1 Introduction

New Insights into Properties of Large- Holographic Thermal QCD at Finite Gauge Coupling at (the Non-Conformal/Next-to) Leading Order in Karunava Sil111email: krusldph@iitr.ac.in and Aalok Misra222e-mail: aalokfph@iitr.ac.in

Department of Physics, Indian Institute of Technology, Roorkee - 247 667, Uttarakhand, India


It is believed that large- thermal QCD laboratories like strongly coupled QGP (sQGP) require not only a large t’Hooft coupling but also a finite gauge coupling [1]. Unlike almost all top-down holographic models in the literature, holographic large- thermal QCD models based on this assumption, therefore necessarily require addressing this limit from M theory. This was initiated in [3] which presented a local M-theory uplift of [2]’s string theoretic dual of large- thermal QCD-like theories at finite gauge/string coupling ( as part of the ‘MQGP’ limit of [3]). Understanding and classifying the properties of systems like sQGP from a top-down holographic model assuming a finite gauge coupling, has been entirely missing in the literature. In this paper we largely address the following two non-trivial issues pertaining to the same. First, up to LO in (the number of -branes), by calculating the temperature dependence of the thermal (and electrical) conductivity and the consequent deviation from the Wiedemann-Franz law, upon comparison with [4], we show that remarkably, the results qualitatively mimic a 1+1-dimensional Luttinger liquid with impurities. Second, by looking at respectively the scalar, vector and tensor modes of metric perturbations and using [5]’s prescription of constructing appropriate gauge-invariant perturbations, we obtain the non-conformal corrections to the conformal results (but at finite ) respectively for the speed of sound, the shear mode diffusion constant and the shear viscosity (and ). The new insight gained is that it turns out these corrections show a partial universality in the sense that at NLO in the same are given by the product of and , being the number of flavor -branes and the number of fractional branes = the number of colors = 3 in the IR after the end of a Seiberg duality cascade. On the Math side, using the results of [6], at LO in we finish our argument of [7] and show that for a predominantly resolved (resolution deformation - this paper) or deformed (deformation resolution - [7]), resolved warped deformed conifold, the local of [3] in the MQGP limit, is the -invariant special Lagrangian three-cycle of [6] justifying the construction in [3] of the delocalized Strominger-Yau-Zaslow type IIA mirror of the type IIB background of [2].

1 Introduction

The AdS/CFT correspondence or in general the gauge/gravity duality has proved to be a very useful tool in understanding the properties of super Yang-Mills theory at large t’Hooft coupling. According to the correspondence, physics of SYM theory in the large limit can be obtained from type superstring theory on geometry, where is the five dimensional anti-de Sitter space and is the five sphere. The SYM theory is a conformal field theory which means its gauge coupling does not run with the energy scale. On the other hand QCD is non-conformal. QCD with gauge group, where is the number of quark colors, is an asymptotically free theory so that the gauge coupling is scale dependent and vanishes logarithmically with large characteristic momentum or with short distance. So to deal with QCD-like theories using Gauge/Gravity duality we need to generalize the AdS/CFT correspondence and incorporate a running coupling in the theory. Building up on the Klebanov-Witten [8], Klebanov-Nekrasov [9] and Klebanov-Tseytlin [10] models, a logarithmic RG flow just like QCD was obtained in the non-conformal Klebanov-Strassler model [11] by considering fractional branes along with branes in a conifold geometry wherein the IR geometry was modified resulting in a deformed conifold.

So far we have not talked about the temperature at all. In fact the AdS/CFT correspondence mentioned above is valid at zero temperature. At finite temperature the situation is different on the gravity side of the correspondence. On the other hand the field theory in question i.e. thermal QCD, is an IR-confining theory at and becomes non-confining at , where is the only scale that we have here. It possesses a phase transition from confining phase to a non-confining phase at , where . At sufficiently high temperature i.e. at , the interaction strength and hence the theory is weakly coupled. However the thermal gauge theory we want to understand is not in the weak coupling regime. In particular, to explore the physics of QCD at , we have to take a look at the strongly coupled regime where . So we cannot apply perturbative methods any more. In lattice gauge theory using numerical simulations the equilibrium properties of the strongly coupled hot QCD can be explored. But interesting non-equilibrium properties such as hydrodynamic behavior or the real time dynamics cannot be seen from the equilibrium correlation functions. So the lack of non-perturbative methods to study hot QCD forces us to look for either a different theory/model or a different limit of a known theory/model.

At finite temperature the equilibrium or non-equilibrium properties of the Euclidean theory are studied requiring time to have periodicity . Thus, at non-zero temperature, the Euclidean space-time looks like a cylinder with the topology . The AdS/CFT correspondence tells us that at the SYM theory defined on is dual to string theory on AdS space with as the boundary of the same. So at zero temperature we can think of the field theory as living on the boundary of AdS space. However, the prime interest is to investigate the finite temperature aspects of the dual field theory from the physics of supergravity. Hence at finite temperature, the space-time of the gravitational description somehow has to be changed such that one gets a geometry of the boundary which is equivalent to and not . In other words one needs to find some bulk geometry which has a boundary with the topology . One possible answer is the AdS-BH space-time with the following metric sometimes called black-brane metric given as: with Minkowskian signature. Here is the radial coordinate and , dependent on the horizon radius , is a ‘black-hole function’. By construction, the time coordinate is defined to be periodic with period which is inverse of temperature and is related to the horizon radius .

Now, let us go back to the Klebanov-Strasslar model where the temperature is turned on in the field theory side effected by introducing a black hole in the dual geometry. Interestingly the KS background with a black hole has the geometry equivalent to the AdS-BH spacetime in the large limit. Moreover the embedding of -branes in KS model via the holomorphic Ouyang embedding [12] and finally the M-theory uplift of the whole set up keeps the background geometry as required provided we consider some limiting values of the parameters in the theory. The details about this, based on [2], [12], will be reviewed in Section 2.

This paper, apart from providing important evidence validating construction in [3] of delocalized Strominger-Yau-Zaslow (SYZ) mirror of [2]’s type IIB holographic dual of large thermal QCD, we believe, fills in a pair of important gaps in the literature pertaining to a top-down holographic study of large- thermal QCD.

  • First, all such large- holographic models cater to the large t’Hooft-coupling limit while keeping the gauge coupling vanishingly small. However, in systems such as sQGP, it is believed that not only should the t’Hooft coupling be large, but even the gauge/string coupling should also be finite [1]. A finite gauge coupling would imply a finite string coupling which necessitates addressing the limit from an M-theory point of view. Also, for a realistic thermal QCD computation, the number of colors should be set to three. This can be realized in the IR after the end of a Seiberg duality cascade and in the MQGP limit of (2.2). This study was initiated in [3, 7] wherein a large- limit, referred to as the ‘MQGP limit’ (2.2), was defined in which the gauge coupling was kept to be slightly less than unity and hence finite. By studying some transport coefficients in this paper, we obtain even at the leading order in , a remarkable result that holographic large- thermal QCD at finite gauge coupling for (Ouyang embedding parameter) mimics qualitatively Luttinger liquid with impurities close to ‘-doping’; for one is able to reproduce the expected linear large- variation of DC electrical conductivity characteristic of most strongly coupled gauge theories with five-dimensional gravity duals with a black hole [13].

  • Second, in the context of top-down holographic models of large- thermal QCD at finite gauge coupling, there are no previous results that we are aware of pertaining to evaluation of the non-conformal corrections to hydrodynamical quantities such as the shear viscosity (as well as the shear-viscosity-entropy-density ratio ), shear mode diffusion constant and the speed of sound . These non-conformal corrections at finite gauge coupling, determined for the first time in this paper in the given context, are particularly relevant in the IR and in fact also encode the scale-dependence of aforementioned physical quantities, and hence are extremely important to be determined for making direct contact with sQGP. The main non-trivial insight gained via such computations is the realization that at NLO in there is a partial universality in these corrections determined by and apart from .

The following is a section-wise description of the sets of issues addressed and the new insights obtained in this paper.

  • Sec. 3 - Identification of sLag in a (predominantly) resolved conifold up to LO in : Up to leading order in and in the UV-IR interpolating region/UV, using the results of [6], we show that the local of [3] is a -invariant special Lagrangian three-cycle in a resolved conifold. This, together with the results of [7], shows that for a (predominantly resolved or deformed) resolved warped deformed conifold, the local of [3] in the MQGP limit of [3], is the -invariant special Lagrangian three-cycle of [6], justifying the construction in [3] of the delocalized SYZ type IIA mirror of the type IIB background of [2]. This was a crucial step missing in [3, 7] in construction of the delocalized SYZ mirror of the top-down type IIB holographic dual of large- thermal QCD of [2], at finite gauge coupling.

  • Transport Coefficients up to (N)LO in : We study some transport coefficients of large- thermal QCD leading to evaluation of various transport coefficients up to (next-to) leading order in . This boils down to evaluating various retarded Green’s functions, but computed from the gravity dual as prescribed in [14]. In order to study the transport phenomenon from the gravity picture, we need to consider a perturbation of the given modified OKS-BH metric - the type IIB string dual of large- thermal QCD as given in [2]. In response to this perturbation the BH will emit gravitational waves with a long period of damping oscillation. The modes associated with this kind of gravitational radiation are called quasinormal modes. Quasinormal modes are the solutions to the linearized EOMs that one gets by considering fluctuations of gravitational background satisfying specific boundary conditions both at the black hole horizon and at the boundary. At the horizon, the quasinormal modes satisfy a pure incoming-wave boundary condition and at the spatial infinity the perturbative field or some gauge invariant combinations of the fields vanishes, that means it follows the Dirichlet boundary condition at infinity. It was shown in [15] that the quasinormal frequency associated with the quasinormal modes defined above in an asymptotically AdS spacetime exactly matches with the pole of the two point correlation function involving operators in the field theory dual to different metric perturbations. Hence evaluating the quasinormal frequency as a function of the special momentum , gives the thermodynamic and hydrodynamic behavior of the plasma.

    Analogous to [11], the non-conformality in [2] is introduced via number of fractional -branes, the latter appearing explicitly in and after construction of a delocalized SYZ type IIA mirror (resulting in mixing of with the metric components after taking a triple T-dual of [2]) as well as its local M-theory uplift, also in the metric. In the context of a (local) M-theory uplift of a top-down holographic thermal QCD dual such as that of [2] at finite gauge coupling, to the best of our knowledge, we estimate for the first time, the non-conformal corrections appearing at the NLO in to the speed of sound , shear mode diffusion constant , the shear viscosity and the shear viscosity - entropy density ratio . The main new insight gained by this set of results is that the non-conformal corrections in all the aforementioned quantities are found to display a partial universality in the sense that at the NLO in the same are always determined by , being the number of flavor -branes. Thus, we see that the same are determined by the product of the very small - part of the MQGP limit (2.2) - and the finite (also part of (2.2)). Of course, the leading order conformal contributions though at vanishing string coupling and large t’Hooft coupling were (in)directly known in the literature. It is interesting to see the conformal limit of our results at finite obtained by turning off of - which encodes the non-conformal contributions - reduce to the known conformal results for vanishing .

    • Sec. 4 - (Thermal and electrical) Conductivity, Wiedemann-Franz law and Luttinger liquid at LO in : As a thermal gradient corresponding to a gauge field fluctuation also turns on vector modes of metric fluctuations, we consider turning on simultaneously gauge and vector modes of metric fluctuations, and evaluate the thermal () and electrical () conductivities, and the Wiedemann-Franz law (). The new insight gained is that for (Ouyang embedding parameter), the temperature dependence of and the consequent deviation from the Wiedemann-Franz law, all point to the remarkable similarity with Luttinger liquid with impurities at ‘-doping’; for one is able to reproduce the expected linear large- variation of DC electrical conductivity for most strongly coupled gauge theories with five-dimensional gravity duals with a black hole [13].

    • Sec. 5 - Speed of sound: For the metric fluctuations in the sound channel the corresponding quasinormal frequency is given by with defined as the speed of sound and as the damping constant of the sound mode. Again for the sound channel the pole of the correlations of longitudinal momentum density gives the same dispersion relation in the conformal limit. From the knowledge of quasinormal modes associated with the scalar modes of metric perturbations, we have computed the next-to-leading order correction to the speed of sound () at finite gauge coupling (part of the MQGP limit). Up to LO in , we calculate using four routes:
      (i) (subsection 5.1.1) the poles appearing in the common denominator of the solutions to the individual scalar modes of metric perturbations (the pure gauge solutions and the incoming-wave solutions),
      (ii) (subsection 5.1.2) the poles appearing in the coefficient of the asymptotic value of the time-time component of the scalar metric perturbation in the on-shell surface action,
      (iii) (subsection 5.2.1) the dispersion relation obtained via a Dirichlet boundary condition imposed on an appropriate gauge-invariant combination of perturbations - using the prescription of [5] - at the asymptotic boundary,
      and
      (iv) (subsection 5.2.2) the pole structure of the retarded Green’s function calculated from the on-shell surface action written out in terms of the same single gauge invariant function.

      The third approach (of solving a single second-order differential equation for a single gauge-invariant perturbation using the prescription of [5]) is then extended to include the non-conformal corrections to the metric and obtain for the first time in the context of a top-down large- holographic thermal QCD at finite gauge coupling uplifted to M theory, an estimate of the non-conformal corrections to up to NLO in .

      Sec. 6 - Shear mode diffusion constant: The quasinormal frequency for the vector modes of black brane metric fluctuation reads , where is the shear mode diffusion constant . This dispersion relation also follows from the pole structure of the correlations of transverse momentum density. From the knowledge of quasinormal modes associated with the vector modes of metric perturbations obtained by imposing Dirichlet boundary condition at the asymptotic boundary, on an appropriate gauge-invariant perturbation constructed using the prescription of [5], we have computed for the first time in the context of the same top-down large- holographic thermal QCD at finite gauge coupling uplifted to M theory, the non-conformal corrections to the shear mode diffusion constant up to NLO in .

    • Sec. 7 - Shear viscosity(-to-entropy density ratio): We have also evaluated for the first time in the context of the aforementioned M-theory uplift corresponding to finite , the non-conformal temperature-dependent correction at the NLO in , to the shear viscosity and shear viscosity - entropy density ratio from the two point energy-momentum tensor correlation function corresponding to the tensor mode of metric perturbation.

    The results for the NLO (in ) corrections are particularly important as they suggest a scale dependance to the above mentioned quantities and hence leads to a non-conformal nature of the field theory in the IR. We have commented on this issue even in section 8.

The paper is organized as follows. First, we briefly review the supergravity dual background of large strongly coupled QCD like theories. The whole discussion, for the sake of clear understanding of the reader is presented stepwise through first three subsections in section 2. In 2.1, the type supergravity background of [2] dual to large- thermal gauge theory which is UV complete and closely resembles thermal QCD, is briefly reviewed. In 2.2, the ’MQGP Limit’ of [3] and its motivation, in particular to address the properties of strongly coupled QGP medium, is briefly reviewed. In 2.3, using the ’MQGP Limit’ we review briefly the delocalized SYZ type mirror via three T dualities along a -invariant special lagrangian fibered over a large base in a predominantly warped resolved conifold - this serves as a precursor to the material of Sec. 3. In the same sub-section, we discuss it’s local uplift to M-theory, where in the large limit the spacetime is given by . In 2.4, following [14] we review the recipe to calculate two-point correlation function with Minkowskian signature. In 2.5, following [5] the gauge invariant variables for vector, scalar and tensor modes of background metric perturbations are discussed - this will be useful to obtain the results of (sub-)sections 5.3, 6 and 7. In section 3, we show that in the MQGP limit of [3], the local of [3] is the -invariant special Lagrangian three-cycle of a resolved conifold as given in [6]. This together with the result reviewd in 2.3, shows that in the MQGP limit, the local of [3] is the -invariant sLag of [6] for both, a predominantly resolved (resolution deformation) or predominantly deformed (deformation resolution), resolved warped deformed conifold. This is important for SYZ mirror construction to work. In section 4, we compute the temperature dependance of thermal (electrical) conductivity via Kubo formula at finite temperature and finite baryon density up to LO in . The same and deviations from the Wiedemann-Franz formula, upon comparison with [4], mimic remarkably a Luttinger liquid with impurities. In section 5, through four subsections we present the calculation of speed of sound both at leading order (and NLO) in in the ’MQGP Limit’ in four different ways. We then show that the leading order result as obtained from the quasinormal modes of scalar metric perturbation is consistent with that obtained from the two point correlation function. In section 6, we evaluate the NLO correction to the shear mode diffusion constant again from the quasinormal modes of the vector metric perturbations. Section 7, is devoted to the NLO correction to the shear viscosity and shear viscosity - entropy density ratio . Section 8 has a summary of the main results of the paper. The technical details of sections 3 - 7 are relegated to eight appendices.

2 The Background

In this section, via five sub-sections we will:

  • provide a short review of the type IIB background of [2] which is supposed to provide a UV complete holographic dual of large- thermal QCD, as well as their precursors in subsection 2.1,

  • discuss the ’MQGP’ limit of [3] and the motivation for considering the same in subsection 2.2,

  • briefly review issues pertaining to construction of delocalized S(trominger) Y(au) Z(aslow) mirror and approximate supersymmetry in subsection 2.3,

  • review the recipe of [14] to evaluate Minkowskian-signature space correlators in subsection 2.4,

  • briefly discuss the vector, tensor and scalar modes of metric perturbations and construction of gauge-invariant variables in subsection 2.5

2.1 Type IIB Dual of Large- Thermal QCD

In this subsection, we will discuss a UV complete holographic dual of large- thermal QCD as given in Dasgupta-Mia et al [2]. As partly mentioned in Sec. 1, this was inspired by the zero-temperature Klebanov-Witten model [8], the non-conformal Klebanov-Tseytlin model [10], its IR completion as given in the Klebanov-Strassler model [11] and Ouyang’s inclusion [12] of flavor in the same 333See [19] for earlier attempts at studying back-reacted geometry at zero temperature; we thank L. Zayas for bringing [19, 18] to our attention., as well as the non-zero temperature/non-extremal version of [16] (the solution however was not regular as the non-extremality/black hole function and the ten-dimensional warp factor vanished simultaneously at the horizon radius), [17] (valid only at large temperatures) of the Klebanov-Tseytlin model and [18] (addressing the IR), in the absence of flavors.
(a) Brane construction

In order to include fundamental quarks at non-zero temperature in the context of type IIB string theory, to the best of our knowledge, the following model proposed in [2] is the closest to a UV complete holographic dual of large- thermal QCD. The KS model (after a duality cascade) and QCD have similar IR behavior: gauge group and IR confinement. However, they differ drastically in the UV as the former yields a logarithmically divergent gauge coupling (in the UV) - Landau pole. This necessitates modification of the UV sector of the KS model apart from inclusion of non-extremality factors. With this in mind and building up on all of the above, the type IIB holographic dual of [2] was constructed. The setup of [2] is summarized below.

  • From a gauge-theory perspective, the authors of [2] considered black -branes placed at the tip of six-dimensional conifold, -branes wrapping the vanishing two-cycle and -branes distributed along the resolved two-cycle and placed at the outer boundary of the IR-UV interpolating region/inner boundary of the UV region.

  • More specifically, the are distributed around the antipodal point relative to the location of branes on the blown-up . If the separation is given by , then this provides the boundary common to the outer UV-IR interpolating region and the inner UV region. The region is the UV. In other words, the radial space, in [2] is divided into the IR, the IR-UV interpolating region and the UV. To summarize the above:

    • (), (): the IR/IR-UV interpolating regions with : deep IR where the gauge theory confines

    • : the UV region.

  • -branes, via Ouyang embedding, are holomorphically embedded in the UV (asymptotically ), the IR-UV interpolating region and dipping into the (confining) IR (up to a certain minimum value of corresponding to the lightest quark) and -branes present in the UV and the UV-IR interpolating (not the confining IR). This is to ensure turning off of three-form fluxes, constancy of the axion-dilaton modulus and hence conformality and absence of Landau poles in the UV.

  • The resultant ten-dimensional geometry hence involves a resolved warped deformed conifold. Back-reactions are included, e.g., in the ten-dimensional warp factor. Of course, the gravity dual, as in the Klebanov-Strassler construct, at the end of the Seiberg-duality cascade will have no -branes and the -branes are smeared/dissolved over the blown-up and thus replaced by fluxes in the IR.

The delocalized S(trominger) Y(au) Z(aslow) type IIA mirror of the aforementioned type IIB background of [2] and its M-theory uplift had been obtained in [3, 7], and newer aspects of the same will be looked into in this paper.

(b) Seiberg duality cascade, IR confining gauge theory at finite temperature and

  1. IR Confinement after Seiberg Duality Cascade: Footnote numbered 3 shows that one effectively adds on to the number of -branes in the UV and hence, one has color gauge group (implying an asymptotic ) and flavor gauge group, in the UV: . It is expected that there will be a partial Higgsing of to at [20]. The two gauge couplings, and flow logarithmically and oppositely in the IR:

    (1)

    Had it not been for , in the UV, one could have set constant (implying conformality) which is the reason for inclusion of -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 flavor -branes which needs to be annulled via -branes. The gauge coupling flows towards strong coupling and the gauge coupling flows towards weak coupling. Upon application of Seiberg duality, in the IR; assuming after repeated Seiberg dualities or duality cascade, decreases to 0 and there is a finite , one will be left with gauge theory with flavors that confines in the IR - the finite temperature version of the same is what was looked at by [2].

  2. Obtaining , and Color-Flavor Enhancement of Length Scale in the IR: So, in the IR, at the end of the duality cascade, what gets identified with the number of colors is , which in the ‘MQGP limit’ to be discussed below, can be tuned to equal 3. One can identify with , where and (the being dual to , wherein is an asymmetry factor defined in [2], and ) where [21]. The effective number of -branes varies between in the UV and 0 in the deep IR, and the effective number of -branes varies between 0 in the UV and in the deep IR (i.e., at the end of the duality cacade in the IR). Hence, the number of colors varies between in the deep IR and a large value [even in the MQGP limit of (2.2) (for a large value of )] in the UV. Hence, at very low energies, the number of colors can be approximated by , which in the MQGP limit is taken to be finite and can hence be taken to be equal to three. However, in this discussion, the low energy or the IR is relative to the string scale. But these energies which are much less than the string scale, can still be much larger than . Therefore, for all practical purposes, as regard the energy scales relevant to QCD, the number of colors can be tuned to three.

    In the IR in the MQGP limit, with the inclusion of terms higher order in in the RR and NS-NS three-form fluxes and the NLO terms in the angular part of the metric, there occurs an IR color-flavor enhancement of the length scale as compared to a Planckian length scale in KS for , thereby showing that quantum corrections will be suppressed. Using [2]:

    (2)

    it was argued in [22] that the length scale of the OKS-BH metric in the IR will be given by:

    (3)

    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 and upon inclusion of of terms in and in (2), in the MQGP limit involving , implying that the stringy corrections are suppressed and one can trust supergravity calculations. As a reminder one will generate higher powers of and in the double summation in in (2), e.g., from the terms higher order in in the RR and NS-NS three-form fluxes that become relevant for the aforementioned values of .

  3. Further, the global flavor group in the UV-IR interpolating and UV regions, due to presence of and -branes, is , which is broken in the IR to as the IR has only -branes.

Hence, the following features of the type IIB model of [2] make it an ideal holographic dual of thermal QCD:

  • the theory having quarks transforming in the fundamental representation, is UV conformal and IR confining with the required chiral symmetry breaking in the IR and restoration at high temperatures

  • the theory is UV complete with the gauge coupling remaining finite in the UV (absence of Landau poles)

  • the theory is not just defined for large temperatures but for low and high temperatures

  • (as will become evident in Sec. 3) with the inclusion of a finite baryon chemical potential, the theory provides a lattice-compatible QCD confinement-deconfinement temperature for the right number of light quark flavors and masses, and is also thermodynamically stable; given the IR proximity of the value of the lattice-compatible , after the end of the Seiberg duality cascade, the number of quark flavors approximately equals which in the ‘MQGP’ limit of (2.2) can be tuned to equal 3

  • in the MQGP limit (2.2) which requires considering a finite gauge coupling and hence string coupling, the theory was shown in [3] to be holographically renormalizable from an M-theory perspective with the M-theory uplift also being thermodynamically stable.

(d) Supergravity solution on resolved warped deformed conifold

The working metric is given by :

(4)

’s are black hole functions in modified OKS(Ouyang-Klebanov-Strassler)-BH (Black Hole) background and are assumed to be: where is the horizon, and the () dependence come from the corrections. The ’s are expected to receive corrections of [20]. We assume the same to also be true of the ‘black hole functions’ . The compact five dimensional metric in (4), is given as:

(5)

for , i.e. in the UV/IR-UV interpolating region. The ’s appearing in internal metric as well as are not constant and up to linear order depend on are given as below:

(6)

One sees from (2.1) and (2.1) that one has a non-extremal resolved warped deformed conifold involving an -blowup (as ), an -blowup (as ) and squashing of an (as is not strictly unity). The horizon (being at a finite ) is warped squashed . In the deep IR, in principle one ends up with a warped squashed being the deformation parameter. Assuming and given that [20], in the IR and in the MQGP limit, ; we have a confining gauge theory in the IR.

The warp factor that includes the back-reaction, in the IR is given as:

(7)

where, in principle, are not necessarily the same as ; we however will assume that up to , they are. Proper UV behavior requires [20]:

(8)

In the IR, up to and setting , the three-forms are as given in [2]:

The asymmetry factors in (2.1) are given by: . As in the UV, , we will assume the same three-form fluxes for .

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 [22], wherein in particular, assuming an F-theory uplift involving, locally, an elliptically fibered , it was shown that UV conformality and the Ouyang embedding are mutually consistent.

2.2 The ‘MQGP Limit’

In [3], we had considered the following two limits:

(10)

(the limit in the first line though not its realization in the second line, considered in [2]);

(11)

Let us enumerate the motivation for considering the MQGP limit which was discussed in detail in [22]. There are principally two.

  1. Unlike the AdS/CFT limit wherein such that is large, for strongly coupled thermal systems like sQGP, what is relevant is and . From the discussion in the previous paragraphs specially the one in point (c) of sub-section 2.1, one sees that in the IR after the Seiberg duality cascade, effectively which in the MQGP limit of (2.2) can be tuned to 3. Further, in the same limit, the string coupling . The finiteness of the string coupling necessitates addressing the same from an M theory perspective. This is the reason for coining the name: ‘MQGP limit’. In fact this is the reason why one is required to first construct a type IIA mirror, which was done in [3] a la delocalized Strominger-Yau-Zaslow mirror symmetry, and then take its M-theory uplift.

  2. From the perspective of calculational simplification in supergravity, the following are examples of the same and constitute therefore the second set of reasons for looking at the MQGP limit of (2.2):

    • In the UV-IR interpolating region and the UV,

    • Asymmetry Factors (in three-form fluxes) in the UV-IR interpolating region and the UV.

    • Simplification of ten-dimensional warp factor and non-extremality function in MQGP limit

With denoting the boundary common to the UV-IR interpolating region and the UV region, for is required to ensure conformality in the UV. Near the -branch, assuming: as and as and for all components except ; in the MQGP limit and near -branch, So, the UV nature too is captured near -branch in the MQGP limit. This mimics addition of -branes in [2] to ensure cancellation of .

2.3 Approximate Supersymmetry, Construction of the Delocalized SYZ IIA Mirror and Its M-Theory Uplift in the MQGP Limit

A central issue to [3, 7] has been implementation of delocalized mirror symmetry via the Strominger Yau Zaslow prescription according to which the mirror of a Calabi-Yau can be constructed via three T dualities along a special Lagrangian fibered over a large base in the Calabi-Yau. This sub-section is a quick review of precisely this.

To implement the quantum mirror symmetry a la S(trominger)Y(au)Z(aslow) [23], one needs a special Lagrangian (sLag) fibered over a large base (to nullify contributions from open-string disc instantons with boundaries as non-contractible one-cycles in the sLag). Defining delocalized T-duality coordinates, valued in [3]:

(12)

using the results of [6] it was shown in [7] that the following conditions are satisfied:

(13)

for the -invariant sLag of [6] for a deformed conifold. It will be shown in the Section 3 that (2.3) is also satisfied for the -invariant sLag of [6] for a resolved conifold, implying thus: . Hence, if the resolved warped deformed conifold is predominantly either resolved or deformed, the local of (12) is the required sLag to effect SYZ mirror construction.

Interestingly, in the ‘delocalized limit’ [24] , under the coordinate transformation:

(14)

and , the term becomes , , i.e., one introduces an isometry along in addition to the isometries along . This clearly is not valid globally - the deformed conifold does not possess a third global isometry.

To enable use of SYZ-mirror duality via three T dualities, one also needs to ensure a large base (implying large complex structures of the aforementioned two two-tori) of the fibration. This is effected via [25]:

(15)

for appropriately chosen large values of . The three-form fluxes remain invariant. The fact that one can choose such large values of , was justified in [3]. The guiding principle is that one requires the metric obtained after SYZ-mirror transformation applied to the non-Kähler resolved warped deformed conifold is like a non-Kähler warped resolved conifold at least locally. Then needs to vanish [3]. This is shown to be true anywhere in the UV in Appendix C.

The mirror type IIA metric after performing three T-dualities, first along , then along and finally along , utilizing the results of [24] was worked out in [3]. We can get a one-form type IIA potential from the triple T-dual (along ) of the type IIB in [3] and using which the following metric was obtained in [3] ():

(16)

Further, in the UV: