Delocalized SYZ Mirrors and Confronting Top-Down $SU(3)$-Structure Holographic Meson Masses at Finite $g$ and $N_c$ with P(article) D(ata) G(roup) Values

Meson spectroscopy at finite gauge coupling - whereat any perturbative QCD computation would break down - and finite number of colors, from a top-down holographic string model, has thus far been entirely missing in the literature. This paper fills in this gap. Using the delocalized type IIA SYZ mirror (with $SU(3)$ structure) of the holographic type IIB dual of large-$N$ thermal QCD of arXiv:hep-th/0902.1540 as constructed in arXiv:hep-th/1306.4339 at finite coupling and number of colors ($N_c$ = Number of $D5(\overline{D5}$)-branes wrapping a vanishing two-cycle in the top-down holographic construct of arXiv:hep-th/0902.1540 = ${\cal O}(1)$ in the IR in the MQGP limit of arXiv:hep-th/1306.4339 at the end of a Seiberg duality cascade), we obtain analytical (not just numerical) expressions for the vector and scalar meson spectra and compare our results with previous calculations of hep-th/0412141, arXiv:1409.0559 [hep-th], and obtain a closer match with the Particle Data Group (PDG) results (Chin. Phys. C, 38, 090001 (2014) and 2015 update). Through explicit computations, we verify that the vector and scalar meson spectra obtained by the gravity dual with a black hole for all temperatures (small and large) are nearly isospectral with the spectra obtained by a thermal gravity dual valid for only low temperatures; the isospectrality is much closer for vector mesons than scalar mesons. The black hole gravity dual (with a horizon radius smaller than the deconfinement scale) also provides the expected large-$N$ suppressed decrease in vector meson mass with an increase of temperature.


Introduction
The AdS/CFT [6] correspondence and its non-conformal generalizations, conjecture the equivalence between string theory on a ten dimensional space-time and gauge theory living on the boundary of space-time. A generalization of the AdS/CFT correspondence is necessary to explore more realistic theories (less supersymmetric and non conformal) such as QCD. The original AdS/CFT conjecture [6] proposed a duality between maximally supersymmetric N = 4 SU(N) SYM gauge theory and type IIB supergravity on AdS 5 × S 5 in the low enegry limit. Different generalized versions of the AdS/CFT have thus far been proposed to study non-supersymmetric field theories. One way of constructing gauge theories with less supersymmetry is to consider stacks of Dp branes at the singular tip of a Calabi-Yau cone. In this paper we use a large N top down holographic dual of QCD [1] to obtain the meson spectrum from type IIA perspective. Embedding of additional D-branes(flavor branes) in the near-horizon limit gives rise to a modification of the original AdS/CFT correspondence which involves field theory degrees of freedom that transform in the fundamental representation of the gauge group. This is useful for describing field theories like QCD, where quark fields transform in the fundamental representation. Mesons operator or a gauge invariant bilinear operator corresponds to the bound state of anti-fundamental and fundamental field.
In the past decade, (glueballs and) mesons have been studied extensively to gain new insight into the non-perturbative regime of QCD. Various holographic setups such as soft-wall model, hard wall model, modified soft wall model, etc. have been used to obtain the glueballs' and mesons' spectra and obtain interaction between them. In the following two paragraphs a brief summary of the work is given that has been done in past decades.
Most of existing literature on holographic meson spectroscopy is of the bottom-up variety based often on soft/hard wall AdS/QCD models. Here is a short summary of some of the relevant works. Soft-wall holographic QCD model was used in [7] and [8] to obtain spectrum and decay constants for 1 −+ hybrid mesons and to study the scalar glueballs and scalar mesons at T = 0 respectively. In [7] no states with exotic quantum numbers were observed in the heavy quark sector. Comparison of the computed mass with the experimental mass of the 1 −+ candidates π 1 (1400), π 1 (1600) and π 1 (2015), favored π 1 (1400) as the lightest hybrid state. In [9] an IR-improved soft-wall AdS/QCD model in good agreement with linear confinement and chiral symmetry breaking was constructed to study the mesonic spectrum. The model was constructed to rectify inconsistencies associated with both simple soft-wall and hard-wall models. The hard-wall model gave a good realization for the chiral symmetry breaking, but the mass spectra obtained for the excited mesons didn't match up with the experimental data well. The soft-wall model with a quadratic dilaton background field showed the Regge behaviour for excited vector mesons but chiral symmetry breaking phenomena cannot be realized consistently in the simple soft-wall AdS/QCD model. A hard wall holographic model of QCD was used in [10], [11] and [12] to analyze the mesons. In [13] a two-flavor quenched dynamical holographic QCD(hQCD) model was constructed in the graviton-dilaton framework by adding two light flavors. In [3] the mesonic spectrum was obtained for a D4/D8(−D8)-brane configuration in type IIA string theory; in [14] massive excited states in the open string spectrum were used to obtain the spectrum for higher spin mesons J ≥ 2. NLO terms were obtained by taking into account the effect of the curved background perturbatively which led to corrections in formula J = α 0 + α ′ M 2 . The results obtained for the meson spectrum were compared with the experimental data to identify a 2 (1320), b 1 (1235), π(1300), a 0 (1450) etc. first and second excited states. In [15] a holographic model was constructed with extremal N c D4-branes and D6-flavor branes in the probe approximation. The model gave a good approximation for Regge behaviour of glueballs but failed to explain mesonic spinning strings because the dual theory did not include quark in the fundamental representations.
To our knowledge, the only top-down holographic dual of large-N thermal QCD which is IR confining, UV conformal and UV-complete (e.g. the holographic Sakai-Sugimoto model [3] does not address the UV) with fundamental quarks is the one given in [1] involving N D3-branes, M D5/(D5) branes wrapping a vanishing two-cycle and N f D7(D7) flavor-branes in a warped resolved conifold at finite temperature in the brane picture (and M D5-branes and D7(D7)-branes with a black-hole and fluxes in a resolved warped deformed conifold gravitational dual). In [4], the authors (some also part of [1]) obtained the vector and scalar mesonic spectra by taking a single T-dual of the holographic type IIB background of [1]. Comparison of the (pseudo-)vector mesons with PDG results, provided a reasonable agreement. One of the main objectives of our work is to see if by taking a mirror of the type IIB background of [1] via delocalized Strominger-Yau-Zaslow's triple-T-duality prescription -a new tool in this field -at finite gauge coupling and with finite number of colors -a new limit and one which is closest to realistic strongly coupled thermal QCD -one can obtain a better agreement between the mesonic spectra so obtained and PDG results than previously obtained in [3] and [4], and in the process gain new insights into a holographic understanding of thermal QCD.
In [16], we initiated top-down G-structure holographic large-N thermal QCD phenomenology at finite gauge coupling and finite number of colors, in particular from the vantage point of the M theory uplift of the delocalized SYZ type IIA mirror of the top-down UV complete holographic dual of large-N thermal QCD of [1], as constructed in [2]. We calculated up to (N)LO in N , masses of 0 ++ , 0 −− , 0 −+ , 1 ++ and 2 ++ glueballs, and found very good agreement with some of the lattice results on the same. In this paper, we continue exploring top-down G-structure holographic large-N thermal QCD phenomenology at finite gauge coupling by evaluating the spectra of (pseudo-)vector and (pseudo-)scalar mesons, and in particular comparing their ratios for both types with P(artcile) D(ata) G(roup) results.
The rest of the paper is organized as follows. In Section 2, via four sub-sections, we briefly review a UV-complete top-down type IIB holographic dual of large-N thermal QCD (subsection 2.1) as given in [1] and its M theory uplift in the 'MQGP limit' as worked out in [2] (sub-section 2.2); sub-section 2.3 has a discussion on the construction in [2] of the delocalized Strominger-Yau-Zaslow (SYZ) type IIA mirror of the aforementioned type IIB background of [1] and sub-section 2.4 has a brief review of SU (3) and G 2 structures relevant to [1] (type IIB) and [2,17] (type IIA and M theory). Section 3 is on the construction of the embedding of D6-branes via delocalized SYZ type IIA mirror of the embedding of D7-branes of type IIB. Via five sub-sections, Section 4 is on obtaining the (pseudo-)vector meson spectra in the framework of [2] at finite coupling assuming a black hole gravity dual for all temperatures, small and large. The (pseudo-)vector mesons correspond to gauge fluctuations about a background gauge field along the world volume of the D6 branes. Unlike [4], the gravity dual involves a black-hole (r h = 0) and consequently, while factorizing the gauge fluctuations along R 3 × S 1 -radial direction into fluctuations along R 3 × S 1 and eigenmode fluctuations along the radial direction, there are two types of eigenmodes along the radial direction -one denoted by α Finally, using the WKB quantization prescription, the vector meson spectrum corresponding to the α {i} n eigenmodes was worked out (in the small and large mass-limits) in 4.4, and that corresponding to the α {0} n eigenmodes (in the small and large mass-limits) in 4.5. In Section 5, we obtain the scalar meson spectrum by considering fluctuations of the D6-branes orthogonal to their world-volume in the absence of any background gauge fields in a black hole background for all temperatures, small and large. In the same vein as vector meson spectroscopy, after obtaining the EOM for the radial eigenfunction mode, the following is an outline of what is done in section 5, devoted to scalar meson spectroscopy. First, in 5.1, assuming an IR-valued scalar meson spectrum, the same is obtained by solving the EOMs near the horizon. Next, by converting the EOMs to a Schrödinger-like EOMs, the scalar meson spectrum is worked out for in 5.2 (in the IR limit in 5.2.1 and the UV limit in 5.2.2). Finally, using the WKB quantization prescription, the scalar meson spectrum was worked out (in the small and large mass-limits) in 5.3. In Section 6, we obtain the (pseudo-)vector meson spectrum in 6.1 (and the three sub-sub-sections therein) and (pseudo-)scalar meson spectrum in 6.2 (and the three sub-sub-sections therein) using a thermal background, and hence verify that the mesonic spectra of Sections 4 and 5 are nearly isospectral with 6. Section 7 has a discussion on the new insights and results obtained in this work and some future directions. There are three supplementary appendices.

Background: A Top-Down Type IIB Holographic Large-N Thermal QCD and its M-Theory Uplift in the 'MQGP' Limit
Via four sub-sections, in this section, we will: • provide a short review of the type IIB background of [1], a UV complete holographic dual of large-N thermal QCD, in subsection 2.1, • discuss the 'MQGP' limit of [2] and the motivation for considering this limit in subsection 2.2, • briefly review issues as discussed in [2], [18], [17] and [19], pertaining to construction of delocalized S(trominger) Y(au) Z(aslow) mirror and approximate supersymmetry, in subsection 2.3, • briefly review the new results of [17] and [19] pertaining to construction of explicit SU (3) and G 2 structures respectively of type IIB/IIA, and M-theory uplift,

Type IIB Dual of Large-N Thermal QCD
In this subsection, we will discuss a UV complete holographic dual of large-N thermal QCD as given in Dasgupta-Mia et al [1]. In order to include fundamental quarks at non-zero temperature in the context of type IIB string theory, [1] considered at finite temperature, N D3-branes, M D5-branes wrapping a vanishing two-cycle and M D5-branes distributed along a resolved two-cycle and placed at the outer boundary of the IR-UV interpolating region/inner boundary of the UV region. The D5/D5 separation is given by R D5/D5 . The radial space, in [1] is divided into the IR, the IR-UV interpolating region and the UV. N f D7-branes, via Ouyang embedding, are holomorphically embedded in the UV, the IR-UV interpolating region and dipping into the (confining) IR (up to a certain minimum value of r corresponding to the lightest quark) and N f D7-branes present in the UV and the UV-IR interpolating (not the confining IR). This is to ensure turning off of three-form fluxes. The resultant ten-dimensional geometry is given by a resolved warped deformed conifold. In the gravity dual D3branes and the D5-branes are replaced by fluxes in the IR. The finite temperature resolves 4 and IR confinement deforms the conifold. Back-reactions are included in the warp factor and fluxes. One has SU (N + M ) × SU (N + M ) color gauge group and SU (N f ) × SU (N f ) flavor gauge group, in the UV. It is expected that there will be a partial Higgsing of SU (N + M )× SU (N + M ) to SU (N + M )×SU (N ) at r = R D5/D5 [21]. The two gauge couplings, g SU (N +M ) and g SU (N ) flow logarithmically and oppositely in the IR: 4π 2 Had it not been for S 2 B 2 , in the UV, one could have set g 2 SU (M +N ) = g 2 SU (N ) = g 2 Y M ∼ g s ≡ constant (implying conformality) which is the reason for inclusion of M D5-branes at the common boundary of the UV-IR interpolating and the UV regions, to annul this contribution. In fact, the running also receives a contribution from the N f flavor D7-branes which needs to be annulled via N f D7-branes. Under an NVSZ RG flow, the gauge coupling g SU (N +M ) -having a larger rank -flows towards strong coupling and the SU (N ) gauge coupling flows towards weak coupling. Upon application of Seiberg ) weak in the IR; assuming after duality cascade, N decreases to 0 and there is a finite M , one will be left with SU (M ) gauge theory with N f flavors that confines in the IR -the finite temperature version of the same is what was looked at by [1].
So, in the IR, at the end of the duality cascade, number of colors N c is identified with M , which in the 'MQGP limit' can be tuned to equal 3. One can identify N c with N eff (r) + M eff (r), where N eff (r) = Base of Resolved Warped Deformed Conifold F 5 and M eff = S 3F3 (the S 3 being dual to e ψ ∧ (sin θ 1 dθ 1 ∧ dφ 1 − B 1 sin θ 2 ∧ dφ 2 ), wherein B 1 is an asymmetry factor defined in [1], and e ψ ≡ dψ + cos , α ≫ 1 [22]. The number of colors N c varies between M in the deep IR and a large value [even in the MQGP limit of (10) (for a large value of N )] in the UV. Hence, at very low energies, the number of colors N c can be approximated by M , which in the MQGP limit is taken to be finite and can hence be taken to be equal to three. In [1], the effective number of D3-branes, D5-branes wrapping the vanishing two-cycle and the flavor D7-branes, denoted respectively by N eff (r), M eff (r) and N eff f (r), are given as: It was argued in [17] that the length scale of the OKS-BH metric in the IR after Seiberg-duality 4 The non-zero resolution parameter 'a' is also there to introduce a separation R D5/D5 between the D5 and D5 branes, which as in [19] we assume is √ 3a (as r > √ 3a lies in the large-r region in a resolved conifold [20]).
cascading away almost the whole of N eff , will be given by: which implies that in the IR, relative to KS, there is a color-flavor enhancement of the length scale in the OKS-BH metric. Hence, in the IR, even for N IR c = M = 3 and N f = 2 (light flavors) upon inclusion of n, m > 1 terms in M eff and N eff f in (1), L OKS−BH ≫ L KS (∼ L Planck ) in the MQGP limit involving g s ∼ < 1, implying that the stringy corrections are suppressed and one can trust supergravity calculations. Further, the global flavor group SU (N f ) × SU (N f ), is broken in the IR to SU (N f ) as the IR has only N f D7-branes.
Hence, the type IIB model of [1] makes it an ideal holographic dual of thermal QCD because, it is UV conformal and IR confining with required chiral symmetry breaking in the IR. The quarks present in the theory transform in the fundamental representation, plus theory is defined for full range of temperature both low and high.

(d) Supergravity solution on resolved warped deformed conifold
The metric in the gravity dual of the resolved warped deformed conifold with g i 's: is given by : The compact five dimensional metric in (3), is given as: wherein we will assume r ≫ a, h 5 ∼ (deformation parameter) 2 r 3 ≪ 1 for r ≫ (deformation parameter) 2 3 . The h i 's appearing in internal metric up to linear order depend on g s , M, N f are given as below: One sees from (4) and (5) that one has a non-extremal resolved warped deformed conifold involving an S 2 -blowup (as h 4 − h 2 = a 2 r 2 ), an S 3 -blowup (as h 5 = 0) and squashing of an S 2 (as h 3 is not strictly unity). The horizon (being at a finite r = r h ) is warped squashed S 2 × S 3 . In the deep IR, in principle one ends up with a warped squashed S 2 (a) × S 3 (ǫ), ǫ being the deformation parameter. Assuming ǫ 2 3 > a and given that a = O gsM 2 N r h [21], in the IR and in the MQGP limit, N eff (r ∈ IR) = warped squashed S 2 (a)×S 3 (ǫ) F 5 (r ∈ IR) ≪ M = S 3 (ǫ) F 3 (r ∈ IR); we have a confining SU (M ) gauge theory in the IR.
The warp factor that includes the back-reaction, in the IR is given as: where, in principle, M eff /N eff f are not necessarily the same as M/N f ; we however will assume that up to O gsM 2 N , they are. Proper UV behavior requires [21]: In the IR, up to O(g s N f ) and setting h 5 = 0, the three-forms are as given in [1]: The asymmetry factors in (8)  , we will assume the same three-form fluxes for h 5 = 0. With R D5/D5 denoting the boundary common to the UV-IR interpolating region and the UV region,F lmn , H lmn = 0 for r ≥ R D5/D5 is required to ensure conformality in the UV. Near the θ 1 = θ 2 = 0-branch, assuming: θ 1,2 → 0 as ǫ γ θ >0 and r → R UV → ∞ as ǫ −γr<0 , lim r→∞Flmn = 0 and lim r→∞ H lmn = 0 for all components except H θ 1 θ 2 φ 1,2 ; in the MQGP limit and near θ 1,2 = π/0-branch, So, the UV nature too is captured near θ 1,2 = 0-branch in the MQGP limit. This mimics addition of D5-branes in [1] to ensure cancellation ofF 3 . Further, to ensure UV conformality, it is important to ensure that the axion-dilaton modulus approaches a constant implying a vanishing beta function in the UV. This was discussed in detail in appendix B of [17], wherein in particular, assuming the F-theory uplift involved, locally, an elliptically fibered K3, it was shown that UV conformality and the Ouyang embedding are mutually consistent.

The 'MQGP Limit'
In [2], we had considered the following two limits: (i)weak(g s )coupling − large t ′ Hooft coupling limit : (the limit in the first line though not its realization in the second line, considered in [1]); (ii)MQGP limit : The motivation for considering the MQGP limit which was discussed in detail in [17] is: 1. Unlike the AdS/CFT limit wherein g YM → 0, N → ∞ such that g 2 YM N is large, for strongly coupled thermal systems like sQGP, what is relevant is g YM ∼ O(1) and N c = 3. From the discussion in the previous paragraphs one sees that in the IR after the Seiberg duality cascade, effectively N c = M which in the MQGP limit of (10) can be tuned to 3. Further, in the same limit, the string coupling g s cos In (14), [λ 1 : λ 2 ] are the homogeneous coordinates of the blown-up CP 1 = S 2 ; λ 2 Hence, if the resolved warped deformed conifold is predominantly either resolved or deformed, the local T 3 of (11) is the required sLag to effect SYZ mirror construction.
To enable use of SYZ-mirror duality via three T dualities, remembering that SYZ mirror symmetry is in fact a quantum mirror symmetry, one also needs to ensure a large base (implying large complex structures of the aforementioned two two-tori) of the T 3 (x, y, z) fibration, ensuring the disc instantons' contribution is very small [23]. This is effected via [27]: for appropriately chosen large values of f 1,2 (θ 1,2 ) = ± cot θ 1,2 [17]. The three-form fluxes remain invariant. The guiding principle behind choosing such large values of f 1,2 (θ 1,2 ), as given in [2], is that one requires the metric obtained after SYZ-mirror transformation applied to the non-Kähler resolved warped deformed conifold to be like a non-Kähler warped resolved conifold at least locally. For completenes, we summarize the Buscher triple-T duality rules [28], [2] in appendix A. A single T-duality along a direction orthogonal to the D3-brane world volume, e.g., z of (11), yields D4 branes straddling a pair of N S5-branes consisting of world-volume coordinates (θ 1 , x) and (θ 2 , y). Further, T-dualizing along x and then y would yield a Taub-NUT space from each of the two N S5-branes [29]. The D7-branes yield D6-branes which get uplifted to Kaluza-Klein monopoles in M-theory [30] which too involve Taub-NUT spaces. Globally, probably the eleven-dimensional uplift would involve a seven-fold of G 2 -structure, analogous to the uplift of D5-branes wrapping a two-cycle in a resolved warped conifold [31]. We obtained a local G 2 structure in [17], which is summarized in 2.4.

G-Structures
In this sub-section, we give a quick overview of G = SU (3), G 2 -structures and how the same appear in the holographic type IIB dual of [1], its delocalized type IIA SYZ mirror and its M-theory uplift constructed in [2].
Any metric-compatible connection can be written in terms of the Levi-Civita connection and the contorsion tensor κ ( [32] and references therein). Metric compatibility requires κ ∈ Λ 1 ⊗ Λ 2 , Λ n being the space of n-forms. Alternatively, in d complex dimensions, since Λ 2 ∼ = so(d), κ also be thought of as Λ 1 ⊗ so(d). Given the existence of a G-structure, one can decompose so(d) into a part in the Lie algebra g of G ⊂ SO(d) and its orthogonal complement g ⊥ = so(d)/g. The contorsion κ splits accordingly into κ = κ 0 + κ g , where κ 0 -the intrinsinc torsion -is the part in Λ 1 ⊗ g ⊥ . One can decompose κ 0 into irreducible G representations providing a classification of G-structures in terms of which representations appear in the decomposition. Let us consider the decomposition of T 0 in the case of SU (3)-structure. The relevant representations are Λ 1 ∼ 3 ⊕3, g ∼ 8, g ⊥ ∼ 1 ⊕ 3 ⊕3. Thus the intrinsic torsion, an element of Λ 1 ⊕ su(3) ⊥ , can be decomposed into the following SU (3) representations [32] : The SU (3) structure torsion classes [33] can be defined in terms of J, Ω, dJ, dΩ and the contraction operator : The torsion classes are then defined in the following way: (3,0) , given by real numbers (2,2) 0 2 Ω + dΩ + (the subscript 0 indicative of the primitivity of the respective forms).
In [18], it was shown that the five SU (3) structure torsion classes, in the MQGP limit, satisfied (schematically): in the UV-IR interpolating region/UV, implying a Klebanov-Strassler-like supersymmetry [34]. Locally around θ 1 ∼ 1 , the type IIA torsion classes of the delocalized SYZ type IIA mirror metric were shown in [17] to be: Further, indicative of supersymmetry after constructing the delocalized SYZ mirror. The mirror type IIA metric after performing three T-dualities, first along x, then along y and finally along z, utilizing the results of [26] was worked out in [2]. The type IIA metric components were worked out in [2].
Apart from quantifying the departure from SU (3) holonomy due to intrinsic contorsion supplied by the NS-NS three-form H, via the evaluation of the SU (3) structure torsion classes, to our knowledge for the first time in the context of holographic thermal QCD at finite gauge coupling and for finite number of colors [in fact for N c = 3 in the IR] in [17]: (i) the existence of approximate supersymmetry of the type IIB holographic dual of [1] in the MQGP limit near the coordinate branch θ 1 = θ 2 = 0 was demonstrated, which apart from the existence of a special Lagrangian three-cycle (as shown in [18,17]) is essential for construction of the local SYZ type IIA mirror; (ii) it was demonstrated that the large-N suppression of the deviation of the type IIB resolved warped deformed conifold from being a complex manifold, is lost on being duality-chased to type IIA -it was also shown that one further fine tuning γ 2 = 0 in W IIA 2 can ensure that the local type IIA mirror is complex; (iii) for the local type IIA SU (3) mirror, the possibility of surviving approximate supersymmetry was demonstrated which is essential from the point of view of the end result of application of the SYZ mirror prescription.
We can get a one-form type IIA potential from the triple T-dual (along x, y, z) of the type IIB F 1,3,5 in [2] and using which the following D = 11 metric was obtained in [2] (u ≡ r h r ): Let us now briefly discuss G 2 structure. We will be following [35,37,36]. If V is a seven-dimensional real vector space, then a three-form ϕ is said to be positive if it lies in the GL (7, R) orbit of ϕ 0 , where ϕ 0 is a three-form on R 7 which is preserved by G 2 -subgroup of GL(7, R). The pair (ϕ, g) for a positive 3-form ϕ and corresponding metric g constitute a G 2 -structure. The space of p-forms are known to decompose as following irreps of G 2 [35]: The subscripts denote the dimension of representation and components of same representation/dimensionality, are isomorphic to each other. Let M be a 7-manifold with a G 2 -structure (ϕ, g). Then the components of spaces of 2-, 3-, 4-, and 5-forms are given in [35,37]. The metric g defines a reduction of the frame bundle F to a principal SO (7)-sub-bundle of oriented orthonormal frames. Now, g also defines a Levi-Civita connection ∇ on the tangent bundle T M , and hence on F . However, the G 2 -invariant 3-form ϕ reduces the orthonormal bundle further to a principal G 2 -subbundle Q. The Levi-Civita connection can be pulled back to Q. On Q, ∇ can be uniquely decomposed as where∇ is a G 2 -compatible canonical connection, taking values in the sub-algebra g 2 ⊂ so (7), while T is a 1-form taking values in g ⊥ 2 ⊂ so (7); T is known as the intrinsic torsion of the G 2 -structurethe obstruction to the Levi-Civita connection being G 2 -compatible. Now so (7) splits under G 2 as Hence T can be represented by a tensor So, the torsion is determined by ∇ϕ. Now, from the Lemma 2.24 of [36]: Due to the isomorphism between the Λ a=1,...,5 7 s, ∇ϕ lies in the same space as T AB and thus completely determines it. Equation (27) is equivalent to: where T AB is the full torsion tensor. Equation (28) can be inverted to yield: The tensor T M A , like the space W, possesses 49 components and hence fully defines ∇ϕ. In general T AB cab be split into torsion components as where T 1 is a function and gives the 1 component of T . We also have T 7 , which is a 1-form and hence gives the 7 component, and, T 14 ∈ Λ 2 14 gives the 14 component. Further, T 27 is traceless symmetric, and gives the 27 component. Writing T i as W i , we can split W as From [39], we see that a G 2 structure can be defined as: where A, B, C = 1, ..., 6, 10; a, b, c, = 1, ..., 6 and f ABC are the structure constants of the imaginary octonions. Using the same and [38]: the G 2 -structure torsion classes were worked out around θ 1 ∼ 1 in [17] to: Hence, the approach of the seven-fold, locally, to having a G 2 holonomy ( As stated earlier, the global uplift to M-theory of the type IIB background of [1] is expected to involve a seven-fold of G 2 structure (not G 2 -holonomy due to non-zero G 4 ). It is hence extremely important to be able to see this, at least locally. It is in this sense that the results of [2] are of great significance as one explicitly sees, for the first time, in the context of holographic thermal QCD at finite gauge coupling, though locally, the aforementioned G 2 structure having worked out the non-trivial G 2 -structure torsion classes.

SYZ Mirror of Ouyang Embedding
To start off our study of meson spectroscopy, we first need to understand how the flavor D6-branes are embedded in the mirror (constructed in [2]) of the resolved warped deformed conifold of [1], i.e., the delocalized SYZ mirror of the Ouyang embedding of the flavor D7-branes in [1].
One can show that the delocalised type IIA mirror metric of the resolved warped deformed conifold metric as worked out in [2], for fixed θ 1 = can be written as: where the T 3 (x, y, z) metric is given by: Interestingly, one can diagonalize the local T 3 metric to: where: Now, from the Buscher triple T-duality results for the NS-NS B as given in (A11), one sees that for small φ 1,2 (which ensures decoupling of M 6 (θ 1,2 , φ 1,2 , ψ, x 10 ) from M 5 (R 1,3 , r) in the M-theory uplift in the MQGP limit): There is an important message we must take in from (39). As one realizes from (A11) and therefore (39), B IIA is independent of M even up to NLO in N even though B IIB is proportional to M . This will be important in obtaining the mesonic spectra in the subsequent sections and obtaining a good match with [5] without having to invoke O gsM 2 N -corrections which the authors of [4] had to use (and set to 0.5 -and yet consider the same to 'small' -to get a reasonable match with [5]).
The complete 10 dimensional type IIA metric in large N limit is given as: To obtain the pullback metric on the D6 branes, we choose the first branch of the Ouyang embedding where (θ 1 ,x) = (0, 0) and we consider thez coordinate as a function of r, i.e.z(r). We then use the equation of motion of the field to find the explicit functional dependence. The coordinates for D6 brane are x µ = x (0,1,2,3) , r, θ 2 ,ỹ. The pull-back of the metric is given by: Near θ 1 = α θ 1 N −1/5 and θ 2 = α θ 2 N −3/10 the type IIA metric components upto NLO are given as following: One will assume that the embedding of the D6-brane will be given by i : ,ỹ ֒→ M 1,9 , effected by:z =z(r). The pull back of the B-field along the directions of the D6-branes is given by: where B θ 2x , B θ 2ỹ , B θ 2z can be read off from (117). Now, one can show that: where the embedding functions Σ 0,1 (r; g s , N f , M, N ) are given in (B1). Thus, the Euler-Lagrange equation of motion yields: Like [4],z =constant, is a solution of (45). Alternatively, (45) is equivalent to: Hence, the Euler-Lagrange equation for thez(r) from the DBI action can be written in the following form: where K is an arbitary constant while C 1 (r; g s , N f , N ), C 2 (r; g s , N f , N ) and C 3 (r; g s , N f , N ) up to NLO-in-N after largeN expansion have the following forms: Substituting the values of C 1 (r; g s , N f , N ), C 2 (r; g s , N f , N ) and C 3 (r; g s , N f , N ) in the differential equation presented above and keeping terms only up to NLO-in-N after taking a large N expansion the differential equation acquires the following form: Analogous to [4], from (45), one sees thatz =constant, is a valid solution and by choosingz = ±C π 2 , one can choose the D6/D6-branes to be at "antipodal" points. Using a very similar computation for a thermal background with no black-hole (r h = 0), one can show that this constant embedding of D6-branes, is still valid.

Vector Meson Spectroscopy in a Black-Hole Background for All Temperatures
Equipped with the embedding of the flavor D6-branes in the delocalized SYZ mirror of resolved warped deformed conifold of [1] from section 3, we now proceed to obtaining the spectra as the Kaluza-Klein modes of the massless sector of open strings in type IIA at finite gauge coupling. In this and the next section, we do not worry about the issues like the black hole gravity dual is not considered at low temperatures wherein one must consider a thermal background. Happily, in section 6, we will via an explicity computation verify that the mesonic spectra obtained in sections 4 (for [pseudo-]vector mesons) and 5 (for [pseudo-]scalar mesons), are nearly isospectral with one obtained by working with a thermal background without a black-hole valid at only low temperatures.
We evaluate the masses of the (pseudo-)vector and (pesudo-)scalar mesons separately -the former by considering gauge fluctuations of a background gauge field along the world volume of the embedded flavor D6-branes and the latter (without turning on a background gauge field) by looking at fluctuations of the embedding transverse to the world volume of the embedded D6-branes.
As done in [4], let us redefine (r,z) in terms of new variables (Y, Z): and the constant embedding of D6(D6)-branes can correspond toz = π 2 for C = 1 for D6-branes and z = − π 2 for C = −1 for D6-branes, both corresponding to Y = 0. Now, consider turning on a gauge field fluctuationF σ 3 2 about a small background gauge field F 0 σ 3 2 and the backround i * (g + B). This implies: Concentrating on the terms quadratic inF : whereG µν are the unwarped R 1,3 metric components. Substituting: one obtains: where V 1,2 are given in (C1).
After integrating by parts once, and utilizing the EOM for B (n) µ , one rewrites (53) as: which yields the following equations of motion: Writing a = r h 0.6 + 4 gsM 2 N (1 + log r h ) , m =m r h √ 4πgsN , one hence obtains the following EOMs: and We will now proceed to obtaining the (pseudo-)vector meson spectrum by three routes. The first will cater exclusively to an IR computation where we solve the α {i} n (Z) and α {0} n (Z) EOMs near the horizon. Imposing Neumann boundary condition at the horizon results in quantization of the (pseudo-)vector meson masses and via N f -and M -dependent contributions, we extract the temperature dependence of the (pseudo-)vector meson spectrum. We will see that up to LO in N, in the IR, there is a near isospectrality in the (pseudo-)vector meson spectrum obtained by solving the α {i} n (Z) and α {0} n (Z) EOMs. The second route will be to convert the α {i} n (Z) and α {0} n (Z) EOMs into Schr"odinger-like EOMs and to solve the same in the IR and UV separately and obtain (pseudo-)vector mass quantization by imposing Neumann boundary conditions at the horizon (IR)/asymptotic boundary (UV). It turns out the former yields a result, which up to LO in N , is of the same order as the IR results of route one. The UV computations satisfy Neumann and/or Dirichlet boundary conditions without any mass quantization condition. The third route catering to the IR-UV interpolating region and what gives us our main results that are directly compared with PDG results, is obtaining the (pseudo-)vector meson masses via WKB quantization condition. We also show that an IR WKB quantization (pseudo-)vector meson spectroscopy is nearly isospectral with the results of route one.

Vector Meson Spectrum from Solution of EOMs near r = r h
The α i n (Z) EOM, near the horizon, i.e., Z = 0(Y = 0), is of the form: whose solution is given by: One sets c 2 = 0 as satisfying the Neumann boundary condition for the associate Laguerre function will not be feasible. From (55), one notes that the differential operator is even under Z → −Z -relevant to parity and charge conjugation [3] -and therefore one can think of solutions that are even or odd under Z → −Z. As one knows from [3] that α Setting c 2 = 0, one sees: One therefore sees that one can impose the Neumann/Dirichlet boundary condition α i n ′ (r = r h ) = 0 provided the following condition is imposed: Given that one is solving the EOM near the horizon, i.e., the IR, one expects the masses to be small, something verified by (65). Now, the EOM (57), near r = r h , can be written as: whose solution is given by: As d dZ vanishes at |Z| → 0 + only for c 2 = 0, one sets c 2 = 0 at the very outset. Similar to (60): Now: Hence, one can successfully impose Neumann/Dirichlet boundary condition at the horizon: α {0} n ′ (r = r h ) = 0 by demanding: In the context of (57): One sees that (70) can be satisfied for only a single value ofm α {0} n n -which we declare to be the ground state (largely due to the proximity of its value to (64)) -which satisfies the following condition: Now, from (65) and (73), disregarding O log r h (log N ) 2 terms, one sees that Hence, from (65) where This potential for α {i} n (Z) can be easily worked out but due to the cumbersome nature of the expression so obtained, we will not be giving its analytical expression.

IR
The potential V (α {i} n ), performing first a large-N and then a small-|Z| expansion, for a = r h 0.6 + 4 gsM 2 N (1 + log r h ) [19], is given by: In the IR, (77) yields: The solution to (77) is given in terms of Whittaker functions: One can show that: One can also show that (84)
but not Dirichlet boundary condition, is identically satisfied, therefore not providing values form.

IR
One can show that: The solution to: is given by: One can show that near r = r h : Hence by requiring: one can impose Neumann boundary condition at the horizon, r = r h . With:

UV
In the UV disregarding the M and N f (as there is no net D7-brane and D5-brane charge in the UV in [1] and therefore in their mirror in [2]): The solution to: One can show that ψ 0 n (Z) does not satisfy the Dirichlet boundary condition in the UV but the Neumann boundary condition: is identically satisfied in the UV and hence one does not obtain any quantization condition on the massesm. Performing first a large-N expansion, one obtains the following:

α
(100) (a) Large-m expansion : UV regime, i.e., r > 0.6 √ 3r h or |Z| > 0.04 One notes from (100) that √ V ∈ R in the UV for largem: or |Z| ∈ log 1.039 + 0.561 Thus, after performing a large-N expansion, followed by a large-m expansion and then a large-|Z| expansion, one obtains: Finally, the WKB quantization condition:

Spectroscopy from WKB Quantization
Writing m =m r h √ 4πgsN s , a = r h 0.6 + 4 gsM 2 N (1 + log r h ) , one can obtain the Schrödinger-like potential for α {0} n (Z) -due to its cumbersome form, we will not be giving the explicit form of its analytical expression.
After retaining terms up LO in N in the potential, the square root of the Schrödinger-like potential for α {0} n (Z) after a large-(log)N expansion yields: (107)

Large-m Expansion
One can show that V (α or which will be the turning points for the WKB quantization condition implementation. One obtains the following large-m expansion from (107): The WKB quantization condition: yields a cubic of the form: a + bm + c m + d m 2 = g where: The only real root up to O   One hence notes a near isospectrality between the (pseudo-)vector meson spectra from Tables 1 and 2, and as will be seen in Table 4, upon comparison with PDG, it is the results of Table 2 that are slightly more closer to the PDG values than those of Table 1.
The WKB quantization does not work for α {0} n (Z) for smallm as it can be easily shown that in the large-N limit there are no turning points of the potential.

Scalar Meson Spectroscopy using a Black-Hole Background for All Temperatures
Unlike vector meson spectroscopy, the scalar meson spectrum will be obtained by considering fluctuation of the D6-brane world volume along Y by switching off any D6-brane world-volume fluxes as in [4]. Now, Y = 0 and the D6-brane metric (41), using (49) and the embedding: is therefore: where: wherein: B IIA N S−N S [2] in diagonal basis (θ 2 ,x,ỹ,z) is given by: Thus, its pull-back on D 6 is given by: where: Now, B θ 2z and B θ 2ỹ are as given in (39). Therefore: where: , and retaining terms in the following up to O Y 2 ,Ẏ 2 , ∂ µ Y ∂ ν Y (indicated by a tilde below), one obtains: Finally, one thus obtains the following DBI action for N f D6-branes (setting the tension to unity): where S 1,2,3 are defined in (C2). Now, similar to [4], we make the KK ansatz: together with the following identifications, normalization and EOM: Making a field redefinition: Z n (Z) = |Z|G n (Z), one obtains the following EOM for G(Z): Analogous to obtaining the (pesudo-)vector meson spectrum in Section 4, we will now proceed to obtaining the (pseudo-)scalar meson spectrum by three routes. The first will cater exclusively to an IR computation where we solve the G n (Z) EOM near the horizon. Imposing Neumann boundary condition at the horizon results in quantization of the (pseudo-)scalar meson masses. The second route will be to convert the G n (Z) EOM into Schrödinger-like EOM and to solve the same in the IR and UV separately and obtain (pseudo-)scalar mass quantization by imposing Neumann boundary conditions at the horizon (IR)/asymptotic boundary (UV). It turns out the former yields a result, which up to LO in N , is of the same order as the IR results of route one. The UV computations satisfy Neumann and/or Dirichlet boundary conditions without any mass quantization condition. The third route catering to the IR-UV interpolating region and what gives us our main results that are directly compared with PDG results, is obtaining the (pseudo-)scalar meson masses via WKB quantization condition.
s One sees that the n = 0 value 2.39 − 0.51 log N is close to 2.51 of (133), and not too far from 3.08 − 7.61 log N of (134).

UV
In the UV, we will assume M and N f to be quite small and hence approximate the potential by: The solution to the Schrödinger-like equation Φ ′′ (Z) + V (Z)Φ(Z) = 0 is given by: where: x 1 = 1.08 +m 2 , One can show that in the UV: which tells us that the Neumann boundary condition is identically satisfied and one does not obtain any mass quantization condition in the UV from this approach.

Scalar Mass Spectrum via WKB Quantization Condition
For the purpose of simplification of this calculation, we will be disregarding N f and M because one can show that the WKB quantization condition integral fails to converge for IR-valuedm and in the UV, M and N f are very small. One will hence work in the large-m/UV limit. From (138), one thus obtains: One sees that √ V ∈ R for |Z| ∈ 0.0385, 0.5 log 0.54 + 1.013×10 9 m 2 which we will approximate as |Z| ∈ 0.0385, 0.5 log(1.013 × 10 9 ) ≈ 10.368 . The WKB quantization condition: yields the following scalar meson spectrum: m n = α 2 θ 1 (157.08n + 78.5398) + 0.5 α 4 θ 1 (98696n 2 + 98696n + 2.08747 × 10 9 ) − 1.04372 × 10 9 α 2 θ 1 α 2 One can argue that Y (x µ , Z) is even under parity: (x 1,2,3 , Z) → (−x 1,2,3 , −Z). The idea is the following. The type IIB setup of [1] includes D3-branes with world-volume coordinates x 0,1,2,3 and D7branes with world-volume coordinates (x 0,1,2,3 , r,x, θ 1 ,z) 5 , which after three T-dualities alongx,ỹ,z yield two sets of D6-branes, one set with world-volume coordinates (x 0,1,2,3 ,x,ỹ,z) (obtained from a triple T-dual of the D3-branes) and the other set with world-volume coordinates (x 0,1,2,3 , r, θ 2 ,ỹ) (obtained from a triple T-dual of the D7-branes). One hence sees that the two sets of D6-branes are separated in r or Y . In the type IIB setup of [1], the flavor D7-branes never touch the D3-branes which in the SYZ or triple T-dual picture implies that the two sets of D6-branes never touch each other. This, like [3], [4], implies one can construct a C 5 ∼ Y dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ∧ dZ which vanishes precisely when the two sets of D6-branes touch. From this C 5 , one can construct a Chern-Simons action: w.v.of D6 F 2 ∧ C 5 where F 2 = dA 1 correponds to a gauge field on the D6-brane world volume. If one demands the Cherns-Simons action be invariant under parity -which includes Z → −Z -given that F 2 is even, one sees that Y is even under parity. Similarly, under charge conjugation -which includes Z → −Z -and noting that F 2 is charge-conjugation odd implies that Y is charge-conjugation even. From (123), under 5D parity, m n=1 result to match PDG values)! This is effected by imposing the following condition on α θ 1 , α θ 2 : m n=3 m n=1 = 549.7787α 2 θ 1 + 0.5 2.08767 × 10 9 α 4 θ 1 − 1.04372 × 10 9 α 2 θ 1 α 2 θ 2 5 √ N 235.6194α 2 + 0.5 2.08747 × 10 9 α 4 θ 1 − 1.04372 × 10 9 α 2 θ 1 α 2 which is satisfied by α θ 1 = 0.70765N 1 10 α θ 2 . Having done so, the ratio m n=5 m n=1 is not too far off of the PDG value -see Table 8 in Section 7! The (pseudo-)scalar meson (0 −− )0 ++ masses are listed in Table  4 below. (The entries against 0 −− are blanks as there are, as of now, no known candidates with this J, P, C assignment.) In this section we show an interesting near isospectrality of the (pseudo-) vector meson spectrum (in 6.1 and its sub-sub-sections) and (pseudo-)scalar meson spectrum (in 6.2 and its sub-sub-sections) obtained using a thermal background which is valid for low temperatures, with the corresponding results of sections 4 and 5 obtained using a black-hole background (expected to be valid/stable at high temperatures) for all temperatures.
As the techniques are similar to and in fact simpler than the ones used in sections 4 and 5, we will only be presenting the main results to substantiate our claim. Writing r = r 0 e √ Y 2 +Z 2 -r 0 being an IR cut-off 6 -and defining m =m r 0 √ 4πgsN , setting r h = 0 and introducing a bare resolution parameter a = γr 0 (to ensure that R D5/D5 = √ 3a = 0), one can show that the α N (Z) EOM -there is no need to attach a superscript to α N anymore as r h = 0 -near the horizon can be written in the form:

Vector Meson Spectroscopy in a Thermal Background
where up to O 1 log N : whose solution is given by: By imposing Neumann boundary condition at r = r 0 by assuming c 2 = 0, numerically, e.g., for N = 6000, γ = 0.6 (similar to a(r h = 0) = 0.6 + O gsM 2 N r h ), one sees one obtains as a rootcorresponding to the lightest vector meson -m ≈ 1.04 -of the same order as the LO value in (65). One gets only the rootm = 0 if c 1 = 0.

Schrödinger-like EOM
One can rewrite the EOM as a Schrödinger-like equation in terms of ψ n (Z) = C 1 (Z)α n (Z) where ψ n (Z) satisfies: ψ ′′ n (Z) + V (Z)ψ n (Z) = 0, where: and Near r = r 0 -the IR -the EOM can written as ψ ′′ n (Z) + (a + b|Z|)ψ n (Z), where: the solution to which is given by: In the large | log r 0 |-limit and setting γ = 0.6, one can show that one obtains the value:m = 0.36.
In the UV, V (Z) = −1+e −2|Z|m2 +O e −4|Z| , and the solution to the EOM is given by J 1 e −|Z|m and Y 1 e −|Z|m which satisfy the Neumann/Dirichlet boundary condition, identically, in the UV and do not providem quantization.

Schrödinger-like EOM
With: and the potential in the Schrödinger-like EOM (analogous to (150)) given by: the a, b, analogous to (151), are given by: One gets a solution analogous to (152); only for c 2 = 0 we can show numerically that for γ = 0.6, m ≈ 0.85 -not too far off of the smallest root in 6.2.1 and about the same order as the results of 5.2.1 -one can approximately satisfy the Neumann boundary condition at r = r 0 .
Hence, disregarding n = 0, the following spectrum is generated: For a low-temperature thermal gravity

Summary, New Insights into Thermal QCD and Future Directions
A top-down finite-gauge-coupling finite-number-of-colors holographic thermal QCD calculation pertaining to meson spectroscopy 7 , has thus far, been missing in the literature. This paper fills in this gap. We should keep in mind that even though lattice QCD is a good tool to deal with IR Physics but it is hard to include fundamental fermions in the same. However, incorporation of fermions is easily taken care of in the top-down type IIB construct of [1] and its type IIA mirror in [2]. In this paper, we have calculated (pseudo-)vector and (pseudo-)scalar meson spectra from the delocalized type IIA SYZ mirror (constructed in [2]) of the UV-complete top-down type IIB holographic dual of large-N thermal QCD (constructed in [1]), at finite coupling and with finite number of colors (part of the 'MQGP' limit), and compared our results with [3], [4] and [5]. We first do a computation with a black hole background assuming the same to be valid for all temperatures, low and high (similar in spirit to the computations in [43]). We then repeat the computation in a thermal background with no black hole which is valid for low temperatures. What we learn about QCD is that the mirror of [1] when considered in the 'MQGP limit' -involving finite gauge/string coupling and finite N c = M (at the end of a Seiberg duality cascade) and not just a large t'Hooft coupling -can, almost without any fine tuning -generate the low-lying vector and scalar meson spectra from the massless string sector. An analytical finite-gauge-coupling computation in perturbative (thermal) QCD is very hard if not undoable. This however, is easily done as a classical supergravity computation in our setup.  The authors of [4] obtain a variety of values by adjusting the values of and working up to first order in δ, as well as a constant appearing in a 'squashing factor' in the metric. Their best values for (pseudo-)vector meson mass ratios are quoted in Table 5 column 5. But they need to do a lot of fine tuning, incorporate contributions to the results from the O(δ) terms and choose δ = 0.5, which in fact can not justify disregarding terms of higher powers of δ as δ = 0.5 is not very small to warrant the same. Our results, specially coming from the WKB quantization condition applied to V (α {0} n ) for the BH gravitational dual or V (α n ) for the thermal gravitational background, working even up to LO in N without having to explicitly numerically compute the O(δ) (δ ∼ 0.001 for our calculations and thereby justifying dropping higher powers of δ) contribution, display the following features: . The temperature-dependent meson massm will have the following form: δ 2 > 0. The temperature, assuming the resolution to be larger than the deformation in the resolved warped deformed conifold in the type IIB background of [1] in the MQGP limit, and utilizing the IR-valued warp factor h(r, θ 1 ∼ N − 1 5 , θ 2 ∼ N − 3 10 ), is [19]: Using (164) and the arguments of [19], one can invert (164) and express r h in terms of T [42] in the MQGP limit. Assuming log r h in (163) to be in fact log r h Λ , Λ > r h being the scale at which confinement occurs, one sees that, as per expectations, the vector meson masses decrease with temperature [43] with the same being large-N suppressed [44] (and references therein). 4. On comparing scalar meson mass ratios obtained from (144) using a black hole gravitational dual WKB quantization and PDG values, we obtain   Table 8, does a good job though. 6. The 0 −− pseudo-scalar mesons in Table 4 do not, thus far, have corresponding entries in the PDG tables. This is one point of difference between our results and PDG tables. 7. There are three main insights we gain into thermal QCD by evaluating mesonic (vector/scalar) spectra and comparing with PDG values. (a) The first is the realization that from a type IIA perspective, meson spectroscopy in the mirror of top-down holographic type IIB duals of large-N thermal QCD at finite coupling and number of colors 9 (closer to a realistic QCD calculation) which are UV completewe know of only [1] that falls in this category for which the mirror was worked out in [2] -will give results closer to PDG values rather than a single T-dual of the same. Even though obtaining the mirror requires a lot of work, but once obtained, one can obtain very good agreement with PDG tables already at O gsM 2 N 0 (for vector mesons, without any fine tuning). This is a major lesson we learn from our computation. There are two major reasons for the same.
-As noted in section 3, the mirror of [1] picks up sub-dominant terms in N of O(N 0 ) in B IIA which are therefore bigger than the O( gsM 2 N ) contributions, and were missed in [4]. This is the reason why the authors of [4] had to set gsM 2 N = 0.5 -not a small enough fraction to warrant disregarding O gsM 2 N 2 contributions which they didto obtain a reasonable match with [5]. -In the context of [1], this is expected to be related to the following 10 . The brane construct of [1] involves N D3-branes, M D5-branes wrapping the vanishing S 2 , M D5branes wrapping the same S 2 but placed at the antipodal points of the resolved S 2 (a) relative to the D3, D5-branes, N f flavor D7-branes wrapping an S 3 and radially extending all the way into the IR starting from the UV and N f D7-branes wrapping the same S 3 but going only up to the IR-UV interpolating region. The mirror of this results in D6 branes in a deformed conifold. Now, the gravity dual of the above picture -which is what we work with -involves a resolved warped deformed conifold with a black hole and D5, D7 branes and D7 branes (plus fluxes) on the type IIB side and the delocalized mirror yields a non-Kähler warped resolved conifold with a black hole and D6, D6 branes (plus fluxes) on the type IIA side. The latter (warped resolved conifolds) are more easier to deal with computationally than the former(resolved warped deformed conifolds).
(b) (related to (a) above) There is an intimate connection between Strominger-Yau-Zaslow mirror of resolved warped deformed conifolds and thermal QCD at strong coupling and finite number of colors; hence, delocalized Strominger-Yau-Zaslow mirror construction is an entirely new technique used for hadron spectroscopy.
(c) The third is that a BH gravitational dual and a thermal gravity dual yield nearly isospectral spectra for the light vector mesons; the same is partially true for the lightest scalar mesons too. Explicit computations reveal that a BH type IIA gravity dual obtained by delocalized SYZ mirror transform of the type IIB holographic dual of [1] is not only able to provide a good match with PDG values for the lightest vector and scalar mesons, it is also able to thereby obtain an explicit temperature-dependence of the (pseudo-)vector masses as a bonus and realize the log r h -dependence in the same appears at the sub-dominant O 1 (log N ) 2 with the desired feature of a small large-N suppressed decrease with increase of temperature.
Using Witten's prescription of going from type IIA to M-theory we could work back and using the aforementioned M-theory metric perturbations, work out the type IIA metric perturbations which hence would yield (in the followingG IIA αβ = G IIA αβ + h αβ ; α, β = 0, 1... where µ, ν = x, y, z. As shown in [28], the form of the metric of the mirror manifold after performing three T-dualities, first along x, then along y and finally along z: The various components of the metric after three successive T-dualities along x, y and z respectively, can be written as [28]: In the above formulae we have denoted the type IIB B fields as b IIB mn . For the generic case we will switch on all the components of the B field: , (A14)

B Vector Meson Embedding Functions appearing in the DBI action for D6-branes
In this appendix we give the embedding functions Σ 0,1 (r; g s , N f , M, N ) relevant to the embedding of D6-branes in the delocalized SYZ type IIA mirror of the type IIB construct of [1] that appear in (44)