QCD-Compatible Supermassive Inert Top-Down Holographic Mesinos at Intermediate Coupling

A longstanding problem with the popular Sakai-Sugimoto holographic dual of thermal QCD is that the"mesinos", the (non-supersymmetric) fermionic partners of the mesons, are nearly isospectral with mesons and have an unsuppressed mesino-meson interaction, both being in contradiction with actual QCD. We solve this problem in a UV-complete (and ${\it different}$) type IIA string dual ${\it at\ intermediate\ coupling}$ of realistic thermal QCD, in which the mesinos are shown to be much heavier than and non-interacting with mesons (the wave-function/mass/interaction terms receiving no ${\cal M}$-theory ${\cal O}(R^4)$ corrections). In particular we derive a large-$N$ enhancement of the KK mass scale $M_{KK}$ (from $M_{KK}$ to $M_{KK}^{\rm eff}\sim N^{1 + \frac{1}{{\cal O}(1)}}M_{KK}$) arising from the construction of the type IIA mirror \cite{MQGP} of the type IIB dual \cite{metrics} of thermal QCD-like theories, as well as the generation of a one-parameter family of $M_{KK}$-independent mass scale at ${\cal O}(R^4)$ in the ${\cal M}$-theory uplift \cite{OR4} wherein the parameter can be made appropriately large. We also show that the mesino-mesino-single-($\rho/\pi$)meson interactions, vanish identically in the aforementioned type IIA holographic dual.


Introduction
One can construct gauge theories from a stack of D-branes and various configurations of the same.In this context, in the spirit of (non-conformal, non-supersymmetric) gauge-gravity duality (inspired by [4]), mostly bosonic fluctuations on the world volume of D-branes have been considered.The type IIA dual inclusive of the O(R 4 ) corrections -to explore the finite-N -limit/intermediate coupling regime of QCD -of the type IIB dual [2] of thermal QCD-like theories, was worked out in [1] and [3].As a recent example, inclusive of higher derivative corrections to address the finite-N /intermediate coupling regime (as worked out in [3]), phenomenologically-compatible low energy coupling constants up to NLO in the chiral expansion in SU (3) chiral perturbation theory (in the chiral limit) were obtained from the DBI action on flavor D6-branes in [5].Dirac-like action for the supersymmetric partners of mesons, the mesinos, has been obtained from a top-down approach on Dp-branes [6], see [7] for the bottom-up approach.However, using the same for the Sakai-Sugimoto type IIA dual [8] of thermal QCD, it was shown that one runs into a problem -the mesinos and mesons turn out to be approximately isospectral and their interaction is not large-N suppressed [9] -both in contradiction with real QCD.This serves as the main motivation for this paper -to see if this issue can be resolved in the type IIA mirror [1] at intermediate coupling [3] of the non-supersymmetric UV-complete type IIB dual [2] of thermal QCD-like theories.In this paper, we explicitly consider the mesino action on flavor D6-branes in the aforementioned type IIA dual.We also see the effect of higher derivative terms on the fermions relevant to holographic thermal QCD in this paper which was missing in [6].In short, we will show that the mesinos are supermassive and do not interact with the vector/π mesons, which is why we refer to them as W(eakly) I(nteracting) S(upermassive) P(articles), thereby not being in conflict with realistic QCD.
The following serves as a brief summary of the main results of this paper.
-Either by looking at the SU (3) and the "transverse" SU (3) structures on M 6 (= S1 t × w T , × w implying a warped product, S 1 t being the thermal circle and T -deformed T 1,1 -being the base of a warped non-Kähler squashed resolved conifold)/ M6 (=non-Kähler warped squashed resolved conifold), or when considering the embedding of the D6-brane world volume Σ (7) ∼ = S 1 t × w (R 3 × R ≥0 ) × w S 2 squashed in M 10 considered either as (S 1 t × w R 3 ) × w M6 or R 3 × w (R ≥0 × M 6 ), one is therefore guaranteed the existence of a pair of globally defined spinors.Using the same, and imposing anti-periodic boundary conditions along S 1 t , the ansatz (26) was made for the mesino spinor, and the radial profile functions therein, were solved for.
-For the thermal background (5) dual to thermal QCD for T < T c , as well as the black-hole background (4) dual to thermal QCD for T > T c , we found that Dirichlet/Neumann boundary condition at r = r 0 (IR cut-off in the thermal background)/r = r h permitted supermassive mesinos.
-Enhacement of mass scale: * Starting from the D = 11 supergravity Einstein's field equations in the presence of four-form G fluxes of M-theory, we explicitly show the generation of an N -enhanced (≡ "N -hanced") mass scale, thereby providing the mechanism of generation of supermassive mesinos.* Replacing the resolution parameter "a" of the blown-up S 2 by a(r), substituting an ansatz: a(r) = b + c β 0 (r − r 0 ) + βA β (r) into the Einstein's equations and estimating r 0 ∼ e −κr 0 N 1/3 [23], near the ψ = 2nπ, n = 0, 1, 2-coordinate patches, we therefore see that: • Vanishing mesino-mesino-meson interaction (Sec.5): Considering fluctuations of the vector mesons µ=t being the only non-zero background value) in the fermionic flavor D6-brane action and retaining terms linear in the same, performing a KK expansion of the field strength fluctuation along with decomposition of the positive-chirality Majorana-Weyl mesino spinor along M 5 (t, x 1,2,3 , r) and M5 (θ 1,2 , φ 1,2 , ψ), we were able to show that no mesino-mesino-ρ/π-meson vertex is generated.
• Non-renormalization of the mesino wave function and mass (Sec.3, and appendices B and D): With the aim of studying the effect of O(R 4 ) terms on the fermions relevant to holographic thermal QCD which was missing in [6], leads us to a non-renormalization of the mesino wave function and mass in the sense that both turn out to be independent of the O(R 4 ) terms up to (l 6 p /N α ), α ≥ 1 1 , l p being the Planckian length.
The paper is organized as follows.In section 2, we discuss the type IIA string dual construction of thermal QCD-like theories at intermediate coupling.In section 3, we show that fermionic superpartner of mesons, i.e., mesinos, are superheavy due to the generation of N -enhanced mass scale discussed in section 4. Section 5 provides further evidence of superheavy mesinos because of the absence of mesinomesino-meson interaction in type IIA string dual.In section 6, we discuss the non-renormalization of the product of quark mass and quark condensate up to O(R) 4 .Section 7 has a discussion of wave-function universality in the context of glueball, meson, and graviton wave-functions.The summary of the paper has been provided in section 8.
There are five appendices.Appendix A contains the discussion of quark chemical potential.Appendix B consists of quantities appearing in the mesino EOMs of section 3.In Appendix C, we compute the embedding of flavor D6-branes in type IIA string theory inclusive of O(R 4 ) corrections.We list the constants appearing in the wave function for the black hole background in appendix D. Finally, we summarize the top-down holographic QCD results obtained by our group in appendix E.
2 Type IIA String Dual of Thermal QCD-Like Theories Inclusive of O(R 4 ) Corrections Thermal QCD-like theories refer to the equivalence class of theories that are IR confining and UV conformal with the "quarks" transforming in the fundamental representation of the symmetry groups(color and flavor).The UV-complete type IIB string dual of such large-N thermal QCD-like theories was constructed in [2].The brane picture consists of N space-time filling D3-branes at the tip of a warped resolved conifold, M space-time filling D5 branes also at the tip of the conifold as mentioned above wrapping the vanishing squashed S2 and at the North Pole of the resolved squashed S 2 of radius a (resolution parameter), and space-time filling D5-branes also at the tip of the conifold wrapping the abovementioned vanishing squashed S 2 (θ 1 , φ 1 ) and at the South Pole of the resolved squashed S 2 (θ 2 , φ 2 ).In addition, there are N f space-time filling flavor D7-branes wrapping the vanishing squashed S 3 (θ 1 , φ An equal number of D7 wrapping the vanishing squashed S 3 (θ 1 , φ 1 , ψ) and at the South Pole of the blown-up squashed S 2 (θ 2 , φ 2 ), are also present.Equal number of D5/D7-branes and D5/D7-branes in the UV ensure UV conformality.The presence of N f flavor D7 and D7-branes in the UV, implies a flavor gauge group SU (N f )×SU (N f ) in the UV which is broken to SU (N f ) due to absence of D7-branes in the IR 2 (analog of chiral symmetry breaking in this brane setup).The brane construct in the type IIB dual is summarized in the table 1: IR confinement in the gravity dual is affected by deforming the vanishing squashed S 3 in the conifold.Since we are interested in finite temperature QCD, the same is effected via the black hole (T > T c ) and thermal (T < T c ) backgrounds on the gravity dual side.Due to finite temperature and finite separation of Table 1: The Type IIB Brane Construct of [2] (NP and SP respectively denote the North Pole and South Pole of the blown-up S 2 ).
D5 and D5-branes on the brane side, the conifold further needs also to possess an S 2 -blow-up/resolution (with radius/resolution parameter a).Additionally, the ten-dimensional warp factor and fluxes include the effect of back-reaction.Therefore, we conclude that string dual of thermal QCD-like theories in the large-N limit involves a warped resolved deformed conifold.The additional advantage of the type IIB dual of [2] is that in the IR, at the end of a Seiberg-like duality cascade, the number of colors N c gets identified with M , which in the intermediate-N MQGP limit [1], [10] can be tuned to equal 3; given that one is working in the vanishing-Ouyang-modulus limit (|µ Ouyang | ≪ 1 in (2)) of the embedding of the flavor D7-branes, N f can be set to either 2 or 3 corresponding to the lightest quark flavors [5].Now, to explore the intermediate coupling regime, the O(R 4 ) terms in eleven-dimensional supergravity action were considered in [3].M-theory uplift was obtained in two steps: the type IIA Strominger-Yau-Zaslow (SYZ) mirror of type IIB setup was first obtained, and then the former was uplifted to M-theory.To obtain type IIA SYZ mirror of type IIB setup, a triple T-duality was performed along a local special Lagrangian (sLag) T 3 (x, y, z) where (x, y, z) are the toroidal analogs of (φ 1 , φ 2 , ψ) -which could be identified with the T 2 -invariant sLag of [11] -in the large-complex structure limit effected by making the base B(r, θ 1 , θ 2 ) (of a T 3 (φ 1 , φ 2 , ψ)-fibration over B(r, θ 1 , θ 2 )) large [1], [12].Hence, all the color and flavor D-branes get T-dualized to color and flavor D6-branes.The M-theory uplift metric [1], [3] (finite-but-large-N /intermediate coupling) of [2] (UV-complete type IIB holographic dual of large-N thermal QCD-like theories) is expressed in the following form: where the type IIA RR 1-forms, A IIA are obtained from type IIB F IIB 1,3,5 fluxes via the SYZ mirror of type IIB string dual [2], g(r) = 1 − r 4 h r 4 , and φ IIA is the type IIA dilaton profile.For low temperatures, i.e., T < T c , the thermal gravitational dual is given by: where g(r) = 1 − r 4 0 r 4 .One notes that t → x 3 , x 3 → t in (4) followed by a Double Wick rotation in the new x 3 , t coordinates obtains (5); h(r, θ 1,2 ) is the ten-dimensional warp factor [1,2].This is equivalent to: −g BH tt (r h → r 0 ) = g x 3 x 3 Thermal (r 0 ), g BH x 3 x 3 (r h → r 0 ) = −g tt Themal (r 0 ) in the results of [13], [3] (see [14] in the context of Euclidean/black D4-branes in type IIA).In (5), we will assume the spatial part of the solitonic M 3 brane (which, locally, could be interpreted as solitonic M 5-brane wrapped around a homologous sum of S 2 squashed [15]) and their world volume given by R 2 (x 1,2 ) × S 1 (x 3 ) with the period of S 1 (x 3 ) given by a very large: 2π M KK , where the very small M KK is given by 2r , r 0 being the very small IR cut-off in the thermal background (see also [16]) and L = (4πg s N ) ), thereby recovering 4D physics.The working metric for the thermal background corresponding to T < T c will involve setting g(r) to unity in (5).
Eleven dimensional supergravity action including O(R 4 ) terms used in [3] is: where: The equations of motion for metric and three form potential C are: where [17]: 6)/( 8) are elven-dimensional Riemann curvature tensor, Ricci tensor, and the Ricci scalar.To solve (8), the following ansatz was made: M N , C M N P = C (0) EOM for C M N P symbolically can be written as: It was shown in [3], that, C M N P = 0 up to O(β).Therefore only the metric receives O(R 4 ) corrections defined as: In general, the M theory metric has the following form including O(R 4 ) corrections: The EOMs for f M N (r) were solved in [3].The type IIA metric inclusive of O(R 4 ) corrections were obtained from the M-theory metric by descending back to type IIA string theory, which has the following form: The type IIB dual of large-N thermal QCD-like theories as constructed in [2] and its type IIA mirror as constructed in [1], [12] were successfully used to study a variety of issues in Condensed Matter Physics, lattice/PDG-compatible particle phenomenology, doubly holographic extension and Page curves of associated eternal black holes and G/(Almost)Contact(3)Metric structure classification of underlying six-, seven-and eight-folds in differential geometry (see E).

Supermassive Mesinos in Type IIA String Theory
The fermionic sector of type IIA holographic dual of QCD as constructed in [8] has the following problems.Not only are the mesinos approximately isospectral with the mesons, the single-mesonmesino-mesino interaction terms are not large-N suppressed [9] (see also [18] for mesino spectroscopy degenerate with mesons in the context of [8] and [14]).Evidently, this is in contradiction with QCD/PDG as no mesino at the EW scale has thus far been observed.What we show in this section is that Dirichlet/Neumann boundary condition at the IR cut-off (for the gravity dual corresponding to T < T c ) or the horizon radius (for the gravity dual corresponding to T > T c ) is consistent with having a supermassive mesino.Further, we show an N -enhancement of the Kaluza-Klein mass scale via an N -enhancement of the resolution parameter for the thermal background (T < T c ), hence providing the mechanism of generation of the aforementioned supermassive mesino.Even though we have not been able to provide in 3 an analog of the N -enhancement of the resolution parameter (that was seen in the thermal background corresponding to T < T c ) for the black hole background corresponding to T > T c , the following should be noted.In 3, what we were able to show for the gravity duals of both the low and high-temperature QCD-like theories is that Dirichlet/Neumann boundary condition at the IR cut-off, horizon radius respectively in the gravity duals for T < T c , T > T c , do not fix the mesino mass.We can hence take the same to be large, and via the aforementioned N -enhancement of the resolution parameter in the former, we had explicitly shown the mechanism of obtaining supermassive mesinos in the thermal background.Given that we were able to show the vanishing of meson-mesino-mesino interaction in 5, even if the mesinos were of the EW scale, there still will be no contradiction with real QCD.
The DBI action for the fermions on flavor D6-branes has the following structure [6]: where Φ IIA is the type IIA dilaton.We can define: such that B IIA α 1 α 2 and F IIA α 1 α 2 are NS-NS B field and gauge field restricted to the world volume of D6-branes.
Further, Γ D 6 and L D 6 appearing in (15) are defined as3 : where where covariant derivative is defined as: F mn and F mnpq are field strength tensors corresponding to type IIA A n and A npq , and For flavor D6-branes in type IIA string theory The Dirac equation for the DBI action for the fermions on flavor D6-branes appearing in type IIA string dual of thermal QCD-like theories turns out to be: where In ( 21), F mnpq is the type IIA RR four-form field strength.This in our computation is set to zero as one can show that the same can not be generated by a triple T dual of the RR F IIB 1,3,5 [1].Type IIA NS-NS B is given by [19]: When we restrict to the world-volume of D6-branes, then only the non-trivial component that survives will be B IIA θ 2 ỹ .The induced metric on the world volume of D6-branes can be obtained from the target space metric as given below: Typically, type IIA metric is not diagonal in the basis (x, y, z).Since we need the metric component along ỹ-direction therefore, we are writing the metric in diagonal basis in subspace (x, ỹ, z) [19]: ds 2 5 in ( 23) is non-compact metric listed along (t, x 1,2,3 , r) subspace, and from (24), g IIA .
Consider the DBI action on the world volume of flavor D6-branes: [the embedding of the flavor D6-branes in the ten-dimensional background involving a warped squashed resolved conifold] in the ψ = 2nπ, n = 0, 1, 2coordinate patches and vanishingly small Ouyang embedding parameter in the parent type IIB dual.
Using the induced metric on the flavor D6-branes as given in (23), NS-NS B IIA as given in (22) and turning on a quark chemical potential (by looking at the DBI action in the UV and solving for A t (r) -see (A1)) corresponding to U (1) sub-group of U (N f ) with the associated field strength F rt = A ′ t (r), the background A t (r) can be obtained (see appendix A).In the IR, L D6 DBI, on−shell , for N ∼ 10 2 , can be shown to be infinitesimal.
The most dominant spin-connection terms in the IR are contained in E r 5 Γ 5 D r Θ, in particular ω 7 10 r /ω 8 10 r respectively for the thermal("TH"), black-hole ("BH") backgrounds.Consequently, substituting (26) into Θ's EOM (details given in this section and B), the same at O(β) is: with J ≡ ω .Note, we have disregarded all O β N α , α ≥ 1 terms (see footnote 1) and therefore there are no β corrections in . One thus sees that the only consistent solution for f i (θ 2 ) is f i (θ 2 ) = 0 for the TH/BH backgrounds.
What we now address in section 4 is how an N -enhancement (≡ N -hancement) of the mass scale M KK = r 0 √ 4πgsN [5] is obtained which therefore explains how one could obtain supermassive M Mesino .

Generation of N -hanced Mass Scale for T < T c
In this section, starting from the D = 11 supergravity Einstein's field equations in the presence of four-form G fluxes of M-theory5 -the first in (8) (also given in E) -we explicitly show the generation of an N -enhanced (≡ "N -hanced") mass scale, thereby providing the mechanism of generation of supermassive mesinos.

EOM
where, Defining, a(r) is given by: We therefore see that the "bare resolution parameter" b given by: One hence can not obtain an r 0 -independent "b".One thus sees an N -hancement of the effective KK mass scale M KK (from O(1) M KK ) arising from the construction of SYZ type IIA mirror of the non-Kähler type IIB dual [2] of thermal QCD-like theories, as well as the generation of a one-parameter (C) family of r 0 /M KK -independent bare resolution parameter at O(R 4 ) in the M-theory uplift involving a G 2 -structure wherein C can be made appropriately large.These are the pair of reasons for generating super-massive mesinos in the fermionic sector in the string/M theory duals of thermal QCD at finite N in [1], [3].

Non-Interacting Mesinos
Given that we have seen in 3 that supermassive mesinos, unlike [8] (see [9]), are permissible in the type IIA holographic dual [1] at intermediate coupling [3] of realistic thermal QCD-like theories, this already explains why mesinos have thus far not been observed near the EW scale.In this section, we will further show that mesino-mesino-single-(ρ/π)meson interactions, unlike [8] (see [9]), vanish identically in the aforementioned type IIA holographic dual.Considering fluctuations of the vector mesons A µ,r → A (0) µ,r + δA µ,r with A (0) µ=t being the only non-zero background value (see 3) which can be shown to be tunable so that |F (0) rt | ≪ 1, implying one need only consider terms linear in F (0) IIA 6 which are contained (recalling from section 3, −det(i * g IIA + F (0) IIA ) ≪ 1, FIIA = i * BIIA + F in the large-N limit) in : Considering fluctuations in the background gauge field in (51) and retaining terms linear in the same yields: The next step is to perform the KK expansion of δF IIA αβ and decompose spinors along M 4 and internal directions, and by integrating over the θ 2 and ỹ we will get mesino-mesino-meson interaction action with couplings given in terms of radial integrals of the radial profile functions of the mesino and mesons.The usual KK expansion ansatz [5] is: and implies and We will keep the n = 1 term for the vector fluctuation and n = 0 for the A r (x µ , r); hence, the degrees of freedom are ρ vector meson and π meson.Using the KK decomposition of δF µν and δF µr , (52) simplified as follows: Using the decomposition of the ten-dimensional gamma matrices [24]: with The ten-dimensional chirality matrix is defined as: The positive-chirality ten-dimensional Θ can hence be decomposed into where . Looking at the second fermionic bilinear in (57): Now, the non-vanishing ΘΓ X 1 ....X p Θ involving Majorana-Weyl spinor Θ requires p = 3, 7 [9].One can further show that the most dominant spin-connection component of the type ω ab r is ω 79 r and only non-vanishing spin-connection component of the type ω ab t is ω x 0 r t .Therefore, using (58): Also, as µ, ν ∈ x 1,2,3 and thus using (58): Hence, no mesino-mesino-ρ/π-meson vertex is generated.Together with what was argued earlier that one could have a supermassive mesino, this suggests the "WISP"(Weakly Interacting Supermassive Particle)y nature of the non-supersymmetric mesino, and consequently resolves the tension between actual QCD and top-down holographic QCD [9].
6 Top-Down m quark qq Non-Renormalization up to O(R 4 ) The O(R 4 ) corrections to the M-theory dual's metric are vanishing small in the UV [25].The EOM of the flavor D6-branes' embedding, z = z(r) in the IR arising from the DBI action for the flavor D6branes with world volume product] effected by z = z(r) in a non-Kähler warped squashed resolved conifold M 6 in the type IIA mirror of the UV-complete type IIB dual [2] of thermal QCD-like theories, using the induced metric on flavor D6-branes of ( 23), NS-NS B IIA of (22), can be shown to yield: z =constant, inclusive of O(β) corrections.
The DBI action in the UV is given by (disregarding overall r-independent factors, and hence the ∼): and consequently the z(r) EOM: C being a constant (subsuming g s -and N -dependent factors).One hence obtains 7 : As z(r) is dimensionless, C 1 will hence also be so, and C 2 will have a mass dimension of four (in units of R D5/D5 = D5 − D5-separation).By looking at fluctuations: z → z + δz in the DBI action (no mass term (δz) 2 is generated) one can show that in the UV and in the ψ = 2nπ, n = 0, 1, 2-coordinate patches and by working near, e.g., (θ 1 , θ 2 ) = [12], [23] (consistent with the µ Ouyang | ≪ 1-limit of the flavor D7-branes in the parent type IIB dual [2]): Again, we see that the mass dimension of the coefficient C 2 of 1 r 4 is four (and c 1 is dimensionless).Given that one obtains an AdS 5 in the UV, the coefficient of 1 r 4 for the massless fluctuation δz is identified with a chiral condensate [26], we conjecture that C 2 is the top-down holographic analog of the massdimension-four m q qq .As the O(R 4 ) corrections are vanishingly small in the UV [5], c 1 and C 2 receive no O(R 4 ) corrections.This is the top-down holographic analog of the RG-invariance of m q qq [27].
7 Universality in Particle Wave Functions in the IR An intriguing universality in the wave functions of the following particle spectroscopies is noticed.
• 1 ++ glueball: The EOM for the radial profile function of the vector-type M-theory metric perturbation µν ≈ 0, R µν denoting the first-order fluctuations in the Ricci tensor as a consequence of linear metric perturbations.
Mesons [19]: Working with the redefined radial variable Z : r = r h e Z , after integrating out the blownup S 2 squashed in the DBI action of the flavor type IIA D6-branes and KK reduction of the gauge field The solution of α {i} is given in terms of the Tricomi Hypergeometric and associated Laguerre functions.
Solutions to the EOMs for the aforementioned field fluctuations/radial profile function are given in terms of the Tricomi Hypergeometric and associated Laguerre functions.The reason is that the relevant near-r h EOMs for 0 −+ , 0 −− , 1 ++ -glueballs [22], and the radial profile function of the graviton wave function [28] are all of the type: A Finite Quark Chemical Potential We explicitly show the generation of a finite quark chemical potential.From equation (27) , one obtains: , being the incomplete elliptic integral of the first kind, and Π(ν; φ|µ) being the incomplete integral of the fourth kind, generating a finite quark chemical potential:

B EOM-Related for Massive Mesinos
The EOM for the radial profile R 2,n (r) of the Mesino Θ, as defined in equation (26), is given by: with a = 7, 8 respectively for the TH, BH backgrounds with suitable aforementioned definitions for J (r).

D Constants appearing in the Solution to the Mesino Wave-Function for T > T c
The parameters µ 1,2,3 , Λ in equation ( 33), are defined as follows (in the following, terms of O β N α , α ≥ 1 have been dropped as the same were sub-dominant as compared to the order considered in the [3]):   E Summary of Applications of Top-Down Holographic QCD [1], [3] One of the authors (AM) has been working on the top-down holographic QCD for the past few years.The holographic dual of finite N QCD was first constructed in [1] and then O(R 4 ) corrections to [1] were obtained in [3].Following is the summary of results obtained in this direction.
• Summary of Applications of [1]: In [15], transport coefficients such as shear viscosity, diffusion constant, electrical conductivity, charge susceptibility, etc., of black M 3-banes (black M 5-branes wrapping a homologous sum of two cycles) in the MQGP limit were obtained, and it was found that the ratio of shear viscosity-to-entropy density is 1/4π.In [12], deconfinement temperature and mass scale of the first generation quarks were obtained without the inclusion of O(R 4 ) corrections relevant to thermal QCD.Further, thermodynamic stability and G 2 structure of [1] and temperature dependence of electrical conductivity and charge susceptibility were also discussed in [12], [29].In this process, Einstein's law was verified by computing the ratio of electrical conductivity to charge susceptibility.For the discussion on Wiedemann-Franz law by calculating the thermal and electrical conductivities up to LO in N and NLO in N correction to the aforementioned transport coefficients and speed of sound from the gauge invariant metric perturbations, see [30].The glueball and meson spectra of finite N QCD have been obtained in [22] and [19], a being the frames: E M a g M N E N b = η ab ) for the TH background; for the BH background, Γ 15678 in the second line of (27) is to be replaced by Γ 156 with J ≡ ω
) C z =Constant Embedding of Flavor D6-Branes Inclusive of O(β) Corrections