${\mathscr {M}}$cTEQ (${\mathscr {M}}$ ${\bf c}$hiral perturbation theory-compatible deconfinement ${\bf T}$emperature and ${\bf E}$ntanglement Entropy up to terms ${\bf Q}$uartic in curvature) and FM (${\bf F}$lavor ${\bf M}$emory)

A holographic computation of $T_c$ at ${\it intermediate\ coupling}$ from M-theory dual of thermal QCD-like theories, has been missing in the literature. Filling this gap, we demonstrate a novel UV-IR mixing, (conjecture and provide evidence for) a non-renormalization beyond 1 loop of ${\bf M}-{\bf c}$hiral perturbation theory arXiv:2011.04660[hep-th]-compatible deconfinement ${\bf T}$emperature, and show equivalence with an ${\bf E}$ntanglement (as well as Wald) entropy arXiv:0709.2140[hep-th] computation, up to terms ${\bf Q}$uartic in curvature. We demonstrate a ${\bf F}$lavor-${\bf M}$emory (FM) effect in the M-theory uplifts of the gravity duals, wherein the no-braner M-theory uplift retains the"memory"of the flavor D7-branes of the parent type IIB dual in the sense that a specific combination of the aforementioned quartic corrections to the metric components precisely along the compact part of the non-compact four-cycle"wrapped"by the flavor D7-branes, is what determines, e.g., the Einstein-Hilbert action at O$(R^4)$. The same linear combination of O$(R^4)$ metric corrections, upon matching the phenomenological value of the coupling constant of one of the SU(3) NLO ChPT Lagrangian, is required to have a definite sign. Interestingly, in the decompactification limit of the spatial circle, we ${\it derive}$ this, and obtain the values of the relevant O$(R^4)$ metric corrections. Further, equivalence with Wald entropy for the black hole at ${\cal O}(R^4)$ imposes a linear constraint on the same linear combination of metric corrections. Remarkably, when evaluating $T_c$ from an entanglement entropy computation in the thermal gravity dual, due to a delicate cancelation between the ${\cal O}(R^4)$ corrections from a subset of the abovementioned metric components, one sees that there are no corrections to $T_c$ at quartic order supporting the conjecture referred to above.


Contents
2 Review of UV Complete Top-Down Type IIB String Dual of [6], Type IIA Mirror of [6] and its M -Theory Uplift at Intermediate 't Hooft Coupling

Introduction
AdS/CFT correspondence [7], in its original form, was a duality between strongly coupled N = 4 SU (N ) supersymmetric Yang-Mills theory and weakly coupled type IIB string theory on AdS 5 × S 5 . All the theories are not conformal therefore it is better to call this duality as gauge/gravity duality to include non-conformal theories. Strong coupling dynamics of non-abelian gauge theories at finite temperature have been studied via gauge/gravity duality. In later years, this duality has been generalised to many branches of Physics (e.g. Particle Physics, Condensed Matter Physics, Cosmology etc.). We are interested in thermal QCD, which is a non-conformal theory because its gauge coupling runs with energy. There are various proposals to study QCD using gauge/gravity duality (e.g. AdS/QCD) but almost all involve a conformal AdS background. There are two approaches to construct holographic duals of thermal QCD-like theories -the bottom-up and the top-down approach. In this paper we will work with the latter.
Gauge/gravity duality also allows us to compute the corrections to the infinite-'t Hooftcoupling limit as done in [8], but again working with an AdS background. These authors included higher-derivative terms on gravity side using terms quartic in the Weyl tensor. On the gauge theory side one has N = 4 SU (N ) supersymmetric Yang-Mills plasma at intermediate 't Hooft coupling. They explained the transport peak in the small frequency region of the stress-energy tensor specatral function at zero spatial momentum. which is a generic feature of perturbative plasma.
QCD at strong coupling has been studied by various bottom-up models constructed from gauge/gravity duality(from both, bottom-up and top-down approaches) and at weak coupling from perturbation theory. A popular top-down holographic type IIA dual, though catering only to the IR, is the Sakai-Sugimoto model [9]. The only UV-complete (type IIB) top-down holographic dual of QCD-like theories at strong coupling that we are aware of is [6], and its M -theory uplift [3]. Authors in [10] have studied QCD thermodynamic functions at intermediate coupling based on hard-thermal-loop perturbation theory (HTLpt). One can also study intermediate coupling regime of QCD from gauge/gravity duality. In this direction, two of the authors (VY and AM) worked out O(l 6 p ) corrections to the M -theory metric in [11]. Starting with M -theory dual of thermal QCD-like theories in the 'MQGP' limit as constructed in [3] and incorporating higher derivative terms in eleven dimensional supergravity action which are quartic in Riemann curvature tensor i.e. O(R 4 ), the corrections to the supergravity background were then worked out in [11] and in fact, successfully used in [1] in obtaining phenomenologically-compatible values of the NLO LECs in SU (3) χPT Lagrangian of [5].
Wald proposed a method to calculate black hole entropy in general theories of gravity [12]. He considered classical theory of gravity in n dimensions which arises from diffeomorphism invariant Lagrangian. In these theories of gravity Noether charge is an (n − 2)-form and black hole entropy will be given by 2π times integral over bifurcate killing horizon of the (n − 2)-form Noether charge. Therefore entropy is Noether Charge for stationary black holes with bifurcate killing horizons. For dynamical black holes a proposal to calculate entropy was given in [13].
Every physical phenomenon is characterised by their energy/distance scale. One goes from UV to IR via RG group flow. But, e.g., in noncommutative field theories and string theory, short distance physics becomes related to long distance Physics, which is known as UV/IR mixing. The consequence of UV/IR mixing [14] is, e.g., that UV divergences of real φ 4 theory defined on commutative space are transformed into infrared poles in the same theory defined on noncommutative space. There are other more examples explained nicely in [14].
Gravitational theories are nonlocal. Therefore UV/IR mixing could appear in such theories. In gauge/gravity duality radial coordinate in gravitational theory corresponds to energy scale in gauge theory side. In [15], authors have studied "I.R.-U.V." connection in the context of AdS/CFT correspondance in which they showed that infrared effects in bulk theory are transformed into ultravoilet effects in the boundary theory. In particular, infrared regulator in the bulk theory plays the role of ultraviolet regulator in N = 4 super Yang Mills theory. It is interesting that a certain form of UV-IR connection manifests itself in our work when matching at the deconfinement temperature, M -theory actions dual to the thermal and black-hole backgrounds at the UV-cut-off and obtain a relationship between the O(R 4 ) metric corrections in the IR In this paper we have calculated deconfinement temperature of QCD-like theory at intermediate coupling from gauge/gravity duality using a semiclassical computation as first discussed in [16]. In our case gravity dual is M -theory uplift of type IIB string dual [6] which includes O(R 4 ) corrections [11] to the MQGP background [3]. One can also discuss confinement deconfinement phase transition in large N c gauge theories from entanglement entropy point of view based on [2]. In this process one is required to calculate entanglement entropy between two regions by dividing one of spatial coordinates into segment of length l and its complement. There is a prescription to calculate entanglement entropy from AdS/CFT correspondence given by Ryu and Takayanagi in [17]. As discussed in [2], there are two surfaces -connected and disconnected. There is a critical value of l which is denoted by l crit . If one is below the critical value of l i.e. l < l crit then it is the connected surface that dominates the entanglement entropy and if one is above the critical value of l i.e. l > l crit then it is the disconnected surface that dominates entanglement entropy; l < l crit corresponds to confining phase of large N c gauge theories whereas l > l crit corresponds to deconfining phase of the same. So we can interpret this as confinement deconfinement phase transition in large N c gauge theories.
The rest of the paper is organized as follows. In Section 2, we review the basic setup and results of [3,6,11] inclusive of a summary of the results of [18,19] on the conjectured SL(2, Z) completion of the D = 11 supergravity action by studying four-graviton scattering in a Dinstanton background and a non-renormalization beyond one loop at O(R 4 ) in the zero-instanton sector. Section 3, divided into two sub-sections 3.1 and 3.2, have to do with the holographic renormalization of the on-shell action corresponding respectively to the gravity duals of the black-hole and thermal backgrounds. Section 4 is on the computation of the deconfinement temperature via a semiclassical computation, as well as on the comparison of the black hole entropy computed with the Wald entropy up to O(R 4 ). Section 5 is on the computation of the deconfinement temperature from entanglement entropy between an interval along one of the non-compact non-radial spatial coordinates and its complement, and comparison with the semiclassical computation of the same in Section 4. Section 6 is on the derivation of the sign and the individual values of the contributions that figure in a linear combination of the same arising from the O(R 4 ) corrections to the M -theory uplift [3] of [6]. Sections 4 and 6 also include a discussion on the "Flavor Memory" effect. Section 7 has a summary of the results obtained in the paper. There is a supplementary appendix on the EOMs and solutions to the O(R 4 ) metric corrections to the MQGP background of [3,4], as well as a discussion on taking the decompactification-limit of a periodic spatial direction in the thermal background M -theory uplift used in [1] to obtain the M -theory thermal background in this paper, and deriving the sign of a linear combination of constants of integration in the solutions to the EOMs of the O(R 4 )Mtheory metric corrections as required from matching of results of [1] with phenomenological values of the LECs at NLO in the SU (3) χPT Lagrangian of [5].
2 Review of UV Complete Top-Down Type IIB String Dual of [6], Type IIA Mirror of [6] and its M -Theory Uplift at Intermediate 't Hooft Coupling In this section we will start with a brief review type IIB string dual of large N QCD-like theory at finite temperature constructed by McGill group [6]. Then we will briefly discuss type IIA SYZ mirror of [6] and its M -theory uplift as constructed in [3]. We will also discuss the origin of O(R 4 ) terms in eleven dimensional supergravity action and non-renormalization of type IIB action up to O(R 4 ) beyond one loop in the zero-instanton sector. In the last part of this section we will explain that how the M -theory metric of [3,4] will be modified by incorporating higher derivative terms in eleven dimensional supergravity action.
To the best of our knowledge the only UV-complete type IIB string dual of large N thermal QCD-like theories from a top-down approach was given in [6] in the infinite-'t Hooft-coupling limit. For strongly coupled systems like QGP, the relavant coupling is finite/intermediate as explained in [20]. Therefore, it is required to construct string dual(s) of large N thermal QCDlike theories at finite/intermediate coupling. In this direction authors in [3,4] considered a particular limit which they called as 'MQGP limit' which uses finite/intermdiate coupling. One can also study intermediate coupling regime of gauge theory using gauge/gravity duality. To study thermal QCD at intermediate coupling higher derivative corrections on the gravity side, need to be incorporated - [8,11].
Brane Setup used in [6] • The authors considered N D3-branes placed at tip of the six-dimensional conifold, M D5branes wrapping the vanishing S 2 (θ 1 , φ 1 ), referred to as fractional D3-branes and M D5branes distributed along the resolved S 2 a (θ 2 , φ 2 ) placed at antipodal points relative to the M D5-branes. Let average seperation between D5/D5-branes will be represented by R D5/D5 . Then one can characterise the various regions using radial coordinate on gravity side.
• To introduce quarks in the fundamental representation of the flavor group, N f D7-branes were introduced via Ouyang embedding [21] in the resolved conifold geometry, "smeared"/delocalized along the angular directions θ 1,2 . The flavor D7-branes are embedded in the UV all the way into the IR up to a certain minimum separation from the color D3-branes determined by the modulus of the Ouyang embedding parameter |µ Ouyang |. To ensure UV conformality in the theory N f D7-branes were introduced in the UV and the UV-IR interpolaring region but not the IR (to effect chiral symmetry breaking). The embedding equation for the flavor D7-branes in the resolved conifold geometry is: effected by (32) for vanishingly small |µ Ouyang |.
• In the UV, the color gauge group is SU (N + M ) × SU (N + M ) and the flavor gauge group is SU (N f ) × SU (N f ). When one goes RG flows from the UV to the IR, two things happen, -The color gauge group SU (N +M )×SU (N +M ) is partially Higgsed down to SU (N + M ) × SU (N ) because in the IR D5-branes are not present due to which rank of one of the product gauge groups(SU (N +number of D5−branes)×SU (N +number of D5− branes)) decreases to (SU (N + number of D5 − branes) × SU (N ).
-Flavor gauge group SU (N f ) × SU (N f ) breaks to the diagonal subgroup SU (N f ) because of absense of D7-branes in the IR. This is the analagoue of chiral symmetry breaking in this brane setup.
• The pair of couplings corresponding to SU (N + M ) and SU (N ) gauge groups flow in opposite directions in the IR.
From the above equation it is clear that S 2 B 2 is the reason for introduction of nonconformality. If flux term on r.h.s. vanishes then couplings for the both gauge groups will be same which is indication of a conformal theory. This is the reason to include M D5branes in [6] to cancel the net D5-brane charge in the UV. Therefore we have a holographic theory which, like QCD, is UV conformal.
• Under the UV-to-IR RG flow, the higher rank gauge group(SU (N + M )) flows towards strong coupling and lower rank gauge group(SU (N )) flows towards the weak coupling.
after performing repeated Seiberg-like dualities, in the IR the number of colors N c gets identified with M , which in the 'MQGP limit' can be taken to be 3 (See [22]). 'MQGP limit' have been defined later in this paper.
• Holographic gravity dual of the brane setup of [6]: If one is above the deconfinement temperature i.e. T > T c then finite temperature on the gauge/brane side corresponds to a black hole in the gravitational dual and if one is below the deconfinement temperature i.e. T < T c then finite temperature on the gauge/brane side corresponds to thermal background in the gravitational dual [16]. Finite temperature on the brane/gauge side and finite seperation between the M D5-branes and M D5-branes (which is denoted by R D5/D5 ) requires to having a non-zero resolution parameter of the conifold in gravitational dual side. Similarly, IR confinement on the gauge theory side corresponds to deformation of the conifold singular à la the Klebanov-Strassler model. Therefore holographic dual of thermal QCD-like theories in this model [6] involves a resolved warped deformed conifold; D3-branes and the D5-branes are replaced by fluxes in the IR, and the back-reactions are included in the warp factor and fluxes.
• Color-Flavor Enhancement of Length Scale in the IR:In the MQGP limit (10), there is color-flavor enhancement of length scale as compared to a Planckian length scale in Klebanov-Strassler(KS)-like model even for O(1) M , in the IR . This is true when one includes terms higher order in g s N f in the RR and NS-NS three-form fluxes and the NLO terms in N in the metric. which indicates suppression quantum corrections and validity of supergravity calculations. For a detailed discussion of this issue, see [4,22]. Now, the effective number of color branes N eff (r) is given by: Type IIB axion is , N eff (r 0 ∈ IR) = 0 and writing the ten-dimensional warp factor h ∼ L 4 r 4 , the length scale L in the IR will be given by the following equation: (4) implies enhancement of color-flavor length scale in the IR as compared to KS. Hence if we consider number of colors N IR c = M = 3 and number of flavors N f = 2(u/d) + 1(s), inclusion of n, m > 1 terms in M eff and N eff f in (3) implies that L L KS (∼ L Planck ) in the MQGP limit (10), which indicates validity of supergravity calculations.
• Obtaining N c = 3: Now based on [22] we will briefly summarize how to identify number of colors N c with M which in the 'MQGP limit' (10) can be tuned to equal 3. One can write N c as sum of the effective number N eff of D3-branes and the effective number M eff of D5-branes as: where N eff (r) is defined via the following relation, where, F 5 ≡ N eff × Vol(Base of Resolved Warped Deformed Conifold). Similarly, M eff is defined via the following relation, In the above equation, S 3 being dual to e ψ ∧ (sin θ 1 dθ 1 ∧ dφ 1 − B 1 sin θ 2 ∧ dφ 2 ), B 1 is an 'asymmetry factor' defined in [6]; e ψ ≡ dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 ) and [23]: Let us denote UV and IR values of the resolution parameter "a" as [a U V , a IR ]. In this notation, From equations (5) and (8), it is clear that This implies that in the IR number of colors i.e. N c is M which in the MQGP limit we have taken to equal 3.
Therefore, we see that after application of repeated Seiberg-like duality N D3-branes are cascaded away in the IR and we have finite M corresponding to a strongly coupled IRconfining SU (M ) gauge theory; authors in [6] considered finite temperature version this theory. The holographic dual of large N thermal QCD-like theories constructed in [6] exhibits UV conformality (no Landau poles), IR confinement, quarks transform in the fundamental representation of flavor and color groups, and is valid at all temperatures.
• Type IIA Strominger-Yau-Zaslow (SYZ) Mirror of [6], M -Theory Uplift and the MQGP Limit Type IIA SYZ mirror of [6] and its M -theory uplift using Witten's prescription at intermediate gauge coupling but large 't Hooft coupling, have been worked out in [3,4]. The O(R 4 ) corrections to the MQGP metric for the black hole background corresponding to intermediate 't Hooft coupling, were worked out by two of the authors (VY, AM) in [11], and for the thermal background the O(R 4 ) corrections to the MQGP metric are given in (B2). Let us discuss how to obtain M -theory uplift at intermediate coupling.
Therefore we see that in SYZ type IIA mirror we have N D6-branes, M D6(D6)branes and N f flavor D6(D6)-branes. Keep in mind that in type IIA mirror ranges for the presence of anti-branes also matter as in [6]. In actual calculation as done in [3], T-dualities are performed along T 3 (x, y, z), where (x, y, z) are toroidal analagoue of (φ 1 , φ 2 , ψ). For the convention we have just written (φ 1 , φ 2 , ψ). Upon uplifting the SYZ type IIA mirror as obtained above to M -theory using Witten's prescription, D6branes will be converted into KK monopoles(variants of Taub-NUT spaces). Hence, all the branes will be converted into geometry and fluxes. One can show that the M -theory uplift involves a G 2 -structure manifold [4,11].
-MQGP Limit MQGP limit is defined as [3,4] wherein one considers intermediate/finite string coupling. This hence necessitates addressing the MQGP limit from M -theory.
• In [11], two of the authors(VY and AM) worked out O(l 6 p ) corrections to the MQGP metric for blackhole background by incorporating O(R 4 ) terms in eleven dimensional supergravity action. Since in gravitational dual side we have higher derivative corrections which on gauge theory side corresponds to intermediate gauge/'t Hooft coupling. Therefore using [11] we can explore the intermediate coupling regime of thermal QCD-like theories. The SU (3)/G 2 /SU (4)/Spin(7)-structure torsion classes of the relevant six-, seven-and eightfolds associated with the M -theory uplift worked out in [11].
There are two ways of understanding the origin of the O(R 4 )-corrections to the N = 1, D = 11 supergravity action. One is in the context of the effects of D-instantons in IIB supergravity/string theory via the four-graviton scattering amplitude [18]. The other is D = 10 supersymmetry [19]. Let us briefly discuss both.
• Let us first look at interactions that are induced at leading order in a D-instanton (closedstring states of type IIB superstring theory in which the whole string is localized at a single point in superspace) background in both, type IIB supergravity and the string descriptions, including a one-instanton correction to the tree-level as well as one-loop R 4 terms [18]both having the same tensorial structure. The bosonic zero modes are parameterised by the coordinates corresponding to the position of the D-instanton. The fermionic zero modes are generated by the broken supersymmetries. The Grassmann parameters are fermionic supermoduli corresponding to zero modes of the dilatino and must be integrated over together with the bosonic zero modes. The simplest open-string world-sheet that arises in a D-brane process is the disk diagram. An instanton carrying some zero modes corresponds, at lowest order, to a disk world-sheet with open-string states attached to the boundary. The one-instanton terms in the supergravity effective action can be deduced by considering on-shell amplitudes in the instanton background. The integration over the fermionic moduli absorbs the independent fermionic zero modes. Consider now amplitudes with four external gravitons. The leading term in supergravity is one in which each graviton is associated with four fermionic zero modes. Integration over the bosonic zero modes generates a nonlocal four-graviton interaction. In the corresponding string calculation the world-sheet consists of four disconnected disks to each of which is attached a single closed-string graviton vertex and four fermionic open-string vertices. Writing the graviton polarization tensor as ζ µrνr = ζ (µrζ νr) , and evaluating the fermionic integral, it was shown in [18] that the four-graviton scattering amplitude can be expressed in terms of: t 8 symbol defined as: (N i being valued in the 8 v of SO(8) or covariantized to 10D (or 11D to be used later in M -theory)), from [26] the light-cone 8D "zero mode" tensor t 8 is generalized to 10D: , and the overall factor of e 2πiτ 0 , which is characteristic of the stringy D-instanton, is evaluated at χ = τ 0 = 0 in the stringy calculation.
• In [19], it is shown that the eleven-dimensional O(R 4 ) corrections have an independent motivation based on supersymmetry in ten dimensions. This was shown to follow from its relation to the term C (3) ∧ X 8 in the M -theory effective action which is known to arise from a variety of arguments, e.g. anomaly cancellation [27,28] 2 . The expression X 8 is the eight-form in the curvatures that is inherited from the term in type IIA superstring theory [29] which is given by where There are two independent ten-dimensional N = 1 super-invariants which contain an oddparity term ( [26] and previous authors): I 3 = t 8 tr R 4 − 1 4 10 Btr R 4 and: it follows that the particular linear combination, contains both the ten-form B ∧ X 8 and t 8 t 8 R 4 .
• We will now summarize the argument of [18] that an SL(2, Z) completion of the effective R 4 interactionleads to an interesting non-renormalization theorem that forbids perturbative corrections to this term beyond one loop in the zero-instanton sector. The term in (11) bilinear in the tensort has precisely the same form as terms that arise in the zero instanton sector that come both from the one-loop four-graviton amplitude and from an (α ) 3 effect at tree level [30]. In the Einstein frame one hence obtains the following expression for the complete effective R 4 action in the Einstein's frame, that can be expressed in the form, where a, b are known numerical constants, R 4 denotes the contractionsttR 4 in (11) and · · · indicates possible perturbative and nonperturbative corrections to the coefficient of R 4 ; the 'constant' c can depend on τ = C RR + ie −φ (type IIB) andτ . The first term in (16) represents the (α ) 3 tree-level contribution and the second the one-loop relative to the former.
However, the complete expression for S R 4 must be invariant under SL(2, Z) transforma- , which provides very strong constraints on its structure. As the R 4 factor is already invariant, f (τ,τ ) in (16) is a scalar under the SL(2, Z) transformations, which hence implies a sum over all instantons and anti-instantons.
There are some strong constraints which this term must satisfy and are spelt out in [18], in which there is a simple function proposed by the authors that satisfies all these criteria, namely, where the sum indicates the sum is over all positive and negative values of p, n except p = n = 0. Now: The sum over p, using the Poisson resummation formula where performing a perturbative expansion in 1 τ 2 of the non-perturbative instanton contribution of charge mn, one uses the asymptotic expansion for K 1 (z) for large z in (19).
The perturbative terms in (19) terminate after the one-loop term as suggested earlier. The non-perturbative terms have the form of a sum over single multiply-charged instantons and anti-instantons with action proportional to |mn|. Identifying p with the discrete momentum (euclidean energy) of a compactified D-particle of charge n then the Poisson resummation exchanges it with the winding number of the world-line and the result is what is expected by T-duality from type IIA in nine dimensions. The terms in parenthesis in (19) represent the infinite sequence of perturbative corrections around the instantons of charge mn. It is hence evident that in the zero-instanton sector, there are no perturbative corrections figuring in the action up to O(R 4 ), i.e., S R 4 , beyond one loop. This will be important in demonstrating the "non-renormalization" beyond one-loop of the deconfinement temperature T c later in the paper.
• The N = 1, D = 11 supergravity action inclusive of O(l 6 p ) terms (promoting the 10Dt to 11D), is hence given by: where: κ 2 11 being related to the eleven-dimensional Newtonian coupling constant, and G = dC with C being the M -theory three-form potential with the four-form G being the associated fourform field strength. The EOMS are: where [31]: (22) being respectively the elven-dimensional Riemann curvature tensor, Ricci tensor and the Ricci scalar.
Now, one sees that if one makes an ansatz: then symbolically, one obtains: One can see that one can find a consistent set of solutions to (25) wherein C (1) M N P = 0 up to O(β). Assuming that one can do so, henceforth we will define: no summation implied.
• Let us discuss how one can controls the higher derivative contributions to the MQGP limit result of [3]. This was explained in [11] and we review the same here. It is evident from (A1) in the ψ = 2nπ, n = 0, 1, 2-coordinate patch that in the IR: r = χr h , χ ≡ O(1), and up to O(β): where , implying there is a competition between Planckian and large-N suppression and infra-red enhancement arising from m, n = 0 in (27). Choosing a heirarchy: hole with the metric given by where IIA are the type IIA RR 1-forms obtained from the triple T/SYZ-dual of the type IIB F IIB 1,3,5 fluxes in the type IIB holographic dual of [6], and g(r) = 1 − r 4 h r 4 . For low temperatures, i.e., T < T c , is given by the thermal gravitational dual: (28) following by a Double Wick rotation in the new x 3 , t coordinates obtains (29); h(r, θ 1,2 ) is the ten-dimensional warp factor [3,6]. This amounts to: in the results of [11,33] (See [34] in the context of Euclidean/black D4-branes in type IIA).
In (29), 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 two-spheres [35]) world volume to be given by , r 0 being the very small IR cut-off in the thermal background (See also [36]) and L = (4πg s N ) , thereby recovering 4D Physics. The working metric used in this section and section 4 for the thermal background corresponding to T < T c will involve settingg(r) to unity in (29). Now, if β BH,Th are respectively the periodicities of the thermal circle in the black and thermal M -theory backgrounds then at r = R UV , β BH G BH tt = β Th G Th tt . Now at T = T c [16], where / excludes the coordinate integral with respect to x 0 , implying:

Black Hole Background Uplift (Relevant to T > T c ) and Holographic Renormalization of On-Shell D = 11 Action
The O(β)-corrected M -theory metric of [3] in the MQGP limit near the ψ = 2nπ, n = 0, 1, 2branches (whereat there is a decoupling of M 5 (t, x 1,2,3 , r) and M 6 (θ 1,2 , x, y, z, for some of the off-diagonal components along the delocalized T 3 (x, y, z)] -the components which do not receive an O(β) corrections, are not listed in (A1)was worked out in [11] and is given in (A1).
Restricting to the Ouyang embedding of the flavor D7-branes in type IIB can, for an extremely small (modulus of the) Ouyang embedding parameter, may be effected by working near [4]: Einstein-Hilbert term in the IR will be given by the following expression: Now performing angular integration of the above equation, implying: Using equation (35) and performing radial integration of equation (33), we obtained the following expression for the action density (for Einstein-Hilbert term) in the IR: where r h being the resolution parameter of the blown up S 2 [11]. Further, there is one more term appearing in on-shell action as given in equation (60), simplified form of that term is: Similarly, Einstein-Hilbert term in the UV is: where, for simplification of the computation, we have dropped the O 1 log N -correction terms. As Hence, one obtains: The UV-divergent contribution of the EH action is: Now for the boundary metric: and counter term to cancel the UV divergence appearing in Einstein-Hilbert term is, The UV-finite boundary Gibbons-Hawking-York contribution up to O(β) turns to be: The UV-divergent contribution to the GHY term is: One can further show that up to O(β 0 ) contribution from higher derivative term will be: using: − dr r 3 log(r) (2 log(r) + 1) 7 = 1 360 r 4 256 log 5 (r) + 704 log 4 (r) + 800 log 3 (r) + 488 log 2 (r) + 184 log(r) + 19 (2 log(r) + 1) 6 − 16Ei(4 log(r) + 2) e 2 ; − dr r log(r) 4 log 2 (r) + 15 log(r) + 9 (2 log(r) + 1) 7 = 2r 2 7504 log 5 (r) + 22512 log 4 (r) + 30016 log 3 (r) + 23384 log 2 (r) + 8799 log(r) + 833 2880(2 log(r) + 1) 6 − 469Ei(2 log(r) + 1) 5760e ; − dr log(r) 45a 4 64 log 3 (r) + 208 log 2 (r) + 212 log(r) + 57 − 8r h 4 (2 log(r) + 5) one can show that UV finite part coming from higher derivative term will be: Use has been made of the following. The most dominant term in √ 6 sin 6 θ 1 +6642 sin 2 θ 2 sin 3 θ 1 −40 √ 6 sin 4 θ 2 sin 7 θ 1 sin 4 θ 2 . As: , one therefore notes that the UV-finite contribution to: 1 + Also UV divergent contribution from higher derivative term will be: We further note that at constant r surface counter term to cancel UV divergence appearing from higher derivative term will be given by the following expression: From: one notes that the dominant contribution arises from small values of θ 1,2 , i.e., along Ouyang embedding for small Ouyang embedding paramter('s modulus). Hence, Therefore simplified form of the counter term is: (52) Also, By comparing (48) with (49), one imposes: implying: In κ 2 11 = 1-units, the D = 11 supergravity action inclusive of O(R 4 ) terms is given by: which yields as an EOM: Taking the trace of (57) one obtains: where the approximation is justified in [11]. Writing R = R (0) + βR, K = K (0) + βK, C 3 = C (0) 3 + βC 3 , etc., in [11] it is also shown that one can self consistently set C 3 = 0. One thus notes: Therefore, the on-shell action up to O(β) using X 8 (M 11 ) = 0 [3], becomes: where . To see the holographic renormalization of (60), its UV-divergent portion is: Therefore, the counter terms required to cancel the UV-divergent terms are: Working in the IR and near (32), Einstein-Hilbert term for thermal background is: being the resolution parameter for the blown up S 2 in the thermal background, and κ β m ,n EH, IR are the numerical prefactors -which we have worked out but have not given here as they are not particularly illuminating in the β m (r − r 0 ) n terms in √ −G M R restricted to the Ouyang embedding of the flavor D7-branes in the type IIB dual and in the IR. From (64), one can evaluate Also additional term that appear in on-shell action (59) at O(β) is: Similarly Einstein-Hilbert term in the UV is: implying: Therefore, after performing radial as well as angular integral of the above equation UV finite Einstein-Hilbert term is: UV divergent Einstein-Hilbert term for thermal background is: The UV-finite part of the boundary Gibbons-Hawking-York term for the thermal background up to O(β) turns to be: Further UV divergent part of the Gibbons-Hawking-York term for the thermal background is: One can further show that up to O(β 0 ) contribution from higher derivative term in the IR is: implying: Further, near (32) contribution from higher derivative term in the UV is: wherer , thus after performing angular and radial integration of the above equation the UV-finite part will be: Further, UV-divergent part from equation (74) will be: We note that: One should note that in (36) (87) and (91)) for the supergravity dual with a BH corresponding to T > T c , and similarly (65), (67), (68), (70), (73), (75), (76) and (77) for the supergravity dual with a thermal background corresponding to T < T c , given that the delocalized toroidal coordinates (x, y, z) are defined as [3]: x ∼ h where the 10-D warp factor is given by: , upon promoting to global coordinates, and noting that all the aforementioned equations already have an N −ζ , ζ > 0 suppression implying that up to O 1 N , one can approximate h ≈ L 4 r 4 , one sees that one will generate a sin θ 1 sin θ 2 contribution upon integration w.r.t. x, y, z. As this is not accompanied with an r-dependence, it will only provide a common multiplicative N -suppressed contribution to all the abovementioned integrals. We have hence not considered the same.
One hence sees that the UV-divergent contribution of 1 2 − 7 2 S (0) EH + 2S (0) GHY is given by: From (77), one notices the following: is canceled off by the counter term r=R UV √ −h.

T c Inclusive of O(R ) Corrections
In this section using the results from 3.1, 3.2, we will calculate deconfinement temperature by equating on-shell actions for blackhole and thermal backgrounds. In the same process we will explain a version of UV-IR connection which is arising from the O(R 4 ) terms. We will also calculate entropy of the black hole and discuss its consistentency with the Wald entropy up to O(R 4 ).
The O(β 0,1 ) term in (60) 1 + of the aforementioned BH and thermal actions are given as under: Writing R bh D5/D5 = 1 + √ 3 r h , and using (89), log 1 − r h Let us show the consistency of the BH entropy calculated via (85) and the Wald's method. One notes from (85) that up to O(β 0 ), the BH entropy S BH is given by: Note, there is a multiplicative factor of β BH ∼ Interestingly, the Wald entropy density arising from the O(β 0 ) action is given by: x,y,z,x 10 ) ∂R ∂R mnpq mn pq √ g x 1,2,3 ,θ 1,2 ,x,y,z,x 10 dθ 1 dθ 2 dxdydzdx 10 where V 3 is the R 3 (x 1,2,3 coordinate volume). Upon substituting b = 1 √ 3 + , ∼ (| log r h |) 9 2 N − 9 10 −α yields: Both methods yield that the BH entropy goes like r 3 h . The results match exactly for (for simplicity, we have disregarded in (89)  At O(R 4 ), one can show that to calculate one needs to evaluate the following four class of terms while calculating ∂J 0 ∂Rtrtr : One hence sees that it is the contribution (iii) which dominates in the MQGP limit, and obtains: One notes the identical r 3 h and N − 1 20 dependence in (91) and the corresponding semiclassical result at O(β). For the latter and the Wald entropy result to match exactly at O(β) imposes the following constraint: Hence, one sees that consistency with the Wald entropy computation up to O(R 4 ), imposes a linear constraint on the constants of integration appearing in the solutions to the EOMs of the metric corrections at O(R 4 ) along the directions that encode the "memory" of the compact part of the non-compact four-cycle "wrapped" by the flavor D7-branes in the type IIB dual [3] of the large-N thermal QCD-like theories. This "Flavor Memory"-effect is also displayed by the M -theory uplift of the corresponding thermal background, as seen in 3.2.
After that brief detour, returning back to the computation of T c , the on-shell action corresponding to the thermal background uplift is given below: Now, equality at O(β 0 ) in (31) requires the following equation to be solved: (94) Now, one can show that near, e.g., (θ 1 , θ 2 ) ∼ EH,IR ∼ 10 5 , and hence, one can drop the κ th, β 0 EH,IR term. Therefore, Identifying r 0 L 2 with m 0 ++ 4 [37] and using T c = r h /πL 2 [4], deconfinement temperature will be given by the following expression: Using (85) and (93), the matching at O(β) yields: which yields: We thus see that by matching the thermal and black-hole M -theory background actions at the UV cut-off, respectively dual to large-N QCD-like theories at low and high temperatures, (98) gives a relationship between the constants of integration in the IR in the respective metric perturbations as O(l 6 p ) corrections to the M -theory uplift worked out in [3] along respectively the M -theory circle (for thermal background dual to T < T c ) and the S 3 -portion of the four cycle (locally) Σ (4) = R + × S 3 "wrapped" by the flavor D7-branes of the type IIB conifold geometry of [6] (for black hole background dual to T > T c )!! The equations (98) along with (96), are what we refer to as UV-IR connection in this context. We also note that the contributions arising from the O(R 4 ) corrections to the BH background metric retains the "Flavor Memory", referred to earlier, of the compact three-cycle which is part of the non-compact four-cycle "wrapped" by the parent type IIB flavor D7-branes.

Deconfinement from Entanglement Entropy
In this section we will discuss confinement-deconfinement phase transition in QCD 2+1 -like theory from entanglement entropy point of view based on [2]. We will calculate entanglement entropy for "connected" and "disconnected" regions -suitably defined. We will show that at a critical value of l of a spatial interval which we are denoting here by l crit , phase transition will occur from confined phase to the deconfined phase. Contact with results of the previous section is made by looking at the M KK → 0-or equivalently r 0 → 0-limit, i.e., the 4D-limit of results of this section.
Authors in [2] have investigated entanglement entropy between an interval and its complement in the gravity dual of large N c gauge theories and they found that there are two RT surfacesdisconnected and connected. Below a critical value of l connected surface dominates while above that critical value of l disconnected surface dominates. This is analogous to finite temperature deconfinement transition in dual theories.
For AdS d+2 /CF T d+1 correspondence, quantum entanglement entropy between the regions A ≡ R d−1 × l and B ≡ R d−1 × (R − l), l being an interval of length l, is given by the following expression [17], ind is the determinant of the induced string frame metric on co-dimension 2 minimal surface γ. Equation (99) can be generalised to non-conformal theories as below [2]: where φ is the dilaton profile. In an M -theory dual since there is no dilaton, therefore equation (100) will be assumed to be replaced by the following expression, Now, consider gravity dual's string frame metric (29), which can written as: where x µ (µ = 0, 1, 2) represents (2 + 1) Minkowskian coordinates, r is the radial coordinate and x m (m = 3, 5, 6,7,8,9,10) corresponds to x 3 and six angular coordinates(θ 1,2 , φ 1,2 , ψ, x 10 ). Noting g x 3 x 3 (r = r 0 ) = 0, (x 3 , r) form a cigar-like geometry. The volume of the seven-fold is given by: Here we have (2 + 1) dimensional QFT defined on IR 2+1 . Let us define two regions A and B in two dimensions as below, Now, we are going to calculate enganglement entropy between A and B for the metric (102) using equation (101). Induced metric on γ is given below (here we have represented x 1 by x):. Therefore, where, As in [2], H(r = r 0 ) = 0. Equation of motion for x(r) can be obtained from equation (107) as, where, r * is the value of radial coordinate at which dr/dx vanishes. Integrating above equation we obtain, From Equations (107) and (109), entanglement entropy can be simplified as, Here, r ∞ is the UV cutoff.
For the disconnected surface we have r * = r 0 as explained in [2], when l > l max entanglement entropy for the disconnected surface will be given by the following expression, For l < l max , difference of entanglement entropy for connected and disconnected surface have the following form, • Connected solution with large r * has the lower entropy when l is small.
• When l starts increasing, two things can happen. First, connected solution can still remain the lower untill l reaches it's maximum value i.e. l max . Second, there could be a critical value of l which is denoted by l crit and l crit < l max above which disconnected solution becomes dominant one.
• There will be a phase transition at l = l max in the first case and at l = l crit < l max in the second case.

Entropy
The connected region entropy catering to small l and large r is: From (B14), noting the extremely non-trivial cancelation of O(R 4 ) corrections C th zz − 2C th θ 1 z = 0, we note that there are no O (R 4 )-corrections to the connected region (and similarly disconnected region) entanglement entropy.
The disconnected region entropy will be given by the following expression: Writing a = 1 √ 3 + r 0 , the contribution in r ∈ r 0 , R D5/D5 is given by: Noting that all radial distances are measured in units of R th D5/D5 , i.e., log R D5/D5 ≡ 0, one obtains: where, One notes that in the large-N limit, recalling R UV < (4πg s N ) As l increases, i.e., r * decreases and reaches R th D5/D5 , S connected changes from being negative to vanishing and S disconnected stays negative implying disconnected region has lesser entropy. At r * = r criticial , S connected = S disconnected and one sees that r critical is the solution of the equation: For . From (95), one thus obtains by taking the corresponding to the deconfinement temperature. As mentioned below, one would require α 3 θ 1 α 2 θ 2 = 2, which would for N = 100, M = N f = 3, g s = 1 would yield l(r critical ) = 3850 (as obtained numerically below) for α θ 2 ∼ O(10). Let us also discuss the possibility of r * ∈ [r 0 , R D5/D5 ], i.e., IR-valued r * . and splitting , one obtains: and similarly, N α 3 • One hence notes that the "UV-divergent" terms in S

M χPT Compatibility
In [1], we worked out one-loop renormalised LECs of SU (3) chiral perturbation theory Lagrangian up to O(p 4 ) from type IIA string dual of large-N thermal QCD like-theory inclusive of O(R 4 ) corrections. We constrained ourselves to the chiral limit due to which some terms were absent in SU (3) chiral perturbation theory Lagrangian of Gasser and Leutwyler [5]. Therefore we were able to calculate L r 1,2,3,9,10 , F π and g Y M . We match our results with phenomenological values of one-loop renormalised LECs as given in [32]. Let us discuss this in some detail.
The meson sector in the type IIA dual background of top-down holographic type IIB setup is given by the flavor D6-branes action. Restricting to the Ouyang embedding (1) for a vanishingly small |µ Ouyang |, one will assume that the embedding of the flavor D6-branes will be given by ι : Σ 1,6 R 1,3 , r, θ 2 ∼ α θ 2 N 3 10 , y → M 1,9 , effected by: z = z(r). As obtained in [39] one sees that z=constant is a solution and by choosing z = ±C π 2 , one can choose the D6/D6-branes to be at "antipodal" points along the z coordinate. As in [39], we will be working with redefined (r, z) in terms of new variables (Y, Z): Vector mesons are obtained by considering gauge fluctuations of a background gauge field along the world volume of the embedded flavor D6-branes (with world volume Σ 7 (x 0,1,2,3 , Z, θ 2 ,ỹ) = Σ 2 (θ 2 ,ỹ) × Σ 5 (x 0,1,2,3 , Z)). Turning on a gauge field fluctuationF about a small background gauge field F 0 and the backround i * (g + B)[i : Σ 7 → M 10 , M 10 being the ten-dimensional ambient space-time]. Y = 0 is the SYZ mirror of the Ouyang embedding [39]. Picking up terms quadratic inF in the DBI action: whereμ = i(= 0, 1, 2, 3), Z. Expanding five dimensional gauge fields as: . After a gauge transformation given below: . Action simplified as given below: Working in the A Z (x µ , Z) = 0-gauge, integrating out all higher order vector and axial vector meson fields except keeping only the lowest vector meson field [40] V (1) (1) . To introduce external vector V µ and axial vector fields A µ , one could use the Hidden Local Symmetry (HLS) formalism of [40] and references therein, wherein 1 Fπ ∂ µ π →α µ⊥ = 1 Fπ ∂ µ π + A µ − i Fπ [V µ , π] + · · · , and one also works withα [∂ µ π, π] + · · · . Mode expansion of the gauge field contains infinite number of vector meson fields V (n) µ (x µ ) and axial vector meson fields A (n) µ (x µ ), therefore to obtain the low energy effective theory of QCD, we need to truncate the KK spectrum in such a way that we are left with lowest vector meson field (V (1) µ = gρ µ ) and lightest psuedo-scalar meson field (π meson) [40], therefore, Based on [41], as regards a chiral power counting, one notes that . Further, V µν , a µν and v µν are of O(p 2 ). Hence, using (149), (F µν F µν ) m is of O(p 4m ), m ∈ Z + . Defining parity as x i → −x i , i indexing the conformally Minkowskian spatial directions and Z → −Z, given that A µ (x, Z) will be odd, α µ⊥ will be even, α µ|| will be odd and V µ will be odd implies ψ 0 (Z) will be odd and ψ 1 (Z) will be even. As coupling constants are assumed to be scalars and they are given by Z-integrals, the Z-dependent terms in the action must be separately of even-Z parity. As ψ 0 has odd Z-parity and ψ 1 has even Z-parity, therefore at O(p 4 ), terms with (3α µ|| s , 1α µ⊥ or 3α µ⊥ s , 1α µ|| ), are dropped as they involve coefficients of the type ψ 2m+1 , and y i s and z j s are given as radial integrals involving ψ 0 (Z) and ψ 1 (Z) (as reviewed in [1]).
In the chiral limit, the O(p 4 ) SU (3) χPT Lagrangian is given by [5]: where Fπ . SU (3) chiral perturbation theory Lagrangian (152) can be obtained from the HLS Lagrangian (150) by integrating out the ρ mesons as done in [41]. One hence obtains, relationships between the LECs y i , z i of (150) and the L i s of (152). Using these relations we have calculated LECs of SU (3) χPT Lagrangian via gauge-gravity duality in [1].
The parameters L i and H i are renormalized at one-loop level with all vertices in one-loop diagrams arising from the O(p 2 ) terms. Using dimensional regularization and performing renormalizations of the parameters via [5]: where µ is the renormalization point, and Γ i and ∆ i are certain numbers given later; λ(µ) is the divergent part given by: d being the non-radial non-compact space-time dimensionality to be set to four. The constants Γ i and ∆ i for SU (3) χPT theory were worked out in [5]: The analog of the 1-loop renormalization in χPT can be understood on the gravity dual side by noting that the latter requires holographic renormalization as discussed in Section 3.
In [1], we had shown how, in five steps, it is possible to match the phenomenological values of the O(p 4 ) SU (3) χPT Lagrangian [5] one-loop renormalized LECs L r 1,9,10 as well as F 2 π , g YM (Λ QCD = 0.4GeV, Λ = 1.1Gev, µ = M ρ ) exactly, where Λ is the "HLS-QCD" matching scale [41] and µ is the renormalization scale, as well as the order of magnitude and signs of L r 2,3 . On matching LECs computed holographically with their phenomenological/experimental values, it turns out that a particular combination of integration constants, (C th zz − C th θ 1 z + 2C th θ 1 x ), is appearing in all the LECs of SU (3) chiral perturbation theory Lagrangian up to O(p 4 ). Here, C th M N are integration constants appearing in solutions to O(R 4 ) corrections as worked out in [1] to the MQGP metric for the thermal gravitational dual correpsonding to (29). This combination, (C th zz − C th θ 1 z + 2C th θ 1 x ), encodes information about the compact part of non-compact four cycle around which flavor D7-branes are wrapping in type IIB setup. Now, as we are working up to O(β) and further due to the smallness of assuming working up to O( 2 ), where is defined via the relation a = 1 √ 3 + r 0 , simplified form of L r 1,2,3 , as worked out in [1], are given below: where: Further, as the log r 0 in (157) is in fact log r 0 R D5/D5 -R D5/D5 > r 0 being the D5−D5 separation -one sees from (157) that in order to obtain a positive value (as required from phenomenological value of L r 1 ), Ω < 0. Note, as shown in [1], matching with the experimental value of the pion decay constant F π requires an N -suppression in α θ 1 , implying the N enhancement in the last term in (157) is artificial. So, to ensure one does not pick up an O 1 β contribution in L 1 from 2 Ω in (157) and also to ensure that the third term in (157) required to partly compensate the first two negative terms in the same (as explained above) to produce a positive term, is not vanishingly small, from (158), one needs to set: Finally, combining the above observations with the requirement to match the experimental value L exp 1 = 0.64 × 10 −3 , as shown in [1], one therefore requires to implement the following constraint (which follows from the discussion above): We hence see from (159) that provides an expansion parameter connecting the 1 N and β expansions. Also, together with (160), as was noted in [1], this was the first connection between large-N and higher derivative corrections in the context of M-theory dual of large-N thermal QCD-like theories.
Similarly, it was showin in [1] that: where, and ω = 4.6N − 2fr 0 = 6.9 × 10 −3 . Substituting (162) -(163) into (160), one obtains: From equation (164), it is clear that combination of constants of integration required to have definite sign which is negative. In appendix A, we have obtained values of these integration constants explicitly by taking the decompactification limit (i.e. M KK → 0 limit) of the spatial circle S 1 (x 3 ) appearing as part of the M -theory metric used in [1], i.e., in S 1 (x 0 ) × R 2 × S 1 1 M KK to recover 4D QCD-like theory. In fact, using those values we derived that the abovementioned combination of integration constants which appears in all the LECs of SU (3) χPT Lagrangian up to O(p 4 ), indeed can be made to have a negative sign as required by matching with the phenomenological/experimental values of the one-loop renormalised LECs of SU (3) χPT Lagrangian as discussed above. In particular from (B14), we see that close to the Ouyang embedding of the flavor D7-branes in the parent type IIB dual, there occurs a delicate cancelation between the contributions arising from the metric corrections at O(R 4 ) in the M theory uplift along the S 1 (ψ/z)-fiber (considering the S 3 (θ 1 , φ 1 /x, ψ/z) as an S 1 -fibration over the vanishing two-cycle S 2 (θ 1 , φ 1 /x)) and S 2 (θ 1 , ψ/z) resulting in a non-zero contribution only along S 2 (θ, φ 1 /x) surviving : C th zz − 2C th θ 1 z + 2C th which is negative for Σ 1 < 0. Once again, the above discussion too is a manifestation of the "Flavor Memory" effect discussed in Section 4.

Summary
Let us summarise our main results. In this paper, in the context of a top-down holographic evaluation of the deconfinement temperature T c in QCD at intermediate coupling, we obtain T c from the M -theory dual of large-N thermal QCD-like theories (belonging to the class of theories that display IR confinement, UV conformality with quarks in the fundamental representation of the flavor and color symmetry groups) inclusive of the O(R 4 ) corrections. In this process, there are the following conclusions that are arrived upon.
• UV-IR Mixing and Flavor Memory: Performing a semiclassical computation as advocated in [16], by matching the actions at the deconfinement temperature of the M -theory uplifts of the thermal and black-hole backgrounds at the UV cut-off, one sees that one obtains a relationship in the IR between the O(R 4 ) corrections to the M -theory metric along the M -theory circle in the thermal background and the O(R 4 ) correction to a specific combination of the M -theory metric components along the compact part of the four-cycle "wrapped" by the flavor D7-branes of the parent type IIB (warped resolved deformed) conifold geometry -the latter referred to as "Flavor Memory" in the M-theory uplift.
• Non-Renormalization of T c : -Semiclassical computation: We further show that the LO result for T c also holds even after inclusion of the O(R 4 ) corrections. The dominant contribution from the O(R 4 ) terms in the large-N limit arises from the t 8 t 8 R 4 terms (along with the subdominant 11 11 R 4 term), which from a type IIB perspective in the zero-instanton sector, correspond to the tree-level contribution at O ((α ) 3 ) as well as one-loop contribution to four-graviton scattering amplitude and obtained from integration of the fermionic zero modes. As from the type IIB perspective, the SL(2, Z) completion of these R 4 terms [18] suggests that they are not renormalized perturbatively beyond one loop in the zero-instanton sector, this therefore suggests the non-renormalization of T c at all loops in M -theory at O(R 4 ).
-T c from Entanglement entropy: With an obvious generalization of [2] to M -theory, we calculated the entanglement entropy between two regions by dividing one of the spatial coordinates of the thermal M -theory background into a segment of finite length l and its complement. Like [2], there are two RT surfaces -connected and disconnected. There is a critical value of l which is denoted by l crit such that if one is below the critical value l crit then it is the connected surface that dominates the entanglement entropy, and if one is above the critical value l crit then it is the disconnected surface that dominates the entanglement entropy; l < l crit corresponds to confining phase of large N c gauge theories whereas l > l crit corresponds to deconfining phase of the same. This is interpreted as confinement-deconfinement phase transition in large N c gauge theories.
Remarkably, when evaluating the deconfinement temperature from an entanglement entropy computation in the thermal gravity dual, due to an exact and delicate cancelation between the O(R 4 ) corrections from a subset of the abovementioned metric components, one sees that there are consequently no corrections to T c at quartic order in the curvature supporting the conjecture made in Section 4 -and summarized above -on the basis of a semiclassical computation.
• Deriving M χPT-Phenomenology compatibility: As shown in [1], matching the phenomenological value of the 1-loop renormalized coupling constant corresponding to the O(p 4 ) SU (3) χPT Lagrangian term " ∇ µ U † ∇ µ U 2 " with the value obtained from the type IIA dual of thermal QCD-like theories inclusive of the aforementioned O(R 4 ) corrections, required the O(R 4 ) corrections arising from contributions due to the very same abovementioned combination of M -theory metric components evaluated at the IR cut-off, to have a definite sign (negative). The thermal supergravity background dual to type IIB (solitonic) D3-branes at low temperatures, includes R 2 × S 1 ( 1 M KK ). By taking the M KK → 0 limit (to recover a boundary four-dimensional QCD-like theory after compactifying on the base of a G 2 -structure cone), remarkably, we obtain the values of the aforementioned O(R 4 ) corrections to the M -theory uplift's metric and therefore derive the MχPT requirement of the sign of their specific relevant combination evaluated at the IR cut-off. Close to the Ouyang embedding of the flavor D7-branes in the parent type IIB dual, there occurs a delicate cancellation between the contributions arising from the metric corrections at O(R 4 ) in the M theory uplift along the S 1 (ψ/z)-fiber (considering the S 3 (θ 1 , φ 1 /x, ψ/z), the compact part of the four-cycle S 3 × R >0 "wrapped" by the flavor D7-branes in the parent type IIB dual of [6], as an S 1 (ψ/z)-fibration over the vanishing two-cycle S 2 (θ 1 , φ 1 /x)) and S 2 (θ 1 , ψ/z) resulting in a non-zero contribution only along S 2 (θ, φ 1 /x) surviving. This further reinforces the "Flavor Memory" discussed earlier. where Σ 1 is defined as: In this appendix, we discuss the independent EOMs for the metric perturbations f M N of (29) close to the IR cut-off r 0 up to LO in N , their solutions and constraints and values of the same in the decompactification-limit of a spatial direction (that plays a crucial role in providing evidence of an all-loop non-renormalization of T c at O(R 4 )). In this way, we are able to obtain the values of the metric perturbations (in the deep IR) along the three-cycle S 3 (θ 1 , x, z) -the delocalized version of S 3 (θ 1 , φ 1 , ψ) -strictly speaking along the fiber S 1 (z) (the S 3 (θ 1 , x, z) is an S 1 (z) fibration over the vanishing two-cycle S 2 (θ 1 , x)) and along another two-cycle S 2 (θ 1 , z) (which is also an S 1 (z)-fibration). A specific linear combination of the contributions from f M N | S 3 (θ 1 ,x,z) , near the Ouyang embedding in the parent type IIB dual, appears ubiquitously in computations of T c in this work and the LECs of SU (3) χPT Lagrangian at O(p 4 ) from M χPT in [1]. We will prove that the negative sign of this combination, as required by matching with the phenomenological values of the LECs in the χPT Lagrangian at O(p 4 ) in [1], can hence be derived; the bonus is the actual values of the contributions appearing in the aforementioned linear combination which further demonstrates a delicate cancellation with the contribution along the vanishing two-cycle S 2 (θ 1 , x) being the only one surviving. These are manifestations of the "Flavor Memory" as discussed in Sections 4, 6.