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\begin{document}

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\section*{weak gravity conjecture}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{loose_statement}{Loose statement}\dotfill \pageref*{loose_statement} \linebreak
\noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak
\noindent\hyperlink{relation_to_cosmic_censorship}{Relation to cosmic censorship}\dotfill \pageref*{relation_to_cosmic_censorship} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{weak gravity conjecture} is a [[conjecture]] (\hyperlink{ArkaniHamedMotlNicolisVafa06}{Arkani-Hamed, Motl, Nicolis, Vafa 06}) often summarized as saying that for [[gravity]] or [[quantum gravity]] to be consistent, the [[force]] of the field gravity must be weaker than that of any of the [[gauge field]] forces, in suitable units.

\hypertarget{loose_statement}{}\subsection*{{Loose statement}}\label{loose_statement}

More precisely, what is argued in (\hyperlink{ArkaniHamedMotlNicolisVafa06}{AH-M-N-V 06, section 2}) is that there alwayst must be ``[[elementary particles]]'' for which the ratio $m/{\vert q\vert}$ of their [[mass]] over their [[gauge field]] [[charge]] (e.g. [[electric charge]], [[magnetic charge]]) is smaller than one: $m/{\vert q\vert} \lt 1$ (in natural [[units]]).

The argument for this in (\hyperlink{ArkaniHamedMotlNicolisVafa06}{AH-M-N-V 06, section 2.2}) is the following. The [[extremal black hole|extremal]] ([[BPS state]]) bound for [[charged black holes]] is that their mass $M$ \emph{exceeds or equals} their charge ${\vert Q\vert}$, hence $M/{\vert Q\vert} \geq 1$. Now via [[black hole radiation]] any black hole is supposed to eventually decay by [[radiation|radiating]] away elementary particle quanta. If there were no elementary particle with $m/{\vert q\vert} \leq 1$, so the argument goes, then by radiating away particles with $m/{\vert q\vert} \gt 1$ the black hole would eventually lose all its mass, while still having some non-vanishing charge left. But such ``black hole remnants'' have been argued to lead to inconsistency (\hyperlink{Susskind96}{Susskind 96}).

\hypertarget{in_string_theory}{}\subsection*{{In string theory}}\label{in_string_theory}

The weak gravity conjecture was originally motivated from [[string theory]], which is argued to validate the ``weak gravity'' assumption.

In (\hyperlink{ArkaniHamedMotlNicolisVafa06}{AH-M-N-V 06, section 4}) it is argued that the closed [[superstring]] excitation states in [[heterotic string theory]] [[KK-compactification|KK-compactified]] on to 4d satisfy the condition $m/{\vert q\vert} \lt 1$:



\hypertarget{relation_to_cosmic_censorship}{}\subsection*{{Relation to cosmic censorship}}\label{relation_to_cosmic_censorship}

In 2015 [[Cumrun Vafa]] has argued that the weak gravity conjecture implies the [[cosmic censorship hypothesis]]: The latter turns out to be generally false, even in in 4 dimensions (\href{CrisfordSantos17}{Crisford-Santos 17}), but there are arguments that configurations violating it also violate the assumption of ``weak gravity''.

\begin{quote}%
Subsequent calculations by Santos and Crisford supported Vafa's hunch; the simulations they're running now could verify that naked singularities become cloaked in black holes right at the point where gravity becomes the weakest force. (\href{https://www.quantamagazine.org/where-gravity-is-weak-and-naked-singularities-are-verboten-20170620/}{Wolchover, June 20 2017})


\end{quote}
For more on this see \hyperlink{HorowitzSantos19}{Horowitz-Santos 19}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[cosmic censorship hypothesis]]


\item [[landscape of string theory vacua]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The statement is due to

\begin{itemize}%
\item [[Nima Arkani-Hamed]], Lubos Motl, Alberto Nicolis, [[Cumrun Vafa]], \emph{The String Landscape, Black Holes and Gravity as the Weakest Force}, JHEP 0706:060,2007 (\href{https://arxiv.org/abs/hep-th/0601001}{arXiv:hep-th/0601001})

\end{itemize}
with reference to arguments in

\begin{itemize}%
\item [[Leonard Susskind]], \emph{Trouble For Remnants} (\href{https://arxiv.org/abs/hep-th/9501106}{arXiv:hep-th/9501106})

\end{itemize}
Review is in

\begin{itemize}%
\item Eran Palti, \emph{The Swampland: Introduction and Review}, lecture notes (\href{https://arxiv.org/abs/1903.06239}{arXiv:1903.06239})

\end{itemize}
Further discussion includes

\begin{itemize}%
\item Clifford Cheung, Grant N. Remmen, \emph{Naturalness and the Weak Gravity Conjecture}, Phys. Rev. Lett. 113, 051601 (2014) (\href{https://arxiv.org/abs/1402.2287}{arXiv:1402.2287})


\item Yu Nakayama, Yasunori Nomura, \emph{Weak Gravity Conjecture in AdS/CFT}, Phys. Rev. D 92, 126006 (2015) (\href{https://arxiv.org/abs/1509.01647}{arXiv:1509.01647})


\item [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], \emph{Sharpening the Weak Gravity Conjecture with Dimensional Reduction}, JHEP02(2016)140 (\href{https://arxiv.org/abs/1509.06374}{arXiv:1509.06374})


\item Clifford Cheung, Junyu Liu, Grant N. Remmen, \emph{Proof of the Weak Gravity Conjecture from Black Hole Entropy} (\href{https://arxiv.org/abs/1801.08546}{arXiv:1801.08546})


\item Miguel Montero, \emph{A Holographic Derivation of the Weak Gravity Conjecture}, JHEP 03 (2019) 157 (\href{https://arxiv.org/abs/1812.03978}{arXiv:1812.03978})


\item [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], \emph{Repulsive Forces and the Weak Gravity Conjecture} (\href{https://arxiv.org/abs/1906.02206}{arXiv:1906.02206})



\end{itemize}
Relation to [[cosmic censorship hypothesis]]

\begin{itemize}%
\item Toby Crisford, [[Jorge Santos]], \emph{Violating weak cosmic censorship in AdS$_4$}, Phys. Rev. Lett. 118, 181101 (2017) (\href{https://arxiv.org/abs/1702.05490}{arXiv:1702.05490})


\item [[Gary Horowitz]], [[Jorge Santos]], \emph{Further evidence for the weak gravity - cosmic censorship connection} (\href{https://arxiv.org/abs/1901.11096}{arXiv:1901.11096})



\end{itemize}
Relation, via [[AdS/CFT]] and the [[conformal bootstrap]], to the solution of the [[sphere packing problem]] in [[dimensions]] 8 and 24:

\begin{itemize}%
\item [[Thomas Hartman]], [[Dalimil Mazáč]], [[Leonardo Rastelli]], \emph{Sphere Packing and Quantum Gravity} (\href{https://arxiv.org/abs/1905.01319}{arXiv:1905.01319})

\end{itemize}


\end{document}