diff --git a/alpha/GlueonAdS/glueon-poster-v2.tex b/alpha/GlueonAdS/glueon-poster-v2.tex index 34d7d25..040b878 100644 --- a/alpha/GlueonAdS/glueon-poster-v2.tex +++ b/alpha/GlueonAdS/glueon-poster-v2.tex @@ -250,8 +250,9 @@ \section*{\textbf{Proposal:}\texstringonly{\\}Glue-on $\mrm{AdS}_3$ -- beyond th \end{equation*}} \begin{center} \vspace{-1\baselineskip}% + \centering - \includegraphics[width=.7\linewidth]{img/diagram.pdf} + \includegraphics[width=.8\linewidth]{img/diagram.png} \vspace{-.45\baselineskip} \scriptsize\ Top-down view of a constant $t$ slice @@ -331,7 +332,7 @@ \section*{\textbf{Proposal:}\texstringonly{\\}Glue-on $\mrm{AdS}_3$ -- beyond th \end{align} $\leadsto$ correct charges, the modified signal propagation speed $v'_{\pm} \equiv \pm {d\varphi'}/{dt'}$, and \TTbar \textbf{thermodynamics} upon Wick rotation. In particular, \begin{equation*} - \hspace{-.5em} \mu > 0,\ \ \, + \hspace{-.8em} \mu > 0,\ \ \, T_L(\mu)\,T_R(\mu) \le - \frac{1}{4\pi^2 \ell^2 \zeta_c} = \frac{3}{4\pi^2c\mu} = T_H(\mu)^2. \end{equation*} $T_{L,R}$: temperatures associated with $u',v' = \varphi' \pm t'$.\\ @@ -371,8 +372,8 @@ \subsection*{\TTbar partition functions from the bulk} \label{se:partitionfuncti \item \textbf{Torus:} {modular invariance} \& sparseness of the ``light'' spectrum at large $c$ $\leadsto$ universal form: \begin{equation*}\small - \hspace{-2.3em} - \log Z_{T\bar T}(\mu) \approx \left\{ \begin{aligned} + \hspace{-2.8em} + \log Z_{T\bar T}(\mu) \approx \left\{ \begin{aligned} & {-\frac{1}{2}}\,(\beta_L + \beta_R)\, RE_{\text{vac}}(\mu), &\beta_L \beta_R > 1, \\ & {-2 \pi^2 \bigg(\frac{1}{\beta_L} + \frac{1}{\beta_R}\bigg) RE_{\text{vac}}\bigg(\frac{4\pi^2}{\beta_L \beta_R} \mu \bigg)}, &\beta_L\beta_R < 1. %\\[2ex] \end{aligned} \right. %\label{ZTTbar} @@ -402,7 +403,7 @@ \subsection*{\TTbar partition functions from the bulk} \label{se:partitionfuncti $} and the flow equation \eqref{TTbardef} admit the general \mbox{solution} with a $\mu$-\textit{independent} integration constant $a$: \begin{equation*}\small - \hspace{-1.6em} + \hspace{-1.8em} \log Z_{\TTbar}(\mu, a) = \tfrac{c}{3} \log \Big[\tfrac{R}{a} \Big(1+\sqrt{1-\tfrac{c \mu }{3 R^2}}\, \Big) \Big] - \tfrac{R^2}{\mu} \sqrt{1-\tfrac{c \mu }{3 R^2}} + \tfrac{R^2}{\mu}. %\label{Zsol} \end{equation*} \begin{itemize}%[noitemsep] diff --git a/alpha/GlueonAdS/glueon-poster-v3.tex b/alpha/GlueonAdS/glueon-poster-v3.tex new file mode 100644 index 0000000..72f743b --- /dev/null +++ b/alpha/GlueonAdS/glueon-poster-v3.tex @@ -0,0 +1,471 @@ +% !TeX document-id = {c68ea1ac-ebac-4541-a297-17005c6d2297} +% !TeX encoding = UTF-8 +% !TeX spellcheck = en_US +% !TeX TS-program = pdflatex +% !TeX TXS-program:bibliography = biber -l zh__pinyin --output-safechars % + +\documentclass[10pt]{article} + +\input{preamble-poster.tex} + +\usepackage{tikz} +\usepackage{caption} +%\usepackage{snapshot} + +\renewenvironment{frame}[1]% + {\section*{#1}}% + {} + +\newcommand{\citations}[1]{{\footnotesize#1\par}} +\newcommand{\reality}{reality\textsuperscript{TM}} + +%\subtitle{An extended AdS\,/\,\TTbar duality \textit{(beyond infinity)}} + + +\newcommand{\veccol}[1]{\pqty{ + \begin{smallmatrix} + #1 + \end{smallmatrix} +}} + +\addbibresource{glueon-beamer.bib} +%\usepackage{cprotect} + +\usepackage{cancel} +\newcommand{\slot}{{\,\bullet}} + +\usepackage{xspace} +\newcommand{\TTbar}{\texorpdfstring{\ensuremath{T\bar{T}}}{TTbar}\xspace} + +\newcommand{\jokeInfinity}{ + \includegraphics[height=.2\linewidth]{img/smbc-cantor-cropped.png} +\\[-.5ex] + {\footnotesize\url{https://smbc-comics.com/comic/cantor}} +} + +\setlength{\droptitle}{-3.25\baselineskip} +\pretitle{\LARGE\bfseries\noindent} +\title{Glue-on AdS holography for $T\bar T$-deformed CFTs} +\posttitle{\ \footnotesize\textsl{JHEP 06 (2023) 117}} + +\preauthor{\par\large\noindent} +\author{% + Luis Apolo, + Peng-Xiang Hao \textkai{\normalsize 郝鹏翔}, + \textbf{Wen-Xin Lai \textkai{\small 赖文昕}}, + and Wei Song \textkai{\small 宋伟} +} +\postauthor{ + \,\arxiv{2303.04836} + \par%\vspace{.5\baselineskip} + \vspace{-.6\baselineskip} + \noindent\rule{.68\linewidth}{1pt}\par + \vspace{-.1\baselineskip} +% \vspace{.3\baselineskip} +} + +\predate{\noindent\sffamily% + \begin{minipage}{3em} + \includegraphics[width=2.7em]{img/ymsc-logo.pdf} + \end{minipage} + Yau Mathematical Sciences Center YMSC, Tsinghua + --- +} +\date{September 2023} +\postdate{ + @ YITP, Kyoto +% \vspace{0\baselineskip} +} + +\pagestyle{empty} + +\begin{document} + +\maketitle +\thispagestyle{empty} + + +\begin{multicols}{3} + +%\textbf{Abstract:} +\,\\[-2.2\baselineskip] + +\TTbar-deformed CFTs with $\mu < 0$ have been proposed by \mbox{\textcite{McGough:2016lol}} to be holographically dual to Einstein gravity with a finite \mbox{Dirichlet} cutoff. + +We generalize the proposal for $\mu > 0$ by \textit{gluing-on} another patch of $\mrm{AdS}_3$, and \mbox{provide} various evidence for this extended holography, now valid for $\mu \in \mbb{R}$. + +\vspace{.8\baselineskip} +\hrule +\vspace{.3\baselineskip} + +\textbf{Pure Einstein gravity} without matter in $\mrm{AdS}_3$: +\begin{align} +\hspace{-.8em} + ds^2 + &= \ell^2 \bigg( \frac{d\rho^2}{4 \rho^2} + \frac{ \big( du + \rho \, \mathcal {\bar L}(v)\, dv \big) \big( dv + \rho \, \mathcal L(u)\, du \big) }{\rho} \bigg) \notag\\ + &= n_\mu n_\nu dx^\mu dx^\nu + \frac{1}{\zeta} \gamma_{ij}dx^i dx^j, \label{fggauge} +\end{align} + +\begin{itemize} +\item Transverse coordinates: $x^{i} \sim \varphi, t$ \\ + Light-cone coordinates: $u,v = \varphi \pm t$ + + $\mathcal L(u), \bar{\mathcal L}(v)$: arbitrary periodic functions +\item Radial coordinate: $ + \ell^{-2} g_{\varphi\varphi} \equiv r^2 \equiv \zeta^{-1}$\\ + $r$: the ``proper radius'' +\end{itemize} +\begin{center} + \vspace{-.5\baselineskip} + \includegraphics[width=.52\linewidth]{img/foliation.pdf} + + \vspace{-.3\baselineskip}\scriptsize + Image courtesy: \textsl{R. Szalai},\\ + \tiny\texttt{DOI:10.1007/s11071-020-05891-1} + + \vspace{-.5\baselineskip} +\end{center} +\begin{itemize} +\item Structure: foliated by constant $\zeta$ surfaces ${\mathcal N}_\zeta$\\[.6ex] +Asymptotic boundary: $\mcal{N}_0$ is at $\zeta \to 0$ +\\[1ex] +\citations{% + \textcite{Fefferman:2007rka}\\ + \textcite{Banados:1992wn}%\\ +% \textcite{Banados:1998gg} +} +\end{itemize} + +\textbf{``$\thinmspace[.9]\TTbar$\,'' deformation} as the flow of action: +\begin{align} +\hspace{-.1em} + \partial_{\sidenote{\mu}} I &= {8 \pi} \int d^2x \, T\bar T_{(\sidenote{\mu})} = {\pi} \int d^2x\,\big( T^{ij}T_{ij}- (T^i_i)^2 \big)_{(\sidenote{\mu})},\notag\\ + &x, \bar{x} = \varphi' \pm t' + \label{TTbardef} +\end{align} + +\begin{itemize} + +\item \textbf{Irrelevant:} initially $\mrm{CFT}_2$: $T\bar{T}_{(\mu = 0)} = T_{xx} T_{\bar{x}\bar{x}}$, \\ +but conformal symmetry will be broken for $\mu \ne 0$. + +\item \textbf{Solvable:} the deformed spectrum of $\hat{H}(\mu)$ and $\hat{J}(\mu)$ on a cylinder of radius $R$ is a simple function: +\begin{equation} +\hspace{-2em} +\begin{gathered} + E(\mu) = - \frac{R }{ 2\mu } \bigg(1-\sqrt{1 + \frac{4\mu}{R} E(0) + \frac{4\mu^2}{R^4} J(0)^2 } + \,\bigg), \\ J(\mu)=J(0) \\[-1.5ex] +\end{gathered} \label{ttbarspectrum} +\end{equation} +of the undeformed spectrum $E(0)$, $J(0)$.\\ +(\textit{under reasonable assumptions}) + +\citations{ +\textcite{Zamolodchikov:2004ce}\\ +\textcite{Dubovsky:2012wk}\\ +\textcite{Dubovsky:2013ira}\\ +\textcite{Smirnov:2016lqw}\\ +\textcite{Cavaglia:2016oda} \textit{et al}%\\ +%\textcite{Dubovsky:2017cnj} +} + +\vspace{-.8\baselineskip} + +\end{itemize} + +\subsection*{Cutoff $\mrm{AdS}_3$ duality:\texstringonly{\\} Holography within a finite Dirichlet wall} +\vspace{-.2\baselineskip} +\citations{ +\textcite{McGough:2016lol}\\ +\textcite{Kraus:2018xrn} \textit{et al} +} + +\textbf{Dictionary:} radial location $\zeta_c$ of the cutoff surface $\mcal{N}_{\zeta_c}$ gets mapped to the deformation parameter $\mu$: +\begin{equation} + \zeta_c = - \frac{c \mu}{3\ell^2} + \label{dictionary} +\end{equation} +\TTbar flow recast geometrically as the $i,j$ components of the Einstein equations. + +This is a ``non-AdS'' / non-CFT duality.\\ +\textit{A step towards quantum gravity in \reality!} + +\textbf{Caveat:} the duality only admits $\zeta_c > 0$ so $\mu < 0$.\\ +But \TTbar itself admits $\mu > 0$ with nice properties. + +\citations{The related proposal of \textcite{Guica:2019nzm}\\ +admits both signs of $\mu$.} + +\textit{What is the other side of the duality?} + +\columnbreak + +\vspace*{-3.6\baselineskip} +\begin{center} + \mbox{ + \hspace{-1.55em} + \includegraphics[width=1.14\linewidth]{img/ads-cft-cutoff.png} + } + + \vspace{-.8\baselineskip}\small + Cutoff $\mrm{AdS}_3$\,/\,\TTbar-deformed $\mrm{CFT}_2$ + + \vspace{-.3\baselineskip} + \scriptsize\ Image courtesy: \textcite{AldegundePWSep22} +\end{center} + + +%\begin{itemize} +% +%\item[] \textbf{Caveat:} the duality only admits $\zeta_c > 0$ so $\mu < 0$.\\ +%But \TTbar itself admits $\mu > 0$ with cool properties. +% +%\citations{The related proposal of \textcite{Guica:2019nzm}\\ +%admits both signs of $\mu$.} +% +%\textit{What is the other side of the duality?} +% +%\end{itemize} + +\vspace{-1.5\baselineskip} + +\section*{\textbf{Proposal:}\texstringonly{\\}Glue-on $\mrm{AdS}_3$ -- beyond the infinity} +\label{se:glueonproposal} + +\parbox{\linewidth}{ +\vspace{-\baselineskip} +\textbf{\begin{equation*} +\hspace{-2.8em} +\begin{aligned} + \textrm{Cutoff} + \ &/\ % + \text{\textit{glue-on} $\mrm{AdS}_3$ Gravity} + \!\!&\equiv\ % + \textrm{\TTbar deformed $\mrm{CFT}_2$ at $\mcal{N}_{\zeta_c}$} \\ + \zeta_c > 0 + \ &/\ % + \zeta_c < 0 + & \mu \in \mbb{R}\hspace{1em}\, +\end{aligned} +\end{equation*}} +\begin{center} + \vspace{-1\baselineskip}% + + \centering + \includegraphics[width=.8\linewidth]{img/diagram.png} + + \scriptsize\ Top-down view of a constant $t$ slice +\end{center} +%\vspace{-\baselineskip} +} + +\begin{itemize} + +\item The metric \eqref{fggauge} has a simple pole, thus admits a well-defined \textbf{analytic continuation:} +%\begin{equation} +% \rho,\ \zeta,\ \mu\ \in\ \mbb{R} +%\end{equation} +\begin{equation} + \frac{1}{\zeta} \equiv \ell^{-2} g_{\varphi\varphi} \in \mbb{R}, +\ \quad + \zeta_c = - \frac{c \mu}{3\ell^2} \in \mbb{R} +\end{equation} +What are we doing other than copy-pasting?\\ +``Renormalize'' the divergences as $\zeta \to 0^\pm$: + \begin{itemize}[noitemsep]\small + \item introduce $\mathcal N_{\zeta={\pm\epsilon}}$ and \textbf{``glue''} them together; + \item i.e.~exclude the $-\epsilon < \zeta < \epsilon$ region, \\ until finally sending $\epsilon \to 0$. + \end{itemize} + +\item \textbf{Extending the flow:} matching energy momentum (Brown-York) \& the \TTbar flow equations: + \begin{equation} + \begin{gathered} + T_{ij}= \frac{\sigma_{\zeta_c}}{8\pi G} \Big( K_{ij} - K \gamma_{ij} + \frac{1}{\ell |\zeta_c|}\gamma_{ij}\Big), \\ \sigma_{\zeta_c} = \tfrac{\abs{\zeta_c}}{\zeta_c} = \pm 1,\quad + \gamma_{ij} = \zeta_c h_{ij}, \\[-1ex] + \end{gathered} \label{BYtensor} + \end{equation} + $h_{ij}$: the induced metric. + + The field theory metric $\gamma_{ij}$ is always positive-definite, while $h_{ij}$ becomes negative-definite for the glue-on region $\rho,\zeta < 0$. + + This discrepancy is the origin of $\abs{\zeta_c}$. + +%\end{itemize} +% +%\subsection*{\TTbar physics from the glue-on geometry} +% +%\begin{itemize} + +\item Demanding a \textbf{non-singular extended geometry} reproduces bounds on the \TTbar deformed theory, e.g. + \begin{equation} + \zeta_c \ge -1 \quad \Leftrightarrow \quad\mu\le \frac{ 3\ell^2 }{c} \label{reality} + \end{equation} + This is a Killing horizon in the glue-on region of the \mbox{extended} geometry; + $(\det g_{\mu\nu})_{\,\zeta = -1} = 0$. + +\item \textbf{Spectrum from the extended geometry:} they are \mbox{conserved} charges $\mcal{Q}$ of the Killing symmetries, and can be computed with the \textit{covariant formalism}. + +\citations{ + \textsl{\citeauthor{Iyer:1994ys,Barnich:2001jy}} et al\\ + With \TTbar: \textcite{Kraus:2021cwf} +}\vspace{-1.5\baselineskip} + \begin{align*} + \delta E(\mu) &\equiv \ell^{-1} \delta \mathcal Q_{\pd_{t'}}, \quad \delta J(\mu) \equiv \delta \mathcal Q_{\pd_{\varphi'}}. + \end{align*} + This reproduces $E(\mu)$, $J(\mu)$ with $\mu \in\mbb{R}$ in \eqref{ttbarspectrum}.\\ + $(t',\varphi')$: normalized boundary coordinates: + \begin{equation} + ds^2_c = \ell^2 ( -dt'^2 + d\varphi'^2) , \qquad \varphi' \sim \varphi' + 2\pi.\label{cutoffmetric} + \end{equation} + +\columnbreak + +\vspace*{-8\baselineskip} +%\item The covariant charges (variation): + + \item \textbf{\textit{State-dependent} maps of \mbox{coordinates}:} + \begin{align} + dt' &= \sqrt{ \big(1 - \zeta_c (T_u+T_v)^2 \big) \big(1 - \zeta_c (T_u-T_v)^2 \big) } \, dt, \notag \\ + d\varphi' &= d\varphi + \zeta_c \big( T_u^2 - T_v^2 \big)\,dt. + \label{eq:state_dependent_map} + \end{align} + $\leadsto$ correct charges, the modified signal propagation speed $v'_{\pm} \equiv \pm {d\varphi'}/{dt'}$, and \TTbar \textbf{thermodynamics} upon Wick rotation. In particular, + \begin{equation*} + \hspace{-.8em} \mu > 0,\ \ \, + T_L(\mu)\,T_R(\mu) \le - \frac{1}{4\pi^2 \ell^2 \zeta_c} = \frac{3}{4\pi^2c\mu} = T_H(\mu)^2. + \end{equation*} + $T_{L,R}$: temperatures associated with $u',v' = \varphi' \pm t'$.\\ + $T_H$: the \textit{Hagedorn temperature}: exceeding $T_H$ leads to a complex entropy. + +\begin{flushright} +\vspace{-.5\baselineskip} +\citations{ +\textcite{Giveon:2017nie}\\ +\textcite{Apolo:2019zai} +} +\vspace{-.8\baselineskip} +\end{flushright} + +\end{itemize} + +\subsection*{\TTbar partition functions from the bulk} \label{se:partitionfunction} + +\begin{itemize} +\item Via bulk on-shell action of the dominant saddle: + \begin{equation} + Z_{T\bar T} (\mu) = \mathcal Z (\zeta_c) \approx e^{-I(\zeta_c)}. \label{partition2} + \end{equation} +$I(\zeta_c)$ \mbox{diverges} at $\zeta \to 0^\pm$: we need \textit{holographic renormalization}, with the boundary action: + \begin{align} +% I_{\mcal{M}}(\zeta_1, \zeta_2) & \coloneqq - \frac{1}{16\pi G} \int_{\zeta_1}^{\zeta_2} d\zeta \int d^2 x \sqrt{g} \, (R + 2 \ell^{-2}), %\label{bulkaction} \\ +% \\ + I_{\mathcal N_{\zeta}} & \coloneqq - \frac{1}{8\pi G} \int_{\mathcal N_{\zeta}} d^2x \sqrt{h}\,h^{ij} K_{ij} \label{boundaryaction} + \\& \qquad + \frac{\sigma_\zeta}{8\pi G} \int_{\mathcal N_{\zeta}} d^2x \sqrt{h} \, \bigg(\ell^{-1} - \frac{\ell \mathcal{R}[h]}{4} \log |\zeta| \bigg). \notag + \end{align} + \hfill{\footnotesize \textcite{Henningson:1998gx} \textit{et al}} + + $\log |\zeta|$: not diff-invariant, due to the Weyl anomaly. \\ + $I_{\mcal{N}_\zeta}$: consistent with the B-Y stress tensor \eqref{BYtensor}. + + $Z_{\TTbar}$ agrees with the field theory analysis, using \eqref{eq:state_dependent_map}: + +\item \textbf{Torus:} {modular invariance} \& sparseness of the ``light'' spectrum at large $c$ $\leadsto$ universal form: + \begin{equation*}\small + \hspace{-2.8em} + \log Z_{T\bar T}(\mu) \approx \left\{ \begin{aligned} + & {-\frac{1}{2}}\,(\beta_L + \beta_R)\, RE_{\text{vac}}(\mu), &\beta_L \beta_R > 1, \\ + & {-2 \pi^2 \bigg(\frac{1}{\beta_L} + \frac{1}{\beta_R}\bigg) RE_{\text{vac}}\bigg(\frac{4\pi^2}{\beta_L \beta_R} \mu \bigg)}, &\beta_L\beta_R < 1. %\\[2ex] + \end{aligned} \right. %\label{ZTTbar} + \end{equation*} + \begin{flushright} + \vspace{-.5\baselineskip} + \citations{\noindent% + \textcite{Datta:2018thy}\\ + \textcite{Apolo:2023aho}\\ + cf.~\textcite{Hartman:2014oaa} + } + \vspace{-.5\baselineskip} + \end{flushright} + +\item \textbf{Sphere:} maximally symmetric,\\[-1.4\baselineskip] + +\hfill\mbox{\footnotesize \textcite{Donnelly:2018bef}} + \begin{equation} +% \label{sphere:stresstensor} + \langle T_{ij}\rangle = -\frac{1}{4\pi\mu} \bigg(1-\sqrt{1-\frac{c\mu}{3R^2}}\, \bigg) \gamma_{ij}. + \end{equation} +Trace relation: \mbox{$ +% \begin{align*} +%\partial_\mu \log Z_{\TTbar}(\mu)&=8\pi \int d^2x\sqrt{\gamma}\, \langle T\bar{T} \rangle \\ +(-R)\,\partial_R \log Z_{\TTbar}(\mu)=\int d^2x\sqrt{\gamma}\,\langle T^i_i \rangle +% \end{align*} +$} and the flow equation \eqref{TTbardef} +admit the general \mbox{solution} with a $\mu$-\textit{independent} integration constant $a$: + \begin{equation*}\small + \hspace{-1.8em} + \log Z_{\TTbar}(\mu, a) = \tfrac{c}{3} \log \Big[\tfrac{R}{a} \Big(1+\sqrt{1-\tfrac{c \mu }{3 R^2}}\, \Big) \Big] - \tfrac{R^2}{\mu} \sqrt{1-\tfrac{c \mu }{3 R^2}} + \tfrac{R^2}{\mu}. %\label{Zsol} + \end{equation*} + \begin{itemize}%[noitemsep] + \item $a = \sqrt{c|\mu|/3}$: \mbox{recover {\small + \textsl{\citeauthor{Donnelly:2018bef}} + \cite{Donnelly:2018bef} + }} \\ + $a = \epsilon$: also a valid choice, where the RG length scale $\epsilon$ is decoupled from $\mu$. + + \item Enlarge the \textbf{space of \TTbar deformed theories:}\\ + with \mbox{independent} parameters $(\mu,a)$. + + \item The $\log |\zeta|$ in \eqref{boundaryaction} guarantees that $I = -\log Z_{\TTbar}$ \mbox{satisfies} the $T\bar T$ flow \eqref{TTbardef}; not the case for \cite{Donnelly:2018bef}. + \end{itemize} + + + \begin{flushright} + \vspace{-.6\baselineskip} + \citations{ + \textcite{Caputa:2020lpa}\\ + \textcite{Li:2020zjb} + }\vspace{-.8\baselineskip} + \end{flushright} + + \item Future: understand the \textbf{entanglement structure} of \TTbar deformation with the help of bulk geometry. + \begin{flushright} + \vspace{-.5\baselineskip} + \citations{ + \textcite{Lewkowycz:2019xse} + } + \end{flushright} +\end{itemize} +\centering\vspace{-1\baselineskip} +\includegraphics[width=.6\linewidth]{img/RT-AdS.pdf} + +\vspace{-1\baselineskip} +{\footnotesize RT surface for the extended $\mrm{AdS}_3$} + +\vspace{.8\baselineskip} +\hrule +\vspace{.1\baselineskip} + +\hspace{-.7em} +\begin{minipage}[c]{.81\linewidth} +\begin{itemize}[noitemsep,leftmargin=5em]\small +\item[\texttt{Email:}] + \url{bryanlais@gmail.com} +\item[\texttt{Inspire:}] + \href{https://inspirehep.net/authors/2640135}{ + \texttt{inspirehep.net/authors/2640135} + } +\item[\texttt{GitHub:}] + \href{https://github.com/bryango}{ + \texttt{github.com/bryango} + } +\end{itemize} +\end{minipage}% +\hfill% +\begin{minipage}{.17\linewidth} +\includegraphics[width=\linewidth]{img/inspire-qr-cy.png} +\end{minipage}% +~\mbox{} + +\end{multicols} + +\end{document} diff --git a/alpha/GlueonAdS/img/diagram-v1.pdf b/alpha/GlueonAdS/img/diagram-v1.pdf new file mode 100644 index 0000000..00e6bf5 Binary files /dev/null and b/alpha/GlueonAdS/img/diagram-v1.pdf differ diff --git a/alpha/GlueonAdS/img/diagram.pdf b/alpha/GlueonAdS/img/diagram.pdf index 00e6bf5..24dbeb0 100644 Binary files a/alpha/GlueonAdS/img/diagram.pdf and b/alpha/GlueonAdS/img/diagram.pdf differ diff --git a/alpha/GlueonAdS/img/diagram.png b/alpha/GlueonAdS/img/diagram.png new file mode 100644 index 0000000..2386153 Binary files /dev/null and b/alpha/GlueonAdS/img/diagram.png differ diff --git a/alpha/GlueonAdS/img/diagram.svg b/alpha/GlueonAdS/img/diagram.svg new file mode 100644 index 0000000..e7a796d --- /dev/null +++ b/alpha/GlueonAdS/img/diagram.svg @@ -0,0 +1,775 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +