0.85pt

Aspects of Thermal QCD Phenomenology at Intermediate Gauge/’t Hooft Coupling
from String/M-Theory, (HD) Gravity Islands, and Multiverse111Based on author’s
Ph.D thesis successfully defended on October 05, 2023

Gopal Yadav222email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Physics, Indian Institute of Technology, Roorkee- 247 667,

Uttarakhand, India

Chennai Mathematical Institute, SIPCOT IT Park, Siruseri 603103, India

Abstract: The holographic dual of thermal QCD-like theories at intermediate
coupling was constructed in [1] by including the ${\cal O}(R^{4})$ terms in
the eleven dimensional supergravity action. First part of the thesis explored
the application of [1] and the following issues have been addressed in first
part of this thesis.

In [2], we have computed the low energy coupling constants (LECs) of $SU(3)$
chiral perturbation theory in the chiral limit from the type IIA string dual
inclusive of ${\cal O}(R^{4})$ corrections. We matched our results with the
phenomenological data and found the connection between higher derivative terms
and large-$N$ expansion.

In [3], we computed the deconfinement temperature of thermal QCD-like theories
at intermediate coupling from ${\cal M}$ theory dual inclusive of ${\cal
O}(R^{4})$ corrections using Witten’s prescription [4] in the absence of
rotation. In this process, we observed a novel “UV-IR” mixing, “Flavor Memory”
effect (memory of flavor $D7$ branes considered in parent type IIB string dual
was contained in the aforementioned no-braner ${\cal M}$-theory uplift of
large-$N$ thermal QCD) and non-renormalization of deconfinement temperature
$(T_{c})$ beyond one loop in the zero instanton sector from semiclassical
computation and entanglement entropy points of view. We also discussed
confinement-deconfinement phase transition in thermal QCD-like theories from
the entanglement entropy point of view too, where the aforementioned sectors
correspond to entanglement entropies of connected and disconnected RT surfaces
in gravity dual. Further, we showed compatibility of the results obtained with
M$\chi$PT as obtained in [2]. Then in [5], we constructed holographic dual of
rotating quark-gluon plasma (QGP) by performing Lorentz transformations in the
subspace of eleven-dimensional ${\cal M}$ theory metric on the gravity dual
side. We computed the deconfinement temperature ($T_{c}$) of rotating QGP
using semiclassical method [4] and found that the deconfinement temperature of
rotating QGP is inversely proportional to Lorentz factor, which implies that
it decreases when $T_{c}$ increases and vice-versa. Further, “UV-IR” mixing,
“nonrenormanization of $T_{c}$” and “Flavor Memory” effect were again observed
in the holographic study of rotating QGP.

From the applications of the island proposal [6], doubly holographic setup
[7], and wedge holography [8, 9], we explored the following issues.

In [10], we obtained the Page curves of Reissner-Nordström black hole in the
presence of higher derivative terms $({\cal O}(R^{2}))$ in the gravitational
action. We used Island proposal [6] to do the same and found that Page curves
are shifted towards later times or earlier times when Gauss-Bonnet coupling
increases or decreases. Scrambling time is affected when we considered general
${\cal O}(R^{2})$ terms, whereas it was unaffected in Einstein-Maxwell-Gauss-
Bonnet gravity. Based on these results, one can say that the “dominance of the
island” is affected by the presence of higher derivative terms.

In [11], we constructed the doubly holographic setup for the non-conformal
bath (thermal QCD) in ${\cal M}$-theory background with the inclusion of
${\cal O}(R^{4})$ terms. We obtained the Page curve of an eternal neutral
black hole by computing the entanglement entropies of Hartman-Maldacena-like
and Island surfaces. Our results agree with area computation and Dong’s
formula for leading order terms in eleven-dimensional supergravity action.
Apart from obtaining the Page curve, we showed that entanglement entropy of
Hartman-Maldacena-like surface exhibits “Swiss-Cheese” structure at ${\cal
O}(\beta^{0}),\beta\sim l_{p}^{6}$ in the large-$N$ scenario. We showed that
no boundary terms would be required on ETW-“brane” for higher derivative
terms, and the tension of ETW-“brane” turns out to be non-zero, and therefore
it is possible to localize gravity on ETW-“brane” for massless graviton in our
setup. We explicitly showed that obtaining the Page curve of an eternal
neutral black hole for massless graviton in our setup is possible. Further, we
found a relationship between eleven-dimensional Planckian length and black
hole horizon radius to get rid of large-$N$ and IR-enhancement appearing in
the ratio of Island surface entanglement entropy and black hole thermal
entropy.

We obtained the Page curve of the Schwarzschild de-Sitter (SdS) black hole
using the island proposal from a non-holographic approach in [12] for the
black hole and de-Sitter patches separately. This was possible because of the
insertion of thermal opaque membranes on both sides of the black hole and de-
Sitter patches. We obtained the Page curve of the black hole patch by plotting
the entanglement entropy of Hawking radiation in the absence and presence of
the island surface. In the case of the SdS black hole, we found that island is
located inside the black hole in contrast to the universal result that island
is located outside the black hole. We also discussed that “dominance of
islands” and “information recovery” takes more time for the low-temperature
black hole patch compared to the high-temperature black hole patch.

We described the Multiverse in [13] from the application of wedge holography.
Multiverse is described as the setup of $2n$ anti de-Sitter (AdS) or de-Sitter
branes embedded in the corresponding bulk. Due to different bulks for the AdS
and de-Sitter branes, we can’t have the Multiverse made up of AdS and de-
Sitter branes both. Since all the universes can communicate with each other
due to transparent boundary conditions at the defect, one could resolve the
“grandfather paradox” in this setup. This setup is also used to obtain the
Page curve of the Schwarzschild de-Sitter black hole, which has two horizons.

## Acknowledgments

I am very grateful to my supervisor, Prof. Aalok Misra, for guiding me during
my Ph.D. and making me what I am today. I enjoyed all the discussions that we
have together. He trained me so that I can now do independent research. Apart
from the research part, I learned many things from him, which helped me in the
development of my personality in real life and as a researcher. I am very
fortunate to have him as my supervisor. I am very thankful to my supervisor
and his family, who always treated me like a family member, and for their
encouragement, appreciation, and blessings. Without all these, my Ph.D.
journey would not have been so smooth and pleasant.

I would like to thank the very humble members of my research committee known
as “Student Research Committee (SRC)”: Prof. Tashi Nautiyal (chairperson SRC),
Prof. Dibakar Roychowdhury (internal member), Prof. Sugata Gangopadhyay
(external member - Department of Computer Science and Engineering, I.I.T.
Roorkee) and Prof. Arnab Kundu (Saha Institute of Nuclear Physics (SINP),
Kolkata, India, an external member of me as a research fellow of CSIR), for
their appreciation, encouragement, valuable comments, questions and evaluating
me time to time. I am very grateful to Prof. Dibakar Roychowdhury for the
discussions related to the research and post-doctoral position. He always
encouraged me not to get tired and never give up.

I would like to thank my collaborators, my supervisor, Dr. Vikas Yadav and Mr.
Nitin Joshi (Indian Institute of Technology Ropar, Punjab, India) for working
on the projects and publishing the papers with me. I would like to thank my
colleagues from the string theory and high energy physics group Shivam,
Hemant, Jitendra, Salman, Debarshi, Pushpa, Sumit and Abhishek for the
valuable discussions, comments and questions. I am very grateful to the two
former students from our research group Prof. Pramod Shukla (Bose Institute,
Kolkata, India) and Dr. Karunva Sil (University of Cyprus) for their very
helpful suggestions and guidance to me which helped me in getting a post-
doctoral position. I am very grateful to Prof. Keshav Dasgupta for the help
related to academia.

I would like to thank Prof. Juan Maldacena (IAS, Princeton, USA), Prof.
Andreas Karch (The University of Texas, Austin, USA), Prof. Tadashi Takayanagi
(YITP, Kyoto University, Kyoto, Japan), Prof. Kostas Skenderis (University of
Southampton, UK), Prof. Thomas Hartman (Cornell University), Prof. Suvrat Raju
(ICTS, Bangalore), Prof. David Berenstein (University of California, Santa
Barbara, USA), Prof. Xi Dong (University of California, Santa Barbara, USA),
Prof. Arpan Bhattacharyya (IIT Gandhinagar, India), Prof. Christoph Uhlemann
(University of Oxford), Prof. Rong-Xin Miao (Sun Yat-Sen University, China),
Prof. Xuanhua Wang (UCAS, Wenzhou, China), Prof. Xian-Hui Ge (Shanghai
University, China), Prof. Mohsen Alishahiha (IPM, Tehran, Iran), Prof. Sanjay
Siwach (BHU, India), Prof. Chethan Krishnan (IISc Bangalore) for the useful
discussions and comments during my PhD. I would also like to thank Prof.
Suresh Tiwari, Prof. Rudra Prakash Malik, and Prof. Bhabani Prasad Mandal from
the Banaras Hindu University (BHU), India, who helped me to start my journey
into the Theoretical High Energy Physics during my master’s.

I would also like to thank the editors, associate editors, and referees of the
Journal of High Energy Physics (JHEP), Physical Review D (PRD), Physics
Letters B (PLB), and European Physical Journal C (EPJC) for their comments,
criticism, etc. Those comments helped me in learning something new which was
missing in my papers. I am very grateful to the organizers (including speakers
and participants) of the various conferences/workshops/schools for organizing
very interesting events. I enjoyed those events and learned about the many new
research topics.

I would like to thank the members of Indian Physics Association (IPA) Roorkee
chapter. I enjoyed working with them. The members include, Prof. Akhilesh
Kumar Mishra, Prof. Anjani Kumar Tiwari, Ishani, Aalok, Jaikhomba, Nifeeya,
Prem Sagar, Priyanshu, Ritika, Shikha, Tsewang, Vanshaj, Vidisha.

I would like to extend my gratitude to all the faculty members, staff, and to
students, and postdocs in the Department of Physics at I.I.T. Roorkee for
making this a memorable journey. I would like to thank my colleagues from the
research scholar room 2: Shakti, Soumya, Monika, Amarjyoti, Sonu, Pooja,
Pooja, Manish, Veer and Mukesh for bearing me and making the workplace
vibrant. I also thank my colleagues from the all the research scholar rooms. I
thank my friends Shailesh, Rahul, Anurag, Nandeshwar, Sarvesh, Amit, Ramnivas,
Shubhajit for their time to time motivation and encouragement.

I am very thankful to have wonderful seniors who treated me like younger
brother: Shubham, Naveen, Aalok, Dibyendu. I specially thank Prem Sagar who
took care of me always during my stay at IIT Roorkee and Deepak for the
wonderful company. I would like to thank Priyanka, Pooja and Jaikhomba who
helped in various way during pre-synopsis and thesis submission. I am very
thankful to Anil, Ramashish, Amarjyoti, Shakti and Soumya for their real life
knowledge, I feel everyone should have friends like them but please don’t
follow them otherwise you may get bitten at some places. I am also thankful to
Shivam and Shubranshu for their valuable time and discussion on the real life
issues. It was a great pleasure to have Manish, Brij Mohan, Ashutosh, Arun,
Harsh, Ishitwa, Shakti at IIT Roorkee who are the alumni from BHU. I thank my
friends Divya, Pavneet, Ankur, Anshu, Hemant, Anju, Rahul, Ashish, Tejaswini,
Gagan, Uwais for their valuable time and discussions (on various topics)
during my stay at IITR.

I am very grateful to my family for their continuous support and everything. I
am very grateful to my parents (Shri Shyam Narayan Yadav and Shrimati Chinta
Devi) who always left me free to do what I wanted to do. I was very fortunate
to learn many things about life from my grandfather (Late Shri Ghurfekan
Yadav). Seeing me as “Dr. Gopal Yadav” was his dream, I think he will be very
happy for me wherever he will be now. I thank my maternal uncle and aunty,
“Phupha” and “Bua”, and brother in laws who always encouraged me to believe in
myself and for their support. I am very fortunate to have bhaiya and bhabhi,
Rinki di, Priyanka di, Shaloo, Prity, Sandip, Arvind bhaiya, Santosh bhaiya,
Ravi bhaiya, Bunty, Aman, Aditya, Mahesh, Vipul, Pooja, Ranjita di, Anjali,
Akriti, Ankita, Shrija, Devansh, Shreya, Shivansh, Tejas in my life.

I am very grateful to my school teachers, college/university professors, and
all of my gurus in various fields for educating me in various aspects of life.
Education can bring changes and what I am today is the perfect example of it.

Dedicated to my Family and in the loving memory of my grandfather (Late Shri
Ghurfekan Yadav)

“HOWEVER DIFFICULT LIFE MAY SEEM, THERE IS ALWAYS SOMETHING YOU CAN DO AND
SUCCEED AT. IT MATTERS THAT YOU DON’T JUST GIVE UP.” - STEPHEN HAWKING

###### Contents

  1. Acknowledgments
  2. 1 Introduction
    1. 1.1 Introduction
    2. 1.2 AdS-CFT Duality
      1. 1.2.1 Hints for String Dual of QCD
      2. 1.2.2 Maldacena’s Conjecture
      3. 1.2.3 D-branes and Strings
      4. 1.2.4 Open String Perspective
      5. 1.2.5 Closed String Perspective
      6. 1.2.6 Combining Both Perspectives
    3. 1.3 Gauge-Gravity Duality
      1. 1.3.1 Klebanov-Witten Model
      2. 1.3.2 Klebanov-Tsytlin Model
      3. 1.3.3 Seiberg Duality Cascade
      4. 1.3.4 Resolved Conifold and Flavor $D7$ Brane Embedding
    4. 1.4 UV Complete Top-Down Holographic Dual of Thermal QCD
      1. 1.4.1 Type IIB String Dual of Large-$N$ Thermal QCD
      2. 1.4.2 Type IIA SYZ Mirror of [14]
      3. 1.4.3 ${\cal M}$-theory Uplift of Type IIB Setup
      4. 1.4.4 ${\cal M}$-Theory Uplift Inclusive of ${\cal O}(R^{4})$-Corrections
  3. 2 SU(3) LECs from Type IIA String Theory
    1. 2.1 Introduction
    2. 2.2 $SU(3)$ Chiral Perturbation Theory Lagrangian and the Phenomenological Values of LECs
    3. 2.3 Holographic Computation of $SU(3)$ LECs
    4. 2.4 Summary
  4. 3 Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation
    1. 3.1 Introduction
    2. 3.2 Deconfinement Phase Transition in Thermal QCD-Like Theories in the Absence of Rotation
      1. 3.2.1 $T_{c}$ from Semi-classical Method
        1. 3.2.1.1 Black Hole Background Uplift (Relevant to $T>T_{c}$) and Holographic Renormalization of On-Shell $D=11$ Action
        2. 3.2.1.2 Thermal Background Uplift (Relevant to $T<T_{c}$) and Holographic Renormalization of On-Shell $D=11$ Action
        3. 3.2.1.3 $T_{c}$ Inclusive of ${\cal O}(R^{4})$ Corrections
    3. 3.3 Wald Entropy, $T_{c}$ from Entanglement Entropy and M$\chi$PT Compatibility
      1. 3.3.1 Wald Entropy
      2. 3.3.2 Deconfinement from Entanglement Entropy
      3. 3.3.3 ${\cal M}\chi$PT Compatibility
    4. 3.4 Deconfinement Temperature of Rotating QGP at Intermediate Coupling from ${\cal M}$-Theory
      1. 3.4.1 Top-Down Holographic Dual of Rotating QGP
      2. 3.4.2 Deconfinement Temperature of Rotating QGP from ${\cal M}$-Theory Dual
      3. 3.4.3 UV-IR Mixing, Flavor Memory Effect and Non-Renormalization of $T_{c}$ in Rotating QGP
    5. 3.5 Summary
  5. 4 Conclusion and Future Outlook
  6. 5 Introduction
    1. 5.1 Holographic Entanglement Entropy: Ryu-Takayanagi and Dong’s Proposals
    2. 5.2 Hawking’s Information Paradox and Page Curve
    3. 5.3 Resolution of Information Paradox from Holography
      1. 5.3.1 Island Proposal and its Extension to HD Gravity
      2. 5.3.2 Doubly Holographic Setup
      3. 5.3.3 Wedge Holography
  7. 6 Page Curves of Reissner-Nordström Black Hole in HD Gravity
    1. 6.1 Introduction and Motivation
    2. 6.2 Review
      1. 6.2.1 Brief Review of Charged Black Hole in Einstein-Gauss-Bonnet Gravity
      2. 6.2.2 Review of Page Curve of Reissner-Nordström Black Hole
        1. 6.2.2.1 Entanglement Entropy without Island
        2. 6.2.2.2 Entanglement Entropy with Island
        3. 6.2.2.3 Page Curve
    3. 6.3 Page Curves of Charged Black Hole in HD Gravity up to ${\cal O}(R^{2})$
    4. 6.4 Page Curves of Charged Black Hole in Einstein-Gauss-Bonnet Gravity
    5. 6.5 Conclusion and Discussion
  8. 7 Entanglement Entropy and Page Curve from the ${\cal M}$-Theory Dual of Thermal QCD Above $T_{c}$ at Intermediate Coupling
    1. 7.1 Introduction and Motivation
    2. 7.2 Doubly Holographic Setup in ${\cal M}$ Theory
    3. 7.3 ETW-“brane” Embedding
      1. 7.3.1 At ${\cal O}(\beta^{0})$
      2. 7.3.2 No Boundary Terms Generated at ${\cal O}(R^{4})$
    4. 7.4 Page Curve without Higher Derivative Terms
      1. 7.4.1 Entanglement Entropy Contribution from Hartman-Maldacena-like Surface
        1. 7.4.1.1 Hartman-Maldacena-like Surface Analytics/Numerics
      2. 7.4.2 Entanglement Entropy Contribution from Island Surface
        1. 7.4.2.1 Island Surface Analytics
      3. 7.4.3 Page curve at LO from Areas of Hartman-Maldacena-like and Island Surfaces
    5. 7.5 Page Curve with Higher Derivative Terms
      1. 7.5.1 Entanglement Entropy Contribution from Hartman-Maldacena-like Surface
      2. 7.5.2 “Swiss-Cheese” Structure of $S_{\rm EE}^{\beta^{0},{\rm HM}}$ in a Large-$N$-Scenario
      3. 7.5.3 Entanglement Entropy Contribution from Island Surface
      4. 7.5.4 Page Curve and An Exponential(-in-$N$) Hierarchy of Entanglement Entropies up to ${\cal O}(R^{4})$ Before/After the Page Time
    6. 7.6 Massless Graviton - the Physical reason for Exponentially Suppressed Entanglement Entropies
    7. 7.7 Summary
  9. 8 Black Hole Islands in Multi-Event Horizon Space-Times
    1. 8.1 Introduction and Motivation
    2. 8.2 Preliminary
      1. 8.2.1 Schwarzschild de-Sitter Black Hole
      2. 8.2.2 Thermal Opaque Membranes in SdS Space-Time
      3. 8.2.3 Bulk Gravitational Effect Near Thermal Opaque Membranes
    3. 8.3 Black Hole Islands in Schwarzschild de-Sitter Black Hole
      1. 8.3.1 No Island Phase
      2. 8.3.2 Island Phase
      3. 8.3.3 Page Curves
    4. 8.4 Information Problem in SdS Black Hole as a Whole
    5. 8.5 Conclusion
  10. 9 Multiverse in Karch-Randall Braneworld
    1. 9.1 Introduction
    2. 9.2 Emerging Multiverse from Wedge Holography
      1. 9.2.1 Anti de-Sitter Background
      2. 9.2.2 de-Sitter Background
      3. 9.2.3 Braneworld Consists of Anti de-Sitter and de-Sitter Spacetimes
    3. 9.3 Application to Information Paradox
      1. 9.3.1 Page Curve of Eternal AdS Black Holes in $n=2$ Multiverse
      2. 9.3.2 Page Curve of Schwarzschild de-Sitter Black Hole
        1. 9.3.2.1 Schwarzschild patch
        2. 9.3.2.2 de-Sitter patch
    4. 9.4 Application to Grandfather Paradox
    5. 9.5 Conclusion
  11. 10 Conclusion and Future outlook
  12. A
    1. A.1 Coupling Constants $y_{1,3,5,7}$, $z_{1,....,8}$, $F_{\pi}^{2}$ and $g_{YM}^{2}$
  13. B
    1. B.1 ${\cal O}(R^{4})$ Corrections to the ${\cal{M}}$-theory metric of [15] in the MQGP limit near the $\psi=2n\pi,n=0,1,2$-branches
    2. B.2 Thermal $f_{MN}$ EOMs, their Solutions in the IR, $4D$-Limit and ${\cal{M}}\chi$PT Compatibility
    3. B.3 ${\cal M}$-Theory Metric Associated with Rotating Cylindrical Black Hole and Thermal Backgrounds
    4. B.4 ${\cal O}(\beta)$ Contribution to the On-Shell Action Densities for the Rotating QGP $T_{c}$ Calculation
    5. B.5 Holographic Renormalization of Rotating Gravitational Backgrounds
  14. C
    1. C.1 Complex Extrinsic Curvatures and Riemann Tensors
    2. C.2 HM-Like/IS Analytics/Numerics
      1. C.2.1 HM-Like Surface Area
      2. C.2.2 IS Turning Point
    3. C.3 Hartman-Maldacena-like Surface Miscellania
    4. C.4 Island Surface Miscellania
    5. C.5 Possible Terms Appearing in Holographic Entanglement Entropies
      1. C.5.1 Hartman-Maldacena-like Surface
      2. C.5.2 Island Surface

###### List of Figures

  1. 3.1 Plot $S_{\rm connected}$ (blue) and $S_{\rm disconnected}$(orange) versus $l(r^{*})$
  2. 3.2 Plot of $\frac{T_{c}(\gamma)}{T_{c}(0)}$ versus $\omega$ for $l=0.2,0.6,1$.
  3. 5.1 Description of doubly holographic setup. Blue curves are the island surfaces and red curve is the Hartman-Maldacena surface. $\delta M$ is the conformal boundary, $z_{*}$ and $z_{T}$ are the turning points of Hartman-Maldacena and island surfaces.
  4. 5.2 Description of wedge holography. Two $d$-dimensional Karch-Randall branes joined at the $(d-1)$ dimensional defect, Karch-Randall branes are embedded in $(d+1)$-dimensional bulk.
  5. 6.1 Plot of $r_{+}$ versus $\alpha$
  6. 6.2.2 Review of Page Curve of Reissner-Nordström Black Hole
  7. 6.4 Page curve of an eternal Reissner-Nordström black hole by considering only leading order term in $G_{N}$.
  8. 6.5 Page curves of an eternal Reissner-Nordström black hole using a number of different Gauss-Bonnet couplings.
  9. 7.1 Doubly holographic setup in ${\cal M}$-theory dual. ETW-black “brane” is coupled the thermal bath (at the tip of seven-fold of $G_{2}$-structure which is a cone over a warped non-Kähler resolved conifold), where the Hawking radiation is collected, which effectively is thermal $QCD_{2+1}$ along $x^{1,2,3}$ after Wick-rotation along $x^{3}$ at $r=0$. Blue curves correspond to the island surfaces and red curve corresponds to the Hartman-Maldacena-like surface.
  10. 7.2 Page curve up to ${\cal O}(\beta^{0})$ of an eternal neutral black hole in ${\cal M}$-theory dual via doubly holographic setup. The blue line in the figure indicates the area of the Hartman-Maldacena-like surface, and the orange line refers for the area of the island surface; $E$ in (7.39) is set to $1.1$ to obtain the same order of the magnitude of $t_{\rm Page}$ as in Fig. 7.7.
  11. 7.3 Hartman-Maldacena-like surface entanglement entropy as a function of $t_{b_{0}}$.
  12. 7.4 Plot of $\tilde{S}_{\rm EE}^{\rm HM}$ as a bi-function of $\left(|c_{1}|,|c_{2}|\right)$ at $t=t_{\rm Page}=10^{6}$
  13. 7.5 $\frac{S^{\beta^{0},\ IS}_{\rm EE}}{S_{\rm BH}}$-versus-$S_{\rm BH}$ for $N=10^{3.3},M=N_{f}=3,g_{s}=0.1,\kappa_{l_{p}}=0.47$
  14. 7.6 The Page curve obtained from the use of Dong’s proposal to compute entanglement entropies of the Hartman-Maldacena-like (blue line) and island(orange line) surfaces.
  15. 7.7 Page time for $N=10^{3.3},M=N_{f}=3,g_{s}=0.1,c_{2}=-10^{8},c_{1}=-10^{3}$
  16. 7.8 Volcano potential for massless graviton from (7.125)
  17. 7.9 Graviton Wave Function localization at the horizon via the $m=0$-limit of the solution () of (7.115).
  18. 7.10 Localization of the Graviton Wave Function near the horizon from the solution of the Schrödinger-like radial wave equation (7.123).
  19. 8.1 Carter-Penrose diagram of Schwarzschild de-Sitter spacetime. The causal connections between any two of the seven wedges are broken, and the passage of time is reversed for $R,L$ regions in reference to the central region (C). If it serves the purpose, spacetime can have an additional extension made toward both sides that goes on indefinitely. The radiation regions $\mathcal{R}$ have been defined on both sides of the thermally opaque membranes, which are represented by the blue color curves. $H^{\pm}(C^{\pm})$ shows the past and future black hole horizons (cosmological horizons), $i^{\pm}$ depicts the past and future timelike infinities, and $I^{\pm}$ denotes the past and future spacelike infinities.
  20. 8.2 Carter-Penrose diagram of Schwarzschild de-Sitter spacetime. Within the black hole patch, we have included an island surface denoted by the notation $\mathcal{I}$ (the red curve) apart from the radiation regions $\mathcal{R}$.
  21. 8.3 Black hole patch’s Page curves for $3M\sqrt{\Lambda}=0.4$(green) and $3M\sqrt{\Lambda}=0.5$(magenta). Page times for $3M\sqrt{\Lambda}=0.4$, $0.5$ are $t_{P_{1}}\approx 26356.8$ and $t_{P_{2}}\approx 56787.5$ respectively.
  22. 9.1 Multiverse constructed from $2n$ Karch-Randall branes ($Q_{-n,-n+1,...,1,2,...,n-1,n}$) which are $d$-dimensional gravitating objects and these branes are embedded in the $(d+1)$-dimensional bulk. The dimension of defect $P$ is $(d-1)$.
  23. 9.2 Multiverse’s cartoon picture when $n=3$ in AdS spacetimes.
  24. 9.3 The figure describing the multiverse when $n=3$ with de-Sitter branes ($Q_{-1/1,-2/2,-3/3}$) and $(d-1)$-dimensional defect $P$.
  25. 9.4 Braneworld consists of $d$-dimensional anti de-Sitter and de-Sitter spacetimes. AdS spacetimes are embedded in the bulk (5.20) where as de-Sitter spacetimes are embedded in the bulk spacetime with metric (9.7). We have used $n_{1}=n_{2}=3$ to draw this figure.
  26. 9.5 The green and yellow curves indicate island surfaces across $Q_{-n}$ and $Q_{n}$, and $Q_{-1}$ and $Q_{1}$. The red curve depicts the Hartman-Maldacena surface from the defect to its thermofield double partner. The bulk AdS boundary is $\delta M$.
  27. 9.6 Page curve of eternal AdS black holes for $n=2$ multiverse.
  28. 9.7 Schwarzschild de-Sitter black hole realization in wedge holography. Black hole and cosmology island surfaces, or in our instance, black hole and de-Sitter horizons, are denoted by $I_{s}$ and $I_{c}$. Schwarzschild and de-Sitter patches’ Hartman-Maldacena surfaces are represented by the red (${HM}_{s}$) and green (${HM}_{c}$) lines. Schwarzschild and de-Sitter patches make up the branes $Q_{s}$ and $Q_{1}$, respectively. Black hole and cosmological horizons emit Hawking and Gibbons-Hawking radiation, which is collected by the baths $Q_{-s}$ and $Q_{-1}$.
  29. 9.8 Page curve of Schwarzschild patch.
  30. 9.9 Multiple universes where various individuals live is $Q_{-1,-2,-3,1,2,3}$.

###### List of Tables

  1. 1.1 The Type IIB Brane Construct of [14]
  2. 2.1 $SU(3)$ NLO LECs $L_{i}^{r}(i=1,2,...,8)$ from various references.
  3. 2.2 Phenomenological Values of the 1-loop renormalised couplings $L_{i}^{r}(M_{\rho})$ of () [16]. Last column shows the source to extract this information.
  4. 2.3 An Overview of the agreement between the theoretical and experimental Values of one-Loop Renormalized $\chi$PT LECs, $F_{\pi}^{2}$, and $g_{SU(3)}^{2}$.

Part-I
Aspects of Thermal QCD Phenomenology at Intermediate Gauge/’t Hooft Coupling
from String/M-Theory

“I was an ordinary person who studied hard. There are no miracle people. It
happens they get interested in this thing and they learn all this stuff, but
they’re just people.” - Richard Feynman

##  Chapter 1 Introduction

###  1.1 Introduction

Nature has four kinds of interactions: strong, electromagnetic, weak, and
gravitational. These four interactions describe our universe. Strong
interaction governs the behavior of quarks inside the nucleons (neutron and
proton) or, more broadly, the interaction between nucleons inside the nucleus;
electromagnetic interaction mediates the interaction between the charged
particles; the nuclear reaction is happening inside the core of the Sun
because of weak interaction, gravitational interaction is responsible for the
stability of celestial objects. A very nice theoretical framework known as
quantum field theory (QFT) has described the three interactions except for
gravity. The strong and electromagnetic interactions have been studied from
the quantum chromodynamics (QCD) and quantum electrodynamics (QED),
respectively; similarly, one is also able to study the electroweak interaction
(unified theory of electromagnetic and weak interactions) from QFT. But when
we try to study the gravitational interaction from the QFT technique, then we
encounter many divergences in theory.

String theory turns out to be a very nice theory because it unifies all the
interactions of nature. The same can be seen from the spectrum of the
“string”, which contains all the particles of nature, including graviton.
Since graviton is the mediator of gravitational interaction, and hence we can
say the sting theory is the quantum theory of gravity. String theory
originated in the 1960s to study the QCD, but it could not explain QCD
adequately at that time. It was found that this theory unifies all the
interactions. Initially, it contained only bosonic degrees of freedom, and the
resulting theory was named “bosonic string theory” where the dimension of
space-time is 26. Fermions were incorporated in string theory using the idea
of supersymmetry (SUSY); the theory is known as “superstring theory”. Five
versions of superstring theory were proposed: type IIA, type IIB, type I,
Heterotic $E_{8}\times E_{8}$, and Heterotic $SO(32)$. In 1995, it was shown
by Witten that these five theories could be unified into a single theory known
as “${\cal M}$-theory”, and these are related to each other via various
dualities, e.g., T-duality, S-duality, etc.

In [17], a very nice duality was proposed between string theory and gauge
theory; the AdS/CFT correspondence. This duality relates the strongly coupled
gauge theories with weakly coupled gravitational theories; generally, this is
called “gauge-gravity duality”. Using the mapping between the parameters of
these theories, one can access the part of the gauge theories which were not
earlier possible because of the limitation of the gauge theories. This duality
has been very useful in various branches of physics: condensed matter physics,
black holes, cosmology, QCD, etc. A very nice return: a theory that was
disregarded because of its limitation in the 1960s to explain QCD is now able
to explain the many interesting features of QCD very beautifully from the
gauge-gravity duality.

###  1.2 AdS-CFT Duality

In this section, we will discuss the AdS/CFT duality. This will be
accomplished via various subsections. We start with the discussion on the
hints of the string dual of QCD in 1.2.1. Maldacena’s conjecture will be
discussed in 1.2.2. This conjecture will be realized in detail using 1.2.3,
1.2.4, 1.2.5, and 1.2.6.

####  1.2.1 Hints for String Dual of QCD

Heuristically, the desire to reformulate QCD as a string theory comes from the
presence of flux tubes between quark-antiquark pairs in QCD, which aid in the
confinement of these pairs. Regge behaviour is produced when a string is used
as a model for the flux tubes. The relationship between the flux tube’s mass
$M$ and its angular momentum $J$, which is defined as $M^{2}\propto J$,
determines the flux tube’s regge behaviour. Regge behaviour was also observed
in the spectrum of the mesons. Non-confining gauge theories, however, are not
covered by this reasoning. A strong case may be made for the existence of a
string counterpart of quantum gravity or any confining gauge theory by taking
into account ’t Hooft’s large-$N_{c}$ limit. Due to dimensional reduction, the
gauge group of QCD, $SU(3)$, does not have an expansion parameter. By changing
$SU(3)$ to $SU(N_{c})$, taking the limit $N_{c}\rightarrow\infty$, and
expanding in $1/N_{c}$, ’t Hooft generalised QCD. With $i,j=1,...,N_{c}$ and
$a=1,...,N_{f}$, respectively, the quark and gluon fields $q^{i}_{a}$ and
$A^{i}_{\mu}j$ represent the degrees of freedom of the generalised theory. The
number of independent gauge fields is decreased to $N_{c}^{2}-1$ since the
gauge group is now $SU(N_{c})$ rather than $U(N_{c})$, but since the working
limit is $N_{c}\rightarrow\infty$, this discrepancy can be disregarded. The
number of gluons is therefore assumed to be $N_{c}^{2}$. Thus $N_{c}^{2}$ is
considerably larger than $N_{f}$, which represents the number of quark degrees
of freedom. Gluons therefore control dynamics in the large-$N_{c}$ limit. One-
loop self-energy of the gluon Scale for the Feynman diagram is
$g^{2}_{YM}N_{c}$. If, in addition to limit $N_{c}\rightarrow\infty$, one
simultaneously sets a limit on the gauge coupling given as $g_{YM}\rightarrow
0$ while holding the ’t Hooft coupling, $\lambda\equiv g^{2}_{YM}N_{c}$,
constant, the Feynman diagram has a smooth limit. This is analogous to
insisting that the confinement scale, $\Lambda_{QCD}$, in the large-$N_{c}$
limit stay constant. The one-loop $\beta$ function is specified as,

$\displaystyle\mu\frac{d}{d\mu}g^{2}_{YM}\propto-N_{c}g^{4}_{YM}.$ (1.1)

It becomes independent of $N_{c}$ when expressed in terms of $\lambda$.

$\displaystyle\mu\frac{d}{d\mu}\lambda\propto-\lambda^{2}.$ (1.2)

It is simpler to determine the $N_{c}$-scaling of Feynman diagrams by using
double-line notation. This is done by swapping out the line connected to a
gluon for a pair of lines connected to a quark and an antiquark. Feynman
diagrams have the expansion in terms of the powers of $\lambda$ and $1/N_{c}$.
To specifically illustrate this, consider the examples of one-, two-, and
three-loop vacuum diagrams. While they scale to the same power of $N_{c}$,
they do so at a different power of $\lambda^{l-1}$, where $l$ is the number of
loops. Diagram classification is greatly influenced by topology, and non-
planar diagrams are suppressed in the large-$N_{c}$ limit. The Feynman
diagrams’ relationship to string theory is established by the topology-based
classification of the diagrams; the relationship is further clarified by tying
a Riemann surface to each diagram. In a Feynman diagram written in double-line
notation, each closed loop line can be viewed as the edge of a two-dimensional
surface. These surfaces are joined at their boundaries to form a Riemann
surface. One adds the point at infinity to the face connected to the diagram’s
exterior line to produce a compact surface. A non-planar diagram’s Riemann
surface is a torus as opposed to a sphere for a planar diagram. Scaling for a
particular Feynman diagram is as follows: $N^{\chi}_{c}$, where $\chi$ is the
associated Riemann surface’s Euler number.

$\chi=2-2g,$ (1.3)

where the boundaryless, orientable, compact surface is represented by the
genus $g$. For the sphere, $\chi=2$ and for the torus, $\chi=0$. Hence, the
amplitude of Feynman diagrams can be expressed as

$\displaystyle{\cal
A}=\sum^{\infty}_{g=0}N^{\chi}_{c}\sum^{\infty}_{n=0}c_{g,n}\lambda^{n},$
(1.4)

where $c,{g,n}$ are the constants. Every confining gauge theory with Yang-
Mills fields and matter in adjoint representation is covered by the analysis.
A boundary in the Riemann surface that is connected to a Feynmann diagram is
introduced when matter or quarks are added to the theory. The Feynmann
diagram’s power of $N_{c}$, which stays $N^{\chi}_{c}$ in the presence of
matter, is unaffected by the existence of matter, but the Euler number changes
to $\chi=2-2g-b$, where $b$ is the number of boundaries. Open strings are
connected to the boundaries. As a result, for a theory that includes both
closed and open strings, the sum over the number of boundaries is recognised
as an expansion. The genus expansion of a string theory corresponds to the
large-$N_{c}$ expansion of a gauge theory. As a result, the classical limit of
the string theory matches the planar limit of the gauge theory. The AdS/CFT
correspondence is the result of this duality.

####  1.2.2 Maldacena’s Conjecture

According to the Maldacena’s conjecture, commonly referred to as the AdS/CFT
correspondence, gravity theories in AdS are related to conformal field
theories on the boundary of AdS [17]. In the beginning, it was proposed that
the type IIB superstring theory on $AdS_{5}\times S^{5}$ is equivalent to
${\cal N}=4$ Super Yang-Mills theory in four dimensions. The $AdS_{5}/CFT_{4}$
model can be simplified down to its most fundamental version, which says that
${\cal N}=4$ SYM theory with gauge group $SU(N_{c})$ and Yang-Mills coupling
constant $g_{YM}$ is dynamically equivalent to type IIB superstring theory
with string length $l_{s}=\sqrt{\alpha\prime}$ and coupling constant $g_{s}$
on $AdS_{5}\times S^{5}$ with radius of curvature $R$ and $N$ units of
$F_{(5)}$ flux on $S^{5}$. The two free parameters on both sides of the theory
are related in the following ways:

$g^{2}_{\rm YM}=2\pi g_{s}$

and

$2g^{2}_{\rm YM}N=R^{4}/\alpha^{\prime^{2}}.$

The conjectured duality yields the equivalence of two theories, which
indicates that there is a precise mapping between the gauge invariants and
local operators of the gauge side and the states and fields of the string
theory. This mapping can be thought of as an exact one-to-one correspondence.
Due to the fact that a complete quantum treatment of the superstring cannot be
carried out, which limits the use of duality in its most powerful form, one
must instead work with the more moderate form of duality, which can be
attained by assuming appropriate approximations. Consideration of the large-’t
Hooft coupling limit is made for a less robust variant of duality:
$N_{c}\rightarrow\infty$ and $g_{\rm YM}^{2}=g_{s}\rightarrow 0$ such that
$\lambda=g^{2}_{YM}N_{c}$ is very large. In this limit, $1/N$ expansion of the
Feynman diagrams in gauge theory are related with the expansion in terms of
the string coupling ($g_{s}$) on the string theory side.

The weaker limit of the conjecture that was mentioned earlier continues to
present some difficulties to deal with, and we need to go still deeper in
order to reach a tractable setting. We are only left with one free parameter
for the new limit, and that is $\lambda$; as a result, we are able to
investigate the behaviour at both extremes of the parameter range, whether
$\lambda$ is extremely tiny or very high. These limits appear naturally in the
D-brane picture, which is what inspired the correspondence; as a result, we
will provide a quick introduction to this duality from the perspective of this
picture in the following section.

####  1.2.3 D-branes and Strings

In superstring theory, extended objects such as strings are not the only ones
that may be specified. The theory also incorporates a wide variety of non-
perturbative higher dimensional objects that are referred to as D-branes. For
the application of $D3$-brane, see [18]. D-branes can be understood from two
distinct vantage points, which are referred to as open string and closed
string, respectively. The significance of the string coupling, denoted by
$g_{s}$, which determines the strength of the interaction that takes place
between open and closed strings is what establishes which viewpoint is
correct.

  * •

Open string perspective: In this scenario, D-branes can be seen as higher-
dimensional objects onto which open strings can terminate. This is one way to
think about them. This viewpoint is true for tiny values of the coupling
constant, such as $g_{s}<<1$, for both closed and open string. For low
energies $E<<\alpha^{\prime^{-1/2}}$, when massive string excitations are
neglected, the dynamics of the open strings can be explained by a
supersymmetric gauge theory that is based on the world volume of the D-branes.
Scalar field $\phi$ corresponds to open string excitations that are transverse
to the D-branes, while gauge field $A_{\mu}$ corresponds to open string
excitations that are parallel to the D-branes. The product $g_{s}N$ is the
effective coupling constant for a stack of it $N$ coincident D-branes with
gauge group $U(N)$, and the open string perspective is reliable for
$g_{s}N<<1$.

  * •

Closed string perspective: When viewed from this angle, D-branes can be
interpreted as solitary solutions to the problem of supergravity (low energy
limit of superstring theory). D-branes are the source of the gravitational
field that is responsible for the curvature of the spacetime that is all
around them. It is important that the length scale $L$ be somewhat big in
order to guarantee the correctness of the supergravity approximation and
maintain a low curvature. The expression $L^{4}/\alpha^{\prime^{2}}\propto
g_{s}N$ appear in the case of a stack of $N$ coincident D-branes. The closed
string perspective is only applicable for the case where $g_{s}N>>1$ is being
considered.

The $AdS_{5}/SCFT_{4}$ correspondence is produced when these two perspectives
are applied to a stack of $N$ $D3$-branes that are placed in Minkowski
spacetime. The stack of $N$ $D3$-branes extends along the Minkowskian
spacetime directions, but it is perpendicular to the other six spatial
directions.

####  1.2.4 Open String Perspective

The two types of strings in type IIB perturbative string theory for
$g_{s}N<<1$ are:

  * •

Open strings can be thought of as the excitation of a (3+1)dimensional
hyperplane, starting and ending on the D3-branes.

  * •

Closed strings are thought of being the excitation of flat spacetime in the
(9+1) dimensions.

For $N$ $D3$-branes in flat spacetime at energies $E<<\alpha^{\prime^{-1/2}}$,
one solely considers massless excitations and disregards all other stringy
excitations. The massless closed string states are organised into a ten-
dimensional $\cal{N}=1$ supergravity multiplet, whereas the massless open
string excitations are organised into a four-dimensional $\cal{N}=$4 multiplet
made up of a gauge field $A_{\mu}$, six real scalar fields $\phi^{i}$, and
fermionic superpartners. According to the transformation properties of
$D3$-branes under their world volume coordinates, the gauge field $A_{\mu}$
emerges from bosonic massless open string excitations along the $D3$-branes,
and six real scalar fields $\phi^{i}$ emerge from bosonic massless open string
excitations transverse to the $D3$-branes. For all massless string modes, the
whole effective action is written as

$\displaystyle S=S_{\rm closed}+S_{\rm open}+S_{\rm int}.$ (1.5)

In ten-dimensional supergravity action, $S_{\rm closed}$ is the closed string
modes contribution, $S_{\rm open}$ is the open string modes, and $S_{\rm int}$
is the interactions between the closed and open string modes. From the DBI
action and the Wess-Zumino term, we can deduce the action for $S_{\rm open}$
and $S_{\rm int}$. For a single $D3$-brane, we have the following Dirac-Born-
Infeld action:

$\displaystyle S_{DBI}=-\frac{1}{(2\pi)^{3}\alpha^{\prime 2}g_{s}}\int
d^{4}xe^{-\phi}\sqrt{-det({\cal P}[g]+2\pi\alpha^{\prime}F),}$ (1.6)

where the pullback metric on the $D3$-brane is denoted by the ${\cal P}[g]$.
In $\alpha^{\prime}$, expanding all the way up to the leading order

$\displaystyle S_{open}=-\frac{1}{2\pi g_{s}}\int
d^{4}x(\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}+{\cal
O}(\alpha^{\prime})),$ $\displaystyle S_{int}=-\frac{1}{8\pi g_{s}}\int
d^{4}x\phi F_{\mu\nu}F^{\mu\nu}+...$ (1.7)

In the scenario when there are $N$ coincident D3 branes, the $U(N)$ valued
gauge and scalar fields are, $A_{\mu}=A^{a}_{\mu}T_{a}$, and
$\phi^{i}=\phi^{ia}T_{a}$ respectively. It is possible to guarantee gauge
invariance by performing a trace over the gauge group when generalising the
low-energy-effective actions $S_{\rm open}$ and $S_{\rm int}$ for $N$
coincident $D3$-branes. Covariant derivatives take the role of partial
derivatives, and one must add a scalar potential V to the action $S_{\rm
open}$, with the value of V being provided as

$\displaystyle V=\frac{1}{2\pi g_{s}}\sum_{i,j}Tr[\phi^{i},\phi^{j}]^{2}.$
(1.8)

In the $\alpha^{\prime}\rightarrow 0$ (or Maldacena) limit, $S_{\rm open}$ is
reduced to the ${\cal N}=4$ SYM theory bosonic component provided $2\pi
g_{s}=g^{2}_{YM}.$ As a result of the fact that
$\kappa\propto\alpha^{\prime}\rightarrow 0$, one realises that $S_{\rm
closed}$ is equivalent to the free supergravity action in Minkowski spacetime
of dimension ten. When one performs canonical normalisation by rescaling the
dilation $\phi$ by $\kappa$, one discovers that $S_{\rm int}$ disappears and
results in the decoupling of open and closed strings.

####  1.2.5 Closed String Perspective

In the strong coupling limit $g_{s}N\rightarrow\infty$, the $N$ $D3$-branes
must be analysed using a closed string model. In this view, they are massive
charged objects that generate different forms of type IIB supergravity, making
string theory of this type possible. It can be shown that the ten-dimensional
supergravity solution of $N$ $D3$-branes, which preserves the $SO(3,1)\times
SO(6)$ isometries of spacetime, and half of the supercharges of type IIB
supergravity, is provided by:

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
H(r)^{-1/2}\eta_{\mu\nu}dx^{\mu}d^{\nu}+H(r)^{1/2}\delta_{ij}dx^{i}dx^{j},$
(1.9) $\displaystyle e^{2\phi(r)}$ $\displaystyle=$ $\displaystyle g^{2}_{s},$
(1.10) $\displaystyle C_{(4)}$ $\displaystyle=$
$\displaystyle(1-H(r)^{-1})dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}+...$
(1.11)

where $i,j=4,...,9$ and $\mu,\nu=0,...,3$, $r^{2}=\sum_{i=4}^{9}x^{2}_{i}$.
The type IIB supergravity’s eoms and (1.10) indicate that

$H(r)=1+\left(\frac{L}{r}\right)^{4},$ (1.12)

where $L^{4}=4\pi g_{s}N\alpha^{\prime^{2}}$. Due to counting the number of
coincident $D3$-branes, the flux $F_{(5)}$ through the sphere $S^{5}$ is
quantized. According to the value of r relative to L, the background can be
split into two regions:

  * •

$r\gg L$: $H(r)$ can be roughly estimated in this region by 1. The metric
(1.9) reduces to 10-dimensional flat spacetime for this value of $H(r)$.

  * •

$r\ll L$: A rough approximation of $H(r)$ in this region is $L^{4}/r^{4}$. The
metric can be roughly described as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{L^{2}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{L^{2}}{r^{2}}\delta_{ij}dx^{i}dx^{j}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2})+L^{2}ds^{2}_{S^{5}},$
(1.13)

where $z=L^{2}/r$, and $\delta_{ij}dx^{i}dx^{j}=dr^{2}+r^{2}ds^{2}_{S^{5}}$.
In (1.13), first term is the $AdS_{5}$ metric in the second line. Therefore,
there are two types of closed strings based on the region of interest.

  * •

Closed strings moving through a 10-dimensional spacetime that is flat, which
is the region corresponding to the limit $r\gg L$.

  * •

Closed strings that are able to propagate in the near horizon region
($AdS_{5}\times S^{5}$) in the limit $r\ll L$.

Both of the types of closed strings that were discussed before are decoupled
when the low energy limit is imposed. In conclusion, there are two regions: an
asymptotically flat region and a near-horizon region. Both of these regions
are distinct from one another. The dynamics of closed strings are described by
type IIB supergravity modes in 10-dimensional flat spacetime in asymptotically
flat spacetime; however, in the near-horizon region, the dynamics of closed
strings are described by string excitations about the solution of type IIB
supergravity. Both classes of strings get decoupled in the low energy limit.

####  1.2.6 Combining Both Perspectives

We obtain two decoupled effective theories in the low-energy limit from both
the open and the closed string viewpoints.

  * •

Closed string perspective: Type IIB supergravity in asymptotically flat
$\mathbb{R}^{9,1}$ spacetime and type IIB supergravity on $AdS_{5}\times
S^{5}$ background.

  * •

Open string perspective: ${\cal N}=4$ super Yang-Mills (SYM) theory in four
dimensional Minkowski spacetime and type IIB supergravity on flat
$\mathbb{R}^{9,1}$ spacetime.

The conclusion that the two views are dual to one other-exactly what the
Maldacena conjectured-is made possible by the fact that physics is unaffected
by the perspective chosen and by the existence of type IIB supergravity modes
on $\mathbb{R}^{9,1}$ in both perspectives.

###  1.3 Gauge-Gravity Duality

The duality proposed by Maldacena in [17] is a special case, and it relates
the ${\cal N}=4$ supersymmetric Yang-Mills (SYM) theory and type IIB string
theory on the $AdS_{5}\times S^{5}$ background. But the gauge theories can be
non-conformal and non-supersymmetric, e.g., QCD. Therefore to construct the
gravity dual of such theories, one is required to break the conformal symmetry
and supersymmetry. The gauge-gravity duality incorporates all such theories;
conformal, non-conformal, supersymmetric, non-supersymmetric, etc.

In 1.2, we argued that one could state the duality in the low energy limit
where the open and closed string degrees of freedom decouple. In [17], this
decoupling is responsible for the duality between type IIB superstring theory
and the maximally supersymmetric Yang-Mills theory. We can construct the
holographic dual from the supergravity solution of the string theories in the
absence of stringy corrections. But the problem with the supergravity versions
is the absence of decoupling, as mentioned above, of the string degrees of
freedom (open and closed). To achieve the same, we have to include the stringy
corrections to the supergravity solutions, and it is not an easy task; see
related work in [19, 20, 21, 22, 23, 24, 25].

See [26],[27],[28],[29],[30],[31] and [32] where many techniques have been
discussed to obtain the non-conformal and non-supersymmetric holographic dual
from the configuration of $D$-branes. We can break the SUSY from the stack of
$D$-branes at the conical singularity of the conifold geometry. In these kinds
of setups, the gravity dual involves an internal manifold as a conifold in the
Calabi-Yau space. For the literature on the large-$N$ QCD defined on $S^{3}$
and $S^{1}\times S^{3}$ where the authors have discussed the various features
of QCD, see [33, 34, 35, 36, 37]. For the holographic description of QCD, see
[38, 39, 40, 41, 42, 43]. The $D$-branes have been used to study the inflation
in [44, 45].

####  1.3.1 Klebanov-Witten Model

To break the SUSY, authors in [26] placed $N$ $D3$-branes at the tip of the
conifold where the conifold is a cone in six dimensions whose base is five-
dimensional ($T^{1,1}$ with topology $S^{2}\times S^{3}$, $T^{1,1}$ has
$(SU(2)\times SU(2))/U(1)$ symmetry). The conifold generates a superpotential
on the gauge theory side as a result of breaking of SUSY to ${\cal N}=1$. The
gauge group in this setup is $SU(N)\times SU(N)$ and this is coupled to four
chiral fields $A_{1},A_{2},B_{1},B_{2}$, transformation of $A_{1},A_{2}$ and
$B_{1},B_{2}$ is as $(N,\bar{N})$ and $(\bar{N},N)$ respectively, under the
aforementioned gauge group. The stack of $N$ $D3$ branes on the gauge theory
side give rise to the geometry $AdS_{5}\times T^{1,1}$ space-time with the
metric on the gravity dual side

$ds^{2}_{10}=h^{-1/2}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+h^{1/2}ds^{2}_{6}$ (1.14)

where the warp factor $h$ is defined as: $h(r)=1+\frac{L^{4}}{r^{4}}$ with
$L^{4}=4\pi g_{s}N\alpha^{\prime^{2}}$. The conifold has the following metric

$ds^{2}_{6}=dr^{2}+r^{2}d^{2}_{T^{1,1}},$ (1.15)

where

$ds^{2}_{T^{1,1}}=\frac{1}{9}(d\psi+\cos\theta_{1}d\phi_{1}+\cos\theta_{2}d\phi_{2})^{2}+\frac{1}{2}\sum^{2}_{i=1}(d\theta^{2}_{i}+\sin^{2}\theta_{i}d\phi^{2}_{i}).$
(1.16)

It has been discussed that there is a relationship of the couplings $g_{1}$
and $g_{2}$ related with the product gauge group $S(N)\times SU(N)$ and the
integral of $NS-NS$ $B_{2}$ and R-R two forms $C_{2}$ over the two-cycle
$S^{2}$ of $T^{1,1}$. If we consider only $D3$ branes, then dilaton and
$B_{2}$ are constant; hence this implies the conformal nature of the gauge
theory.

####  1.3.2 Klebanov-Tsytlin Model

The attempt to break the conformal symmetry and supersymmetry was done in [46]
by including the additional $M$ fractional $D3$ branes apart from the $N$ $D3$
branes at the tip of the conifold. Fractional $D3$ branes are the $D5$ branes
wrapping $S^{2}$ of the $T^{1,1}$ space mentioned in the previous subsection.
Although these $M$ branes break the conformal symmetry in the gauge theory, we
still have ${\cal N}=1$ supersymmetry in theory. Therefore we have a
supersymmetric non-conformal gauge theory. Due to the addition of the extra
$M$ fractional $D3$ branes into the setup, the gauge group changes to
$SU(N+M)\times SU(N)$, and accordingly, the chiral superfields now have the
transformations under the product gauge group as, $A_{i}$ transforms as
$(N+M,\overline{N})$ and $B_{i}$ transforms as $(\overline{N+M},N)$. The
supergravity solution of this setup has three and five-form fluxes written
below

$\displaystyle F_{3}=M\omega_{3};\ B_{2}=3g_{s}M\omega_{2}\ln(r/r_{0});\
H_{3}=dB_{2}=3g_{s}M\frac{1}{r}dr\wedge\omega_{2},$ (1.17)
$\displaystyle\tilde{F}_{5}={\cal F}_{5}+*_{10}{\cal F}_{5};\ {\cal
F}_{5}=\left(N+\frac{3}{2\pi}g_{s}M^{2}\ln(r/r_{0})\right){\rm vol}(T^{1,1}),$

where $\omega_{2}$ and $\omega_{3}$ are the basis of the conifold’s base, five
form flux includes the backreaction of $F_{3}$. The effective number of $D3$
branes is now defined as

$N_{eff}(r)=N+\frac{3}{2\pi}g_{s}M^{2}\log\left(\frac{r}{r_{0}}\right)$ (1.18)

(1.18) implies the decrease of $N_{eff}$ when $r\rightarrow 0$ In the
Kelbanov-Tsytlin setup, the 10-dimensional warp factor has the following form
in the near-horizon limit.

$h(r)=\frac{L^{4}}{r^{4}}\left(1+\frac{3g_{s}M^{2}}{2\pi N}\log r\right)$
(1.19)

From (1.17), we see that $B_{2}$ is not constant anymore, and it depends upon
the logarithmic of the radial coordinate. Therefore as discussed in 1.3.1, due
to the non-vanishing nature of $B_{2}$, the gauge couplings now run with the
radial coordinate, and hence we now have a non-conformal theory. Equations
(1.17) and (1.19) implying that ${\cal F}_{5}$ and $h(r)$ will be zero at some
radial distance and KT solution is singular in the infra-red (IR).

####  1.3.3 Seiberg Duality Cascade

The problem with the Klebanov-Strassler (KS) solution is that it is singular
in the IR; this singularity should be removed. The KS model also described the
confinement on the gauge theory side. As one goes from the UV to IR, the flux
through the vanishing $S^{2}$ $(\int_{S_{2}}B_{2})$ decreases by one unit when
r decreases, whereas the five form flux decreases by $M$ units, and hence
$N_{\rm eff}\rightarrow N_{\rm eff}-M$. This starts the Seiberg duality in
this setup, and gauge group $SU(N+M)\times SU(N)$ changes to $SU(N)\times
SU(N-M)$. By performing the repeated Seiberg duality cascade, in the end, we
have $SU(M)$ gauge theory in the IR. The series of Seiberg duality performed
as above is known as the “Seiberg Duality Cascade”.

One can interpret this from the gauge theory perspective as follows. When the
Seiberg duality cascade ends, $U(1)_{R}$ symmetry of the chiral fields is
broken into $Z_{2M}$ in the presence of $M$ fractional $D3$-branes. In the IR,
$Z_{2M}$ symmetry is now spontaneously broken into the group $Z_{2}$ because
of gaugino condensation, which results the transformation of the singular
conifold into a deformed conifold. The Klebanov-Strassler solution is a
deformed conifold in the IR, which behaves as the Klebanov-Tsytlin solution in
the UV. One can remove the singularity of the Klebanov-Tsytlin solution in the
IR from the KS solution, but the KS solution is still singular in the UV. That
is why one is required to modify the KS solution in the UV, and by doing so,
one ends up with a UV complete theory.

####  1.3.4 Resolved Conifold and Flavor $D7$ Brane Embedding

The singularity appearing in the KT solution has been removed in the KS
solution via the deformation of the three cycle ($S^{3}$) in the IR, which
results in a deformed conifold. One can remove the singularity from the
resolution of the two cycle ($S^{2}$) as well, and the geometry is known as
the resolved conifold, metric for the same is given below.

$\displaystyle ds^{2}_{res}$ $\displaystyle=$
$\displaystyle\kappa(r)^{-1}dr^{2}+\frac{\kappa(r)}{9}r^{2}(d\psi+\cos\theta_{1}d\phi_{1}+\cos\theta_{2}d\phi_{2})^{2}+\frac{r^{2}}{6}(d\theta_{1}^{2}+\sin^{2}\theta_{1}d\phi_{1}^{2})$
(1.20)
$\displaystyle+\frac{r^{2}+6a^{2}}{6}(d\theta_{2}^{2}+\sin^{2}\theta_{2}d\phi_{2}^{2})$

where $\kappa(r)=\frac{r^{2}+9a^{2}}{r^{2}+6a^{2}}$, “$r$” is the radial
coordinate, and “$a$” being the resolution parameter. In this geometry, metric
vanishes on $S^{2}(\theta_{1},\phi_{1})$ in the limit $r\rightarrow 0$ whereas
the metric on resolved $S^{2}(\theta_{2},\phi_{2})$ remains
finite.$a\rightarrow 0$ limit results in a singular conifold geometry.

Authors in [47] constructed the holographic dual from the branes configuration
of $N$ $D3$-branes at the tip of the conifold and $M$ $D5$-branes wrapping the
blown up $S^{2}$. The supergravity background includes fluxes and dilaton, but
this setup does include the fundamental quarks. One can incorporate the
fundamental quarks in holographic theories by including additional flavor
branes in the probe approximation in the setup, and it was done in [46]. In
[48], the author embedded the stack of flavor $D7$-branes via Ouyang
embedding. We are interested in the holographic dual constructed in [14] where
$N_{f}$ D7-branes were embedded via Ouyang embedding with embedding parameter
$\mu$:

$(r^{2}+9a^{2}r^{4})^{1/4}e^{\frac{\iota}{2}\left(\psi-\phi_{1}-\phi_{2}\right)}\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}=\mu$
(1.21)

The presence of the flavor branes affects the dilaton profile as

$e^{-\phi}=\frac{1}{g_{s}}-\frac{N_{f}}{8\pi}\log(r^{6}+9a^{2}r^{4})-\frac{N_{f}}{2\pi}\log\left(\sin\frac{\theta_{1}}{2}\sin\frac{\theta_{2}}{2}\right)$
(1.22)

###  1.4 UV Complete Top-Down Holographic Dual of Thermal QCD

This section discusses the construction of holographic dual of thermal QCD-
like theories at the intermediate coupling. We will start with the type IIB
string dual of large-$N$ thermal QCD in 1.4.1. We will discuss the type IIA
mirror of [14] in 1.4.2 using which we discuss the ${\cal M}$-theory uplift at
finite coupling in 1.4.3. Finally we will discuss the ${\cal M}$-theory uplift
at intermediate coupling in 1.4.4

####  1.4.1 Type IIB String Dual of Large-$N$ Thermal QCD

The UV complete type IIB string dual of large-$N$ thermal QCD-like theories
has been discussed in this section, and this is based on [14]. This model has
been constructed in such a way that it removes all the drawbacks of [26],
[27], [48], [49] -[51] so that one can study the realistic QCD from the gauge-
gravity duality. Holographic thermal QCD obtained from this model has the
properties: IR confined, fundamental quarks, non-conformality, UV complete,
defined for the confined $(T<T_{c})$ and deconfined $(T>T_{c})$, both phases
of QCD and non-supersymmetric. UV completion problem of the Klebanov-Strassler
model [27] has been cured in this model.

  * •

Brane/Gauge Picture: Branes configuration of [14] involve $N$ color $D3$
branes, $M\ D5$ and $M\ \overline{D5}$-branes, $N_{f}$ flavor $D7$ and
$\overline{D_{7}}$-branes. The details about the world volume coordinates of
the branes appearing in [14] are given in table 1.1. Let us see how one
realizes the features of thermal QCD from these branes configuration.

S. No. | Branes | World Volume
---|---|---
1. | $N\ D3$ | $\mathbb{R}^{1,3}(t,x^{1,2,3})\times\\{r=0\\}$
2. | $M\ D5$ | $\mathbb{R}^{1,3}(t,x^{1,2,3})\times\\{r=0\\}\times S^{2}(\theta_{1},\phi_{1})\times{\rm NP}_{S^{2}_{a}(\theta_{2},\phi_{2})}$
3. | $M\ \overline{D5}$ | $\mathbb{R}^{1,3}(t,x^{1,2,3})\times\\{r=0\\}\times S^{2}(\theta_{1},\phi_{1})\times{\rm SP}_{S^{2}_{a}(\theta_{2},\phi_{2})}$
4. | $N_{f}\ D7$ | $\mathbb{R}^{1,3}(t,x^{1,2,3})\times\mathbb{R}_{+}(r\in[|\mu_{\rm Ouyang}|^{\frac{2}{3}},r_{\rm UV}])\times S^{3}(\theta_{1},\phi_{1},\psi)\times{\rm NP}_{S^{2}_{a}(\theta_{2},\phi_{2})}$
5. | $N_{f}\ \overline{D7}$ | $\mathbb{R}^{1,3}(t,x^{1,2,3})\times\mathbb{R}_{+}(r\in[{\cal R}_{D5/\overline{D5}}-\epsilon,r_{\rm UV}])\times S^{3}(\theta_{1},\phi_{1},\psi)\times{\rm SP}_{S^{2}_{a}(\theta_{2},\phi_{2})}$
Table 1.1: The Type IIB Brane Construct of [14]

    * –

$N$ $D3$-branes are placed at the tip of conifold geometry, $M\ D5$ and $M\
\overline{D5}$-branes have been placed at the antipodal points of the blown-up
$S^{2}$ with the separation ${\cal R}_{D5/\overline{D5}}$ between them and
share the world volume coordinates of vanishing $S^{2}$. UV, UV-IR, and IR
regions are defined as follows in this setup.

      * *

UV: $r>|\mu_{\rm Ouyang}|^{\frac{2}{3}}/{\cal R}_{D5/\overline{D5}}$

      * *

IR/IR-UV interpolating regions: $r_{0}/r_{h}<r<|\mu_{\rm
Ouyang}|^{\frac{2}{3}}/{\cal R}_{D5/\overline{D5}}$; $r\sim\Lambda$
corresponds to the deep IR region with confined $SU(M)$ theory.

    * –

Flavor $D7$-branes are embedded via Ouyang embedding [48] with embedding
equation (1.23). $N_{f}\ D7$-branes exist in the UV, UV-IR interpolating
region, and IR whereas $N_{f}\ \overline{D7}$-branes exist only in the UV and
the UV-IR interpolating regions. Due to this configuration of flavor branes,
three-form fluxes vanish in the UV, axion-dilaton modulus remains constant, UV
conformality, and absence of Landau poles are ensured in theory. This also
implies the realization of chiral symmetry breaking in QCD from
$SU(N_{f})\times SU(N_{f})$ in the UV to $SU(N_{f})$ in the IR as no
$\overline{D7}$-branes exist in the IR.

$\left(r^{6}+9a^{2}r^{4}\right)^{\frac{1}{4}}e^{\frac{i}{2}\left(\psi-\phi_{1}-\phi_{2}\right)}\sin\left(\frac{\theta_{1}}{2}\right)\sin\left(\frac{\theta_{2}}{2}\right)=\mu_{\rm
Ouyang},$ (1.23)

for vanishingly small $|\mu_{\rm Ouyang}|$.

    * –

Similar to the flavor $\overline{D7}$-branes, $\overline{D5}$-branes also
exist only in the UV-IR and UV regions. This implies the existence of
$SU(N+M)\times SU(N+M)$ color gauge group in the UV and at $r={\cal
R}_{D5/\overline{D5}}$, the aforementioned color gauge group changes to
$SU(N+M)\times SU(N)$ as soon as one enters into the IR regime [52]. The
following equation controls the flow of gauge couplings

$4\pi^{2}\left(\frac{1}{g_{SU(N+M)}^{2}}+\frac{1}{g_{SU(N)}^{2}}\right)e^{\phi}\sim\pi;\
4\pi^{2}\left(\frac{1}{g_{SU(N+M)}^{2}}-\frac{1}{g_{SU(N)}^{2}}\right)e^{\phi}\sim\frac{1}{2\pi\alpha^{\prime}}\int_{S^{2}}B_{2}.$
(1.24)

Equation (1.24) implies that one can obtain
$g_{SU(M+N)}^{2}=g_{SU(N)}^{2}=g_{YM}^{2}\sim g_{s}\equiv{\rm constant}$ only
if $\int_{S^{2}}B_{2}=0$, and hence UV conformality. This is the reason to
include the $M$ $\overline{D5}$-branes at the common boundary of the UV-IR
interpolating and the UV regions. The gauge couplings $g_{SU(N+M)}$ and
$g_{SU(N)}$ flow in opposite directions (strong and weak, respectively). The
RG flow receives a contribution from the $N_{f}$ flavor $D7$-branes too, and
hence $N_{f}\ \overline{D7}$-branes have been included in the setup.

    * –

The RG flow from the UV-to-IR triggers the Seiberg-like duality cascade in
theory, and under Seiberg duality, $SU(N+M)_{\rm strong}\xrightarrow{\rm
Seiberg\ Dual}SU(N-(M-N_{f}))_{\rm weak}$ in the IR. The application of many
Seiberg dualities results in the confined $SU(M)$ gauge theory with $N_{f}$
flavors in the IR, the finite temperature has been studied in [14], see
[53],[54], [55],[56] for some related works. The chiral symmetry breaking in
the presence of external magnetic field was discussed in [57] in top-down
Sakai-Sugimoto model.

  * •

Gravity Picture: The metric of the ten-dimensional string background [14],
also known as the modified Ouyang-Klebanov-Strassler black hole (OKS-BH)
background, is written below

$ds^{2}=\frac{1}{\sqrt{h}}\left(-g_{1}dt^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)+\sqrt{h}\biggl{[}g_{2}^{-1}dr^{2}+r^{2}d{\cal
M}_{5}^{2}\biggr{]},$ (1.25)

where the black hole functions $g_{i}$’s with the horizon $r_{h}$ are
$g_{1,2}(r,\theta_{1},\theta_{2})=1-\frac{r_{h}^{4}}{r^{4}}+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right)$. where ($\theta_{1},\theta_{2}$)
dependence arises due to ${\cal O}\left(\frac{g_{s}M^{2}}{N}\right)$
corrections. Compact metric ($d{\cal M}_{5}^{2}$) appearing in (1.25) has the
following form:

$\displaystyle d{\cal M}_{5}^{2}=h_{1}(d\psi+{\rm
cos}~{}\theta_{1}~{}d\phi_{1}+{\rm
cos}~{}\theta_{2}~{}d\phi_{2})^{2}+h_{2}(d\theta_{1}^{2}+{\rm
sin}^{2}\theta_{1}~{}d\phi_{1}^{2})+$
$\displaystyle+h_{4}(h_{3}d\theta_{2}^{2}+{\rm
sin}^{2}\theta_{2}~{}d\phi_{2}^{2})+h_{5}~{}{\rm
cos}~{}\psi\left(d\theta_{1}d\theta_{2}-{\rm sin}~{}\theta_{1}{\rm
sin}~{}\theta_{2}d\phi_{1}d\phi_{2}\right)+$ $\displaystyle+h_{5}~{}{\rm
sin}~{}\psi\left({\rm sin}~{}\theta_{1}~{}d\theta_{2}d\phi_{1}+{\rm
sin}~{}\theta_{2}~{}d\theta_{1}d\phi_{2}\right),$ (1.26)

$r\gg a,h_{5}\sim\frac{({\rm deformation\ parameter})^{2}}{r^{3}}\ll 1$ for
$r\gg({\rm deformation\ parameter})^{\frac{2}{3}}$, i.e. in the UV/IR-UV
interpolating region. The $h_{i}$’s of (• ‣ 1.4.1) and $M,N_{f}$ are not
constant. They have dependence on $g_{s},M,N_{f}$ up to linear order as given
below:

$\displaystyle\hskip-32.52127pth_{1}=\frac{1}{9}+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right),\ h_{2}=\frac{1}{6}+{\cal
O}\left(\frac{g_{s}M^{2}}{N}\right),\ h_{4}=h_{2}+\frac{a^{2}}{r^{2}},$
$\displaystyle h_{3}=1+{\cal O}\left(\frac{g_{s}M^{2}}{N}\right),\ h_{5}\neq
0.$ (1.27)

Equations (• ‣ 1.4.1) and (• ‣ 1.4.1) indicate the existence of non-extremal
resolved warped deformed conifold involving an $S^{2}$-blowup (as
$h_{4}-h_{2}=\frac{a^{2}}{r^{2}}$), an $S^{3}$-blowup (as $h_{5}\neq 0$) and
squashing of an $S^{2}$ (because $h_{3}$ is not strictly unity). The horizon
($r_{h}$) is a warped squashed $S^{2}\times S^{3}$. The ten-dimensional warp
factor that includes back-reaction has the following form in the IR

$\displaystyle\hskip-32.52127pth=\frac{L^{4}}{r^{4}}\Bigg{[}1+\frac{3g_{s}M_{\rm
eff}^{2}}{2\pi N}{\rm log}r\left\\{1+\frac{3g_{s}N^{\rm
eff}_{f}}{2\pi}\left({\rm log}r+\frac{1}{2}\right)+\frac{g_{s}N^{\rm
eff}_{f}}{4\pi}{\rm log}\left({\rm sin}\frac{\theta_{1}}{2}{\rm
sin}\frac{\theta_{2}}{2}\right)\right\\}\Biggr{]},$ (1.28)

where $L=\left(4\pi g_{s}N\right)^{\frac{1}{4}}$, and $M_{\rm eff}/N_{f}^{\rm
eff}$ are given as below.

$\displaystyle\hskip-32.52127ptM_{\rm eff}/N_{f}^{\rm eff}=M/N_{f}+\sum_{m\geq
n}(a/b)_{mn}(g_{s}N_{f})^{m}(g_{s}M)^{n}.$ (1.29)

Three form fluxes up to ${\cal O}(g_{s}N_{f})$ are given by the following
expressions in the IR (with $h_{5}=0$) [14]

$\displaystyle\hskip-28.90755pt(a){\widetilde{F}}_{3}=2M{A_{1}}\left(1+\frac{3g_{s}N_{f}}{2\pi}~{}{\rm
log}~{}r\right)~{}e_{\psi}\wedge\frac{1}{2}\left({\rm
sin}~{}\theta_{1}~{}d\theta_{1}\wedge d\phi_{1}-{B_{1}}~{}{\rm
sin}~{}\theta_{2}~{}d\theta_{2}\wedge d\phi_{2}\right)$
$\displaystyle\hskip-21.68121pt-\frac{3g_{s}MN_{f}}{4\pi}{A_{2}}~{}\frac{dr}{r}\wedge
e_{\psi}\wedge\left({\rm cot}~{}\frac{\theta_{2}}{2}~{}{\rm
sin}~{}\theta_{2}~{}d\phi_{2}-{B_{2}}~{}{\rm
cot}~{}\frac{\theta_{1}}{2}~{}{\rm sin}~{}\theta_{1}~{}d\phi_{1}\right)$
$\displaystyle\hskip-21.68121pt-\frac{3g_{s}MN_{f}}{8\pi}{A_{3}}~{}{\rm
sin}~{}\theta_{1}~{}{\rm sin}~{}\theta_{2}\left({\rm
cot}~{}\frac{\theta_{2}}{2}~{}d\theta_{1}+{B_{3}}~{}{\rm
cot}~{}\frac{\theta_{1}}{2}~{}d\theta_{2}\right)\wedge d\phi_{1}\wedge
d\phi_{2},$
$\displaystyle\hskip-28.90755pt(b)H_{3}={6g_{s}{A_{4}}M}\Biggl{(}1+\frac{9g_{s}N_{f}}{4\pi}~{}{\rm
log}~{}r+\frac{g_{s}N_{f}}{2\pi}~{}{\rm log}~{}{\rm
sin}\frac{\theta_{1}}{2}~{}{\rm sin}\frac{\theta_{2}}{2}\Biggr{)}\frac{dr}{r}$
$\displaystyle\hskip-21.68121pt\wedge\frac{1}{2}\Biggl{(}{\rm
sin}~{}\theta_{1}~{}d\theta_{1}\wedge d\phi_{1}-{B_{4}}~{}{\rm
sin}~{}\theta_{2}~{}d\theta_{2}\wedge
d\phi_{2}\Biggr{)}+\frac{3g^{2}_{s}MN_{f}}{8\pi}{A_{5}}\Biggl{(}\frac{dr}{r}\wedge
e_{\psi}-\frac{1}{2}de_{\psi}\Biggr{)}$ $\displaystyle\wedge\Biggl{(}{\rm
cot}~{}\frac{\theta_{2}}{2}~{}d\theta_{2}-{B_{5}}~{}{\rm
cot}~{}\frac{\theta_{1}}{2}~{}d\theta_{1}\Biggr{)},$ (1.30)

where asymmetry factors appearing in (• ‣ 1.4.1) are: $A_{i}=1+{\cal
O}\left(\frac{a^{2}}{r^{2}}\ {\rm or}\ \frac{a^{2}\log r}{r}\ {\rm or}\
\frac{a^{2}\log r}{r^{2}}\right)+{\cal O}\left(\frac{{\rm deformation\
parameter}^{2}}{r^{3}}\right),$ $B_{i}=1+{\cal O}\left(\frac{a^{2}\log r}{r}\
{\rm or}\ \frac{a^{2}\log r}{r^{2}}\ {\rm or}\ \frac{a^{2}\log
r}{r^{3}}\right)+{\cal O}\left(\frac{({\rm deformation\
parameter})^{2}}{r^{3}}\right)$.

As we discussed, the black hole on the gravity dual side is responsible for
the finite temperature on the thermal QCD side. This is true when $T>T_{c}$ in
QCD, where $T_{c}$ is the deconfinement temperature. When $T<T_{c}$, then the
gravitational dual involves a thermal background. We also discussed that we
had resolved $S^{2}$ and the deformation of $S^{3}$ (responsible for the IR
confinement) in this setup. Therefore, we found that gravity dual of large-$N$
thermal QCD as constructed in [14] is a resolved warped deformed conifold
where fluxes in the IR have replaced branes. The warp factor and fluxes
contain the information about back-reactions.

####  1.4.2 Type IIA SYZ Mirror of [14]

The Strominger-Yau-Zaslow (SYZ) mirror symmetry is triple T-duality along the
three isometry directions [58]. The requirement to apply the SYZ mirror
symmetry is that one must have a special Lagrangian (sLag) three-cycle,
$T^{3}$, fibered over a large base to cancel the contributions from open-
string disc instantons with boundaries as non-contractible one-cycles in the
sLag. Due to absence of the isometry along $\psi$ direction in the metric of
[14], we have to work with the coordinates $(x,y,z)$ defined in $T^{3}(x,y,z)$
which are toroidal analogue of $(\phi_{1},\phi_{2},\psi)$, and these are
related by the following equations [15]:

$x=\sqrt{h_{2}}h^{\frac{1}{4}}sin\langle\theta_{1}\rangle\langle
r\rangle\phi_{1},\
y=\sqrt{h_{4}}h^{\frac{1}{4}}sin\langle\theta_{2}\rangle\langle
r\rangle\phi_{2},\ z=\sqrt{h_{1}}\langle r\rangle h^{\frac{1}{4}}\psi,$ (1.31)

based on [59] authors in [61, 60] obtained the following results for the
$T^{2}$-invariant sLag of [59]:

$\displaystyle\hskip-14.45377pt\left.i^{*}J\right|_{\rm RC/DC}\approx 0;\
\left.\Im m\left(i^{*}\Omega\right)\right|_{\rm RC/DC}\approx 0;\ \left.\Re
e\left(i^{*}\Omega\right)\right|_{\rm RC/DC}\sim{\rm volume\
form}\left(T^{3}(x,y,z)\right),$ (1.32)

for a deformed conifold. Therefore, local $T^{3}$ of (1.31) is the sLag to
implement the SYZ mirror symmetry when resolved warped deformed conifold is
predominantly either resolved or deformed. It is interesting to note that in
the “delocalized limit” [62] $\psi=\langle\psi\rangle$, performing the
following coordinate transformation:

$\left(\begin{array}[]{c}sin\theta_{2}d\phi_{2}\\\
d\theta_{2}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}cos\langle\psi\rangle&sin\langle\psi\rangle\\\
-sin\langle\psi\rangle&cos\langle\psi\rangle\end{array}\right)\left(\begin{array}[]{c}sin\theta_{2}d\phi_{2}\\\
d\theta_{2}\end{array}\right),$ (1.33)

and
$\psi\rightarrow\psi-\cos\langle{\bar{\theta}}_{2}\rangle\phi_{2}+\cos\langle\theta_{2}\rangle\phi_{2}-\tan\langle\psi\rangle
ln\sin{\bar{\theta}}_{2}$, the term $h_{5}$ will be changed to as
$h_{5}\left[d\theta_{1}d\theta_{2}-sin\theta_{1}sin\theta_{2}d\phi_{1}d\phi_{2}\right]$,
and $e_{\psi}\rightarrow e_{\psi}$, i.e., One adds a new isometry along $\psi$
to the existing isometries along $\phi_{1,2}$. It is abundantly evident that
this does not hold true on a global scale; the deformed conifold does not have
a third global isometry. One must additionally make sure that the
$T^{3}(x,y,z)$ fibration has a large base, as described above (implicating
large complex structures of the two two-tori specified above), in order to
enable usage of SYZ-mirror duality via three T dualities. This is accomplished
through [63]:

$\displaystyle d\psi\rightarrow
d\psi+f_{1}(\theta_{1})\cos\theta_{1}d\theta_{1}+f_{2}(\theta_{2})\cos\theta_{2}d\theta_{2},$
$\displaystyle d\phi_{1,2}\rightarrow
d\phi_{1,2}-f_{1,2}(\theta_{1,2})d\theta_{1,2},$ (1.34)

for suitably chosen big values of $f_{1,2}(\theta_{1,2})$. The three-form
fluxes continue to remain invariant. It was explained in [15] why one might
select such large values of $f_{1,2}(\theta_{1,2})$. The guiding premise is
that one must have a metric that, at least locally, resembles a non-Kähler
warped resolved conifold after applying the SYZ-mirror transformation to the
non-Kähler resolved warped deformed conifold. Let’s look at the effect of
three T-dualities along the $(\phi_{1},\phi_{2},\psi)$ directions.

  * •

T-duality along $\psi$: First T-duality along $\psi$ directions implies that
$N\ D3$-branes are converted into $N\ D4$-branes wrapping the $\psi$ circle,
$N_{f}$ flavour $D7(\overline{D7})$-branes are transformed into $N_{f}$
flavour $D6(\overline{D6})$-branes, and $M$ fractional
$D3(\overline{D3})$-branes are transformed into $M\ D4(\overline{D4})$-branes
straddling a pair of orthogonal $NS5$-branes111See [64, 65] for where $NS5$
branes have been studied. ${NS5}_{1}(x^{0,1,2,3},\theta_{1},\phi_{1})$ and
${NS5}_{2}(x^{0,1,2,3},\theta_{2},\phi_{2})$.

  * •

T-duality along $\phi_{1}$: Second T-duality in the $\phi_{1}$ direction does
not change
$NS5_{1}(x^{0,1,2,3},\theta_{1},\phi_{1})$, but it does convert
$NS5_{2}(x^{0,1,2,3},\theta_{2},\phi_{2})$ into a Taub-NUT space
$(r,\psi,\theta_{2},\phi_{1})$. In addition, $N\ D4$-branes, $M\
D4(\overline{D4})$-branes, and $N_{f}$ flavor $D6(\overline{D6})$-branes will
be turned into the $N\ D5$-branes, the $M\ D5(\overline{D5})$-branes, and the
$N_{f}$ flavor $D5(\overline{D5})$-branes.

  * •

T-duality along $\phi_{2}$: The
${NS5}_{2}(x^{0,1,2,3},\theta_{2},\phi_{2})$-brane remains unchanged by the
third T-duality along the $\phi_{2}$ direction, whereas the
${NS5}_{1}(x^{0,1,2,3},\theta_{1},\phi_{1})$ is transformed into a Taub-NUT
space$(r,\psi,\theta_{1},\phi_{2})$. Further, $N\ D5$-branes, $M\
D5(\overline{D5})$-branes and $N_{f}$ flavor $D5(\overline{D5})$-branes will
be transformed into $N\ D6$-branes, $M\ D6(\overline{D6})$-branes and $N_{f}$
flavor $D6(\overline{D6})$-branes wrapping the non-compact three-cycle
$\Sigma^{(3)}(r,\theta_{1},\phi_{2})$.

As a result of this, we are able to see that the SYZ type IIA mirror has $N\
D6$-branes, $M\ D6(\overline{D6})$-branes and $N_{f}$ flavor
$D6(\overline{D6})$-branes. Remember that, just like in [14], ranges for the
presence of anti-branes are also important in the type IIA mirror. T-dualities
were performed along the $T^{3}(x,y,z)$ with $(x,y,z)$ as the toroidal
analogue of $(\phi_{1},\phi_{2},\psi)$. For simplicity, we have written
$(\phi_{1},\phi_{2},\psi)$.

####  1.4.3 ${\cal M}$-theory Uplift of Type IIB Setup

From the triple T-dual of the type IIB $F_{1,3,5}$ in [15], we can derive a
one-form type IIA potential, from which the following $D=11$ metric
($u\equiv\frac{r_{h}}{r}$) was obtained:

$\displaystyle
ds^{2}_{11}=e^{-\frac{2\phi^{IIA}}{3}}\left[g_{tt}dt^{2}+g_{\mathbb{R}^{3}}\left(dx^{2}+dy^{2}+dz^{2}\right)+g_{uu}du^{2}+ds^{2}_{IIA}({\theta_{1,2},\phi_{1,2},\psi})\right]$
$\displaystyle+e^{\frac{4{\phi}^{IIA}}{3}}\Bigl{(}dx_{11}+A^{F_{1}}+A^{F_{3}}+A^{F_{5}}\Bigr{)}^{2}\equiv\
{\rm Black}\ M3-{\rm Brane}+{\cal O}\left(\left[\frac{g_{s}M^{2}\log
N}{N}\right]\left(g_{s}M\right)N_{f}\right),$ $\displaystyle{\rm where}:$
$\displaystyle g_{uu}=\frac{3^{2/3}(2\sqrt{\pi
g_{s}N})}{u^{2}(1-u^{4})}\left(1-\frac{3g_{s}^{2}M^{2}N_{f}\log(N)\log\left(\frac{r_{h}}{u}\right)}{32\pi^{2}N}\right)$
$\displaystyle g_{tt}=\frac{3^{2/3}(u^{4}-1)r_{h}^{2}}{u^{2}(2\sqrt{\pi
g_{s}N})}\left(\frac{3g_{s}^{2}M^{2}N_{f}\log(N)\log\left(\frac{r_{h}}{u}\right)}{32\pi^{2}N}+1\right)$
$\displaystyle g_{\mathbb{R}^{3}}=\frac{3^{2/3}r_{h}^{2}}{u^{2}(2\sqrt{\pi
g_{s}N})}\left(\frac{3g_{s}^{2}M^{2}N_{f}\log(N)\log\left(\frac{r_{h}}{u}\right)}{32\pi^{2}N}+1\right).$
(1.35)

Moreover, in the UV:

$\displaystyle
G_{{x}{x}}^{M}=G_{{y}{y}}^{M}=G_{{z}{z}}^{M}=\frac{3^{2/3}r_{h}^{2}}{g_{s}^{2/3}u^{2}(2\sqrt{\pi
g_{s}N})}\left(\frac{3g_{s}^{2}M^{2}N_{f}\log(N)\log\left(\frac{r_{h}}{u}\right)}{32\pi^{2}N}+1\right)$
$\displaystyle G_{\phi_{1}r}^{M}\sim$
$\displaystyle\frac{2{g_{s}}^{4/3}{N_{f}}^{2}{\sin^{2}\phi_{1}}2\sin\left(\frac{\psi}{2}\right)\sin^{2}({\theta_{1}})\left(9\sin^{2}({\theta_{1}})+6\cos^{2}({\theta_{1}})+4\cos({\theta_{1}})\right)\sqrt[4]{{g_{s}}N\left(1-\frac{3{g_{s}}^{2}M^{2}{N_{f}}\log(N)\log(r)}{16\pi^{2}N}\right)}}{3^{5/6}\pi^{7/4}(\cos(2{\theta_{1}})-5)^{2}}$
$\displaystyle\times\left(9{h_{5}}\sin({\theta_{1}})+4\cos^{2}({\theta_{1}})\csc({\theta_{2}})-2\cos({\theta_{1}})\cot({\theta_{2}})+6\sin^{2}({\theta_{1}})\csc({\theta_{2}})\right)$
$\displaystyle G_{11\
r}^{M}\sim\frac{3^{\frac{3}{2}}g_{s}^{\frac{4}{3}}N_{f}\sin\phi_{1}\left(-8\cos\theta_{1}+3(-5+\cos(2\theta_{1}))\right)\sin\theta_{1}}{\pi\left(-5+\cos(2\theta_{1})\right)}.$
(1.36)

Near
$\theta_{1}\sim\frac{1}{N^{\frac{1}{5}}},\theta_{2}\sim\frac{1}{N^{\frac{3}{10}}}$
coordinate patch, one is able to see that:

$\displaystyle
G_{\phi_{1}r}\sim\frac{10g_{s}^{\frac{19}{12}}\sin^{2}\phi_{1}\sin\left(\frac{\psi}{2}\right)N^{\frac{3}{20}}}{23^{\frac{5}{6}}\pi^{\frac{7}{4}}}+{\cal
O}\left(\frac{1}{N^{\frac{1}{4}}}\right)\ll 1\ {\rm for}\
\psi\sim\frac{1}{N^{\alpha\gg\frac{3}{20}}},$ $\displaystyle G_{11\
r}\sim{\cal O}\left(\frac{1}{N^{\frac{1}{5}}}\right).$ (1.37)

The five-dimensional subspace $M_{5}(t,x_{1,2,3},u)$ from the six dimensional
internal space $M_{6}(\theta_{1,2},\phi_{1,2},\psi,x_{10})$ in the MQGP limit
around
$\theta_{1}\sim\frac{1}{N^{\frac{1}{5}}},\theta_{2}\sim\frac{1}{N^{\frac{3}{10}}}$.
It has been shown that ${\cal M}$-theory uplift is equipped with a
$G_{2}$-structure manifold [15, 66].

  * •

MQGP Limit: The following limit is taken into consideration in [14]:

$\displaystyle\hskip-12.28577pt{\rm weakstring}(g_{s}){\rm coupling-large\
t^{\prime}Hooft\ coupling\ limit}:$ $\displaystyle\hskip-12.28577ptg_{s}\ll
1,g_{s}N_{f}\ll 1,\frac{g_{s}M^{2}}{N}\ll 1,g_{s}M\gg 1,g_{s}N\gg 1.$ (1.38)

The “MQGP” limit is defined as [15]:

$g_{s}\sim\frac{1}{{\cal O}(1)},M,N_{f}\equiv{\cal O}(1),\ g_{s}N_{f}<1,\ N\gg
1,\ \frac{g_{s}M^{2}}{N}\ll 1,$ (1.39)

The motivation for taking into account the MQGP limit mostly stems from two
reasons.

    1. 1.

In contrast to AdS/CFT duality where $g_{\rm YM}\rightarrow
0,N\rightarrow\infty$ such that $g_{\rm YM}^{2}N$ is huge, for strongly
coupled thermal QCD like sQGP, we have $g_{\rm YM}\sim{\cal O}(1)$ and
$N_{c}=3$ and hence $g_{\rm YM}^{2}N=g_{s}$ is finite, i.e.,
$g_{s}\stackrel{{\scriptstyle<}}{{\sim}}1$. $N_{c}=M$ has been derived in [15]
from the Seiberg duality cascade and it is allowed to take $N_{c}=3$. Due to
finite string coupling, one is required to address this issue from ${\cal M}$
theory, and the “M” in the “MQGP” limit implies ${\cal M}$-theory. See [67]
where the authors have studied ${\cal M}$-theory without any uplift. There are
also papers available in the related context of ${\cal M}$-theory, e.g., [68,
69, 70].

    2. 2.

The following are examples of calculational simplification in supergravity,
and they make up the second importance of the justifications for considering
the MQGP limit (1.39):

       * –

In the region of UV-IR interpolation and the UV, $(M_{\rm eff},N_{\rm
eff},N_{f}^{\rm eff})\xrightarrow{\rm MQGP}{\approx}(M,N,N_{f})$

       * –

Asymmetry Factors $A_{i},B_{j}$(in three-form fluxes)$\xrightarrow{MQGP}1$ in
the region where the UV and IR are interpolated, in addition to the UV.

       * –

The ten-dimensional warp factor and the non-extremality function in the MQGP
limit have the simple forms.

  * •

IR Color-Flavor Enhancement of Length Scale: Even with ${\cal O}(1)$ $M$,
there is color-flavor enhancement of the length scale in the MQGP limit (1.39)
compared to a Planckian length scale in KS-like model in the IR. This is valid
if one takes into account terms of higher order in $g_{s}N_{f}$ in the RR and
NS-NS three-form fluxes, and NLO in $N$ in the ${\cal M}$-theory metric. It
suggests the validity of supergravity calculations and suppresses quantum
corrections. This issue has been discussed in detail in [66, 71]. Now, the
effective number of color branes, denoted by the notation$N_{\rm eff}(r)$,
will be given by

$\displaystyle N_{\rm eff}(r)=N\left[1+\frac{3g_{s}M_{\rm eff}^{2}}{2\pi
N}\left(\log r+\frac{3g_{s}N_{f}^{\rm eff}}{2\pi}\left(\log
r\right)^{2}\right)\right],$ $\displaystyle M_{\rm
eff}(r)=M+\frac{3g_{s}N_{f}M}{2\pi}\log r+\sum_{m\geq 1}\sum_{n\geq
1}N_{f}^{m}M^{n}f_{mn}(r),$ $\displaystyle N^{\rm
eff}_{f}(r)=N_{f}+\sum_{m\geq 1}\sum_{n\geq 0}N_{f}^{m}M^{n}g_{mn}(r).$ (1.40)

The type IIB axion $C_{0}$ is given as $C_{0}=N_{f}^{\rm
eff}\frac{\left(\psi-\phi_{1}-\phi_{2}\right)}{4\pi}$, $N_{\rm
eff}(r_{0}\in\rm IR)=0$. The ten-dimensional warp factor is being written as
$h\sim\frac{L^{4}}{r^{4}}$. The following equation will provide the length
scale $L$ in the IR.

$\displaystyle\hskip-65.04256ptL\sim\sqrt[4]{M}N_{f}^{\frac{3}{4}}\sqrt{\left(\sum_{m\geq
0}\sum_{n\geq 0}N_{f}^{m}M^{n}f_{mn}(r_{0})\right)}\left(\sum_{l\geq
0}\sum_{p\geq 0}N_{f}^{l}M^{p}g_{lp}(r_{0})\right)^{\frac{1}{4}}L_{\rm KS},$
(1.41)

$L_{KS}\sim\sqrt[4]{g_{s}M}\sqrt{\alpha^{\prime}}$. Equation (1.41) suggests
that, in comparison to KS, the colour-flavor length scale is enhanced in the
IR. This implies the validity of supergravity calculations with $N_{c}^{\rm
IR}=M=3$ and $N_{f}=2(u/d)+1(s)$ provided one should include the $n,m>1$ terms
in $M_{\rm eff}$ and $N_{f}^{\rm eff}$ in (• ‣ 1.4.3).

  * •

Obtaining ${\bf N_{c}=3}$: We will now briefly summarise how to identify the
number of colours $N_{c}$ with $M$ that can be taken as 3 in the “MQGP limit”
(1.39). This is based on the work done in [71]. Let us write $N_{c}$ as

$N_{C}=N_{\rm eff}(r)+M_{\rm eff}(r),$ (1.42)

where $N_{\rm eff}(r)$ appears in the following equation,

$\tilde{F}_{5}\equiv dC_{4}+B_{2}\wedge F_{3}={\cal F}_{5}+*{\cal F}_{5},$
(1.43)

via ${\cal F}_{5}\equiv N_{\rm eff}\times{\rm Vol}({\rm Base\ of\ Resolved\
Warped\ Deformed\ Conifold})$. One can also define $M_{\rm eff}$ as

$M_{\rm eff}=\int_{S^{3}}\tilde{F}_{3}.$ (1.44)

In (1.44), $S^{3}$ being dual to $\
e_{\psi}\wedge\left(\sin\theta_{1}d\theta_{1}\wedge
d\phi_{1}-B_{1}\sin\theta_{2}\wedge d\phi_{2}\right)$, $B_{1}$ is an
‘asymmetry factor’ defined in [14]; $e_{\psi}\equiv d\psi+{\rm
cos}~{}\theta_{1}~{}d\phi_{1}+{\rm cos}~{}\theta_{2}~{}d\phi_{2}$) and [72]:
$\tilde{F}_{3}(\equiv F_{3}-\tau H_{3})\propto M(r)\equiv
M\frac{1}{1+e^{\alpha(r-{\cal R}_{D5/\overline{D5}})}},\alpha\gg 1$. If we
define the UV and IR values of a quantity “A” as $[A_{UV},A_{IR}]$ then we can
define the UV and IR values of $N_{\rm eff}$ and $M_{\rm eff}$ as

$N_{\rm eff}\in[N,0]\hskip 14.45377ptand\hskip 14.45377ptN_{\rm eff}\in[0,M].$
(1.45)

Using (1.42) and (1.45), one can derive the UV and IR values of $N_{c}$ as
given below

$N_{c}\in[N,M].$ (1.46)

Therefore, from (1.46), we can easily see that $N_{c}=M$ in the IR and it was
taken as $N_{c}=M=3$ in the MQGP limit [14]. From the various discussions in
this section, we found that ${\cal M}$-theory dual constructed from the type
IIB string dual of [14] has finite number of colors, it is IR confined, it has
fundamental quarks, and this dual is valid for all temperatures (both
$T>T_{c}$ and $T<T_{c}$).

####  1.4.4 ${\cal M}$-Theory Uplift Inclusive of ${\cal
O}(R^{4})$-Corrections

This section is based on [1]. The first part of the thesis (Part-I) and
chapter 7 are based on this holographic model. We will discuss the inclusion
of higher derivative terms in the eleven-dimensional supergravity action and
solutions of the metric EOMs. With the inclusion of higher derivative terms on
the gravity dual side, we are able to explore the intermediate coupling regime
of thermal QCD, which was not possible earlier. The
$SU(3)/G_{2}/SU(4)/Spin(7)$-structure torsion classes of the relevant six-,
seven- and eight-folds associated with the ${\cal M}$-theory uplift have been
computed in [1].

The origin of ${\cal O}(R^{4})$-corrections to the ${\cal N}=1,D=11$
supergravity action has been discussed in [73]. This kind of term has been
used in the effective one-loop heterotic action [74]. The other kind of higher
derivative term constructed from Weyl tensor was used in [75] to compute the
correction to the viscosity-to-entropy density ration. There are two ways of
understanding the origin of the ${\cal O}(R^{4})$-corrections to the ${\cal
N}=1,D=11$ supergravity action. One is in the context of the effects of
$D$-instantons in IIB supergravity/string theory via the four-graviton
scattering amplitude [73]. The other is $D=10$ supersymmetry [76]. Let us
briefly discuss these before going into the details of the construction of
${\cal M}$-theory dual at intermediate coupling.

  * •

Let us initially consider interactions generated at leading order in a
$D$-instanton (closed-string states of type IIB superstring theory where the
entire string is confined at one point in superspace) background within both
type IIB supergravity and string descriptions, which includes a one-instanton
correction to the tree-level along with the one-loop $R^{4}$ terms [73] \- the
two containing an identical tensorial structure. The coordinates pertaining to
the location of the $D$-instanton are used to parameterize the bosonic zero
modes. The broken supersymmetries yield the fermionic zero modes. The
Grassmann parameters are fermionic supermoduli that correspond to dilatino
zero modes and should be integrated out with the bosonic zero modes. The disk
diagram is the most basic open-string world-sheet that occurs in a D-brane
procedures. At the lowest order, an instanton with some zero modes refers to a
disk world-sheet along with open-string states connected to the boundary.
Consider on-shell amplitudes in the instanton background to infer the one-
instanton terms in the supergravity effective action. The separate fermionic
zero modes are absorbed by the integration over the fermionic moduli. Consider
four external graviton amplitudes. In supergravity, the leading term refers to
the situation where each graviton is connected with the four fermionic zero
modes. A nonlocal four-graviton interaction is produced by integrating over
the bosonic zero modes. The world-sheet in the associated string computation
is made up of four unconnected disks, each with a single closed-string
graviton vertex and four fermionic open-string vertices. By expressing the
graviton polarization tensor via
$\zeta^{\mu_{r}\nu_{r}}=\zeta^{(\mu_{r}}\tilde{\zeta}^{\nu_{r})}$ and
calculating the fermionic integral, the authors in [73] showed that the four-
graviton scattering amplitude is able to be represented via the equation:

$\displaystyle Ce^{2i\pi\tau_{0}}\int d^{10}ye^{i\sum_{r}k_{r}\cdot y}$
$\displaystyle\times\left(\hat{t}^{i_{1}j_{1}\cdots
i_{4}j_{4}}\hat{t}_{m_{1}n_{1}\cdots m_{4}n_{4}}-{1\over
4}\epsilon^{i_{1}j_{1}\cdots j_{4}j_{4}}\epsilon_{m_{1}n_{1}\cdots
m_{4}n_{4}}\right)R_{i_{1}j_{1}}^{m_{1}n_{1}}R_{i_{2}j_{2}}^{m_{2}n_{2}}R_{i_{3}j_{3}}^{m_{3}n_{3}}R_{i_{4}j_{4}}^{m_{4}n_{4}};$

$t_{8}$ is specified by:

$\displaystyle\hat{t}_{8}^{N_{1}\dots
N_{8}}=\frac{1}{16}\biggl{(}-2\left(G^{N_{1}N_{3}}G^{N_{2}N_{4}}G^{N_{5}N_{7}}G^{N_{6}N_{8}}+G^{N_{1}N_{5}}G^{N_{2}N_{6}}G^{N_{3}N_{7}}G^{N_{4}N_{8}}+G^{N_{1}N_{7}}G^{N_{2}N_{8}}G^{N_{3}N_{5}}G^{N_{4}N_{6}}\right)$
$\displaystyle+8\left(G^{N_{2}N_{3}}G^{N_{4}N_{5}}G^{N_{6}N_{7}}G^{N_{8}N_{1}}+G^{N_{2}N_{5}}G^{N_{6}N_{3}}G^{N_{4}N_{7}}G^{N_{8}N_{1}}+G^{N_{2}N_{5}}G^{N_{6}N_{7}}G^{N_{8}N_{3}}G^{N_{4}N_{1}}\right)$
$\displaystyle-(N_{1}\leftrightarrow N_{2})-(N_{3}\leftrightarrow
N_{4})-(N_{5}\leftrightarrow N_{6})-(N_{7}\leftrightarrow N_{8})\biggr{)}$
(1.48)

($N_{i}$ belongs to the ${\bf 8}_{v}$ of $SO(8)$ or covariantized to $10D$ (or
$11D$ that will be used later in M-theory)), from [77] the light-cone 8D “zero
mode” tensor $t_{8}$ is generalized to $10D$:
$\hat{t}_{8}=t_{8}-1/4B\epsilon_{10}$ where if one assumes $B_{\rm
LCdirections}=1$, implies
$\hat{t}^{i_{1}j_{1}...i_{4}j_{4}}=t^{i_{1}j_{1}...i_{4}j_{4}}-\frac{1}{2}\epsilon^{i_{1}j_{1}...i_{4}j_{4}}$,
and an overall factor of $e^{2\pi i\tau_{0}}$, the signature of stringy
D-instanton, is computed at $\chi=\Re\tau_{0}=0$ in the stringy calculation.

  * •

Green and Vanhove showed that the eleven-dimensional ${\cal O}(R^{4})$
adjustments possess a separate motive based on ten-dimensional supersymmetry
[76]. The above has been shown by its relationship with the term
$C^{(3)}\wedge X_{8}$ in the ${\cal M}$-theory effective action, that is
believed to originate through a number of reasons, including anomaly
cancellation [78, 79]. The formula $X_{8}$ represents the eight-form in the
curvatures obtained via the term in type IIA superstring theory [80]:

$-\int d^{10}xB\wedge X_{8}=-{1\over 2}\int
d^{10}x\sqrt{-g^{A(10)}}\epsilon_{10}BX_{8},$ (1.49)

where

$X_{8}={1\over 192}\left({\rm tr}\ R^{4}-{1\over 4}({\rm tr}\
R^{2})^{2}\right).$ (1.50)

There exist two distinct ten-dimensional $N=1$ super-invariants with an odd-
parity term ([77] and previous authors): $I_{3}=t_{8}{\rm tr}\ R^{4}-{1\over
4}\epsilon_{10}B{\rm tr}\ R^{4}$ and: $I_{4}=t_{8}({\rm tr}\
R^{2})^{2}-{1\over 4}\epsilon_{10}B({\rm tr}\ R^{2})^{2}.$ Utilizing that
$t_{8}t_{8}R^{4}=24t_{8}{\rm tr}(R^{4})-6t_{8}({\rm tr}\ R^{2})^{2},$ as a
result, the specific linear combination,

$I_{3}-{1\over 4}I_{4}={1\over 24}t_{8}t_{8}R^{4}-48\epsilon_{10}B\ X_{8}$
(1.51)

includes the ten-form $B\wedge X_{8}$ as well as $t_{8}t_{8}R^{4}$.

Now let us discuss the ${\cal M}$-theory uplift in the presence of ${\cal
O}(R^{4})$ terms based on [1]. The working action of eleven-dimensional
supergravity with ${\cal N}=1$ supersymmetry inclusive of ${\cal O}R^{4})$
terms is written below:

$\displaystyle\hskip-57.81621ptS=\frac{1}{2\kappa_{11}^{2}}\int_{M}\sqrt{-g}\left[{\cal
R}*_{11}1-\frac{1}{2}G_{4}\wedge*_{11}G_{4}-\frac{1}{6}C\wedge G\wedge
G\right]+\frac{1}{\kappa_{11}^{2}}\int_{\partial M}d^{10}x\sqrt{h}K$
$\displaystyle\hskip-57.81621pt+\frac{1}{(2\pi)^{4}3^{2}2^{13}}\left(\frac{2\pi^{2}}{\kappa_{11}^{2}}\right)^{\frac{1}{3}}\int
d^{11}x\sqrt{-g}\left(J_{0}-\frac{1}{2}E_{8}\right)+\left(\frac{2\pi^{2}}{\kappa_{11}^{2}}\right)\int
C_{3}\wedge X_{8},$ (1.52)

where $\kappa_{11}^{2}$ is the eleven-dimensional Planckian length, and four-
form field strength is defined as $G=dC$ for the three-form potential $C$.
$J_{0}$ and $E_{8}$ are define below:

$\displaystyle\hskip-57.81621ptJ_{0}=3\cdot
2^{8}(R^{HMNK}R_{PMNQ}{R_{H}}^{RSP}{R^{Q}}_{RSK}+{1\over
2}R^{HKMN}R_{PQMN}{R_{H}}^{RSP}{R^{Q}}_{RSK})$
$\displaystyle\hskip-57.81621ptE_{8}={1\over 3!}\epsilon^{ABCM_{1}N_{1}\dots
M_{4}N_{4}}\epsilon_{ABCM_{1}^{\prime}N_{1}^{\prime}\dots
M_{4}^{\prime}N_{4}^{\prime}}{R^{M_{1}^{\prime}N_{1}^{\prime}}}_{M_{1}N_{1}}\dots
R^{M_{4}^{\prime}N_{4}^{\prime}}_{M_{4}N_{4}},$
$\displaystyle\hskip-57.81621pt\kappa_{11}^{2}=\frac{(2\pi)^{8}l_{p}^{9}}{2};$
(1.53)

Equations of motion of the ${\cal M}$-theory metric and three-form potential
are given below.

$\displaystyle R_{MN}-\frac{1}{2}g_{MN}{\cal
R}-\frac{1}{12}\left(G_{MPQR}G_{N}^{\
PQR}-\frac{g_{MN}}{8}G_{PQRS}G^{PQRS}\right)$
$\displaystyle=-\beta\left[\frac{g_{MN}}{2}\left(J_{0}-\frac{1}{2}E_{8}\right)+\frac{\delta}{\delta
g^{MN}}\left(J_{0}-\frac{1}{2}E_{8}\right)\right],$ $\displaystyle
d*G=\frac{1}{2}G\wedge G+3^{2}2^{13}\left(2\pi\right)^{4}\beta X_{8},$ (1.54)

where $\beta$ is defined as [81]:

$\beta\equiv\frac{\left(2\pi^{2}\right)^{\frac{1}{3}}\left(\kappa_{11}^{2}\right)^{\frac{2}{3}}}{\left(2\pi\right)^{4}3^{2}2^{12}}\sim
l_{p}^{6},$ (1.55)

$R_{MNPQ},R_{MN},{\cal R}$ in (1.4.4)/(1.4.4) being the Riemann curvature
tensor, the Ricci tensor, and the Ricci scalar, respectively, in the elven-
dimensions. Ansatz made by authors to solve (1.4.4) in [1] are written as

$\displaystyle\hskip-57.81621ptg_{MN}=g_{MN}^{(0)}+\beta g_{MN}^{(1)},$
$\displaystyle\hskip-57.81621ptC_{MNP}=C^{(0)}_{MNP}+\beta C_{MNP}^{(1)}.$
(1.56)

Equation of motion for the three-form potential symbolically written as [1]

$\displaystyle\beta\partial\left(\sqrt{-g}\partial
C^{(1)}\right)+\beta\partial\left[\left(\sqrt{-g}\right)^{(1)}\partial
C^{(0)}\right]+\beta\epsilon_{11}\partial C^{(0)}\partial C^{(1)}={\cal
O}(\beta^{2})\sim 0[{\rm up\ to}\ {\cal O}(\beta)].$

Authors found that $C^{(1)}_{MNP}=0$ for the consistent set of solutions to
(1.4.4) up to ${\cal O}(\beta)$. Therefore, ${\cal M}$-theory receives the
${\cal O}(R^{4})$ corrections but not the three-form potential. The correction
term of the metric is written as follows.

$\displaystyle\delta g_{MN}=\beta g^{(1)}_{MN}=G_{MN}^{\rm MQGP}f_{MN}(r),$
(1.58)

without summation in the indices.

Let’s discuss how to control the higher derivative contributions to the MQGP
metric [15]. This was discussed in [1], and we are going to review it here. It
is obvious from (B.1) in the $\psi=2n\pi,n=0,1,2$-coordinate patch that in the
IR: $r=\chi r_{h},\chi\equiv{\cal O}(1)$, and up to ${\cal O}(\beta)$:

$f_{MN}\sim\beta\frac{\left(\log{\cal R}_{h}\right)^{m}}{{\cal
R}_{h}^{n}N^{\beta_{N}}},\ m\in\left\\{0,1,3\right\\},\
n\in\left\\{0,2,5,7\right\\},\ \beta_{N}>0,$ (1.59)

where ${\cal R}_{h}\equiv\frac{r_{h}}{{\cal R}_{D5/\overline{D5}}}$. Now,
$|{\cal R}_{h}|\ll 1$ and as estimated in [82] that $|\log{\cal R}_{h}|\sim
N^{\frac{1}{3}}$ implies the competition between Planckian and large-$N$
suppression and infra-red enhancement arising from $m,n\neq 0$ in (1.59). Let
us choose a hierarchy: $\beta\sim
e^{-\gamma_{\beta}N^{\gamma_{N}}},\gamma_{\beta},\gamma_{N}>0:\gamma_{\beta}N^{\gamma_{N}}>7N^{\frac{1}{3}}+\left(\frac{m}{3}-\beta_{N}\right)\log
N$ such that IR-enhancement does not overpower Planckian suppression that is
why authors considered the ${\cal O}(\beta)$ correction to $G^{\cal M}_{yz}$,
which had the largest IR enhancement, to set a lower bound on
$\gamma_{\beta,N}$/Planckian suppression. Authors concluded that one will go
beyond ${\cal O}(\beta)$ if $\gamma_{\beta}N^{\gamma_{N}}\sim
7N^{\frac{1}{3}}$. Therefore, they restricted themselves at ${\cal O}(\beta)$.

##  Chapter 2 SU(3) LECs from Type IIA String Theory

This chapter is based on the paper [2]. Some portion of [2] is already present
in a thesis of one of the co-authors (VY) [83]. Detailed calculations of the
relevant quantities appearing in this chapter are given in [83]. Hence, we
will quote those results in appendix A without going into the details and use
those results to calculate the low energy coupling constants (LECs) of $SU(3)$
chiral perturbation theory in the chiral limit.

###  2.1 Introduction

The low energy regime of quantum chromodynamics (QCD) is described by chiral
perturbation theory ($\chi$PT), which includes the hadrons (mesons and
baryons) as degrees of freedom. One constructs the Lagrangian of $\chi$PT
based on the symmetry of the theory, and the symmetries are chiral symmetry
$SU(N_{f})_{L}\times SU(N_{f})_{R}$, charge conjugation, and parity symmetry.
The theory turns out to be renormalized order-by-order in momentum expansion.
One obtains the $(N_{f}^{2}-1)$ (pseudo-)Goldstone bosons from the spontaneous
symmetry breaking of the chiral symmetry mentioned above, where $N_{f}$ is the
number of QCD flavors. In this chapter, we focused on the $SU(3)$ mesonic
chiral perturbation theory, where degrees of freedom considered is $\rho$
vector meson and $\pi$-meson.
The procedure to include $\rho$ vector meson is the HLS (Hidden Local
Symmetry) approach with gauge group is $G_{global}\times H_{local}$, where
$G_{global}=SU(N_{f})_{L}\times SU(N_{f})_{R}$ and $H=SU(N_{f})_{V}$ is the
HLS [84]. Gauge bosons of HLS are the $\rho$ vector meson and its flavor
partners. Expansion of generating functional of QCD is possible in terms of
$p/\Lambda_{\chi}$ or $m/\Lambda_{\chi}$, where $\Lambda_{\chi}\sim 4\pi
F_{\pi}\sim 1.1GeV$ [85] is the chiral symmetry breaking scale. The $\chi$PT
Lagrangian is constructed order by order in the derivative expansion and
should be consistent with the symmetries mentioned in the first paragraph. The
leading order Lagrangian is $O(p^{2})$. At leading order, there are two
coupling constants $F_{\pi}^{2}$ and $F_{\sigma}^{2}$(Where $F_{\pi}$ and
$F_{\sigma}$ are decay constants of $\pi$ and $\sigma$). $SU(2)$ and $SU(3)$
$\chi$PT Lagrangians were worked out by Gasser and Leutwyler in [86] and [87]
respectively. The NLO $SU(3)$ $\chi$PT Lagrangian contains 12 coupling
constants “$(L_{i=1,2,..10},H_{1},H_{2})$” known as low energy constants
(LECs). $L_{4},L_{5},L_{6},L_{8}$ were calculated from the Lattice simulation
in [88], $L_{i=1,2,..10}$ were obtained at the scale $\mu=M_{\rho}$ in [16],
see [89] for the most update values.
Holographic computation of $SU(3)$ LECs was done in [84] from the top-down
Sakai-Sugimoto model [90]. The Sakai-Sugimoto model is not UV complete. The
Lagrangian obtained in [84] is different from the $SU(3)$ $\chi$PT Lagrangian
given in [87]. The paper [91] contains the relationship between the $SU(3)$
LECs and the coupling constants obtained in [84]. We obtained these LECs from
a UV complete top-down holographic dual. The gravity dual is type IIA string
theory in the presence of higher derivative terms. Type IIA background had
been obtained by descending back from the ${\cal M}$-theory to type IIA string
theory.

###  2.2 $SU(3)$ Chiral Perturbation Theory Lagrangian and the
Phenomenological Values of LECs

The term appearing in the $SU(3)\ \chi$PT Lagrangian at ${\cal O}(p^{4})$ with
one-loop renormalized low energy constants (LECs),$L_{i}$ and $H_{i}$, in
chiral limit are written as follows [87]:

$\displaystyle L_{1}\left({\rm
Tr}(\nabla_{\mu}U^{\dagger}\nabla^{\mu}U)\right)^{2}+L_{2}\left({\rm
Tr}(\nabla_{\mu}U^{\dagger}\nabla_{\nu}U)\right)^{2}+L_{3}{\rm
Tr}\left(\nabla_{\mu}U^{\dagger}\nabla^{\mu}U\right)^{2}$ $\displaystyle-
iL_{9}Tr\left({\cal L}_{\mu\nu}\nabla^{\mu}U\nabla^{\nu}U^{\dagger}+{\cal
R}_{\mu\nu}\nabla^{\mu}U\nabla^{\nu}U^{\dagger}\right)+L_{10}Tr\left(U^{\dagger}{\cal
L}_{\mu\nu}U{\cal R}^{\mu\nu}\right)+H_{1}Tr\left({\cal L}_{\mu\nu}^{2}+{\cal
R}_{\mu\nu}^{2}\right),$

where $\nabla_{\mu}U\equiv\partial_{\mu}U-i{\cal L}_{\mu}U+iU{\cal R}_{\mu},\
U=e^{\frac{2i\pi}{F_{\pi}}}$. LECs, $L_{i}$ and $H_{i}$ are defined via the
dimensional regularization and renormalizations of the parameters as follow
[87]:

$L_{i}=L_{i}^{r}(\mu)+\Gamma_{i}\lambda(\mu)\ ,\qquad
H_{i}=H_{i}^{r}(\mu)+\Delta_{i}\lambda(\mu)\ ,$ (2.2)

where $\mu$ is the renormalization scale, and for $SU(3)$ $\chi$PT,
$\Gamma_{i}$ and $\Delta_{i}$ are given as [87]:

$\begin{array}[]{ccccc}\Gamma_{1}=\frac{3}{32}\ ,&\Gamma_{2}=\frac{3}{16}\
,&\Gamma_{3}=0\ ,&\Gamma_{4}=\frac{1}{8}\ ,&\Gamma_{5}=\frac{3}{8}\ ,\\\
\Gamma_{6}=\frac{11}{144}\ ,&\Gamma_{7}=0\ ,&\Gamma_{8}=\frac{5}{48}\
,&\Gamma_{9}=\frac{1}{4}\ ,&\Gamma_{10}=-\frac{1}{4}\ ;\\\
\Delta_{1}=-\frac{1}{8}\ ,&\Delta_{2}=\frac{5}{24}\ .&&&\end{array}$ (2.3)

The divergent piece, $\lambda(\mu)$ appearing in (2.2) is given below.

$\lambda(\mu)=-\frac{1}{2\left(4\pi\right)^{2}}\left[\frac{1}{\bar{\epsilon}}-\ln\mu^{2}+1\right]\
,$ (2.4)

where

$\frac{1}{\bar{\epsilon}}=\frac{2}{4-d}-\gamma_{E}+\ln 4\pi\ ,$ (2.5)

where $d=4$. The phenomenological values of the one-loop renormalized $SU(3)$
LECs appearing in (2.2) are listed in the table 2.1 [89]. In table 2.2 sources
from where the LECs have been extracted is given for the column titled GL 1985
[87].

LECs |  GL 1985 [87] | NLO 2014 | NNLO free fit |  NNLO BE14 [92]
---|---|---|---|---
$10^{3}$ $L_{1}^{r}$ | 0.7(3) | 1.0(1) | 0.64)06 | 0.53(06)
$10^{3}$ $L_{2}^{r}$ | 1.3(7) | 1.6(2) | 0.59(04) | 0.81(04)
$10^{3}$ $L_{3}^{r}$ | -4.4(2.5) | -3.8(3) | -2.80(20) | -3.07(20)
$10^{3}$ $L_{4}^{r}$ | -0.3(5) | 0.0(3) | 0.76(18) | 0.3
$10^{3}$ $L_{5}^{r}$ | 1.4(5) | 1.2(1) | 0.50(07) | 1.01(06)
$10^{3}$ $L_{6}^{r}$ | -0.2(3) | 0.0(4) | 0.49(25) | 0.14(05)
$10^{3}$ $L_{7}^{r}$ | -0.4(2) | -0.3(2) | -0.19(08) | -0.34(09)
$10^{3}$ $L_{8}^{r}$ | 0.9(3) | 0.5(2) | 0.17(11) | 0.47(10)
Table 2.1: $SU(3)$ NLO LECs $L_{i}^{r}(i=1,2,...,8)$ from various references. i |  $L_{i}^{r}(M_{\rho})$ $10^{3}$ | Source
---|---|---
1 | 0.4 $\pm$ 0.3 |  $K_{e4}$, $\pi\pi\rightarrow\pi\pi$
2 | 1.4 $\pm$ 0.3 |  $K_{e4}$, $\pi\pi\rightarrow\pi\pi$
3 | -3.5 $\pm$ 1.1 |  $K_{e4}$, $\pi\pi\rightarrow\pi\pi$
4 | -0.3 $\pm$ 0.5 | Zweig rule
5 | 1.4 $\pm$ 0.5 | $F_{K}:F_{\pi}$
6 | -0.2 $\pm$ 0.3 | Zweig rule
7 | -0.4 $\pm$ 0.2 | Gell-Mann-Okubo, $L_{5}$, $L_{8}$
8 | 0.9 $\pm$ 0.3 |  $M_{K0}-M_{K_{+}}$, $L_{5}$, $(m_{s}-\hat{m}):(m_{d}-m_{u})$
9 | 6.9 $\pm$0.7 | $<r^{2}>_{V}^{\pi}$
10 | -5.5 $\pm$ 0.7 | $\pi\rightarrow e\nu\gamma$
Table 2.2: Phenomenological Values of the 1-loop renormalised couplings
$L_{i}^{r}(M_{\rho})$ of (2.2) [16]. Last column shows the source to extract
this information.

In the chiral perturbation theory that is used in gauge-gravity duality, the
holographic renormalization acts as the analogue of the one-loop
renormalization. The eleven dimensional on-shell supergravity action including
${\cal O}(\beta)$ correction was worked out in [3] and written below:

$S_{D=11}^{\rm on-shell}=-\frac{1}{2}\Biggl{[}-{2}S_{\rm EH}^{(0)}+2S_{\rm
GHY}^{(0)}+\beta\left(\frac{20}{11}S_{\rm
EH}-{2}\int_{M_{11}}\sqrt{-g^{(1)}}R^{(0)}+2S_{\rm
GHY}-\frac{2}{11}\int_{M_{11}}\sqrt{-g^{(0)}}g_{(0)}^{MN}\frac{\delta
J_{0}}{\delta g_{(0)}^{MN}}\right)\Biggr{]}.$ (2.6)

UV divergent terms that exist in the (2.6) have the following forms:

$\displaystyle\left.\int_{M_{11}}\sqrt{-g}R\right|_{\rm UV-divergent},\
\left.\int_{\partial M_{11}}\sqrt{-h}K\right|_{\rm UV-divergent}\sim r_{\rm
UV}^{4}\log r_{\rm UV},$
$\displaystyle\left.\int_{M_{11}}\sqrt{-g}g^{MN}\frac{\delta J_{0}}{\delta
g^{MN}}\right|_{\rm UV-divergent}\sim\frac{r_{\rm UV}^{4}}{\log r_{\rm UV}}.$
(2.7)

These UV divergences had been cancelled by using the boundary counter terms in
a suitable linear combination and the counter terms are: $\left.\int_{\partial
M_{11}}\sqrt{-h}K\right|_{r=r_{\rm UV}}$ and $\left.\int_{\partial
M_{11}}\sqrt{-h}h^{mn}\frac{\partial J_{0}}{\partial h^{mn}}\right|_{r=r_{\rm
UV}}$.

###  2.3 Holographic Computation of $SU(3)$ LECs

This section will discuss the computation of $SU(3)$ LECs holographically. We
considered the gravity dual as type IIA string dual inclusive of ${\cal
O}(R^{4})$ corrections. Since chiral perturbation theory is the low energy
regime of QCD ($T<T_{c}$), we worked with the thermal metric in our
calculations. Mesons are the fluctuations of the flavor $D6$-branes in type
IIA string dual of the type IIB string dual constructed from a top-down
approach [14]. We started with the ${\cal M}$-theory uplift constructed in [1]
in the presence of ${\cal O}(R^{4})$ terms and then descent back to type IIA
string theory. Flavor $D6$-branes is given as:
$\iota:\Sigma^{1,6}\Bigg{(}R^{1,3},r,\theta_{2}\sim\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}},y\Bigg{)}\hookrightarrow
M^{1,9}$ near the Ouyang embedding (1.23) for a vanishingly small $|\mu_{\rm
Ouyang}|$. The embedding is parametrized by $z=z(r)$. Similar to [93], it
turns out that the embedding solution is $z$=constant, and if we choose
$z=\pm{\cal C}\frac{\pi}{2}$, then $D6/\overline{D6}$-branes will be located
at “antipodal” points along the z coordinate. We worked with the redefined
radial coordinates $(r,z)$ in terms of the new coordinates $(Y,Z)$ as used in
[93].

$r=r_{0}e^{\sqrt{Y^{2}+Z^{2}}};\ z={\cal C}\arctan\frac{Z}{Y}.$

Fluctuations of the $D6$-branes along the world volume correspond to the
vector mesons, and fluctuations orthogonal to the $D6$-branes correspond to
the scalar mesons111See [94, 95] where the mesons spectrum have been studied
for $N=2$ super Yang-Mills theory with fundamental matter using AdS/CFT
correspondence. See [96], where the mesonic excitations have been studied from
a non Abelian T-dual of $AdS_{5}\times S^{5}$ background.. The world volume
coordinates of the $D6$-branes are
${\Sigma}_{7}(x^{0,1,2,3},Z,\theta_{2},\tilde{y})={\Sigma}_{2}(\theta_{2},\tilde{y})\times{\Sigma}_{5}(x^{0,1,2,3},Z)$.
We considered $\tilde{F}$ as the gauge field fluctuation about a small
background gauge filed $F_{0}$ and induced background
$i^{*}(g+B)[i:\Sigma_{7}\hookrightarrow M_{10}$, where $M_{10}$ is the type
IIA geometry]. The SYZ mirror of the Ouyang embedding appears to be written as
$Y=0$ [93]. We found the following DBI action after retaining the terms up to
quadratic order in $\tilde{F}$:

${\rm
S}^{IIA}_{D6}=\frac{T_{D_{6}}(2\pi\alpha^{\prime})^{2}}{4}\left(\frac{\pi
L^{2}}{r_{0}}\right){\rm
Str}\int\prod_{i=0}^{3}dx^{i}dZd\theta_{2}dy\delta\Bigg{(}\theta_{2}-\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}}\Bigg{)}e^{-\Phi}\sqrt{-{\rm
det}_{\theta_{2}y}(\iota^{*}(g+B))}\sqrt{{\rm
det}_{{\mathbb{R}}^{1,3},Z}(\iota^{*}g)}g^{\tilde{\mu}\tilde{\nu}}\tilde{F}_{\tilde{\nu}\tilde{\rho}}g^{\tilde{\rho}\tilde{\sigma}}\tilde{F}_{\tilde{\sigma}\tilde{\mu}},$
(2.8)

where $\tilde{\mu}=i(=0,1,2,3),Z$. Gauge fields in five dimensions are
expanded as:
$A_{\mu}(x^{\nu},Z)=\sum_{n=1}^{\infty}\rho^{(n)}_{\mu}(x^{\nu})\psi_{n}(Z)$
and $A_{Z}(x^{\nu},Z)=\sum_{n=0}^{\infty}\pi^{(n)}(x^{\nu})\phi_{n}(Z)$. If we
consider the lowest lying scalar mode in the aforementioned expansions, which
is the pseudo-scalar $\pi$ meson and after the application of gauge
transformation:$\rho_{\mu}^{(n)}\rightarrow\rho_{\mu}^{(n)}+{\cal
M}_{(n)}^{-1}\partial_{\mu}\pi^{(n)}.$ We obtained the following action
written in terms of the $\pi$ meson and $\rho_{\mu}^{(n)}$ vectors mesons.

$\displaystyle{\rm S}^{IIA}_{D6}\sim-\int d^{3}x\ \ {\rm
tr}\left[\frac{1}{2}\partial_{\mu}\pi^{(0)}\partial^{\mu}\pi^{(0)}+\sum_{n\geq
1}\left(\frac{1}{4}\tilde{F}^{(n)}_{\mu\nu}\tilde{F}^{(n)\mu\nu}+\frac{m_{n}^{2}}{2}\rho^{(n)}_{\mu}\rho^{(n)\mu}\right)\right].$
(2.9)

Now we integrate out the all higher order vector and axial vector mesons and
keep only the lowest vector mesons ($\rho$ meson) in the gauge
$A_{Z}(x^{\mu},Z)=0$ [84]:
$V_{\mu}^{(1)}(x^{\mu})=g\rho_{\mu}(x^{\mu})=\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\left(\rho_{\mu}^{0}+\omega_{\mu}\right)&\rho_{\mu}^{+}&K_{\mu}^{*+}\\\
\rho_{\mu}^{-}&-\frac{1}{\sqrt{2}}\left(\rho_{\mu}^{0}-\omega_{\mu}\right)&K_{\mu}^{*0}\\\
K_{\mu}^{*-}&{\bar{K}}_{\mu}^{*0}&\phi_{\mu}\end{array}\right)$ and the
lightest pseudo-scalar meson $\pi$ is defined as:
$\pi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}&K^{0}\\\
K^{-}&{\bar{K}}^{0}&-\frac{2}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}\end{array}\right)$.
The expression for the gauge field $A_{\mu}(x^{\nu},Z)$ up to ${\cal O}(\pi)$
is as follows:

$\displaystyle
A_{\mu}(x^{\nu},Z)=\frac{\partial_{\mu}\pi}{F_{\pi}}\psi_{0}(Z)-V_{\mu}(x^{\nu})\psi_{1}(Z),$
(2.10)

where
$\psi_{0}(z)=\int^{Z}_{0}dZ^{\prime}\phi_{0}(Z^{\prime}),V_{\mu}^{(1)}(x^{\nu})=\rho^{(1)}_{\mu}-\frac{1}{{\cal
M}_{(1)}}\partial_{\mu}\pi^{(1)}$. Our motivation was to compute the LECs of
$SU(3)$ chiral perturbation theory and we followed the Hidden Local Symmetry
(HLS) approach of [84]. To do so, we introduced external vector ${\cal
V}_{\mu}$ and axial vector ${\cal A}_{\mu}$ fields in addition to the $\pi$
and $\rho$ vector mesons similar to [84] where we defined
$\frac{1}{F_{\pi}}\partial_{\mu}\pi\rightarrow\hat{\alpha}_{\mu\perp}=\frac{1}{F_{\pi}}\partial_{\mu}\pi+{\cal
A}_{\mu}-\frac{i}{F_{\pi}}[{\cal V}_{\mu},\pi]+\cdots$ , and
$\hat{\alpha}_{\mu||}\equiv-V_{\mu}+{\cal
V}_{\mu}-\frac{i}{2F_{\pi}^{2}}[\partial_{\mu}\pi,\pi]+\cdots$. Therefore
$\hat{\alpha}_{\mu\perp}$ and $\hat{\alpha}_{\mu||}$ act as the degrees of
freedom in this approach. As we discussed earlier that there are infinite
number of fields in the mode expansion gauge and scalar fields. Hence, we
truncated the spectrum in such a way that the remaining fields are lowest
vector meson $(V_{\mu}^{(1)}=g\rho_{\mu})$ and lightest psuedo-scalar meson
($\pi$ meson)[84]. In terms of the $\hat{\alpha}_{\mu\perp}$ and
$\hat{\alpha}_{\mu||}$ , the gauge field is written as:

$A_{\mu}(x^{\mu},z)=\hat{\alpha}_{\mu\perp}(x^{\mu})\psi_{0}(z)+(\hat{\alpha}_{\mu||}(x^{\mu})+V_{\mu}^{(1)}(x^{\mu}))+\hat{\alpha}_{\mu||}(x^{\mu})\psi_{1}(z),$
(2.11)

implying

$\displaystyle
F_{\mu\nu}=-V_{\mu\nu}\psi_{1}+v_{\mu\nu}(1+\psi_{1})+a_{\mu\nu}\psi_{0}-i[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}]\psi_{1}(1+\psi_{1})+i[\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu\perp}](1+\psi_{1}-\psi_{0}^{2})$
$\displaystyle-i([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])\psi_{1}\psi_{0},$
(2.12)

where $\psi_{0}(\equiv\psi_{0}(Z)$ and $\psi_{1}(\equiv\psi_{1}(Z)$ are the
profile functions for the psuedo-scalar $\pi$ meson and $\rho$ vector meson
given in appendix A. Based on [97], as concerns a chiral power counting, one
notices that $M_{\rho}\equiv{\cal O}(p)$ implying
$\hat{\alpha}_{\nu||}\equiv\frac{{\cal O}(p^{3})}{M_{\rho}^{2}}\equiv{\cal
O}(p),\hat{\alpha}_{\nu\perp}\equiv{\cal O}(p)$.
Further,$V_{\mu\nu},a_{\mu\nu}$ and $v_{\mu\nu}$ are of ${\cal O}(p^{2})$.
Therefore, when using (2.3), $\left(F_{\mu\nu}F^{\mu\nu}\right)^{m}$ is of
${\cal O}(p^{4m}),m\in\mathbb{Z}^{+}$. Because of this, one must take into
account the kinetic term ($m=1$) at ${\cal O}(p^{4})$, which results in the
expression shown below:

$\displaystyle
F_{\mu\nu}F^{\mu\nu}=\psi_{1}^{2}V_{\mu\nu}V^{\mu\nu}-\psi_{1}(1+\psi_{1})V_{\mu\nu}v^{\mu\nu}-\psi_{0}\psi_{1}V_{\mu\nu}a^{\mu\nu}+i\psi_{1}^{2}(1+\psi_{1})V_{\mu\nu}[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{||}]$
$\displaystyle-i\psi_{1}(1+\psi_{1}-\psi_{0}^{2})V_{\mu\nu}[\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{\perp}]+i\psi_{0}\psi_{1}^{2}V_{\mu\nu}([\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{||}]+[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{\perp}])-\psi_{1}(1+\psi_{1})v_{\mu\nu}V^{\mu\nu}$
$\displaystyle+(1+\psi_{1})^{2}v_{\mu\nu}v^{\mu\nu}+\psi_{0}(1+\psi_{1})v_{\mu\nu}a^{\mu\nu}-i\psi_{1}(1+\psi_{1})^{2}v_{\mu\nu}[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{||}]$
$\displaystyle+i(1+\psi_{1})(1+\psi_{1}-\psi_{0}^{2})v_{\mu\nu}[\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{\perp}]-i\psi_{0}\psi_{1}(1+\psi_{1})v_{\mu\nu}([\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{||}]+[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{\perp}])$
$\displaystyle-\psi_{0}\psi_{1}a_{\mu\nu}V^{\mu\nu}+\psi_{0}(1+\psi_{1})a_{\mu\nu}v^{\mu\nu}+\psi_{0}^{2}a_{\mu\nu}a^{\mu\nu}-i\psi_{0}\psi_{1}(1+\psi_{1})a_{\mu\nu}[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{||}]$
$\displaystyle+i\psi_{0}(1+\psi_{1}-\psi_{0}^{2})a_{\mu\nu}[\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{\perp}]-i\psi_{0}^{2}\psi_{1}a_{\mu\nu}([\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{||}]+[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{\perp}])$
$\displaystyle+i\psi_{1}^{2}(1+\psi_{1})[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}]V^{\mu\nu}-i\psi_{1}(1+\psi_{1})^{2}[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}]v^{\mu\nu}$
$\displaystyle-i\psi_{0}\psi_{1}(1+\psi_{1})[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}]a^{\mu\nu}-\psi_{1}^{2}(1+\psi_{1})^{2}[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}][\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{||}]$
$\displaystyle+\psi_{1}(1+\psi_{1})(1+\psi_{1}-\psi_{0}^{2})[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}][\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{\perp}]-\psi_{0}\psi_{1}^{2}(1+\psi_{1})[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu||}]([\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{||}]+[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{\perp}])$
$\displaystyle-i\psi_{1}(1+\psi_{1}-\psi_{0}^{2})[\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu\perp}]V^{\mu\nu}+i(1+\psi_{1})(1+\psi_{1}-\psi_{0}^{2})[\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu\perp}]v^{\mu\nu}$
$\displaystyle+\psi_{1}(1+\psi_{1})(1+\psi_{1}-\psi_{0}^{2})[\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu\perp}][\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{||}]-(1+\psi_{1}-\psi_{0}^{2})^{2}[\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu\perp}][\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{\perp}]$
$\displaystyle+\psi_{0}\psi_{1}(1+\psi_{1}-\psi_{0}^{2})[\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu\perp}]([\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{||}]+[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{\perp}])$
$\displaystyle+i\psi_{0}\psi_{1}^{2}([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])V^{\mu\nu}-i\psi_{0}\psi_{1}(1+\psi_{1})([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])v^{\mu\nu}$
$\displaystyle-i\psi_{0}^{2}\psi_{1}([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])a^{\mu\nu}-\psi_{0}\psi_{1}^{2}(1+\psi_{1})([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{||}]$
$\displaystyle+\psi_{0}\psi_{1}(1+\psi_{1}-\phi_{0}^{2})([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])[\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{\perp}]$
$\displaystyle-\psi_{0}^{2}\psi_{1}^{2}([\hat{\alpha}_{\mu\perp},\hat{\alpha}_{\nu||}]+[\hat{\alpha}_{\mu||},\hat{\alpha}_{\nu\perp}])([\hat{\alpha}^{\mu}_{\perp},\hat{\alpha}^{\nu}_{||}]+[\hat{\alpha}^{\mu}_{||},\hat{\alpha}^{\nu}_{\perp}]).$
(2.13)

Parity is defined as: $x^{i}\rightarrow-x^{i}$, and $Z\rightarrow-Z$ where $i$
are the conformally Minkowskian spatial directions. When $A_{\mu}(x,Z)$ is odd
then $\alpha_{\perp}$ is even and $\alpha_{||}$ is odd. Further, when
$V_{\mu}$ is odd then $\psi_{0}(Z)$ and $\psi_{1}(Z)$ are odd and even
respectively. Since the coupling constants should be scalar and even-$Z$
parity. We found that the terms with $(3\hat{\alpha}_{||}s\ ,\
1\hat{\alpha}_{\perp}\ {\rm or}\ 3\hat{\alpha}_{\perp}s\ ,\
1\hat{\alpha}_{||})$ in (2.3) are odd under the Parity transformation.
Similarly, term like $tr(\hat{\alpha}_{\mu\perp}\hat{\alpha}_{||}^{\mu})$ has
been dropped because of odd-$Z$ parity at ${\cal O}(p^{2})$. At ${\cal
O}(p^{2})$, we have the following Lagrangian [2, 83]:

$\displaystyle\hskip-32.52127pt{\mathcal{L}}_{(2)}\ni F_{\pi}^{2}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\mu}_{\perp}]+aF_{\pi}^{2}\,{\rm
tr}[{\hat{\alpha}}_{\mu||}{\hat{\alpha}}^{\mu}_{||}]-\frac{1}{2g_{YM}^{2}}\,{\rm
tr}[V_{\mu\nu}V^{\mu\nu}].$ (2.14)

At ${\cal O}(p^{4})$, $SU(3)$ chiral perturbation theory Lagrangian is
obtained as [2, 83]:

$\displaystyle\hskip-32.52127pt{\mathcal{L}}_{(4)}\ni y_{1}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\nu}_{\perp}]+y_{2}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{\perp}]+y_{3}\,{\rm
tr}[{\hat{\alpha}}_{\mu||}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\nu}_{||}]+y_{4}\,{\rm
tr}[{\hat{\alpha}}_{\mu||}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]$
$\displaystyle+y_{5}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\nu}_{||}]+y_{6}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]+y_{7}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\nu}_{||}{\hat{\alpha}}^{\mu}_{||}]$
$\displaystyle+y_{8}\,\left\\{{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\nu}_{||}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\mu}_{||}]+{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}_{\nu\perp}{\hat{\alpha}}^{\nu}_{||}\right\\}+y_{9}\,{\rm
tr}[{\hat{\alpha}}_{\mu\perp}{\hat{\alpha}}_{\nu||}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{||}]$
$\displaystyle+z_{1}\,{\rm tr}[v_{\mu\nu}v^{\mu\nu}]+z_{2}\,{\rm
tr}[a_{\mu\nu}a^{\mu\nu}]+z_{3}\,{\rm tr}[v_{\mu\nu}V^{\mu\nu}]+iz_{4}\,{\rm
tr}[V_{\mu\nu}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{\perp}]$
$\displaystyle+iz_{5}\,{\rm
tr}[V_{\mu\nu}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]+iz_{6}\,{\rm
tr}[v_{\mu\nu}{\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{\perp}]+iz_{7}\,{\rm
tr}[v_{\mu\nu}{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{||}]-iz_{8}\,{\rm
tr}\left[a_{\mu\nu}\left({\hat{\alpha}}^{\mu}_{\perp}{\hat{\alpha}}^{\nu}_{||}+{\hat{\alpha}}^{\mu}_{||}{\hat{\alpha}}^{\nu}_{\perp}\right)\right].$
(2.15)

where

$\hat{{\alpha}}_{\mu{||/\perp}}=\frac{i}{2}\left(\xi_{R}D_{\mu}\xi^{\dagger}_{R}\pm\xi_{L}D_{\mu}\xi^{\dagger}_{L}\right);D_{\mu}\xi^{\dagger}_{R/L}=\partial_{\mu}\xi^{\dagger}_{R/L}-i({\cal
R/\cal L})_{\mu}\xi^{\dagger}_{R/L}+i\xi^{\dagger}_{R/L}V_{\mu}.$
$v_{\mu\nu}=\frac{1}{2}\left(\xi_{R}{\cal
R}_{\mu\nu}\xi^{\dagger}_{R}+\xi_{L}{\cal
L}_{\mu\nu}\xi_{L}^{\dagger}\right)\hskip 14.22636pt{\rm and}\hskip
14.22636pta_{\mu\nu}=\frac{1}{2}\left(\xi_{R}{\cal
R}_{\mu\nu}\xi^{\dagger}_{R}-\xi_{L}{\cal
L}_{\mu\nu}\xi_{L}^{\dagger}\right),$

${\cal L}_{\mu\nu}=\partial_{[\mu}{\cal L}_{\nu]}-i[{\cal L}_{\mu},{\cal
L}_{\nu}]$ and ${\cal R}_{\mu\nu}=\partial_{[\mu}{\cal R}_{\nu]}-i[{\cal
R}_{\mu},{\cal R}_{\nu}]$ and ${\cal L}_{\mu}={\cal V}_{\mu}-{\cal A}_{\mu}$
and ${\cal R}_{\mu}={\cal V}_{\mu}+{\cal A}_{\mu}$, and
$\xi_{L}^{\dagger}(x^{\mu})=\xi_{R}(x^{\mu})=e^{\frac{i\pi(x^{\mu})}{F_{\pi}}}$.
Coupling constants appearing in (2.14) and (2.3) are given by the following
radial integrals.

$\displaystyle F_{\pi}^{2}=-\frac{{\cal
V}_{\Sigma_{2}}}{4}<<\dot{\psi}_{0}^{2}>>,$ $\displaystyle
aF_{\pi}^{2}=-\frac{{\cal V}_{\Sigma_{2}}}{4}<<\dot{\psi}_{1}^{2}>>,$
$\displaystyle\frac{1}{g_{\rm YM}^{2}}=\frac{{\cal
V}_{\Sigma_{2}}}{2}<\psi_{1}^{2}>,$ $\displaystyle y_{1}=-y_{2}=-\frac{{\cal
V}_{\Sigma_{2}}}{2}<(1+\psi_{1}-\psi_{0})^{2}>,$ $\displaystyle
y_{3}=-y_{4}=-\frac{{\cal V}_{\Sigma_{2}}}{2}<\psi_{1}^{2}(1+\psi_{1})^{2}>,$
$\displaystyle y_{5}=-{\cal V}_{\Sigma_{2}}<\psi_{0}^{2}\psi_{1}^{2}>,$
$\displaystyle y_{6}=-y_{7}=-{\cal
V}_{\Sigma_{2}}<\psi_{1}(1+\psi_{1})(1+\psi_{1}-\psi_{0}^{2})>,$
$\displaystyle y_{8}=-y_{9}=-{\cal V}_{\Sigma_{2}}<\psi_{0}^{2}\psi_{1}^{2}>,$
$\displaystyle z_{1}=-\frac{{\cal V}_{\Sigma_{2}}}{4}<(1+\psi_{1})^{2}>,$
$\displaystyle z_{2}=-\frac{{\cal V}_{\Sigma_{2}}}{4}<\psi_{0}^{2}>,$
$\displaystyle z_{3}=\frac{{\cal V}_{\Sigma_{2}}}{2}<\psi_{1}(1+\psi_{1})>,$
$\displaystyle z_{4}={\cal V}_{\Sigma_{2}}<\psi_{1}(1+\psi_{1}-\psi_{0}^{2}),$
$\displaystyle z_{5}=-{\cal V}_{\Sigma_{2}}<\psi_{1}^{2}(1+\psi_{1})>,$
$\displaystyle z_{6}=-{\cal
V}_{\Sigma_{2}}<(1+\psi_{1})(1+\psi_{1}-\psi_{0}^{2})>,$ $\displaystyle
z_{7}={\cal V}_{\Sigma_{2}}<\psi_{1}(1+\psi_{1})^{2}>,$ $\displaystyle
z_{8}={\cal V}_{\Sigma_{2}}<\psi_{0}^{2}\psi_{1}>,$ (2.16)

where we have used the following notations.

$<<A>>=\int_{0}^{\infty}{\mathcal{V}_{1}(z)}Adz,$
$<A>=\int_{0}^{\infty}{\mathcal{V}_{2}(z)}Adz,$

and

$\hskip 115.63243pt{\cal
V}_{\Sigma_{2}}=-\frac{T_{D_{6}}(2\pi\alpha^{\prime})^{2}}{4}\int
dyd\theta_{2}\delta\Bigg{(}\theta_{2}-\frac{\alpha_{\theta_{2}}}{N^{\frac{3}{10}}}\Bigg{)}.$

The simplified expressions of the radial integrals appearing in (2.3) are
given in the appendix A. Relation between SU(3) LECs of [87] (equation (2.2))
and couplings appearing from holographic computation (equation (2.3)) was
obtained in [91]:

$\displaystyle
L_{1}^{r}=\frac{L_{2}^{r}}{2}=-\frac{L_{3}^{r}}{6}=\frac{1}{32}\left(\frac{1}{g_{YM}^{2}}-z_{4}+y_{2}\right),$
(2.17) $\displaystyle
L_{9}^{r}=\frac{1}{8}\left(\frac{2}{g_{YM}^{2}}-2z_{3}-z_{4}-z_{6}\right),$
(2.18) $\displaystyle
L_{10}^{r}=\frac{1}{4}\left(-\frac{1}{g_{YM}^{2}}+2z_{3}-2z_{2}+2z_{1}\right).$
(2.19)

Matching Holographic Result with the Phenomenological Data: We now discuss the
matching of the one-loop renormalized ${\cal O}(p^{4})$ $SU(3)$ $\chi$PT LECs
obtained by us with the phenomenological data given in the literature. We will
also discuss the matching of $F_{\pi}^{2},g_{\rm YM}(\Lambda_{\rm QCD}=0.4{\rm
GeV},\Lambda=1.1{\rm Gev},\mu=M_{\rho})$ where “HLS-QCD” matching scale is
represented by $\Lambda$ [97], while the renormalization scale is represented
by $\mu$. We showed that $L_{1,9,10}^{r}$ match exactly whereas $L_{2,3}^{r}$
match up to the order of magnitude and signs.

  * •

Step 1: Matching $L_{1,2,3}^{r}$: Using (A.1), (A.12), (A.1), (A.1) and
$y_{2}=-y_{1}$ we obtained:

$\displaystyle\hskip-28.90755ptL_{1}^{r}=\frac{L_{2}^{r}}{2}=-\frac{L_{3}^{r}}{6}=\frac{1}{32}\left(\frac{1}{g_{YM}^{2}}-z_{4}+y_{2}\right)$
$\displaystyle\hskip-28.90755pt=\frac{1}{143360\sqrt{7}({f_{r_{0}}}-1){f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{3}{\cal C}_{\psi_{1}}^{\rm IR}\
{}^{2}\Omega}\Biggl{\\{}3\pi({f_{r_{0}}}+1)N^{7/5}$
$\displaystyle\hskip-28.90755pt\times\Biggl{(}-\frac{4822335\sqrt{2}\sqrt[8]{3}\sqrt[4]{7}\pi^{3}\beta\left({\cal
C}_{zz}^{(1)}-2{\cal C}_{\theta_{1}z}^{(1)}+2{\cal
C}_{\theta_{1}x}^{(1)}\right){\cal C}_{\phi_{0}}^{\rm IR}\
{}^{2}\epsilon^{9/4}{f_{r_{0}}}({f_{r_{0}}}+1){g_{s}}^{4}M^{2}N_{f}^{4}\alpha_{\theta_{1}}^{9}{\cal
C}_{\psi_{1}}^{\rm IR}\
N^{2{f_{r_{0}}}+\frac{7}{5}}}{({f_{r_{0}}}-1)^{3}\left(\log N\right)^{3}}$
$\displaystyle\hskip-28.90755pt+\frac{430080\sqrt{21}\beta\left({\cal
C}_{zz}^{(1)}-2{\cal C}_{\theta_{1}z}^{(1)}+2{\cal
C}_{\theta_{1}x}^{(1)}\right)\epsilon^{3}({f_{r_{0}}}-1){f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}{m_{0}}^{2}N^{3/5}N_{f}^{8}\alpha_{\theta_{1}}^{5}{\cal
C}_{\psi_{1}}^{\rm IR}\ {}^{2}}{\pi({f_{r_{0}}}+1)}$
$\displaystyle\hskip-28.90755pt+\frac{64\sqrt{\frac{7}{\pi}}({f_{r_{0}}}-1)^{2}{g_{s}}^{9}\log
r_{0}M^{4}N_{f}^{8}{\cal C}_{\psi_{1}}^{\rm IR}\
{}^{2}N^{\frac{2{f_{r_{0}}}}{3}-\frac{3}{5}}\Omega}{({f_{r_{0}}}+1)^{2}}+\frac{26880\sqrt{7}({f_{r_{0}}}-1){g_{s}}^{9}\log
r_{0}M^{4}N_{f}^{8}{\cal C}_{\psi_{1}}^{\rm IR}\
{}^{2}N^{\frac{2{f_{r_{0}}}}{3}-\frac{2}{5}}\Omega}{({f_{r_{0}}}+1)^{2}}$
$\displaystyle\hskip-28.90755pt-\frac{8960\sqrt{7}({f_{r_{0}}}-1){f_{r_{0}}}{g_{s}}^{9}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{2}{\cal C}_{\psi_{1}}^{\rm IR}\
{}^{2}N^{\frac{2{f_{r_{0}}}}{3}-\frac{2}{5}}\Omega}{{f_{r_{0}}}+1}$
$\displaystyle\hskip-28.90755pt-\frac{132269760\sqrt{2}\sqrt[8]{3}\sqrt[4]{7}\pi^{3}{\cal
C}_{\phi_{0}}^{\rm IR}\
{}^{2}\epsilon^{17/4}{f_{r_{0}}}({f_{r_{0}}}+1)^{2}{g_{s}}^{4}M^{2}N_{f}^{4}\alpha_{\theta_{1}}^{9}{\cal
C}_{\psi_{1}}^{\rm IR}\
N^{2{f_{r_{0}}}+\frac{7}{5}}}{({f_{r_{0}}}-1)^{3}\left(\log N\right)^{3}}$
$\displaystyle\hskip-28.90755pt-\frac{5160960\sqrt{7}\epsilon^{2}({f_{r_{0}}}-1){f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N^{3/5}N_{f}^{8}\alpha_{\theta_{1}}^{5}{\cal C}_{\psi_{1}}^{\rm IR}\
{}^{2}}{\pi}$
$\displaystyle\hskip-28.90755pt-\frac{195259926456\sqrt[4]{3}\pi^{7}{\cal
C}_{\phi_{0}}^{\rm IR}\
{}^{4}\epsilon^{13/2}{f_{r_{0}}}({f_{r_{0}}}+1)^{4}\alpha_{\theta_{1}}^{15}N^{4{f_{r_{0}}}+\frac{11}{5}}}{({f_{r_{0}}}-1)^{7}\left(\log
N\right)^{7}}\Biggr{)}\Biggr{\\}},$ (2.20)

where:

$\Omega\equiv\left(7\beta\left({\cal C}_{zz}^{(1)}-2{\cal
C}_{\theta_{1}z}^{(1)}+2{\cal
C}_{\theta_{1}x}^{(1)}\right){f_{r_{0}}}^{2}\gamma^{2}{g_{s}}^{2}\left(\log
N\right)^{2}M^{4}+3456\epsilon^{2}({f_{r_{0}}}+1)N^{2}\right).$ (2.21)

Since, we are working up to ${\cal O}(\beta)$ and $\epsilon$ is small and
therefore working up to ${\cal O}(\epsilon^{2})$, we found that (• ‣ 2.3)
simplified as:

$\displaystyle\hskip-28.90755ptL_{1}^{r}=\frac{1}{143360\sqrt{7}({f_{r_{0}}}-1){f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{3}}$
$\displaystyle\hskip-28.90755pt\times\Biggl{\\{}3\pi({f_{r_{0}}}+1)N^{7/5}\Biggl{(}\frac{64\sqrt{\frac{7}{\pi}}({f_{r_{0}}}-1)^{2}{g_{s}}^{9}\log
r_{0}M^{4}N_{f}^{8}N^{\frac{2{f_{r_{0}}}}{3}-\frac{3}{5}}}{({f_{r_{0}}}+1)^{2}}$
$\displaystyle\hskip-28.90755pt+\frac{26880\sqrt{7}({f_{r_{0}}}-1){g_{s}}^{9}\log
r_{0}M^{4}N_{f}^{8}N^{\frac{2{f_{r_{0}}}}{3}-\frac{2}{5}}}{({f_{r_{0}}}+1)^{2}}-\frac{8960\sqrt{7}({f_{r_{0}}}-1){f_{r_{0}}}{g_{s}}^{9}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{2}N^{\frac{2{f_{r_{0}}}}{3}-\frac{2}{5}}}{{f_{r_{0}}}+1}$
$\displaystyle\hskip-28.90755pt-\frac{5160960\sqrt{7}({f_{r_{0}}}-1){f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N^{3/5}N_{f}^{8}\alpha_{\theta_{1}}^{5}}{\pi}\left(\frac{\epsilon^{2}}{\Omega}\right)\Biggr{)}\Biggr{\\}}.$
(2.22)

Further, we need to consider: $f_{r_{0}}=1-\kappa,0<\kappa\ll 1$ (from (2.29)
and (2.30), will be discussed later when we match $L_{9}^{r}$ with the
phenomenological value). Hence (• ‣ 2.3) further simplified as:

$\displaystyle\hskip-28.90755ptL_{1}^{r}=\frac{1}{143360\sqrt{7}{f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{3}}\Biggl{\\{}3\pi({f_{r_{0}}}+1)N^{7/5}\Biggl{(}\frac{26880\sqrt{7}{g_{s}}^{9}\log
r_{0}M^{4}N_{f}^{8}}{({f_{r_{0}}}+1)^{2}}$
$\displaystyle\hskip-28.90755pt-\frac{8960\sqrt{7}{f_{r_{0}}}{g_{s}}^{9}\log
NM^{4}N_{f}^{8}\alpha_{\theta_{1}}^{2}}{{f_{r_{0}}}+1}-\frac{5160960\sqrt{7}{f_{r_{0}}}{g_{s}}^{8}\log
NM^{4}N^{\frac{1}{3}}N_{f}^{8}\alpha_{\theta_{1}}^{5}}{\pi}\left(\frac{\epsilon^{2}}{\Omega}\right)\Biggr{)}\Biggr{\\}}.$
(2.23)

In addition, considering that the $\log r_{0}$ in (• ‣ 2.3) is, in reality,
$\log\left(\frac{r_{0}}{{\cal R}_{D5/\overline{D5}}}\right)$ \- ${\cal
R}_{D5/\overline{D5}}>r_{0}$, where ${\cal R}_{D5/\overline{D5}}$ denotes the
separation between $D5-\overline{D5}$-branes; we realised from (• ‣ 2.3) that
in order to obtain a positive value (which is required by the phenomenological
value of $L_{1}^{r}$), $\Omega<0$ is necessary. Note that, as demonstrated
later in (2.26), fitting with the experimentally determined value of the pion
decay constant $F_{\pi}$ demands a $N$-suppression in $\alpha_{\theta_{1}}$,
which implies that the $N$ enhancement in the last term in (• ‣ 2.3) is an
artificial enhancement. Therefore, in order to make certain that one is not
picking up a ${\cal O}\left(\frac{1}{\beta}\right)$ contribution in
$L_{1}^{r}$ from $\frac{\epsilon^{2}}{\Omega}$ in (• ‣ 2.3), in addition to
make sure that the last term in (• ‣ 2.3) required to partially compensate for
the initial two terms that are negative within the same (as clarified earlier)
results in a positive $L_{1}^{r}$, from (2.21), we set:

$\epsilon=\lambda_{\epsilon}\frac{\sqrt{\beta}}{N}.$ (2.24)

In conclusion, when all of these findings are taken into account, along with
the demand that the value of $L_{1}^{\rm exp}=0.64\times 10^{-3}$, we get the
following: Finally, combining the above observations with the requirement to
match the experimental value $L_{1}^{\rm exp}=0.64\times 10^{-3}$, we
implemented the following constraint:

$\displaystyle\left({\cal C}_{zz}^{(1)}-2{\cal C}_{\theta_{1}z}^{(1)}+2{\cal
C}_{\theta_{1}x}^{(1)}\right)=-\frac{493.7(\delta+1)({f_{r_{0}}}+1)\lambda_{\epsilon}^{2}}{{f_{r_{0}}}^{2}\gamma^{2}{g_{s}}^{2}\left(\log
N\right)^{2}M^{4}},$
$\displaystyle\delta=\frac{0.053\alpha_{\theta_{1}}^{3}N^{-\frac{2{f_{r_{0}}}}{3}-1}}{{g_{s}}}.$
(2.25)

Because of this, we may deduce from (2.24) that the parameter $\epsilon$
serves as an expansion parameter combining the $\frac{1}{N}$ and $\beta$
expansions. In addition, this is the first relationship among large-$N$ and
higher derivative corrections in the framework of ${\ calM}$-theory, which is
the dual of large-$N$ thermal QCD-like theories. This connection was made
achievable from the use of (• ‣ 2.3).

When we compare the theoretical values of $L_{2,3}^{r}$ to their experimental
values, as shown in (• ‣ 2.3), we found that it is possible to obtain a match
with the order of magnitude and the sign of the value, but not with the
precise numerical value.

  * •

Step 2: Matching $F_{\pi}^{2}$: Now, using (• ‣ 2.3) and (A.1), we showed that
the difference of (A.1) and the experimental value of
$F_{\pi}^{2}=\frac{0.0037N^{-\frac{2{f_{r_{0}}}}{3}-1}}{{g_{s}}}$222$F_{\pi}^{2}$
can be made to match the experimental value of $92.3MeV$ wherein from
$0^{++}$-glueball mass [71], one identifies:
$\frac{1700}{4}MeV\equiv\frac{r_{0}}{\sqrt{4\pi g_{s}N}}$. is zero when:

$\displaystyle\alpha_{\theta_{1}}=\frac{0.03\sqrt[24]{\beta}{g_{s}}^{11/3}\sqrt[12]{\lambda_{\epsilon}}\left(\log
N\right)^{2/3}\sqrt[3]{M}N_{f}^{2/3}\sqrt[3]{{\cal C}_{\psi_{1}}^{\rm IR}\
}{(1-{f_{r_{0}}})^{2/3}}N^{-\frac{{f_{r_{0}}}}{3}-\frac{13}{60}}}{\sqrt[3]{{\cal
C}_{\phi_{0}}^{\rm IR}\ }\sqrt[3]{{f_{r_{0}}}+1}}.$ (2.26)

  * •

Step 3: Matching $L_{9}^{r}$: From [98], we have:

$L_{9}^{r}=\frac{1}{8}\left(\frac{2}{g_{\rm
YM}^{2}}-2z_{3}-z_{4}-z_{6}\right).$ (2.27)