Decomposing aμ(u) into longitudinal and transverse components according to that can be set to zero by gauge transformations. The surviving transverse components then satisfy an equation that is identical to the equation for a canonical massless scalar Φ. After setting the equation is put in canonical form with an effective potential given by Unfortunately due to the 'pathological' UV behavior there is no spectrum of discrete states associate to the bulk massless vectors. For SU(2) the story is similar. This behavior is analogous to the one found for the fluctuations of the metric in MN solution [4]. Indeed, the transverse traceless components of the metric fluctuations h T T ij (u, x) = e ij (p)f p (u)e ipx decouple from the rest and satisfy a free massless scalar equation of the form 7 For MN solution [4], the relevant equation has been studied in [24,25] and shown to have a continuous spectrum without a mass gap. Longitudinal and radial components of the metric mix with the active scalars and behave better [24,25]. Despite the area-law behavior of the Wilson loop in MN background [4], the presence of massless fluctuations casts some shadow on the holographic interpretation as a dual to a confining theory such as N = 1 SYM.

Conclusions and summary
Let us conclude by summarizing our results and draw lines for future investigation.
We have shown that all RG flows described by the PT ansatz [5] in Type IIB supergravity enjoy exact SU(2) × SU(2) symmetry in that not only the metric but also background p-forms are invariant under diffeomorphisms generated by the six Killing vectors. We have then identified a very general ansatz for the combined fluctuations of metric and p-forms that diagonalizes the resulting equations for the bulk massless vectors. Although derived in the context of holography, our ansatz is expected to have much wider applicability in any flux compactification with isometry. Restricting our attention to the case of backgrounds invariant under world-sheet parity Ω, we have illustrated our procedure in the case of MN solution [4] for wrapped D5-branes. The spectrum of massless vector harmonics in this background -very much as the spectrum of massless scalars and transverse traceless fluctuations of the metric [24,25] -is continuous and has no mass gap. This drawback might be related to the impossibility of fully decoupling KK states from the desired physical modes, which survive in the deep IR. In particular the very presence of an exact SU(2) × SU(2) symmetry is a remnant of the breaking of N = 4 to N = 1 * with common mass for the three chiral multiplets. Since this symmetry is an exact symmetry of any RG flows described by PT ansatz [5], not excluding KS solution [3], we should conclude that holographic SYM is still undelivered [42] at least in a top-down approach. The strictly 5-d bottom-up approach embodied by Holographic QCD [35,36,37,38,39] seems more promising in this respect.