AdS/CFT Duality in the Extraction of the ACConductivity of a SemiMetal
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Published: 27th Oct 2021 in
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Abstract
The AdS/CFT duality has been one of the most celebrated areas of research in the field of string theory, and here we give a brief introduction to it and follow up by utilizing it in an example calculation as a demonstration. Most physical theories involving particle interactions in metals involve weak coupling. Our project uses a way to understand the properties of a system with a large density of charged particles in a limiting case where the interactions between the particles is strong, which, due to the AdS/CFT duality between quantum field theories and solutions of string theory, makes the system more mathematically tractable because the fields in the Antide Sitter become weakly interacting. We will call this system a quantum semimetal. We then proceed to utilize Maxwell's equations to extract the twopoint current correlation function of this object and its ACconductivity, using a probe D_{5}brane.
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Assignment Writing ServicePreface
This is the report to conclude a directed project as part of the requirements to completing the PHYS 349 course at UBC. We present no original work, and this report is mostly a retelling of the story of the AdS/CFT duality from a combination of sources, mostly from the book "Gauge/Gravity Duality" by Martin Ammon and Johanna Erdmenger, "Introduction to the AdS/CFT Correspondence" by Jan de Boer [6], "The AdS/CFT Correspondence" by Veronika E. Hubeny [11] and "AdS/CFT Correspondence in Condensed Matter" by A.S.T Pires [15]. The calculations were directed by Dr. Gordon Walter Semenoff (Department of Physics, The University of British Columbia)
Table of Contents
Table of Contents.......................................................................................... iv
List of Figures................................................................................................ vi
Acknowledgments........................................................................................ vii
1 Introduction................................................................................................ 1
2 Taking the Large N Limit and Holography...................................................... 3
3 Antide Sitter Space..................................................................................... 5
4 Correlation Functions.................................................................................. 7
5 Derivation of the Ads/CFT Correspondence.................................................. 9
5.0.1 Mapping between parameters............................................................. 11
5.0.2 Turning off Quantum Fluctuations, and the weak form ofthe Correspondence 12
6 Probing a Strongly Correlated CFT with a D5Brane..................................... 13
6.0.1 Maxwell's Action on the D5 Brane........................................................ 16
6.0.2 Extracting the ACConductivity............................................................. 21
7 Epilogue.................................................................................................... 25
Bibliography................................................................................................ 27
List of Figures
Figure 6.1 A D5 brane is held perpendicular to a stack of an infinite number of coincident D3 branes that are at r = 0. The stack of D3 branes extends along the spacetime directions x^{0},x^{1},x^{2},x^{3}, and are transversal to the other six spatial directions x^{4},...x^{9} . In the limit depicted in the figure, the strings in the bulk are rigid, and we solve Maxwell's equations on the surface of the D5brane to extract the twopoint current correlation function at the boundary (r = ∞). In essence, we are looking for this particular operator on the "upper edge" of the infinite square depicted as the D5 brane in this figure. . . . . . . . . . . . . . 14
Introduction
The AdS/CFT Correspondence has become very important in the last two decades, with thousands of papers having been published in high energy Physics. While still a conjecture not having been proved, there is substantial evidence to believe that it holds. One reason for its importance is that it can be used to understand strongly interacting field theories by mapping them into classical gravity in one extra dimension. The idea is that largeN gauge theories (strongly coupled conformal field theories) in ddimensions is mapped to a gravitational theory in d+1 dimensions which is asymptotically AdS. This happens to be a strongweak duality where if for example, the field theory is strongly coupled, the gravity theory is weakly coupled and vice versa. The most wellknown form of this that the correspondence takes, originally conjectured by Maldacena [12], is that between 4dimensional N = 4 supersymmetric YangMills theory and type IIB string theory on AdS_{5} ×S^{5}  which is a 10dimensional world where AdS_{5} refers to Anti deSitter space in 5 dimensions and the S^{5} to a 5dimensional sphere. Antide Sitter spaces are maximally symmetric solutions of the Einstein equations with a negative cosmological constant [6]. N = 4 super YangMills theory was known to be conformally invariant (making this quantum field theory a conformal field theory, or a CFT), and it turns out that its group of conformal symmetries matches up exactly with the large symmetry group of 5dimensional Antide Sitter space. Different CFTs in ddimensions will generally correspond to theories of gravity with different field content and different parameters in the bulk (d+1 dimensions). Different arguments have shown that quantum field theories are actually quantum theories of gravity and so we can use them to compute observables of the QFT when the gravity theory is classical [13]. This fact we will use in this report.
Notice that the AdS/CFT correspondence takes a gravity theory in d+1 dimensions and relates it to a nongravitational one in ddimensions. String theory provides a means to compute finite quantum corrections to classical gravity, but its full nonperturbative structure is still not well understood [6]. If the AdS/CFT correspondence is true, this means that the actual quantum degrees of freedom cannot be the same as that of a local field theory that lives on the same space. If gravity had degrees of freedom that were local, we could take arbitrarily large volumes of fixed energy density  but we know from classical gravity that such a thing would eventually collapse to form a black hole. Clues for this gaugegravity duality have been around for some time  in three dimensions in particular . It turns out that in the 3dimensional case, gravity can be described by ChernSimons theory [1, 22].ChernSimons theory is a topological field theory that reduces to a 2dimensional field theory when put in a manifold with a boundary [21] [6]. A precise relation between Hilbert spaces and correlation functions in compact gauge groups could be established  while for noncompact gauge groups this precise relationship is not well understood. Nevertheless this duality between ChernSimmons theory and twodimensional conformal field theory has many similarities to the AdS/CFT duality [6].
Here we will primarily work with a weaker form of the AdS/CFT correspondence by working in the lowenergy limit in the string theory side. At very low energies, it is known that type IIB string theory on AdS_{5} ×S^{5} reduces to type IIB supergravity on AdS_{5} ×S^{5}. Correspondingly, we end up taking a limit on the N = 4 supersymmetric field theory side where the rank of the U(N) gauge group, N (not the same as the earlier N) and g_{YM}^{2} both become large. In this limit, the equivalence between type IIB supergravity on AdS_{5} ×S^{5} and N = 4 gauge theory has been well tested [6].
Taking the Large N Limit and Holography
The AdS/CFT correspondence is associated to two ideas in Physics. The first is that large N gauge theory is equivalent to a string theory [19]. In terms of 1/N and g_{YM}^{2} , the perturbative expansion of a large N gauge theory has the following form:
Z = ∑ N^{2}^{−2g} f_{g}(λ) (2.1)
g≥0
where λ = g_{YM}^{2} N is called the 't Hooft coupling constant. This is reminiscent of the loop expansion in string theory:
Z Z_{g } (2.2)
g≥0
 if we replace the string coupling constant g_{s} with 1/N. Through some mechanism, Feynman diagrams of the gauge theory are turned into surfaces that represent strings [14]  and apparently this is exactly what happens in the AdS/CFT correspondence.
The second idea in Physics is that of "holography" [17, 20]. This idea is related to the thermodynamics of black holes. Hawking and Bekenstein showed that black holes can be seen as thermodynamic entities [4, 5, 9].Thus they have entropies and temperatures. The black body radiation that the black hole emits relate directly to its temperature, while its entropy is given by S = c^{3}A/4Gh¯, where A is the surface area of the event horizon of the black hole, and G the Newton constant. These definitions make the laws of thermodynamics consistent with Einstein's equations of general relativity. The entropy of a system can be viewed as its "information content". It is a measure of the number of degrees of freedom that a theory has  and so it is surprising that the entropy of a black hole scales with its horizon area, and not its volume. Local field theories have their entropies proportional to their volume, so it cannot be that gravity behaves like a local field theory. It turns out that if we assume that gravity in d +1 dimensions is somehow equivalent to a local field theory in d dimensions, we reach a consistent picture. This is where this idea in Physics gets its name of "holography" it is like a hologram where 3dimensional information is encoded in a 2dimensional surface. The AdS/CFT correspondence is therefore holographic, since it states that quantum gravity in 5dimensions (forgetting the 5sphere) is equivalent to a field theory in 4 dimensions [6].
Antide Sitter Space
The Antide Sitter space belongs to a wide class of homogeneous spaces that can be defined as quadric surfaces in flat vector spaces [15]. An example is the ddimensional sphere S^{d} given by:
X (3.1)
embedded in Euclidean d+1 dimensional space. The ddimensional antide Sitter space can be defined:
d−1
X R^{2 } (3.2)
i=1
embedded in flat d+1 dimensional space with the metric:
d−1
dsdX_{i}^{2 } (3.3)
i=1
This has constant negative curvature [15].Of course, this can be written in different coordinate systems, and in the Poincare coordinates´ (r,t,~x),r > 0,~x ∈ R^{d}^{−2} given by [2]:
X_{0} = 1 [1+r^{2}(R^{2} +~x^{2} −t^{2})], X_{d} = Rrt
2r
X^{i} = Rrx^{i}, (i = 1,....,d −2)
X^{d}^{−1} = [1−r^{2}(R^{2} −~x^{2} +t^{2})], 2r the antide Sitter metric then becomes 
(3.4) 
^{1}
ds (3.5)
g_{rr} , 
gxx = gyy = gzz = r2, 
gtt = −r2 
(3.6) 
r
The limit where the radial coordinate r goes to infinity is called the boundary of the antide Sitter space . The boundary is the place where the dual field theory lives [6]. It turns out that string theory excitations extend all the way to the boundary of the space [7]. In this way it is possible to obtain a map from string theory states in the bulk to states in the field theory living in the boundary. It also seems that r can be seen as a dimension of scale of the dual field theory [18].
Correlation Functions
Maldacena's paper [12], proposed a form of the AdS/CFT correspondence but did not yet provide a map between the two sides. In [23] and [8] such a map was given and makes the correspondence clear. To illustrate, we consider a free field with a mass m on the AdS side, propagating in antide Sitter space. We see that the field equation:
(+m^{2})_{φ} = 0 has two linearly independent solutions that behave like [6] : 
(4.1) 
e−∆r, e(∆−4)r as r → ∞, where 
(4.2) 
∆(∆−4) = m^{2} 
(4.3) 
Now let us consider a solution of the supergravity equations of motion , possessing the boundary condition that near r = ∞ , the fields behave as [6]:
e(∆−4)r (4.4)
The map between AdS and CFT quantities is then given by:
exp −S_{sugrav} (4.5)
The left hand side of the above is the supergravity action evaluated on the classical solution given by φ_{i} , with the right hand side being a generating function for correlation functions in YangMills theory [6]. For most applications, the above suffices, but it's important to note that the left hand side is really an approximation  and we should really use the full string theory partition function subject to the relevant boundary conditions, to which our supergravity approximation is a saddlepoint approximation [11]:
Z_{string} (4.6)
We see that there is an operator O_{i} in YangMills theory corresponding to every bulk field φ_{i} (in AdS). We can use symmetries to find which boundary operators (in the field theory side) , map to which fields (in the bulk, that is, the AdS side). The two sides must have the same quantum numbers and Lorentz structure [11].As an example, conserved currents in the field theory side correspond to global symmetries and therefore their corresponding sources act as external background gauge fields, which are boundary values of dynamical gauge fields in the AdS side (the bulk) [11]. We will use the above to find the "two point" current correlation function in the subsequent sections.
Derivation of the Ads/CFT Correspondence
To do the derivation of the AdS/CFT correspondence given in Maldacena's original paper [12], we need the idea of Dbranes (or Dirichlet branes). Polchinski introduced them in [16] as extended objects in string theory. The number of spatial dimensions they have give them a label  D0 branes are like particles, D1 branes are strings , D2 branes are like sheets etc. We have two ways to think about Dbranes . The first one is as solitonic solutions to the equations of motion of low energy string theory , that is, of supergravity . (this perspective is reliable for g_{s}N >> 1 , to be explained shortly)  known as the "closed string perspective" [3], the Dbranes act as sources of a gravitational field which curves the surrounding spacetime, and the characteristic length "R" (we will explain this shortly) is large to ensure the validity of the supergravity approximation [3] and the second way to think about Dbranes is as higher dimensional objects where open strings can end. Open strings have finite tension, and their center of mass cannot be taken arbitrarily far away from a Dbrane [6]. Consequently, the degrees of freedom of the open string are confined to in a direction parallel to the D brane. It's important to note that both the open and closed string perspectives have their advantages, and as we shall see, with a stack of D3 branes, and taking a certain limit the open string description reduces to N = 4 super YangMills theory while the closed string perspective reduces to string theory on AdS_{5} ×S^{5}  thus the AdS/CFT correspondence arises as a consequence of the duality between open and closed strings.
Let us consider N coincident D3 branes (these have three spatial dimensions and a time dimension) in type IIB string theory. g_{s} is the string coupling constant, which governs the strength of the interactions between open and closed strings. The coupling constant is g_{s}N. At weak coupling g_{s}N << 1 the branes live in ta 10dimensional flat spacetime with open strings ending on the D branes and closed strings in the bulk. At strong coupling g_{s}N >> 1 the spacetime is substantially curved by the branes, sourcing the extremal black 3brane geometry [10]:
ds2 −1/2 µ ν + f(r)1/2(dr2 +r2dΩ25), f(r) = 1+ 4πgs4Nls4 (5.1) = f(r) ηµνdx dx
r
where dΩ^{2}_{5} is the metric of a unit 5sphere (S^{5}), l_{s} is the string length and x^{µ}
denotes the 4 coordinates along the D3 brane world volume. The solution has a selfdual 5form field strength which has a flux on the S^{5} here [11].
We have seen two regimes of g_{s}N : g_{s}N << 1 and g_{s}N >> 1, with apparently no overlap. The insight that Maldacena had was to decouple the theory on the branes from gravity  which can be done by taking a low energy limit that simplifies things a lot. The open string sector decouples from the rest of the theory and we end up with an SU(N) YangMills theory that describes the dynamics of the branes. On the other hand, this limit in the blackbrane spacetime in the near horizon geometry  asymptotically, any nearhorizon finite energy excitation is strongly redshifted, while modes that propagate in the asymptotic region of 5.1 decouple from the region near the horizon since its cross section vanishes in this limit [11].
The event horizon r = 0 of the black brane solution 5.1 lies at infinite proper distance from spacelike geodesics  so its embedding diagram has an infinite "throat". The near horizon geometry then simplifies to a product of a sphere and AntideSitter spacetime [11].The nearhorizon geometry here is this 10dimensional AdS_{5} ×S^{5}.
Now in 5.1, if we define R = (4πg_{s}N)^{1}^{/}^{4}l_{s}, we note that as r → 0, that is as we
zoom in on the horizon, f(r) → R^{2}/r^{2}, and we end up getting:
^{2 } r2 R2 2 2 2
ds = R^{2} η_{µν}dx_{µ}dx_{ν} ^{+} r_{2} dr +R dΩ5 (5.2)
The first two terms above describe AdS_{5} (something which we've seen before)a maximally symmetric spacetime of constant negative curvature, with a radius of curvature of R, while the last term describes the sphere S^{5} (also with radius R). The Dbranes are not localized in the geometry anymore, but their effect is manifested in the 5form flux through the sphere S^{5} [11].
Our stack of D3branes in the low energy limit is then described by a 10dimensional string theory with closed strings when g_{s}N >> 1 in AdS_{5} ×S^{5}, and by 4dimensional super YangMills SU(N) field theory when g_{s}N << 1. But the gauge theory is defined at any value of the coupling constant, and so we can conjecture that it holds even when g_{s}N is large  the same regime where the closed string description holds [11]. It was this observation that led Maldacena [12] to conjecture:
(String theory on AdS_{5} ×S^{5}) = (N = 5, SU(N) gauge theory in 4D)
In other words, the two sides describe the same physics  it's a full duality.
5.0.1 Mapping between parameters
The AdS/CFT duality is an example of a weakstrong coupling duality. The correspondence specifies how the parameters on the two sides  the field theory side and the bulk, relate to each other. On AdS_{5} ×S^{5}, string theory has a dimensionless coupling constant g_{s} , which determines the strength of the interactions between strings splitting and joining, and the string length l_{s} that sets the size of the fluctuations of the string world sheet,, and another parameter R which as we've seen before, is the radius of curvature of AdS_{5} ×S^{5}.
4dimensional N = 4 super YangMills theory with gauge group U(N) has of course, the rank N of the gauge group and a dimensionless coupling constant that we've seen before  g_{YM}^{2} . The parameters between the two sides relate to each other in the following way [6]:
, (R/l_{s})^{4} = 4πg_{YM}^{2 } N = 4πλ (5.3)
λ in the above equation is the 't Hoooft coupling constant.
5.0.2 Turning off Quantum Fluctuations, and the weak form of the Correspondence
For our purposes, we turn off quantum fluctuations and suppress any stringy corrections of the geometry, so we take g_{s} << 1, while keeping R/l_{s} constant. To leading order in g_{s} then, the AdS side reduces to classical string theory. The string length l_{s} (which is measured in units of R)is kept constant. This we refer to as the strong form of the AdS/CFT correspondence [3]. Looking at 5.3 we see that on the CFT side this implies taking g_{YM} << 1 while g_{YM}^{2} N stays finite. This means that we have to take the large N limit, N → ∞ for a fixed λ, which is called the 't Hooft limit. In this limit, there is only one free parameter on each side  on the CFT side we have λ, and on the string theory side, we have the radius of curvature R/l_{s} (remembering we kept l_{s} constant). Then 5.3 tells us that these two parameters are related by (R/l_{s}) = λ^{1}^{/}^{4}. We want to work with a strongly correlated material, so we take the limit λ → ∞  that is, the material (on the CFT side ) is infinitely coupled  and we see that on the string theory side this corresponds to the limit R → ∞. The string length is then very small compared to the radius of curvature, and we see that we have obtained the pointparticle limit of type IIB string theory, which is given by type IIB supergravity on AdS_{5} ×S^{5}. Thus we have a strongweak duality  strongly coupled N = 4 Super YangMills is mapped to type IIB supergravity on weakly curved AdS_{5} ×S^{5} space. This special limit is known as the weak form of the AdS/CFT conjecture [3]. We see that due to this duality, we can solve for strongly coupled systems on the CFT side much more easily on the gravity side the mathematics becomes tractable.
Probing a Strongly Correlated CFT with a D5Brane
The AdS/CFT duality tells us that every bulk field φ corresponds to an operator O in the gauge theory. Now we want to take a strongly correlated material (for which we assume λ = ∞, given by a CFT, and wish to extract its ACconductivity. Because of the coupling constant λ showing up in perturbative expansions, they diverge and the mathematics is thus not tractable. However, in chapter 5 we have seen that there is a dual problem we can solve on the AdS side, which, for this value of λ, would be a problem on a weakly curved spacetime with no stringy interactions. We wish to find the twopoint current correlation function for this strongly correlated object, which we will call a semimetal, and its ACconductivity.
We use the weak form of the AdS/CFT duality as talked about in chapter 5, and we set up a "probe" D5brane perpendicular to our stack of infinite number of coincident D3 branes at r = 0 (Figure 6.1).
We will solve Maxwell's equations on the surface of this D5brane, and so we will suppress one spatial direction (without loss of generality, let this suppressed direction be the zdirection). The geometry of this surface is thus AdS_{4} ×S^{2}. The two point currentcurrent correlation function at r = ∞ , < j^{µ}(x)j^{ν}(y) >, where j^{µ} and j^{ν} are currentdensity operators, defined at two points x and y. This implies we need infinite resolution, and as pointed out in chapter 6, since the r dimension is a dimension of scale, this corresponds to our operators living at r = ∞.
Figure 6.1: A D5 brane is held perpendicular to a stack of an infinite number of coincident D3 branes that are at r =0. The stack of D3 branes extends along the spacetime directions x^{0},x^{1},x^{2},x^{3}, and are transversal to the other six spatial directions x^{4},...x^{9} . In the limit depicted in the figure, the strings in the bulk are rigid, and we solve Maxwell's equations on the surface of the D5brane to extract the twopoint current correlation function at the boundary (r = ∞). In essence, we are looking for this particular operator on the "upper edge" of the infinite square depicted as the D5 brane in this figure.
We recall that Maxwell's equations in Lorentz covariant form are:
∂µFµν = jν, Fµν = ∂µAν −∂νAµ (6.1)
And an equivalent form of Maxwell's equations in a curved spacetime is:
µρ
g(6.2)
We have the fields A_{b}(r,x,y,t) on the surface of the D5 brane (the bulk field), and we set the boundary conditions:
lim A_{b}(r,x,y,t) → a_{b}(r), lim A_{b}(r,~x) = 0 (6.3) r→∞ x,y,z→∞
It is known from chapter 4, that we can put a_{b}(r) into equation 4.5 to get correlation functions, which in this case will turn out to be the currentcurrent correlation functions.
< ei^{R} a_{µ} j^{µ} >= e−S[a] (6.4)
where we use the BornInfeld Action, S, from string theory on the right hand side.
The left hand side is expanded as:
< e dx_{1}...dx_{n} a_{µ}
n=0 n!
< eiR aµ jµ
(6.5)
while we expand the right hand side of equation 6.4 as:
^{R } ^{µ } Z
e−S[a] = < ei a_{µ} j > = 1−n aρ∆a_{ρ} +.. (6.6)
where ∆ in the equation above is some operator to be identified. Putting 6.5
and 6.6 together we get:
Comparing 6.6 and 6.7, we see that the operator ∆ is in fact the twopoint current correlator < j^{µ}(x)j^{ν}(y)> , the operator that we seek, and whose coefficient will tell us the ACconductivity of our strongly correlated material. We see that ∆ is obtained from the quadratic term in 6.6. In the low energy limit (the regime we are working in here) , the BornInfeld action can be approximated by the Maxwell action, and it turns out that this approximation for S at the boundary, evaluated with the boundary conditions 6.3, give us the quadratic term when we insert it into 6.6. We will first show this and get the currentcurrent correlation function in the next section, and then fix its coefficient (the ACconductivity) in the section that comes after that.
6.0.1 Maxwell's Action on the D5 Brane
Given the D5 brane metric 3.6, we intend to solve 6.2 in the case where j^{ν} = 0 with the boundary conditions 6.3. So we want to solve:
pµρ νσ = 0
∂µ −detg g g Fρσ
⇒ ∂_{µ}^{p}−detg g^{µµ}g^{νν}F_{ρσ} = 0 (6.8)
For this section, we will ignore the parameter R in the metric, for reasons that will be made clear in the next section, 6.0.2. Choosing ν = r, let µ = a = x,y,t (this "a" has nothing to do with our boundary condition, it is just an index here).
Putting into 6.8:
∂a(r2r2gaaFar) = 0 r4∂a(gaaFar) = 0
r^{2}∂_{a}(∂^{a}A^{r} −∂^{r}A^{a}) = 0 (6.9)
Using our freedom to choose a gauge, we choose one such that A_{r} = A^{r} = 0.
Thus, 6.9 becomes:
r^{2}∂_{r}[∂_{a}(A^{a})] = 0 For ν = x,y,t, we have: 
(6.10) 
p^{µµ bb } = 0
∂_{µ} −detgg g F_{µ}b
∂a −detggaagbbFab +∂rp−detggrrgbbFrb = 0 p (1/r^{2})∂_{a}(F^{ab})+∂_{r}(r^{2}F^{rb}) = 0 (1/r2)∂a(∂aAb −∂bAa)+∂r(r2∂rAb −r2∂bAr) = 0 

(1/r^{2})∂_{a}(∂^{a}A^{b} −∂^{b}A^{a})+∂_{r}(r^{2}∂^{r}A^{b}) = 0 We multiply 6.11 by r^{2}, 
(6.11) 
∂a(∂aAb −∂bAa)+r2∂r(r2∂rAb) = 0 
(6.12) 
Using gauge invariance, we choose ∂_{a}A^{a} = 0 (and it follows from 6.10 that ∂_{a}A^{a} = 0 at r = 0 as well  it has no r dependence) So for a 6= b, 6.12 is:
∂_{a}(∂^{a}A^{b})+(r^{2}∂_{r})(r^{2}∂_{r})(A^{b}) = 0 Using separation of variables, we set the following condition: 
(6.13) 
A_{b}(r,~x) = A_{b}(r,k)e^{ik}^{cx}^{c} 
(6.14) 
where^{~}k is a vector in momentum space having modulus k. Putting this into 6.13:
−k_{a}k^{a}A_{b} +(r^{2}∂_{r})(r^{2}∂_{r})(A^{b}) = 0
∂ −k2A_{b} = 0 (6.15)
A_{b}
We set A_{a}(r,k) at r = 0 to be finite and: 

lim A_{a}(r,k) = a_{a}(k) r→∞ Now let (1/r) = σ, so 6.15 becomes: 
(6.16) 
0 (6.17)
The above equation has the general solution: 

Ab =C1ekσ +C2e−kσ 
(6.18) 
The fact that we set A_{b} to be finite at r = 0 gives us C_{1} = 0 in 6.18, and
6.16 implies C_{2} = a_{b}(k). So we get the solution:
Ab(σ(r),k) = ab(k)e−kσ = ab(k)e(−k/r) Putting the above into 6.14, we obtain: 
(6.19) 
A_{b}(r,~x) = eikcx^{c}a_{b}(k)e−k/r 
(6.20) 
Since 6.17 is a homogeneous equation, a superposition of solutions is still a solution. So we take a superposition of solutions of the form 6.20 :
^{Z } ^{c } −k/r 3
A_{b}(r,~x) = eikcx a_{b}(k)e d k
^{Z ik}^{cx}^{c}−k/ra_{b}(k)d3k (6.21)
= e
We are now in a position to plug the solution 6.21 into the Maxwell action, given by:
Z
drdgµνgρσFµρFνσ
Z
= drdg^{µν}g
Z
= drdgµνg
Z
= drdgµνgρσAgµνg
(6.22)
The second term in 6.22 is just 0, because it is a solution to Maxwell's equations. So 6.22 turns to:
Z
drdgµνg
Z
= drdgµµgρσA_{ρ}F_{µσ}] (6.23)
We take r → ∞, and use Stokes' Law on 6.23 to obtain:
grrg
grrg (6.24)
Since A_{r} = 0, ∂_{σ}A_{r} = 0 in 6.24. Continuing:
grrg
grrg
(6.25)
The above is what the Maxwell action has been reduced to at r → ∞. Finally, we plug 6.21 into 6.25:
#
eikcx^{c } eimcx^{c}−(m/r)a_{ρ}(m)d3m
eik_{c}x^{c}−(k/r)aρ(k)d3k^{Z} eim_{c}x^{c}
" #
Z
= me−k/raρ(k)e−m/raρ(m)d3k d3m ∂3(k+m)
m
where in the last step, we have relabeled m = k for convenience. As promised before, the Maxwell's action does indeed give the quadratic term in 6.6 from which to extract ∆ , which we identified as the currentcurrent correlation function by comparing with 6.7. We note that in our case, we assume that there is no net outflow or inflow of charge, so the conservation of charge tells us ∂_{µ} j_{µ} = 0. Hence we must have ∂_{µ} < j^{µ}(x)j^{ν}(y) >= ∂_{µ}∆^{µν} = 0, which, when Fourier transformed, gives us k_{µ}∆^{µν}(k) = 0. So we need to figure out what operator fits into 6.26 and satisfies these conditions. We see that the following works:
√ k^{α}k^{β}
< j^{α}(k)j^{β}(−k) > = k^{2}(η^{αβ} − k_{2} ) To check that it satisfies the conditions just outlined above: 
(6.27) 
!
k_{µ}
k kµkν
= k_{µ}ηµν − µ
= k^{ν} −
k2
= 0 (6.28)
We also need to check if we can somehow get 6.27 from 6.26 by rewriting the expression in 6.26 (essentially just adding zero to the expression), that is  we need to make sure we have not "changed" the expression. We recall that since we have let the bulk field approach a_{µ}(x^{µ}) (the Fourier transform of a_{µ}(k)) as r → ∞, from 6.10 we see that we must have ∂_{µ}a_{µ} = 0, whose Fourier transform is k^{µ}a_{µ} = 0.
With this piece of information:
d^{3}k
!
a_{ν}(k)d^{3}k
ka^{µ}
k (6.29)
6.29 is exactly 6.26 , just with the up and down indices switched  the last line in 6.29 follows because k^{µ}a_{µ} = 0, making the second term on the third line zero. Thus, 6.27 gives the two point current correlation function for our strongly correlated semimetal (called a semimetal because it does not have a Fermi surface like metals do  but a Fermi point). Now what remains is to fix its coefficient and thus extract the ACconductivity.
6.0.2 Extracting the ACConductivity
We now show that our approximation of the BornInfeld action as the Maxwell action was justified and then we fix the coefficient of 6.27.
The DiracBornInfeld Action is:
d6σqdet(g+2πα0Fµν) (6.30)
where the coefficient in the above expression is the D5brane tension,
and the D5 brane metric (corresponding to g) is:
ds (6.31)
where the other variables are the same as ones considered before, with the last two terms coming from the sphere S^{5}.
6.30 can be rewritten as:
q
d6σ det(g).det(^{1}+2πα0g−1F)
dTr ln (6.32)
The trace of a symmetric matrix times an antisymmetric one is zero, so:
∑ g−µν1 Fνµ = 0 (6.33)
µν
6.32 then becomes:
2Trg−1F g−1F d
Tr g
Tr g^{−1} F g^{−1} F (6.34)
where the term taken in the expansion above leads to the quadratic term in 6.6 as we will see, because it can be shown that the expression is a manifestation of Maxwell's action, which in the last section has been shown to lead to the quadratic term. In 6.34 above, we get a factor of R^{6} from the determinant, and a factor of 1/R^{4} from the trace, which will add an overall factor of R^{2} to 6.34. Furthermore, we can rewrite the trace in 6.34 as:
Tr g−1 F g−1 F
µµ
µ
^{=} ∑ g−1 µν F_{νρ} g−1 ρσ F_{σµ}
µ
= gµµ gρσ FµρFνσ (6.35)
Putting 6.35 into 6.34 and taking out the R^{2} factor to the front gives us:
Rp (g) gµνgρσF_{µρ}F
dtdxdydr det
We see that the rightmost quadruple integral in the expression above is the Maxwell action. So we were justified in approximating the BornInfeld action this way. We also note that the R^{2} factor was taken out to the front in the expression above, so det(g) there is different from the det(g)'s that came before it in the sense that it does not contain any factors of R, and this is the reason we ignored the R factors of the metric in the last section 6.0.1. Continuing to work with 6.36 should therefore give us the coefficient of the currentcurrent correlation function (the ACconductivity of the semimetal). We compute the integrals over θ and φ, giving us a factor of 4π, that we take out to the front:
dtdxdr^{p}det(g) g^{µν}g^{ρσ}F_{µρ}F (6.37)
We recall that λ = g_{YM}^{2} N and g_{s} = (4πN/λ)^{−}^{1}, so we can rewrite the coefficient outside the brackets in 6.37 as:
√
24π3 λα03 4πN
(6.38)
It turns out that if we evaluated the interactions at the boundary in the field theory, we would get a potential that goes like g_{YM}/(4π ~x ), where ~x  is the distance between the two interacting entities. This is like a Coulomb potential, so g_{YM} is like a "charge" on our boundary. Multiplying 6.38 by g_{YM}^{2} , we obtain the ACconductivity:
(6.39)
For weakly correlated materials with a low coupling constant λ, the ACconductivity scales like λ, while for our calculation with the D5 brane, it appears that this con
√
ductivity scales like λ. The slower rate of increase of the conductivity with λ for strongly correlated systems is evidence of some "screening" taking place.
Now that we have fixed the coefficient of the currentcurrent correlation function, 6.27 becomes:
√ α(k)jβ(−k) > = 2 πλ√k2(ηαβ − kαkk2β ) < j 
(6.40) 
The above gives our currentcurrent correlation function in momentum space, so we do an inverse Fourier transform on 6.40 to obtain what we present as our twopoint current correlation function:
2√ y) d_{3}k (6.41)
π
Epilogue
Black holes were once seen as a problematic, potential flaw in general relativity in the first half of the twentieth century. But they have moved on from that status to being astrophysical objects that we now have evidence for, and have become fascinating mathematical objects. The study of black holy entropy has motivated the advances in string theory which provided its basic framework [11], and extremal black branes have supplied the context for the derivation of the AdS/CFT correspondence. This correspondence is profound in that it relates a nongravitational theory to a gravitational one, and it turns out that large black holes in AdS encode the dynamics of fluids and describe a vast array of systems that can be probed experimentally [11]. The AdS/CFT correspondence was used in this report to make a mathematically intractable problem involving a strongly correlated system into a mathematically tractable one, which allowed us to extract the two point currentcurrent correlation function and the ACconductivity for what we called a semimetal. The correspondence can also be used to study finite temperature real time processes, examples of which are response functions and dynamics far from equilibrium in quantum critical points in condensed matter systems [15]. Computing these problems are made much simpler using the correspondence  if the original theory has d dimensions, solving them is reduced to solving classical gravitational problems in d +1 dimensions. The AdS/CFT correspondence is not only a useful playground in solving these problems, but it is also weaved into the exploration of the connections between quantum information and gravity. The most fascinating part of the AdS/CFT correspondence it seems, is that it connects so many apparently disparate ideas together, and it is likely to be a key ingredient in the unraveling of quantum gravity.
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