Fundamentals of Holography and its Applications 

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Fundamentals of Holography and its Applications

Holography, as its name suggests, is the study of making holograms where a one-dimension higher system could be reconstructed by a lower system, specifically in the current context, it is regarded as a synonym of the AdS/CFT correspondence. As an attractive subject which has been developed for more than twenty years, it has become a large field of study, which roughly can be divided into two categories based on the approaches to study the subject. The first one is the top-down approach, where clear descriptions on both sides are available from the UV complete theory perspective (superstring theory or supergravity theory), as well as the existence of precise matching of the symmetries, spectra, coupling constants, amplitudes etc. The first prominent example is given by Maldacena [5], and later by Witten [22] and Gubser, Klebanov and Polyackov [23] stating that 4D N = 4, SU(N) Yang-Mills theory ≡ Type IIB string theory on AdS5 × S5 , (2.1) which is realized through N−stack of coincident D3 branes in the ten-dimensional spacetime whose near horizon geometry is AdS5 × S5, while the low energy dynamics of the brane volumes is governed by the N = 4 U(N) gauge theory1. A quick look at the symmetries on both sides reveals the consistency of the duality, since on the left side, we have the SO(4, 2) global conformal symmetry as well as the global SU(4) R-symmetry, while on the right side, these symmetries correspond to the isometries of the full spacetime, SO(4, 2) ×SO(6), which are equivalent to each other, since SO(6) ∼ SU(4). Notably, the duality relates the parameters on both sides in the following manner, (`/ls)4 ≡ `4/α02 = 4πgsN = gYM2N = λ, (2.2) where λ is the ’t Hooft coupling and gYM is the coupling for the term − 1 TrFµν F µν in the 2gYM2 YM theory, ls is the string length related to the string tension (2πα0)−1 as α 0 = ls2 and gs is the string coupling. To have a classical gravity description requires that N → ∞ and λ → ∞, physically this means that we consider the planar limit of a strongly coupled gauge theory. For more interesting aspects and details on this duality or some other top-down models, one can refer to the thorough AGMOO review [25].
The second approach is the opposite, called bottom-up approach, where we consider an effective AdS gravity theory that is dual to a one dimension lower quantum field theory, with the belief that there exists a UV completed theory in higher dimensions that can make precise the duality, usually a ten-dimensional superstring theory or eleven-dimensional M-theory if can be consistently constructed. This gives an advantage of this approach, since one only needs to consider the necessary relavant bulk ingredients without complicating the system by finding a consistent theory including the compactified directions. Considering its nice features, the bottom-up approach will be the main melody of the current thesis.
In the following sections, I will start with some basic ingredients about the holographic duality, with emphasis on the geometric features of the AdS space and the most useful extrapolating method to obtain the CFT correlation functions. After that, I will introduce the application of the holographic duality on the subjects that have been studied in this thesis.

Basic Ingredients of Holographic Duality

In this part, I only include the most necessary ingredients that will be used in the current thesis. For more details, one can refer to the thorough review [25] by Aharony, Gubser, Maldacena, Ooguri and Oz, or lecture notes by, for example, Skenderis [26] and Polchinski [24].

A Glimpse of the Anti-de Sitter Space

As is known, the AdSd+1 space is a maximally symmetric spacetime, which means that its Rie-mann tensor is fixed by the metric tensor gab up to a total factor related to the AdS scale ` Rabcd = − 1 (gacgbd − gadgbc) , (2.3) where a runs from 0 to d. It has the right scale dimension and is antisymmetric under the change of a ↔ c and b ↔ d and symmetric under the exchange of {ab} ↔ {cd} as expected. Tracing twice of the indices, we have the Ricci scale R = −d(d`+1)2 , which simply tells us that AdS space has a constant negative curvature, resulting in a negative cosmological constant Λ = d(1−2d) when applying the Einstein equation.
The isometry group for the AdS space as aforementioned is SO(d, 2), which can also be looked as the Lorentz group for (d + 2)-Minkowski space with two negative signatures. In fact, one can embed AdSd+1 space as a hyperboloid surface in the (d + 2)-Minkowski space with the following surface equation −X02 X + Xi2 − Xd2+1 = −`2, (2.4) i=1 where XM (M = 0, . . . , d + 1) are the Minkowski coordinates. Through different coordinate choices2, one would get different patches of the AdS space. Notably, there is the global embedding, which is a result of the following relation between the global AdS coordinates with the Minkowski coordinates,
X0 = ` cosh ρ cos τ , Xd+1 = ` cosh ρ sin τ ,
Xi = ` sinh ρ sin θ1 sin θ2 . . . sin θi−1 cos θi (i = 1, . . . , d − 1) , (2.5)
Xd = ` sinh ρ sin θ1 sin θ2 . . . sin θd−2 sin θd−1
where θi ∈ [0, π] for i = 1, . . . , d − 1 and θd−1 ∈ [0, 2π). To cover the hyperboloid once, one can take ρ ≥ 0 and τ ∈ [0, 2π), hence the name global coordinates, which leads to the metric ds2 = `2(− cosh2 ρdτ2 + dρ2 + sinh2 ρdΩ2) (2.6) with Ω being all the spherical coordinates and the asymptotic boundary locates at ρ → ∞. Another embedding worth mentioning is given by the Poincaré coordinate, which is often used with the embedding relation
X0 = 1 + (`2 + ~x2 − t2) , Xd+1 = , 2 z2 z
Xi = `xi (i = 1, . . . , d − 1) , Xd = z 1 − 1 (`2 − ~x2 + t2) , (2.7) z 2 z2
which covers a patch of the AdS space, with the asymptotic boundary located at z → 0. The Poincaré metric is given by ds2 = `2 (dz2 + ηµν dxµdxν ) , (2.8) z2 where one see clearly that the asymptotic boundary is conformally flat. By a coordinate change tan θ = sinh ρ (θ ∈ [0, π/2) for the global coordinate (2.6), it is easy to see the conformal flatness when approaching proposal is self-revealed in this
λ = δ
λ = 0
⇢=0 ⇢=1
Figure 2.1: Massive particles moving in AdS are attracted towards the center, as illustrated by two timelike geodesics in AdS2, one begins at λ = 0 while the other is the shifted one with δ displacement.
As AdS space has a negative cosmological constant, it is an attractor, which can be seen could refer to, for example the appendix of [27] from the trajectory of a massive particle in the global coordinates. To obtain that trajectory, in principle, one has to solve the timelike geodesic equation or the equivalent Euler-Lagrangian equation, however, thanks to the embedding picture, we could use a simpler manner to obtain a glimpse of the trajectory without solving the second-order differential equation. The observation is that in (2.6), a known timelike geodesic is ρ=0, τ =λ, (2.9) where λ is the affine parameter. Then one can use the isometries to obtain all the timelike geodesics. We can see the simplest example explicitly in the AdS2 space, if we perform a boost η in (01) direction in the embedding space, which means
` 0 ` cos λ sinh η = ` sinh ρ , (2.10)
cos λ → cos λ cosh η cosh ρ cos τ
sin λ sin λ cosh ρ sin τ
expressing ρ, τ in terms of λ, η within half a period τ ∈ [0, π] where τ(λ = 0) = 0 gives,
ρ = sinh−1(cos λ sinh η), (2.11)
τ = tan−1 tan λ . (2.12)
cosh η
The above solution gives a set of new timelike geodesics in AdS2. In a full period τ ∈ [0, 2π], one see that the geodesic will emanate from the center ρ = 0, then at some point turn back, go pass the centeral axis to the other side, later turn back again and end on the central axis, as Fig 2.1 shows. We can also get another geodesic from this one by performing a shift δ in τ direction, which amounts to a rotation in (02) directions in the embedding space. Taking advantages of all the isometries, one is able to obtain all the timelike geodesics for an AdS space with arbitrary dimensions, where anyone of those can reveal the attracting property of the AdS space due to its negative cosmological constant.

CFT Correlation functions via the Duality

Holography realizes a deep connection between the low energy supergravity in AdS space and a CFT theory living on its boundary. From a CFT perspective, one might wonder how the correlation functions are encoded in the gravity theory, since those correlators carry the whole CFT data. This piece of hidden information has been dug out by two independent works, Witten [22] and Gubser, Klebanov and Polyakov [23], giving rise to the name GKPW relation. The idea is to identify the generating functional of the CFT on the boundary side to the bulk gravity action at evaluated at the saddle point3, subject to the asymptotical Dirichlet boundary condition for the bulk fields which act as sources for the gauge invariant CFT operators. In practice, the saddle point is approximated by the solution to the equation of motion of the field respecting the chosen boundary condition. In the Euclidean signature4, the relation can be formulated as WCFT[φ0] = − ln exp Z φ0O = −Igravextremal[Φ∂ = φ0] , (2.13) ∂ CFT where φ0 is the asymptotic boundary condition for the bulk field Φ, which carries spin index implicitly. Holographic dictionary shows that the bulk scalar field is dual to the boundary scalar field, the bulk gauge field is dual to the boundary conserved current, the bulk graviton is dual to the boundary stress energy tensor, etc. In this manner, the connected CFT correlation func-tions are given by the functional derivatives of the extremized gravity action with respect to its boundary value, hO(x1)O(x2) • • • O(xn)ic = − δnIgravextremal[Φ∂ = φ0] φ0=0 . (2.14) δφ0 (x1)δφ0 (x2) • • • δφ0(xn)
For higher (than two) correlation functions, the issue becomes more complicated. In the following part of this section, we will consider an example with free massive scalar in the bulk, and get an idea how to obtain its two-point correlator holographically.
An example: two-point function for a massive scalar field in the bulk
Now as an illustration to obtain the two point function holographically, we will consider the canonical example with a free scalar field in the bulk, I = 1 Z ddxdz√ gφ(x)(− + m2)φ(x) + 1 Z∂AdS ddx√ γ∂⊥φ(x) + Ict (2.15) where the second integral is the boundary action and the third one is the counter term needed to make the action finite. Varying the Lagrangian with respect to φ gives the equation of motion, 1 ∂ (gab√ g∂ )φ = m2 φ . (2.16)
A similar equation of the above also appears when considering spin fields like a vector field or graviton in the AdS [28], that’s the reason we will focus on this as an illustration. The back reaction of the scalar field to the geometry can be neglected since it is of order O(G−N1) with GN being the Newton’s constant in (d + 1) dimension. Therefore, we are safe to use the metric for pure AdS. In the following, we will use the Euclidean version of the Poincaré coordinate (2.8) to give the explicit result, substituting into the equation of motion (2.16) gives ` d+1 ∂z z − 1 ∂z ! φ + `2 ∂µ∂µφ − m2 φ=0, (2.17) z ` d z2 whose solution contains two branches, depending on the leading power of z → 0. The full ansatz can be written as a power series of the radial coordinate z φ(z, x) = zΔ−(φ0(x) + z2φ2(x) + . . . ) + zΔ+ (φ2ν (x) + z2φν+2(x) + . . . ) (2.18)
| {z } | {z }
φ− φ+
± ν with ν = q
where Δ± = d d2 + m2`2 and φ2(x) stands for two-derivative term which can actually be obtained explicitly as well as the other higher derivative terms. Let’s assume that ν 6∈Z in the following analysis. One can see that φ− is the dominant branch and contains the non-normalisable mode when approaching the asymptotic boundary if Δ− < 0 < Δ+5, thus most of the time φ0 is identified as the source for the dual CFT operator, as well as the boundary condition for the bulk scalar. It is not hard to see that when substituting the ansatz (2.18) back to the action (2.15) assuming some conterterms like a quadratic term in φ on the boundary [26], one could obtain the expectation value of the boundary dual operator O(x) by using the description (2.14), hO(x)i = − δIon-shell = (2Δ+ − d) `d−1φ2ν (x) , (2.19) δφ 0 ( x ) which shows that, the asymptotically leading power the other branch φ+ works as the response to the source φ0 and provides one-point function in presence of the source.6 Once we have identified the source, in principle, one could write the bulk solution as an integral of the source φ(z, x) = ddyK(x, z; y)φ0(y) (2.20) where K(x, z; y) is the alleged bulk-to-boundary propagator, encoding the way how the boundary source φ0 propagates into the bulk. The explicit form for K(x, z; y) can be obtained by solving the scalar e.o.m7 assuming a localized boundary source φ0(x) = δ(d)(x − x0), given by K(x, z; y) = π−d/2 (Δ+) zΔ+ (2.21)
one sees that when z → 0, K(x, z; y) → 0 as expected. Finally, one arrives the two-point function
x y δhO(x)i d `d−1 δφ2ν (x) d π−d/2 (Δ+) 1 .
hO( )O( )i = = (2Δ+ − ) = (2Δ+ − ) (2.22)
δφ0(y) δφ0(y) (ν) (x − y)2Δ+
One can read the conformal dimension of the scalar operator from the power of the proper distance between the two operators, which is Δ+, the unitarity bound requires that Δ+ is bounded from below, i.e., Δ ≥ d/2 − 1, as in [29, 30].
5 This relation is obtained assuming m2 ≥ 0, in general, one only has to work on the condition m2 ≥ −d/4, known as the Breitenlohner-Freedman stability bound, though the leading asymptotic mode in φ− is no longer necessarily non-normalizable.
6In fact, φ2ν (x) can be expressed as an integral of the source φ0(x) where φ2ν (x) = R ddyF (x, y)φ0(x), with F (x, y) = limz→0 z−Δ+ K(x, z; y).

Holography and Two-Dimensional Defect CFT

In the study of AdS/CFT correspondence, a special case arises when the bulk dimension is “2+1” whose dual boundary theory is the “1+1” conformal field theory. The speciality in this dimension is that there are infinitely many conserved charges due to an enhancement of global SL(2, R) × SL(2, R) symmetry to two sets of local Virasoro symmtries whose generators satisfy the Virasoro algebra, the unique central extension of Witt algebra [31]. In the bulk side, if one choose the asymptotic boundary conditions for the metric as Brown and Henneaux did in [32], the asymptotic symmetries will be enlarged as well to the complete two dimensional conformal symmetries. In this manner, the central charge is identified as the ratio of the AdS radius over Newton’s constant for Einstein gravity, c = 23G` . For a general higher derivative theory of AdS3 gravity taking into account the quantum corrections, the central charge has a more covariant form c = ` gab ∂L [33, 34], one sees it coincides with Brown-Henneaux formula for Einstein gravity.
While the above should be regarded as a bottom-up consideration of the correspondence, the AdS3 geometry could also be constructed from a top-down approach, i.e., have an origin of an underlined string theory. The most well known realization comes from the near horizon geometry of D1-D5 brane system. Supposing we have a type IIB string theory on the background R1,4 × S1 × M4 with N5 D5-branes wrapped on S1 × M4 while N1 D1-branes wrapped on the S1, taking the near horizon limit yields the ten dimensional geometry AdS3 × S3 × M4 with M4 being T 4 or K3 [25, 35, 36]. The central charge can be obtained through an analysis of the anomaly on the field theory living on D1-D5 branes whose IR fixed point is the dual conformal field theory. In [25], the central charge is c = 6(ka + 1) = 6(N1N5 + 1), which is the same as six times the dimension of the instanton (D1 brane) moduli space for a large number of branes. Substituting the values for AdS3 radius ` = (g6N1N5)1/4ls and the three dimensional Newton’s constant G(3) = g62ls4/(4`3), one recovers the Brown-Henneaux central charge. Some recent process on understanding the duality for M4 = S3 × S could be found, for example [37,38] and references therein. However, this is outside the scope of the current thesis and we will not use too much space to talk about that.
In the following part of this section, we will consider only from the bottom-up perspective without questioning if there is a microscopic string theory. To start with, we will make some brief introductions on the holographic Weyl Anomaly and then reach the 2D defect CFT and its energy transport coefficients. A careful holographic consideration on how to obtain those coefficients and the bounds associated will be discussed in Chapter 4.

Holographic Weyl Anomaly

It is known that in conformal field theories, there exists Weyl anomalies in even spacetime dimen-sions (d = 2n) which can be described purely in terms of geometric quantities [39], falling into two types of classes, type A anomaly and type B anomaly.8 Type A is proportional to the Euler density of the dimension E(d) which is a topological term, in the sense that its integral over the spacetime manifold is the Euler characteristic, hence scale invariant. Type B constitutes of all the conformal scalar polynomials constructed by the Weyl tensor9 and its derivatives, denoted as I(d). There is also a third class consisting of total derivatives, however, it is trivial since it can be removed by adding finite local counterterms to the action [39, 42, 43].

Table of contents :

1 Introduction 
2 Fundamentals of Holography and its Applications 
2.1 Basic Ingredients of Holographic Duality
2.1.1 A Glimpse of the Anti-de Sitter Space
2.1.2 CFT Correlation functions via the Duality
2.2 Holography and Two-Dimensional Defect CFT
2.2.1 Holographic Weyl Anomaly
2.2.2 2D Defect CFT and Energy Transport Coefficients
2.3 Quantum Complexity and the Holographic Conjectures
2.3.1 The Two Holographic Conjectures
2.3.2 Field Theory Approach for Complexity: Nielsen Method
2.4 Modular Flow and Bulk Reconstruction
2.4.1 Entanglement Entropy and Modular Hamiltonian
2.4.2 Modular Hamiltonians in Holography: JLMS Relation
2.4.3 Code Subspace and Entanglement Wedge Reconstruction
3 Holographic complexity: “CA” or “CV” ? 
3.1 Warm up with the Defect Toy Model
3.1.1 Two-Dimensional Branes in AdS3
3.1.2 Fefferman-Graham Expansion and the Cutoff Surface
3.1.3 Wheeler-DeWitt Patch in Defect AdS3
3.2 Holographic Complexity with a Defect
3.2.1 CV Conjecture
3.2.2 CA Conjecture
3.3 Holographic Complexity for Subregions
3.3.1 Subregion CV Conjecture
3.3.2 Subregion CA Conjecture
3.4 Complexity in QFT
3.5 Discussion
4 A Careful Consideration of Holographic 2D dCFT 
4.1 Holographic Scattering States and Matching
4.2 Summary and Outlook
5 Revisiting Circuit Complexity in 2d Bosonisation 
5.1 2D Bosonisation
5.1.1 Basic Ingredients
5.1.2 Fermionic Fock space
5.1.3 Correspondence between states
5.2 Fubini-Study Method for Bosonic Coherent States
5.3 Application of Nielsen Method on Bosonic Coherent States
5.3.1 Complexity between Bosonic Ground States
5.3.2 Bosonic Coherent States with One Excited Mode
5.3.3 Complexity for Bosonic Coherent States with Shifts in More Modes
5.4 A class of Fermionic and Bosonic Bi-Gaussian States
5.4.1 A Bosonisation Identity
5.4.2 An Example with One Mode
5.4.3 An Example with Two Modes
5.5 Conclusion
6 Berry Curvature as A Probe of Bulk Curvature 
6.1 Entanglement as a Connection
6.2 Modular Berry Connection
6.2.1 A toy example
6.2.2 Gauging the modular zero modes
6.2.3 Comment on two-sided modular Hamiltonians
6.2.4 Modular Berry holonomy examples
6.3 Entanglement Wedge Connection
6.3.1 Modular zero modes in the bulk
6.3.2 Relative edge-mode frame as a connection
6.3.3 Bulk modular curvature and parallel transport
6.3.4 Example: Pure AdS3
6.4 The Proposal and Implications
7 Conclusions

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