8.1 Effective actions

^[1]

We will first study AdS/CFT, the best-understood example of holography, before the speculative flat space and de Sitter analogs.

The statement of AdS/CFT is that quantum gravity in asymptotically AdS spacetimes is exactly equivalent to a CFT. More precisely, the boundary conditions of the partition function of AdS quantum gravity are given by the partition function of a CFT. Similarly the Hilbert spaces map such that CFT states correspond to different asymptotically AdS geometries. \begin{align}\label{eq:AdS/CFT} Z_{\mathcal{O}} [ \phi_{0} ]{\mathrm{CFT}}&=Z{\phi} [ \phi_{0} ]{\mathrm{s t r i n g}}
\mathcal{H}{\text{CFT states}}&=\mathcal{H}_{\text{Asymptotically AdS geometries}} \end{align} This is an even more radical statement than S-duality or T-duality. Those dualities say 2 different string theories or 2 different QFTs are actually the same theory, but here, both sides use the same framework. But in AdS/CFT, the duality is between 2 different frameworks: string theory and QFT.

[ \begin{tikzcd} & \text{AdS boundary conditions} \arrow[dr,”\text{local map}”,Leftrightarrow]
\text{AdS quantum gravity} \arrow[ur,”“,Leftrightarrow] \arrow[rr,”Holography”,Leftrightarrow] && CFT \end{tikzcd} ]

To make AdS/CFT more digestible, we can divide the radical statement into 2 parts, as shown above.

The first part says that quantum gravity in AdS is completely determined if we know the local observables at the asymptotic conformal boundary. In section NoLocalObservables, I mentioned that there are no local diffeomorphism invariant observables in AdS except at the asymptotic conformal boundary. But there are some diffeomorphism invariant nonlocal observables that are not at the asymptotic conformal boundary. For example, the dual of mutual information in CFT is the area of a surface that is 1) diffeomorphism invariant, 2) UV finite (taking quantum extremal surfaces including $S_{bulk}$ will cancel divergences due to matter fields), 3) IR finite (because its mutual information the diverging areas cancel). AdS/CFT is saying that these non-local diffeomorphism invariant observables are not independent observables. Once we know all the local diffeomorphism invariant observables at the asymptotic conformal boundary, we can get all these observables that are deep in the bulk from them. For the earlier example, we can get the area of that surface by studying the entanglement between observables at the asymptotic conformal boundary. This part is the trivial part and can be argued without using any quantum gravity or string theory, see ^[2].
The second part is the non-trivial part and says that these boundary conditions behave like a well-defined CFT with unitarity and locality.

Relation to Bekenstein–Hawking formula: $S_{BH}$ attributes an entropy to a Black Hole horizon. In general, the same argument works for any other Killing horizon; for example, it’s called the Gibbons-Hawking formula for cosmological horizons. We can generalize this statement in 2 stages to get AdS/CFT:

The first generalization is that we can attribute entropy to any extremal surface as explained in section HEE using the so-called RT/HRT/QES formulas. Killing horizons are also extremal surfaces so they are a subset of this prescription. These formulas for holographic entanglement entropies also naturally define the so-called entanglement wedge.
The second generalization is that not only the boundary von Neuman entropy maps to the QES on the bulk, but all the observables in a given boundary region map to all the observables in the corresponding entanglement wedge, and this is called subregion duality. The statement of subregion duality is essentially the same as the definition eq:AdS/CFT.

\subsection{AdS/CFT} ^[3]

\subsubsection{Dictionary} All the elements of the dictionary are, in principle, extractable from the first one because they contain all the correlations and, therefore, all the information.

\begin{tabular}{||c | c | c ||} \hline $AdS_{d+1}$ & $CFT_d$ & Ref\ [0.5ex] \hline\hline $Z_{AdS}$ & $Z_{CFT}$ &
\hline $\mathcal{H}{AdS}$ & $\mathcal{H}{CFT}$ &
Asymptotically AdS geometry & Quantum state &
Ex: Pure AdS & Vacuum State &
Ex: 2-sided Schwarzschild AdS BH & Thermofield Double State &
\hline Scalar fields & Scalar primary operators & Correlation
$m^2L^2=\Delta(\Delta-d)$ & $\Delta$ &
\hline Dirac fermion & $d=\text{even}\Rightarrow$ chiral fermion & ^[4]
$|m|L=\Delta-\frac{d} {2}$ & $d=\text{odd}\Rightarrow$ Dirac fermion & Correlation
\hline Gauge fields $A_\mu$ & Conserved currents $j_\mu$ $\Delta=d-1$ & ^[5]
Gauge symmetry & Global symmetry & Correlation
\hline $p$ form higher gauge fields $A_{\mu_1\cdots \mu_p}$ & Conserved currents $j_{\mu_1\cdots \mu_p}$ & ^[6]
$m^2L^2=(\Delta-p)(\Delta+p-d)$ & & Correlation
\hline $h_{\mu\nu}$ (perturbed $g_{\mu\nu}$) & $T_{\mu\nu}$ & ^[7]
$m^2L^2=\Delta(\Delta-d)=0$ & $\Delta =d$ & Correlation
\hline Single string excitations & Single trace operators &
\hline Wrapped branes & Baryonic vertex &
\hline $Z_{AdS}$ with a string worldsheet & Wilson loops &
ending at the boundary loop $C$ & &
\hline IR divergence & UV divergences & HolographicRenormalisation
Radial direction & Energy scale &
\hline Branes & Solitons &
D(-1)-instanton & instanton VEV &
\hline RT or QES & von Neumann entropy & HEE
Einstein’s equations & (\delta S_A = \delta \langle K_A \rangle) & gravityfromentanglement
Entanglement wedge & $\rho$ of a subregion& ^[8]
Cosmic Brane & R´enyi entropies & ^[9]
\hline Action & Complexity &
\hline \end{tabular}

\subsubsection{Correlation functions} \subsubsection{Finite temperature} \subsubsection{Solitons} Chapter 13 of ^[10] \& ^[11] \subsubsection{The pp wave correspondence} \subsubsection{Different dimensions} \paragraph{$A d S_{3}$ compactifications}~
^[12] \subsubsection{Orbifolds and orientifolds} \subsubsection{Integrability} ^[13] and ^[14] and chapter 7 of ^[15]. \subsubsection{Holographic renormalisation group} \subsubsection{Supersymmetric localization} ^[16] \subsubsection{Higher spin holography (tensionless limit)} ^[17] and for arbitrary D ^[18]. See also HigherSpinTheory. \subsubsection{Fluid/gravity correspondence} ^[19] \subsubsection{Holographic QCD} ^[20] \paragraph{Bottom-up} \paragraph{Top-down} \subsubsection{Nonrelativistic limit} ^[21] and section nonrelativistic \subsection{It from Qubit AdS/CFT} ^[22] \subsubsection{Holographic entanglement entropy} ^[23]

\paragraph{RT and HRT formulas}

\paragraph{Classical gravity from entanglement} \paragraph{Quantum extremal surfaces} \paragraph{Derivation of RT, HRT, QES from gravity path integral}~
^[24]. Original articles are~^[25]
\paragraph{Holographic entropy cone}~
\paragraph{Quantum focusing conjecture \& QNEC}~
\paragraph{Bit threads}~
\paragraph{Islands}~
^[26]. Criticism about massive islands, fuzzballs etc is discussed in~^[27]^[28] and ^[29] is a reply. \subsubsection{Tensor networks} ^[30] \subsubsection{Error correction} ^[31] \subsubsection{Chaos} ^[32] \subsubsection{Complexity} ^[33]^[34] \subsubsection{JT} ^[35]. Check section 2dstring.

\subsubsection{SYK} ^[36]

The SYK model

The SYK model is a model of N Majorana fermions $\psi_i$ with all-to-all $p$-local interactions defined by \begin{align} &H_{SYK} =i^{p / 2} !!!!!!!!\sum_{1 \leq i_{1} <\cdots < i_{p} \leq N} !!!!!!J_{i_{1} \ldots i_{p}} \psi_{i_{1}} \cdots \psi_{i_{p}},
&\left{\psi_{i}, \psi_{j}\right}=2 \delta_{i j},
&\left\langle J_{I}J_{K}\right\rangle =\frac{\mathcal{J}^2}{ {N \choose p} }\delta_{IK}=\frac{N}{2p^2}\frac{\mathbb{J}^2}{ {N \choose p} }\delta_{IK} \end{align} where $\mathcal{J}$, $\mathbb{J}$ are different normalization conventions and $I=(i_{1}, \ldots ,i_{p})$. Some people also normalize the fermions to remove the factor of $2$. By $p$-local, we mean each term in the Hamiltonian contains $p$ Majorana fermions.

Instead of the Hamiltonian, we could equivalently define the model with the Euclidean ($\tau=i t$) action $S_{\mathrm{S Y K}}=\int d \tau\left( \frac{1} {2} \sum_{i=1}^{N} \psi_{i} \frac{d} {d \tau} \psi_{i}+i^{p / 2} \sum_{I} J_{I} \psi_{I} \right)$.

For the Hamiltonian to be Hermitian, $p$ must be even. Let’s also assume that $N$ is even; then $\psi_i$ are square matrices with size $2^{N/2}$. $\psi_i:\mathbb{R}\to \mathbb{C}^{2^{N/2} \times 2^{N/2}}$.

\todo{

Note2

Lorentzian action is $S_{\mathrm{S Y K}}=\int dt\left( \frac{1} {2} \sum_{i=1}^{N} \psi_{i} i\frac{d} {d t} \psi_{i}-i^{p / 2} \sum_{I} J_{I} \psi_{I} \right)$. Note that since the action is linear in terms of $\frac{d} {d t} \psi_{i}$, there will not be a kinetic term in the Hamiltonian. }

For $N=2r$, we can define the following representation using the Kronecker product of Pauli matrices. For $N=2$, they become $\psi_{1}=\sigma_{1}$ and $\psi_{2}=\sigma_{2}$. \begin{align} \psi_{1}& =\sigma_{1} \otimes1 \otimes\cdots\otimes1
\psi_{2}& =\sigma_{3} \otimes \sigma_{1} \otimes\cdots\otimes1
&\;\;\vdots \notag
\psi_{r}& =\sigma_{3} \otimes\sigma_{3} \otimes\cdots\otimes\sigma_{1}
{\psi_{r+1}} & =\sigma_{2} \otimes1 \otimes\cdots\otimes1\ {\psi_{r+2}} & =\sigma_{3} \otimes\sigma_{2} \otimes\cdots\otimes1
&\;\;\vdots \notag
\psi_{2 r}&=\sigma_{3} \otimes\sigma_{3} \otimes\cdots\otimes\sigma_{2} \end{align}

\etocsettocdepth.toc{subsubsection}

\paragraph{Feynman diagrams and the melonic dominance}

\paragraph{Schwarzian theory} \paragraph{Four point function} \subsubsection{DSSYK} ^[37]
Take the same SYK problem from section SYK. \begin{align} &H =i^{p / 2} !!!!!!!!\sum_{1 \leq i_{1} <\cdots < i_{p} \leq N} !!!!!!J_{i_{1} \ldots i_{p}} \psi_{i_{1}} \cdots \psi_{i_{p}},
&\left{\psi_{i}, \psi_{j}\right}=2 \delta_{i j},
&\left\langle J_{i_{1} \ldots i_{p}}^{2}\right\rangle =\frac{\mathcal{J}^2}{ {N \choose p} } \end{align}

Now, instead of fixing $p$ if we fix $\lambda := 2p^2/N$ then we get DSSYK. We switched the discrete parameter $p$ that is always even (as $H$ is Hermitian) with the continuous $\lambda$. The parameter $\lambda$ is somewhat similar to $G_N$ or $l_P/l_{AdS}$. As $\lambda\to 0$, we get back the original SYK but with large $p$. This limit is a semi-classical limit. When $\lambda$ is not near $0$, quantum gravity effects become significant. So, compared to SYK, this model 1) Probes quantum gravity instead of just semi-classical gravity and 2) Is exactly solvable for large $N$ at all energy scales; SYK is only exactly solvable for large $N$ in the IR limit.

Normally SYK model is solved by introducing the master fields $G,\Sigma$ and then using a saddle point approximation. This method can also be used for DSSYK when $\lambda \to 0$ (large $p$ SYK). But to solve for arbitrary $\lambda$, we need to use “chord diagrams”.

Chord diagrams: Let $q := \exp(-\lambda)$. Holography is defined by $Z_{\mathrm{boundary}}=Z_{\mathrm{bulk}}$ and $\mathcal{H}{\mathrm{boundary}}=\mathcal{H}{\mathrm{bulk}}$. $Z_{\mathrm{boundary}}$ is the most important quantity to calculate.

By Taylor expansion we get, $\displaystyle Z =\langle\langle T r ~ e^{-\beta H} \rangle\rangle=\sum_{k}\frac{(-\beta)^k}{k!}m_k$ where $m_k=\langle\langle T r ~ H^k\rangle\rangle$. We normalize such that $Tr(\mathbb{I})=1$. Let $\mathcal{J}=1$ from now.

Let’s consider $m_4$. From now on, we follow $I \equiv{i_{1}, i_{2},… i_{p} }$ notation $\begin{aligned} m_4 &= \sum\limits_{I_1,I_2,I_3,I_4} \langle\langle J_{I_1}J_{I_2}J_{I_3}J_{I_4}\rangle\rangle Tr(\psi_{I_1}\psi_{I_2}\psi_{I_3}\psi_{I_4})\\ =& \frac{\mathcal{J}^2}{N\choose p}^2\sum\limits_{I_1 I_2} \left\lbrace Tr(\psi_{I_1}\psi_{I_1}\psi_{I_2}\psi_{I_2}) + Tr(\psi_{I_1}\psi_{I_2}\psi_{I_1}\psi_{I_2}) \right. \\ & \left. + Tr(\psi_{I_1}\psi_{I_2}\psi_{I_2}\psi_{I_1})\right\rbrace \end{aligned}$

Here, we are only considering pairwise matching and this can be diagrammatically represented using chord diagrams as shown below.

\resizebox{.9\hsize}{!}{ \begin{tikzpicture} \node[black] at (-2.75,0) {$m_4 = \dfrac{1}{ {N \choose p}^2}\sum\limits_{I_1,I_2}$}; \draw[thick] (0,0) circle(1 cm); \draw[thick,color=black] (-1,0) to[bend right] (0,1); \draw[thick,color=black] (0,-1) to[bend left] (1,0); \node[black] at (1.5,0) {$+$}; \draw[thick] (3,0) circle(1 cm); \draw[thick,color=black] (3,1) to[bend right] (4,0); \draw[thick,color=black] (3,-1) to[bend right] (2,0); \node[black] at (4.5,0) {$+$}; \draw[thick] (6,0) circle(1 cm); \draw[thick,color=black] (5,0) to[bend right] (7,0); \draw[thick,color=black] (6,-1) to[bend right] (6,1); \end{tikzpicture} }

There will be higher-order corrections, such as some pairs and some four $\psi_I$s matching, etc., but those are suppressed in the large $N$ limit. Notice that $\psi_{I}^2=(-1)^{\frac{p}{2}}\mathbb{I}$. So, the first two terms just give $1$ and $\sum\limits_{I_1,I_2}$ cancels ${N \choose p}^2$. $\psi_I\psi_J = (-1)^{|I\cap J|} \psi_J\psi_I$ In the above identity, $|I\cap J|$ is the number of sites in common to $I$ and $J$. In the formula for $m_4$ as we have $\sum\limits_{I_1,I_2}$ we only need to find the average value $(-1)^{|I\cap J|}$ for 3rd diagrams contribution.

Poisson distribution: Let $I$ be of size $p$ and $J$ be of size $p’$. For fixed $I$, the probability of $|I\cap J|=k$ is $\displaystyle\frac{\binom{p} {k} \binom{N-p} {p^{\prime}-k}} {\binom{N} {p^{\prime}}}\approx \frac{1} {k!} \left( \frac{p p^{\prime}} {N} \right)^{k} e^{-p p^{\prime} / N} $. That is a Poisson distribution with parameter $p p^{\prime} / N$.

For our case $p’=p$ $\frac{1}{ {N \choose p}^2}\sum\limits_{I_1,I_2}(-1)^{|I\cap J|}=\sum\limits_{k=0}^{p}(-1)^{k}\frac{(\lambda/2)^{k}}{k!}e^{-\lambda/2}=e^{-\lambda}=q$

So, $m_4=1+1+q$. This method can be generalized to $m_k\neq m_4$. We just need to 1) Find all chord diagrams corresponding to $m_k$ and 2) In each chord diagram, multiply by $q$ for every chord intersection. Now, using these 2 rules and the transfer-matrix method, we will get

\[m_k = \int\limits_0^\pi \frac{d\theta}{2\pi} (q,e^{\pm 2i\theta};q)_{\infty}\left(\frac{2\cos\theta}{\sqrt{1-q}}\right)^k\]

where the q-Pochhammer symbol is defined as $(q,e^{\pm 2i\theta};q)_{\infty}=(q;q)_\infty(e^{ 2i\theta};q)_\infty(e^{-2i\theta};q)_\infty$ $(a;q)_{\infty} = \prod\limits_{k=0}^\infty (1-aq^{k}).$

Transfer-matrix method: At a random point on the chord diagrams, we cut them open. We can think of the number of chords $l$ between 2 nodes as making a state $

l\rangle$ in some auxiliary Hilbert space. Once we know the state before a node, then the state after this node has 1 way of creating a chord and $l$ ways of annihilating a chord as shown in Fig fig:TransferMatrix. If you think of these nodes as some discrete time, then the corresponding Hamiltonian is defined as the transfer matrix $T

l \rangle=

l+1 \rangle+\left( 1+q+\cdots+q^{l-1} \right)

l-1 \rangle=

l+1 \rangle+\frac{1-q^{l}} {1-q}

l-1 \rangle\,$. Now we can solve $Z$ using $\displaystyle\langle\langle T r \, e^{-\beta H} \rangle\rangle=\langle0

e^{-\beta T}

0 \rangle$. Equivalently $T$ can be also be defined using the $q$ deformed harmonic oscillator

$T=A_{q}+A_{q}^{\dagger}, \ A_{q} A_{q}^{\dagger}-q A_{q}^{\dagger} A_{q}=1$ $[ N, A_{q} ]=-A_{q}, \ [ N, A_{q}^{\dag} ]=A_{q}^{\dag}.$

Now that we know how to solve $Z$, we can see how this can be generalized to correlation functions $\mathrm{t r} ( e^{-\tau_{1} H} M_{s} e^{-\tau_{2} H} M_{s} \cdots)$ of matter operators defined as

\[M_{s}=i^{s / 2} \sum_{I} K_{I} \Psi_{I}^{s}.\]

where $K_{I}$ are independent Gaussian random variables defined so that $TrM_{s}^2=1$. In the Double Scaled limit, we let $s\to \infty$ by fixing $\Delta = s/p$. So we should now call it $M_{\Delta}$. Now we can proceed similarly using chord diagrams. We will now have 2 different chords corresponding to $H$ and $M_{\Delta}$. When 2 $M_{\Delta}$ chords intersect we get a factor of $q_{M}=e^{-2 s^{2} / N}=e^{-\lambda\Delta^{2}}$ and when a $H$ chord and $M_{\Delta}$ chord intersect we get a factor of $q_{HM}=e^{-2 ps/ N}=e^{-\lambda\Delta}$. Similarly, we can introduce multiple matter operators $M_1, M_2\cdots$ with $\Delta_1,\Delta_2,\cdots$.

Quantum groups and noncommutative geometry:~^[38] \subsubsection{von Neumann algebras} ^[39] these are holography related. For math~^[40]\

\etocsettocdepth.toc{paragraph} \subsection{Flat space holography bottom-up approaches} \subsubsection{Celestial holography} ^[41] \subsubsection{Carrollian holography} ^[42] \subsection{de Sitter holography approaches} ^[43] \subsubsection{dS/CFT} \subsubsection{Static patch holography} \subsubsection{$T\bar T$ deformation} ^[44] \subsubsection{The dS/dS Correspondence}

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