Advancements in String Theory: Resolving Conifold and Flavor D7 Brane Embedding

Table of Links

Abstract

Acknowledgment

PART I

Chapter 1: Introduction

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

Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation

Chapter 4: Conclusion and Future Outlook

PART II

Chapter 5: Introduction

Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity

Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling

Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times

Chapter 9: Multiverse in Karch-Randall Braneworld

Chapter 10: Conclusion and Future outlook

APPENDIX A

APPENDIX B

APPENDIX C

Bibliography

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 E8 × E8, and Heterotic SO(32). In 1995, it was shown by Witten that these five theories could be unified into a single theory known as “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

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

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 Nc, which stays Nχ c in the presence of matter, is unaffected by the existence of matter, but the Euler number changes to χ = 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-Nc 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

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-toone 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: Nc → ∞ and g 2 YM = gs → 0 such that λ = g 2 Y M Nc 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 (gs) 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 λ; as a result, we are able to investigate the behaviour at both extremes of the parameter range, whether λ 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 gs, 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 gs << 1, for both closed and open string. For low energies E << α′−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 ϕ corresponds to open string excitations that are transverse to the D-branes, while gauge field Aµ corresponds to open string excitations that are parallel to the D-branes. The product gsN 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 gsN << 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/α′ 2 ∝ gsN appear in the case of a stack of N coincident D-branes. The closed string perspective is only applicable for the case where gsN >> 1 is being considered.

The AdS5/SCF T4 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 gsN << 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.

S = Sclosed + Sopen + Sint.

1.2.5 Closed String Perspective

In the strong coupling limit gsN → ∞, 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)×SO(6) isometries of spacetime, and half of the supercharges of type IIB supergravity, is provided by:

• r ≫ 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 ≪ L: A rough approximation of H(r) in this region is L 4/r4 . The metric can be roughly described as

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.

1.3 Gauge-Gravity Duality

The duality proposed by Maldacena in [17] is a special case, and it relates the N = 4 supersymmetric Yang-Mills (SYM) theory and type IIB string theory on the AdS5 × 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 YangMills 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 [19–25].

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 × S 3 where the authors have discussed the various features of QCD, see [33–37]. For the holographic description of QCD, see [38–43]. The D-branes have been used to study the inflation in [44, 45].

1.3.1 Klebanov-Witten Model

1.3.2 Klebanov-Tsytlin Model

From (1.17), we see that B2 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 B2, 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 F5 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

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.

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 Nf D7-branes were embedded via Ouyang embedding with embedding parameter µ:

The presence of the flavor branes affects the dilaton profile as

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 M-theory uplift at finite coupling in 1.4.3. Finally we will discuss the 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 < Tc) and deconfined (T > Tc), both phases of QCD and non-supersymmetric. UV completion problem of the Klebanov-Strassler model [27] has been cured in this 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

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 ψ 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 (ϕ1, ϕ2, ψ), and these are related by the following equations [15]:

1.4.3 M-theory Uplift of Type IIB Setup

Moreover, in the UV:

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

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

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):

• IR Color-Flavor Enhancement of Length Scale: Even with 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 gsNf in the RR and NS-NS three-form fluxes, and NLO in N in the 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 notationNeff(r), will be given by

• Obtaining Nc = 3: We will now briefly summarise how to identify the number of colours Nc 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 Nc as

Using (1.42) and (1.45), one can derive the UV and IR values of Nc as given below

Nc ∈ [N, M].

Therefore, from (1.46), we can easily see that Nc = M in the IR and it was taken as Nc = M = 3 in the MQGP limit [14]. From the various discussions in this section, we found that 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 > Tc and T < Tc).

1.4.4 M-Theory Uplift Inclusive of O(R4

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 elevendimensional 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)/G2/SU(4)/Spin(7)- structure torsion classes of the relevant six-, seven- and eight-folds associated with the Mtheory uplift have been computed in [1].

The origin of O(R4 )-corrections to the 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 O(R4 )-corrections to the N = 1, D = 11 supergravity action. One is in the context of the effects of D-instantons in IIB supergravity/string theory via the four-graviton scattering amplitude [73]. The other is D = 10 supersymmetry [76]. Let us briefly discuss these before going into the details of the construction of 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 R4 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 ζ µrνr = ζ (µr ˜ζ νr) and calculating the fermionic integral, the authors in [73] showed that the four-graviton scattering amplitude is able to be represented via the equation:

• Green and Vanhove showed that the eleven-dimensional O(R4 ) adjustments possess a separate motive based on ten-dimensional supersymmetry [76]. The above has been shown by its relationship with the term C (3)∧X8 in the M-theory effective action, that is believed to originate through a number of reasons, including anomaly cancellation [78, 79]. The formula X8 represents the eight-form in the curvatures obtained via the term in type IIA superstring theory [80]:

Now let us discuss the M-theory uplift in the presence of O(R4 ) terms based on [1]. The working action of eleven-dimensional supergravity with N = 1 supersymmetry inclusive of OR4 ) terms is written below:

Equations of motion of the M-theory metric and three-form potential are given below.

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

[1] See [64, 65] for where NS5 branes have been studied.