Transport Coefficients of Black MQGP M3-Branes

The SYZ mirror, in the `delocalized limit' of [1], of (M)N (fractional)D3-branes, and wrapped N_f flavor D7-branes in the presence of a black-hole resulting in a non-Kaehler resolved warped deformed conifold (NKRWDC) in [2], was carried out in [3] and resulted in black M3-branes. The uplift, if valid globally(like [4] for fractional D3 branes in conifolds), asymptotes to M5-branes wrapping a two-cycle (homologously an (large) integer sum of two-spheres) in AdS_5xM_6. Interestingly, in the MQGP limit, assuming the deformation>resolution, by estimating the five SU(3) structure torsion (\tau) classes W_{1,2,3,4,5} we show that \tau\in W_4+W_5: 2/3 Re(W^3bar_5)=W^3bar_4 in the UV, implying the NKRWDC locally preserves SUSY. Further, the local T^3 of [3] in the large-r limit and the `MQGP' limit of [3], satisfies the same conditions as the maximal T^2-invariant special Lagrangian three-cycle of T^*(S^3) of [6], partly justifying use of local SYZ mirror symmetry in [3]. Using the Ouyang embedding in the DBI action of a D7-brane or by dimensionally reducing the 11-dimensional EH action to five (R^{1,3},r) dimensions, we then calculate a variety of gauge and metric-perturbation-modes' two-point functions using the prescription of [5], and show: (i) diffusion constant D~1/T, (ii) the electrical conductivity \sigma~T, (iii) the charge susceptibility \chi~ T^2, (iv) [using (i) - (iii)] the Einstein's relation \sigma/\chi=D, is indeed satisfied, (v) the R-charge diffusion constant D_R~1/T, and (vi) the possibility of generating \eta/s=1/4pi from solutions to the vector and tensor mode metric perturbations' EOMs, separately. All results are also valid in the limit of [2].


Introduction
Since the formulation of AdS/CFT correspondence [6][7][8], it has been of profound interest to gain theoretical insight into the physics of strongly coupled Quark Gluon Plasma (sQGP) produced in the heavy-ion collision at RHIC by using holographic techniques. Within this context, the holographic spectral functions play an important role because of their ability to provide a framework to compute certain transport properties of QGP such as the conductivity and viscosity of the plasma which otherwise are not computationally tractable. The calculation scheme generally involves Kubo's formulae, which involve the long distance and low frequency limit of two-point Green's functions. The methodology to calculate two-point Green's functions by applying gauge-gravity duality has been developed in the series of papers [5,9,10] by considering D3-brane background. However, this approach is useful for transport coefficients that are universal within models with gravity duals. One of the biggest upshots of the same is the viscosity-over-entropy ratio which takes on the universal value 1 4π , known as KSS bound [11][12][13]. Though the bound was initially obtained by assuming zero charge density, the same was shown to hold to satisfy for wide class of theories including non-zero chemical potential, finite spatial momentum and other different backgrounds (see [14][15][16][17]). In addition to this, it is interesting to analyse the photon production in the context of non-perturbative theories as the same reveals one of the most fascinating signatures of a QGP. The study of photoemission in strongly coupled theories by using AdS/CFT correspondence was initiated in [18] in the presence of vanishing chemical potential, and then continued in [19,20] in the presence of non-zero chemical potential by considering D3/D7 and D4/D6 backgrounds where baryonic U (1) symmetry is exhibited from U (N f ) = SU (N f ) × U (1) global symmetry present on the world volume of stack of coincident flavoured branes. In addition to the finite chemical potential, spectral functions were obtained in [17,21,22] by considering finite spatial momentum, finite electric field and anisotropic plasma respectively. The validity of the results of different transport coefficients obtained in the context of different gravitational backgrounds is manifested on the basis of certain universal bounds. For example, it is known that the results of some transport coefficients should satisfy the famous Einstein relation according to which one can express the diffusion constant as the ratio of conductivity and susceptibility [21]. Based on [5,9], the diffusion coefficient can be extracted from poles in the retarded Green's functions corresponding to density of conserved charges. Also, it was suggested in [23] that similar to viscosity-to-entropy ratio, there can exist a universal bound on the ratio of conductivityto-viscosity in theories saturated by models with gravity duals. Further, based on the Minkowski space prescription of Son and Starinets, two-point functions were obtained in [24,25] for conserved R-symmetry current and different components of the stress-energy tensor for M 2-and M 5-brane in ADS 4 × S 7 and ADS 7 × S 4 backgrounds respectively, using which a variety of transport coefficients including diffusion coefficient, shear viscosity as well as the second speed of sound, were calculated. Now, given that AdS/CFT backgrounds renders the gauge theories conformal, they are insufficient to explain the UV completion of gauge theories. The knowledge of UVcompletion is important to handle the issues related to finiteness of the solution at short distances as well as to capture certain aspects of large-N thermal QCD. Therefore, the first successful attempt to explain the RG flow in the dual background was made by Klebanov and Strassler in [26] by embedding D5-branes wrapped over S 2 in a conifold background which was further extended to Ouyang-Klebanov-Strassler (OKS) background in [27] in the presence of fundamental quarks (by including N f D7-branes wrapped over non-compact four-cycle(s)), and finally, followed by modified OKS-BH background [1] in the presence of a black-hole (BH). It would be interesting to determine the transport coefficients by considering non-conformal modified OKS-BH background, specially in the finite-string coupling limit [28], which motivates a (local) uplift to M-theory and calculation of the transport coefficients in the same.
In this paper we intend to study the behaviour of transport coefficients by using Mtheory uplift obtained in [2] (by considering modified OKS-BH background of [1]) which produces black M 3-brane whose near-horizon geometry near the θ 1,2 = 0, π branches preserves 1 8 supersymmetry in both limits as again described in [2]. One should keep in mind that the second limit could be more suited to demonstrate the characteristics of sQGP [28] by exploiting gauge-gravity duality, and due to the finiteness of the string coupling can meaningfully only be addressed within an M-theoretic framework. Using this background, we have already computed some of the transport coefficients such as viscosity-to-entropy ratio and diffusion constant in [2] by adopting KSS prescription [11]. Given that, so far in the literature, the transport coefficients have been obtained in the context of string/M-theory by using AdS n × S n backgrounds, it would be interesting to study important transport coefficients such as electrical conductivity in the domain of weak to finite string/gauge coupling by using the background of M-theory uplift of non-conformal resolved warped deformed conifold background of [1] which asymptotically, we show, corresponds to black M 5-branes wrapping a sum of two-spheres in AdS 5 × M 6 . An overview of the paper is as follows: In subsection 2.1, we review the framework of our local M-theory uplift of type IIB background of [1] as constructed in [2] using local S(trominger) Y(au) Z(aslow) mirror symmetry. In this section, we also highlight results of various hydro/thermodynamical quantities obtained in the limit of [1] as well as MQGP limit provided in [2], and then review the thermodynamical stability of the M-theory uplift. In subsection 2.2, we obtain magnetic charge on the black M 3-branes due to four-form fluxes G 4 through all (non)compact four-cycles and point out that the black M 3-branes asymptotically can be thought of as M 5-branes wrapped around two-cycles given homologously by integer sum of two-spheres. In subsection 2.3, given that supersymmetry via the existence of a s(pecial) Lag(rangian) is necessary for implementing mirror symmetry as three T-dualities a la SYZ, we show that at least in either limits of (2.4) and assuming that the deformation parameter is larger than the resolution parameter, the local T 3 of [2], in the large-r limit, is the maximal T 2 -invariant sLag of a deformed conifold as defined in [4]. In section 3, we turn toward the discussion of various transport coefficients. A variety of transport coefficients can be obtained by considering fluctuations of U (1) N f gauge field corresponding to N f D7-branes wrapping non-compact four-cycle of (resolved) warped deformed conifold as discussed in [2], the fluctuations around U (1) R gauge field, and the stress energy tensor. In this spirit, in section 3.1, we first consider gauge field fluctuations around non-zero temporal component of gauge field background (non-zero chemical potential as worked out in [2]) in the presence of N f D7-branes wrapping non-compact four-cycle in the resolved warped deformed conifold background and work out the equations of motion (EOM) of the same. Further, we move on to the computation of two-point retarded Green's functions by using the prescription first set out in [5] using which one can calculate the electrical conductivity by utilising Kubo's formula. In section 3.2, we work out the charge susceptibility due to non-zero baryon density which turns out to be such that the ratio of the electrical conductivity and charge susceptibility satisfies the Einstein relation. Following the same strategy as described in 3.2, we compute the correlation function corresponding to U (1) R current in the context of 11-dimensional M-theory background in section 3.3, the pole of which simply reads off "diffusion coefficient". In section 3.4, we determine correlation functions of stress energy tensor modes. After classifying different stress-energy tensor modes, we compute two-point correlation functions of vector and tensor modes of black M 3-brane. Using Kubo's formula, we therefore calculate the shear viscosity and show that is possible to obtain the shear viscosity-to-entropy ratio (η/s) to be 1/4π. We finally summarize our results and remark on the possible extensions of the current work in section 4. There are two appendices. For the paper to be self-contained, in appendix A we provide the simplified expressions of M-theory metric components obtained in [2]. The appendix B include simplified expressions of possible non-zero G µνλρ 's obtained in the limit of [1].
2 The Black M 3-Branes of [2] This section has two sub-sections. In 2.1, we will review the local uplift of type IIB background of [1] to M-theory, as carried out in [2] using local SYZ mirror symmetry, as well as its hydrodynamical and thermodynamical properties. In 2.2, we discuss the charge(s) of the black M 3-branes of 2.1 and show that asymptotically the 11D space-time is a warped product of AdS 5 and a six-fold with M 5-branes wrapping a two-cycle that is homologous to an integer sum of two-spheres.

The Local Uplift
To include fundamental quarks at finite temperature relevant to, e.g., the study of the deconfined phase of strongly coupled QCD i.e "Quark Gluon Plasma", a black hole and N f flavor D7-branes 'wrapping' a (non-compact) four-cycle were inserted apart from N D3branes placed at the tip of six-dimensional conifold and M D5-branes are wrapping a two cycle S 2 , in the context of type IIB string theory in [1] which causes, both, resolution as well as deformation of the two-and three-cycles of the conifold respectively at r = 0 (apart from warping), resulting in a warped resolved deformed conifold. In that background, backreaction due to presence of black hole as well as D7-branes is included in 10-D warp factor h(r, θ 1 , θ 2 ). The metric from [1], is given as under: g i 's demonstrate the presence of a black hole and is given as follows: where r h is the horizon, and the (θ 1 , θ 2 ) dependence come from the O gsM 2 N corrections. In (2.1), . Due to presence of Black-hole, h i appearing in the internal metric (2.1) as well as M, N f are not constant, and up to linear order depend on g s , M, N f as given below: The warp factor that includes the back-reaction due to fluxes as well as black-hole, for finite, but not large r, is given as: In [2], we considered the following two limits: (i) weak (g s ) coupling − large t Hooft coupling limit : Apart from other calculational simplifications, one of the advantages of working with either of the limits in (2.4) is that the ten dimensional warp factor h (2.3) for large r approaches the expression (2.2) for finite/small r. The three-form fluxes, including the 'asymmetry factors', are given by [1]: The asymmetry factors A i , B i encode information of the black hole and of the background and are given in [1]. The O(a 2 /r 2 ) corrections included in asymmetry factors correspond to modified Ouyang background in the presence of black hole. The values for the axion C 0 and the five form F 5 are given by [29]: It can be shown that the deviation from supersymmetry, one way of measuring which is the deviation from the condition of G 3 ≡F 3 −ie −φ H 3 being imaginary self dual: |iG 3 − * 6 G 3 | 2 , is proportional to the square of the resolution parameter a, which (assuming a negligible bare resolution parameter) in turn is related to the horizon radius r h via: . We hence see that despite the presence of a black hole, in either limits of (2.4), SUSY is approximately preserved.
To enable use of SYZ-mirror duality via three T dualities, one needs to ensure that the abovementioned local T 3 is a special Lagrangian (sLag) three-cycle. This will be explicitly shown in section 3. For implementing mirror symmetry via SYZ prescription, 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. This is effected via [31]: for appropriately chosen large values of f 1,2 (θ 1,2 ). The three-form fluxes remain invariant. The fact that one can choose such large values of f 1,2 (θ 1,2 ), was justified in [2]. The guiding principle was that one requires that the metric obtained after SYZ-mirror transformation applied to the resolved warped deformed conifold is like a warped resolved conifold at least locally, then G IIA θ 1 θ 2 needs to vanish [2]. 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 .We therefore can construct the following type IIB gauge field one-form in the local limit [2]: resulting in the following local uplift: which are black M 3-branes.
The simplified expressions for all non-zero 11-dimensional metric components as given in appendix 1 in either limit ((i) small g s and large t'Hooft couplings or (ii) MQGP limit) ∀θ 1,2 ∈ [ θ 1,2 , π − θ 1,2 ], assuming that globally one can replace x, y, z respectively In the small g s , large t'Hooft couplings limit of [1], the G M tt,rr components of the elevendimensional metric take the form: − . In the near-horizon 2 , in conformity with [1] . Similar result are obtained for the MQGP limit.
Now, in both limits, G M 00 , G M rr have no angular dependence and hence the temperature 4π √ G 00 Grr [11,12] of the black M 3-brane then turns out to be given by: To get a numerical estimate for r h , we see that equating T to r h πL 2 ; in both limits one then obtains r h = 1 + , where 0 < < 1.
The amount of near-horizon supersymmetry was determined in [2] by solving for the killing spinor by the vanishing superysmmetric variation of the gravitino in D = 11 supergravity. In weak (g s )-coupling-large-t'Hooft-couplings limit [2] with constraints: and for a constant spinor ε 0 ; g Φ 1,2 Φ 1,2 = −α Φ , g ψψ = −α ψ , g x 10 x 10 = α 10 near θ 1,2 = 0, π. The near-horizon black M 3-branes solution possess 1/8 supersymmetry near θ 1,2 = 0, π. Similar arguments also work in the MQGP limit of [2]. Freezing the angular dependence on θ 1,2 (there being no dependence on φ 1,2 , ψ, x 10 in, both, weak (g s )-coupling-large-t'Hooft-couplings and MQGP limits), noting that G IIA/M 00,rr,R 3 are independent of the angular coordinates (additionally possible to tune the chemical potential µ C to a small value [2], using the result of [11]: In the notations of [11] one can pull out a common Z(r) in the angular-part of the metrics as: Z(r)K mn (y)dy i dy j , (which for the type IIB/IIA backgrounds, is √ hr 2 ) in terms of which: Let us review the thermodynamical Stability of Type IIB Background in Ouyang Limit as discussed in [2]. Assuming µ( = 0) ∈ R in Ouyang's embedding [32]: r 3 2 e i 2 (ψ−φ 1 −φ 2 ) sin θ 1 2 sin θ 2 2 = µ, which could be satisfied for ψ = φ 1 + φ 2 and r 3 2 sin θ 1 2 sin θ 2 2 = µ. Including a U (1)(of U (N f ) = U (1)×SU (N f )) field strength F = ∂ r A t dr ∧dt in addition to B 2 , in the D7-brane DBI action, taking the embedding parameter µ to be less than but close to 1, the chemical potential µ C was calculated up to O(µ) in [2]: One thus sees: T 6 (f 2 + 4lnµ) Let us, towards the end of this section, review the arguments on demonstration of thermodynamical stability from D = 11 point of view by demonstrating the positivity of the specific heat, as shown in [2]. Keeping in mind that as r h ∼ l s , higher order α corrections become important, the action: where T 2 ≡ M 2-brane tension, and: for Euclideanised space-time where M is a volume of spacetime defined by r < r Λ , where the counter-term S ct is added such that the Euclidean action S E is finite [33,34]. The reason why there is no 1 in the action is because of the tacit understanding that as one goes from the local T 3 coordinates (x, y, z) to global coordinates (φ 1 , φ 2 , ψ) via equation (2.7), the (g s N ) large-N (>> M, N f ) limit, can be estimated by [27]: (g s ) coupling -large t'Hooft coupling(s) limit of [1] defined by considering g s → 0, g s M → ∞, g s N → ∞, gsM 2 N → 0, g s 2 M N f → 0, we have (2.13) Using equation (2.5), we have 1 Since the integrand receives the most dominant contribution near θ 1,2 = 0, π, we have introduced a cut-off θ 1,2 ∼ 1 2 where 0.1 in MQGP limit. Using the same, We have 1 The action, apart from being divergent (as r → ∞) also possesses pole-singularities near θ 1,2 = 0, π. We will regulate the second divergence by taking a small θ 1,2 -cutoff θ , θ 1,2 ∈ [ θ , π − θ ], and demanding θ ∼ γ , for an appropriate γ. We will then explicitly check that the finite part of the action turns out to be independent of this cut-off / θ . It was shown in [2] that in [1]'s and MQGP limit, S f inite EH+GHY +|G 4 | 2 +O(R 4 ) ∼ −r 3 h and the counter terms are given by: +a R 4 45 , respectively in the two limits -in both limits the counter terms could be given an ALDgravity-counter-terms [34] interpretation. It was also shown in [2] that the entropy (density s) is positive and one can approximate the same as s ∼ r 3 h -result which is what one also obtains (as shown in [2]) -by calculation of the horizon area. Using the same, therefore,

Black M 3-Branes as Wrapped M 5-branes around Two-Cycle
Let us turn our attention toward figuring out the charge of the black M 3-brane. Utilizing the non-zero expressions of the components of the four-form flux G 4 as given in (B.1), and: (i)assuming the local expressions are valid globally, (ii) using the principal values of the integrals over θ 1,2 , i.e., assuming: to make the base of the local T 3 -fibered resolved warped deformed conifold to be large, will globally be cot θ i , and (iv) believing that the distinction between results with respect to φ 1 and φ 2 arising due to the asymmetric treatment of the same while constructing A IIA from triple T-duals of F IIB 1,3,5 , being artificial and hence ignoring the same, using (B.1) one sees that in the limit of [1]: We first take weak (g s ) coupling − large t Hooft coupling limit in G rθ 2 ψx 10 itself to annul this flux. Also, G rθ 2 ψx 10 ∼ N f r in the weak (g s ) coupling − large t Hooft coupling limit and hence negligible for large r; Again we first take weak (g s ) coupling − large t Hooft coupling limit in G θ 1 θ 2 ψx 10 itself to annul this flux; where we calculate flux of G 4 through various four cycles C I 4 . We have dropped contribution r -suppressed as compared to the ones retained, which is hence, dropped, at large r. From (2.16), one sees that the most dominant contribution to all possible fluxes arises (near ψ = 0, 2π, 4π), in the large-r limit, from G θ 1 θ 2 φ 1 x 10 and G θ 1 θ 2 φ 2 x 10 . Using (A.1), the large-r limit of the D = 11 metric can be written as: Hence, asymptotically, the D = 11 space-time is a warped product of AdS 5 (R 1,3 × R >0 ) and an M 6 (θ 1,2 , φ 1,2 , ψ, x 10 ) where M 6 has the following fibration structure: (2.18) Klebanov-Strassler background corresponding to D5-branes wrapped around a two-cycle which homologously is given by , the black M 3brane metric, asymptotically can be thought of as black M 5-branes wrapping a two cycle homologously given by: for some large n 1,2 , m 1,2 ∈ Z. A similar interpretation is expected to also hold in the MQGP limit of [2] -in the same however, the analogs of (B.1) are very tedious and unmanageable to work out for arbitrary θ 1,2 . But, we have verified that in the MQGP limit, will be very large. Warped products of AdS 5 and an M 6 corresponding to wrapped M 5-branes have been considered in the past -see [35].

Kählerity from Torsion Classes and a Local Warped Deformed Conifold sLag in the large r-Limit
In this subsection, first, by calculating the five SU (3) structure torsion(τ ) classes W 1,2,3,4,5 , we show that in both limits discussed in [1], [2] and assuming the deformation parameter to be larger than the resolution parameter, the non-Kähler resolved warped deformed conifold of [1], in the large-r limit, reduces to a warped Kähler deformed conifold for which τ ∈ W 5 . Then, in either of the two limits in (2.4) and in the large-r limit, we show that the local T 3 of [2] satisfies the same constraints as the one satisfied by a maximal T 2 -invariant S(special) L(agrangian) sub manifold of a T * S 3 so that the application of mirror symmetry as three T-dualities a la SYZ, in the two limits of (2.4), could be implemented on the type IIB background of [1]. The SU (3) structure torsion classes [36], [37] can be defined in terms of J, Ω, dJ, dΩ and the contraction operator : Λ k T ⊗ Λ n T → Λ n−k T , J being given by: and the (3,0)-form Ω being given by The torsion classes are defined in the following way: (3,0) , given by real numbers (2,2) 0 (3,1) 0 : W 5 = 1 2 Ω + dΩ + (the subscript 0 indicative of the primitivity of the respective forms).

Two-Point Functions of D7-Brane Gauge Field, R-charge and Stress
Energy Tensor, and Transport Coefficients In this section, our basic aim is to calculate two-point correlation functions and hence transport coefficients due to gauge field present in type IIB background and U (1) R gauge field and stress energy tensor modes induced in the M-theory background by using the gauge-gravity prescription originally formulated in [5]. Let us briefly discuss the strategy of calculations related to evaluation of retarded two-point correlation functions to be used in further subsections and the basic formulae of different transport coefficients in terms of retarded two-point Green's function.
Step 1: Given that 11-dimensional background geometrically can be represented as AdS 5 ×M 6 asymptotically, we evaluate the kinetic term corresponding to a particular field in the gravitational action by integrating out the angular direction i.e.
Step 2: Solve the EOM corresponding to a particular field in that background by considering fluctuations around the same. Further, the solution can be expressed in terms of boundary fields as: in the momentum space and u = r h r . The solution can be evaluated by using boundary conditions φ(u, q) = φ(0) at u = 0 and incoming wave boundary condition according to which φ(u, q) ∼ e −iwt at u = 1.
Step 3: Evaluate two-point Green's functions by using In general, the transport coefficients in hydrodynamics are defined as response to the system after applying small perturbations. The small perturbations in the presence of external source can be evaluated in terms of retarded two-point correlation functions. For example, the shear viscosity from dual background is calculated from the stress energy tensor involving spatial components at zero momentum i.e. By applying a small perturbation to the metric of curved space, and assuming homogeneity of space, one gets the Kubo's relation to calculate shear viscosity [12]: where the retarded two-point Green's function for stress-energy tensor components is defined as: One can compute the above correlation function for different stress energy tensor modes by using the steps defined above. Similarly, when one perturbs the system by applying an external field, the response of the system to the external source coupled to current is governed by [9]: In the low frequency and long distance limit, the j 0 evolves according to overdamped diffusion equation given as with dispersion relation w = −iDq 2 generally known as Fick's law. The diffusion coefficient 'D can be evaluated from the pole located at w = −iDq 2 in the complex w-plane in the retarded two-point correlation function of j 0 . In (d+1) dimensional SU (N ) gauge theory, in thermal equilibrium, the differential photon emission rate per unit volume and time at leading order in the electromagnetic coupling constant e is given as [19]: where q = (w, q) is the photon momentum and χ µν is the spectral density given as: and is the retarded correlation function of two electromagnetic currents. The trace of spectral function will be given as: The electrical conductivity in terms of trace of spectral function is defined as: (3.13) We utilize the above expression to calculate the conductivity in the next subsection.

D7-Brane Gauge field fluctuations
The gauge field in the type IIB background as described in section 2 appears due to presence of coincident N f D7-branes wrapped around non-compact four-cycle in the resolved warped deformed conifold background. The U (1) symmetry acting on the centre of U (N f ) = SU (N f )×U (1) group living on the world volume of D7-branes wrapped around non-compact four-cycle induces a non-zero chemical potential. The spectral functions in the presence of non-zero chemical potential are obtained by computing two-point correlation functions of gauge field fluctuations about background field [44] which includes only a non-zero temporal component of gauge field.

EOMs
Considering fluctuations of gauge field around non-zero temporal component of gauge field, we haveÂ It is assumed that fluctuations are gauged to have non-zero components only along Minkowski coordinates.
The action for gauge field in the presence of flavor branes is given as: where 'g' corresponds to the determinant of type IIB metric as given in equation (2.1), and i * g gives the pull back of 10D metric onto D7-brane world volume [2]. Introduce the DBI Action will be given by Expanding above in the fluctuations quadratic in field strength, the DBI action as worked out in [44] can be written as : (3.17) and the EOM forÃ µ is: Using above, the DBI action at the boundary will be given as: .

(3.19)
We have already obtained the expression of F rt in [2] and the same is given as (3.20) Considering u = r h r , we have r h C 2 e 2φ(u) + r 6 . (3.21) Looking at the above expression, we see that at u = 0 (boundary), F ut → 0. The action in equation (3.19) will get simplified and given as: x, y, z) and Defining gauge invariant field components the DBI Action in terms of these co-ordinates (using equation (3.51)) will be given as: .

(3.24)
Defining the longitudinal electric field as E x (q, u) = E 0 (q)

Eq(u)
Eq(u=0) , the flux factor as defined in [5] in the zero momentum limit will be given as: , (3.25) and the retarded green function for E x will be G(q, u) = −2F(q, u). The retarded green function for A x is w 2 times above expression and for q = 0, it gives The spectral functions in zero momentum limit will be given as: To obtain spectral functions, we need to determine the expressions for gauge fields. The form of EOMs of gauge fields by considering fluctuations about the non-zero temporal component of gauge field (chemical potential) are: Since we are interested to obtain two-point Green's function in the zero-momentum limit (q = 0), the transverse and longitudinal components of electric field will be simply given as E x = wA x , E y = wA y . In these set of co-ordinates, the above equations take the form as given below: where the prime denotes derivative w.r.t u. The co-ordinates in Minkowski directions are chosen such that the momentum four-vector exhibits only one spatial component i.e incorporating the values of various metric components, the aforementioned equations take the form: where E(u) = E x (u) = E y (u), and w 3 = w πT . Substituting the value of F ut and g 1 , we have From (3.31), one sees that u = 1 is a regular singular point and the roots of the indicial equation about the same are given by: ± iw 3 4 ; choosing 'incoming wave' solution, solutions to (3.31) sought will be of the form: where E(u) is analytic in u. As one is interested in solving for E(u), analytic in u, near u = 0, (3.31) will be approximated by: One converts (3.33) into a differential equation in E. Performing a perturbation theory in powers of w 3 , one looks for a solution of the form: Near u = 0: The solutions are given as under: To get non-zero conductivity, we will required C (0) 1 ∈ C.

Electrical Conductivity, Charge Susceptibility and Einstein Relation
The conductivity, using (3.27),(3.36) and T = r h π √ 4πgsN [2], will be given as: Another physically relevant quantity is the charge susceptibility, which is thermodynamically defined as response of the charge density to the change in chemical potential [45].
where n q = δS DBI δFrt , and the chemical potential is defined as µ = r B r h F rt dr. Using the same, the charge susceptibility will be given as: (3.39) From [2], which using (3.20) implies: For µ = 1 − , → 0 + , C << 1 [2], in both, weak (g s ) coupling-Strong t'Hooft coupling limit [1] and the MQGP limit [2], one sees: χ ∼ The diffusion coefficient corresponding to non-zero charge density appear by demanding the longitudinal component of electric field strength to be zero i.e E y = 0. As given in [45], the condition provides with the expression of diffusion coefficient to be expressed as: Subtituting the values for above metric components as given in equation (3.23), after integration, we get D = L 2 r h + O(C 2 g 2 s /r 8 h ) ∼ 1 T . Using equations (3.38), (3.41) and (3.42), we get σ χ ∼ 1 T ∼ D, hence verifying the Einstein relation.

R-Charge Correlators
The U (1) R -charges are defined in the bulk gravitational background dual to the rank of isometry group corresponding to the spherical directions transverse to the AdS space. Given that the 11-dimensional M-theory background corresponds to black M 3-branes which asymptotically can be expressed as M 5-branes wrapped around two-cycles defined homologously as integer sum of two-spheres (as described in section 2), there will be a rotational (R)-symmetry group dual to the isometry group U (1) × U (1) corresponding to φ 1/2 and ψ in the directions transverse to M 5-branes wrapped around S 2 (θ 1 , φ (2/1) ) + S 2 (θ 2 , x 10 ). To determine the diffusion coefficient due to the R-charge, one needs to evaluate the two-point correlation function of A µ which basically will be a metric perturbation of the form h M µ where M is a spherical direction and µ is an asymptotically AdS direction. As a first step, below we evaluate the EOM of A µ . Under a fluctuation: g µν → g µν + h µν about a non-Minkowski background g µν , the Einstein-Hilbert action up to quadratic order in h µν is given by [46] (setting κ 2 4 = 1): Writing h mν ≡Ã ν , after integration by parts twice, one set of terms is: µν will, after integration by parts twice, yield: . This term is the same as: Hence, combining (3.44) and (3.45), one obtains: The A µ EOM is: By defining u= r h r so that g 1 = g 2 = 1 − u 4 , the black M 3-brane metric of [2] reduces to the form as given below: , and in weak (g s ) coupling-strong t'Hooft coupling/ MQGP limit, the non-zero component include The simplified expressions of aforementioned metric components are given in [2]. u) and working in the A u = 0 gauge, by setting µ = u, x, t, α(∈ R 3 ) in (3.47) one ends up with the following equations: These can be simplified to yield: The simplest to tackle is the last equation as it is decoupled from the previous three equations. One notices that the horizon u = 1 is a regular singular point and the exponents of the indicial equation about the same are given by: ± iw 3 4 ; one chooses the incoming-wave solution and hence writeÃ α (u) = (1 − u) − iw 3 4Ã α (u). The last equation in (3.51) then is rewritten as: We make the following double perturbative ansatz for the solution toÃ α (u)'s EOM (3.52): Plugging (3.53) into (3.52) yields the following set of equations: In the w → 0, q → 0-limit, one can takeÃ α (0) ∼ c 2 . Also, The kinetic terms relevant to the evaluation of two-point correlators of A u,t,α are: Hence, the retarded Green's function G R αα will be given as: (3.58) On comparison with G R αα ∼ iw +2D R q 2 [24], one sees that there is a three-parameter family (which in the c 2 >> c 1,3,4 -limit are c 1,3,4 ) of solutions to the R-charge fluctuationÃ α which would generate D α R ∼ 1 πT 1 .
We now go to theÃ x,t EOMs and observe that one can decoupleÃ x andÃ t , and obtain, e.g., the following third order differential equation forÃ x : Now, rewriting (3.59) as: 60) one notes that u = 1 is a regular singular point of the second order differential equation (3.60) inÃ x . The exponents of the indicial equation are −1 ± iw 3 4 ; choosing the incoming-boundary-condition solution, we writeÃ x and assume a double perturbative series for solution to the second order differential equation satisfied by 1 In c2 >> c1,3,4-limit, D α R = 24c 3 +π 2 c 1 Hence, (3.61) decomposes into the following differential equations: The equations if to be solved exactly, are intractable. We will be content with their solutions near u = 0. They are given as under: where the Tricomi confluent hyergeometric function U (a, b; z) Γ(a−b+1) 1 F 1 (a; b; z), b not being an integer, and L µ ν (z) is the associate Laguerre function. The solutions ofÃ where After some MATHEMATICAlgebra, one can show: A 01 x (u) ≈ where Again after some MATHEMATICAlgebra, one hence obtains: one obtains: As in the differential equation: we requireÃ x (u) to be well-defined at u = 0, one has to impose the following constraint on c 1 , c 2 : a(c 2 ) + w 3ã (c 1 , c 2 ) + q 2 3ã (c 1 , c 2 ) = 0. (3.75) one sees that: (3.77) Substituting (3.73) into (3.74) evaluated at u = 0, one obtains: one realizes that: . This yields: (3.81) Thus, the retarded Green's function G R xx will be given by: The constants of integration c 2 , c 3 must satisfy: Choose: which can be fine tuned to ensure c 3 ≈Ã 0 x . Thus, the retarded Green's function G R xx will be given by: (3.85) Now a 1 = a 2 and if one were to consider the O(w 3 q 0 3 ) term to be iw 3Ã [24], one sees that D x R = 1 πT . By requiring D x R = D α R , one sees that in fact that there is a two-parameter family of solutions that would generate D R = 1 πT as, in the c 2 >> c 1,3,4 -limit as an example, one generates a constraint: 24c 3 +π 2 c 1 12A 0 α = 1.

Stress Energy Tensor Modes
The two point function of stress-energy tensors are obtained by considering small perturbations of the five-dimensional metric, g µν = g µν(0) + h µν . Up to first order in the metric perturbation (dropping G 4 flux contributions) the Einstein equation as given in [24]: where d is dimension of AdS space. To linear order in h µν , Ricci scalar will be given by [47]: The five-dimensional metric in M-theory background in the limits of (2.4), is as follows: We assume the perturbation of metric of M 3-branes to be dependent on x and t only i.e after Fourier decomposing the same, we have h µν (x, t) = d 4 q (2π) 4 e −iwt+iqx h µν (q, w) and choose the gauge where h µu = 0. In case of M 3-branes, there will be rotation group SO(2) acting on the directions transverse to u, t, and x. Based on the the spin of different metric perturbations under this group , the same can be classified into groups as follows: (i) vector modes: h xy , h ty = 0 or h xz , h tz = 0, with all other h µν = 0. We are interested to calculate shear viscosity in the context of M 3-brane by obtaining correlator functions corresponding to vector and tensor modes.

Metric Vector Mode Fluctuations
The vector mode fluctuations will be given by considering non-zero h ty and h xy components with all other h µν = 0 [24]. Since the aforementioned metric is conformally flat near u = 0, one can make a Fourier decomposition at large r such that: where Λ is the cosmological constant arising from |G 4 | 2 and higher order corrections (O(R 4 )).
It is shown in [2] that the higher order corrections are very subdominant as compared to flux term in both limits.
The dominant flux term as calculated in [2] is given by action up to quadratic order in h µν given as: The only first two terms in the action will be relevant to get the kinetic term for vector modes. Solving the same, we get: The very simplified form of 11-dimensional metric in θ i → 0 limit will be given by Using the fact that integrand possess maximum contribution along θ = 0, π, we assume that result of integration along θ 1,2 will be given by sum of the contribution of integrand at θ 1,2 = 0, π. In [2], we have introduced a cut-off θ 1,2 ∼ 1 2 in 'weak (g s ) coupling-strong t'Hooft coupling limit' and θ 1,2 ∼ 3 2 in 'MQGP' limit where ≤ 10 −2 in the former and 0.1 in MQGP in later case. Using the same and equation (3.89), the simplified Action will be given as: weak (g s ) coupling − strong t Hooft coupling limit , (3.100) and According to the Kubo's formula as mentioned in the beginning of section 4, shear viscosity is defined as η = − lim w→0 1 w mG xy,xy . In the q 3 → 0 limit, the H t and H x decouple and the EOM for 'H x ' becomes: where u = 1 is thus seen to be a regular singular point with exponents of the corresponding indicial equation given by: ± iw 3 4 . Choosing the 'incoming boundary condition' exponent, we will look for solutions of the form H x (u) = (1 − u) − iw 3 4 H x (u), H x (u) being analytic in u. Assuming a perturbative ansatz for H x (u) : x +O(w 3 3 ), we obtain: s L 2 Λ. The solution to (3.103) for arbitrary α is given as: x (u = 0, α) = 0 for α = 0 for which we will henceforth write: ; u 4 . Assuming α = 0 henceforth, H (1) (u) will be determined by the following differential equation: Near u = 0, (3.106) is solved to yield:  O(1)β (i)/(ii) ≡ 3 2 / 9 2 for the two limits -(i) for [1] and (ii) for the MQGP limit [2]: , which, writing g s = α in the limit of [1]. If one assumes that the introduction of M fractional D3-branes and N f flavour D7-branes does not have a significant effect on the 10D warp factor h, then in the limit of [1] effected as the first limit of (2.4), one can show that can not be taken to be much smaller than around 0.1. One can choose appropriate α Similarly, in the MQGP limit, one obtains: . Now, in the MQGP limit, is less than but close to unity, hence yet again we can choose α

Metric Tensor Mode Fluctuations
To obtain the correlations function corresponding to tensor mode, we consider a fluctuation of M 3-brane metric of the form h yz = 0 with all other h µν = 0 [24]. By Fourier decomposing the same, Using equations (3.86) and (3.87), the linearized Einstein EOM for φ(u) by will be given as: The horizon u = 1 is a regular singular point and the roots of the indicial equation around this are ± iw 3 4 ; choosing the incoming-wave boundary condition, φ(u) = (1 − u) − iw 3 4 Φ(u) where Φ(u) therefore satisfies: Writing a double perturbative ansatz: (3.111) is equivalent to the following system of differential equations: (3.113) The solutions to (3.113) near u = 0, up to O(u 4 ) are given below: Writing: the boundary condition yields: φ 0 ≈ a 1 + w 3 a 2 c 1,2 >>c 3,4 −→ a 1 . The kinetic term for φ is given by: Hence, using (3.116) and Kubo's formula:

Summary and Outlook
In this paper, we have computed a variety of transport coefficients such as electrical conductivity due to U (1) N f gauge field living on the world volume of N f D7-branes wrapping a non-compact four-cycle in the type IIB modified 'OKS-BH' background of [1], charge diffusion due to U (1) R gauge field and shear viscosity corresponding to different stress energy tensor modes of the black M 3-branes of [2]. The black M 3-branes in [2] were basically obtained as a solution to the local M-theory uplift of resolved warped deformed conifold constructed by using modified 'OKS-BH' background [1] given in the context of Type IIB string theory by N D3-branes placed at the tip of the conifold, N f D7-branes wrapped around a four-cycle in the conifold and M D5-branes wrapping an S 2 inside the conifold in the limits discussed in N . The thermodynamical stability of the M-theory uplift was demonstrated in [2] by showing positivity of specific heat in both limits. Also, it was shown in [2] that the black M 3-branes' near-horizon geometry near the θ 1,2 = 0, branches, preserved 1 8 supersymmetry. By using the KSS prescription [11], we had calculated in [2] the diffusion coefficient to be 1 T in both type IIB and type IIA backgrounds, and the η/s turned out to be 1 4π in the type IIB, Type IIA as well as both limits (and hence in particular also for g s < ∼ 1 as part of the MQGP limit) of M-theory background.
In this work, we have first elaborated upon the geometry of black M 3-branes of [2]. By evaluating the flux(/charge) of(/corresponding to) G 4 by integrating over all (non)compact four cycles, the black M 3-branes, asymptotically, were shown to be black M 5-branes wrapping a two-cycle homologously given by (large)integer sum of two-spheres in AdS 5 × M 6 . As shown in [3], the supersymmetry breaking measured by violation of the ISD condition of the flux G 3 , is proportional to the square of the resolution parameter which in turn (turning off a bare resolution parameter, or assuming it be extremely small) goes like O gsM 2 N r 2 h ; in either limits of [1] or [2], as in [2], we hence disregard the same. By comparing the non-Kähler resolved warped deformed conifold (NKRWDC) metric with the one of [38][39][40] and hence evaluating the five SU (3) torsion(τ ) classes W 1,2,3,4,5 , we show that in [1]'s limit and the MQGP limit of [2], for extremely large radial coordinates, the NKRWDC is (i) is Kähler as τ ∈ W 5 , and (ii) is asymptotically a Calabi-Yau as W 1,2,3,4,5 = 0 (as expected). Further, to permit use of SYZ symmetry, in addition to the large base of the T 3 used for triple T dualities in [2], one requires this three-cycle to be special Lagrangian. We explicitly prove that the local three-torus T 3 of [2] satisfies the constraints satisfied by the maximal T 2 -invariant special Lagrangian submanifold of a deformed conifold of [4], in both limits of [2]. Then, exploiting the aforementioned asymptotic AdS 5 × M 6 background and based on the prescription of [5] , we have evaluated different transport coefficients by calculat-ing fluctuations of metric as well as gauge field corresponding to D7-brane living on the world volume of non-compact four cycle in the (warped) deformed conifold and R-symmetry group present in the M-theory. However, to do the same, we need to extract out the five dimensional AdS metric by integrating out all r-independent angular directions. Going ahead, to calculate the two-point correlator/spectral functions, we evaluate the EOMs for U (1) N f -gauge field, U (1) R gauge field as well as vector and tensor modes and then evaluate the solutions by double perturbative ansatze up to O(w 3 , q 2 3 ). The electrical conductivity, diffusion coefficient and charge susceptibility obtained due to U (1) N f gauge field satisfy Einstein's relation, which is a reasonable check of our results. Similarly, we show that one can calculate shear viscosity from vector and tensor modes by using Kubo's formula such that η/s turns out 2 to be 1 4π , which is expected for any theory obeying gauge-gravity correspondence. It should be noted that our results are also valid for g s < ∼ 1 as part of the MQGP limit.
In contrast to earlier works using AdS/CFT to study transport coefficients, our results are significant in the sense that the same have been obtained in the context of M-theory background [2] that are valid even for finite coupling constant g s , which in fact could be more appealing to studying aspects of 'strongly coupled Quark Gluon plasma plasma' and might bring one closer to the results obtained using experimental data at RHIC.
For the future, it would be interesting to extend these calculations to compute "second speed of sound" by working out two-point correlator functions of scalar modes of stress energy tensor. One can also calculate thermal conductivity corresponding to R-charge correlators and check if the ratio of thermal conductivity and viscosity satisfies Wiedemann-Franz law. Also, one should obtain the holographic spectral function by using non-abelian SU (N f ) gauge field background and produce the expected continuos meson spectra as a function of non-zero chemical potential due to presence of black hole (BH) in the background of [2].