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\begin{document}

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\section*{p-adic string theory}

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

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{open_string_amplitudes}{Open string amplitudes}\dotfill \pageref*{open_string_amplitudes} \linebreak
\noindent\hyperlink{TopologicalWittenGenus}{Closed string 1-loop vacuum amplitudes (topological Witten genus)}\dotfill \pageref*{TopologicalWittenGenus} \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}

What is called \emph{$p$-adic string theory} is the study of a variant of [[scattering amplitudes]] in [[string theory]] (hence [[string scattering amplitudes]]) where the [[worldsheet]] of the [[string]] is regarded not as a [[Riemann surface]] but as an object in [[p-adic geometry]].

\hypertarget{open_string_amplitudes}{}\subsubsection*{{Open string amplitudes}}\label{open_string_amplitudes}

The original observation is that of (\hyperlink{Volovich87}{Volovich 87}, reviewed in \hyperlink{VVZ95}{VVZ 95, section XIV}) that the [[integral]] expression for the [[Veneziano amplitude]] of the [[open string|open]] [[bosonic string]] naturally generalizes from an integral over the [[real numbers]] (which in this case parameterize the [[boundary]] of the [[open string]] [[worldsheet]]) to the [[p-adic numbers]] by \emph{[[adelic integration]]}.

The standard [[Veneziano amplitude]] has the expression

\begin{displaymath}
\int_{\mathbb{R}}
  {\vert x\vert}^{a-1}
  \cdot
  {\vert 1-x\vert}^{b-1}
  d x
\end{displaymath}
where the [[norm]] involved is the usual [[absolute value]], and the proposed $p$-adic version is hence

\begin{displaymath}
\int_{\mathbb{Q}_p}
  {\vert x\vert}_p^{a-1}
  \cdot
  {\vert 1-x\vert}_p^{b-1}
  d x
\end{displaymath}
with the [[p-adic norm]] instead.

The main result here is (\hyperlink{FreundWitten87}{Freund-Witten 87}) that the ordinary [[Veneziano amplitude]] equals the inverse of the product of its $p$-adic versions, for all primes $p$, apparently a version of the \href{group%20of%20ideles#ProductFormula}{idelic product formula}.

With due regularization this result carries over to other [[string scattering amplitudes]], too. When forming these products one also speaks of \emph{[[ring of adeles|adelic]] string theory}.

Since the [[Veneziano amplitude]] concerns the [[bosonic string]] [[tachyon]] [[state]], [[p-adic string theory]] has been discussed a lot in the context of [[tachyon condensation]] and [[Sen's conjecture]] (\hyperlink{Cottrell02}{Cottrell 02}).

Traditionally literature on $p$-adic string theory asserts that the generalization of this from the [[open string]] to the [[closed string]] remains unclear (e.g \hyperlink{CMZ89}{CMZ 89, section 4}, \hyperlink{Cottrell02}{Cottrell 02, section 5}), since it is unclear which adic version of the [[complex numbers]] to use. However, in other parts of the literature adic versions of closed strings are common, this we discuss \hyperlink{TopologicalWittenGenus}{below}.

\hypertarget{TopologicalWittenGenus}{}\subsubsection*{{Closed string 1-loop vacuum amplitudes (topological Witten genus)}}\label{TopologicalWittenGenus}

Generally, the development of string theory has shown that its [[worldsheet]] is usefully regarded as an object in [[algebraic geometry]] (see also at \emph{[[number theory and physics]]}) and mathematically the generalization from [[algebraic varieties]] over the [[complex numbers]] to more general algebraic varieties (or [[schemes]]) is often natural, if not compelling.

For instance when the [[Witten genus]] (essentially the [[partition function]] of the [[superstring]]) is refined to the [[string orientation of tmf]] then the [[elliptic curves]] over the [[complex numbers]] which serve as the [[torus|toroidal]] [[worldsheets]] over the [[complex numbers]] are generalized to [[elliptic curves]] over general [[rings]] and by the [[fracture theorems]] the computations in [[tmf]] in fact typically proceed (see \href{tmf#DecomopositionViaArithmeticSquares}{here}) by decomposing the general problem into that of ellitpic curves over the [[rational numbers]] and over the [[p-adic integers]]. The result refines the [[Witten genus]]

\begin{displaymath}
\Omega^String_{\bullet} \longrightarrow MF_\bullet
\end{displaymath}
(being a [[ring]] [[homomorphism]]) from the [[String structure]] [[cobordism ring]] to that of [[modular forms]] to one of [[E-∞ rings]]

\begin{displaymath}
M String \longrightarrow tmf
\end{displaymath}
from the [[String structure]] [[Thom spectrum]] to [[tmf]]. Notice that $M String$ here classifies String-[[cobordism cohomology theory|cobordism]] and hence parameterizes ordinary (not $p$-adic) [[target space|target]] [[spacetime]] [[manifolds]], while $tmf$ on the right does regard the [[genus of a surface|genus]]-1 [[worldsheet]] as a general [[elliptic curve]], hence in particular possibly as an \href{elliptic%20curve#OverpAdics}{elliptic curve over the p-adic integers}.

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

\begin{itemize}%
\item The interesting aspects of $p$-adic string theory have led people to consider \emph{[[p-adic physics]]} more generally. But it remains noteworthy that in $p$-adic string theory it is exactly only the [[worldsheet]] which is regarded in [[p-adic geometry]], while for instance the [[complex numbers]] as they appear as [[coefficients]] of [[quantum physics]] are not replaced by $p$-adics.


\item [[p-adic AdS-CFT]]


\item [[number theory and physics]]


\item [[p-adic geometry]]


\item [[p-adic homotopy]]


\item [[p-adic cohomology]]



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

The original articles include

\begin{itemize}%
\item Peter Freund, Mark Olson, \emph{Non-archimedean strings}, Physics Letters B 199,2 (1987) ()


\item I. V. Volovich, \emph{p- -   }, , 71:3 (1987)\href{http://www.mathnet.ru/links/704079ec1d28d91ba6eb6d492359b387/tmf4962.pdf}{free Rus. pdf}; transl. \emph{$p$-adic space-time and string theory}, Theor. Math. Phys. \textbf{71}, 574--576 (1987), \href{http://dx.doi.org/10.1007/BF01017088}{eng doi}, \href{http://www.springerlink.com/content/k514t553607324n0/fulltext.pdf}{nonfree Eng. pdf}



\end{itemize}
That the ordinary [[Veneziano amplitude]] is the inverse product of all its $p$-adic versions is due to

\begin{itemize}%
\item [[Peter Freund]], [[Edward Witten]], \emph{Adelic string amplitudes}, Phys. Lett. B 199, 191 (1987). (\href{http://adsabs.harvard.edu/abs/1987PhLB..199..191F}{web record})

\end{itemize}
also

\begin{itemize}%
\item Lee Brekke, [[Peter Freund]], Mark Olson, [[Edward Witten]], \emph{Non archimedean string dynamics}, Nucl. Phys. B302, 3 (1988) ()

\end{itemize}
A detailed discussion of $p$-adic open [[string scattering amplitudes]] is in

\begin{itemize}%
\item L. Chekhov, A. Mironov, A. Zabrodin, \emph{Multiloop calculations in $p$-adic String theory and Bruhat-Tits Trees}, Comm. Math. Phys. 125, 675-711 (1989) (\href{http://projecteuclid.org/euclid.cmp/1104179635}{Euclid})

\end{itemize}
A review of this is in

\begin{itemize}%
\item Lee Brekke and [[Peter Freund]], \emph{$p$-Adic numbers in physics}, Phys. Rep. 233, 1 (1993) (\href{http://adsabs.harvard.edu/abs/1993PhR...233....1B}{web record})

\end{itemize}
and with an eye towards [[AdS-CFT duality]] in

\begin{itemize}%
\item [[Peter Freund]], \emph{$p$-adic Strings Then and Now} (\href{https://arxiv.org/abs/1711.00523}{arXiv:1711.00523})

\end{itemize}
Discussion of [[tachyon condensation]] in $p$-adic string theory includes

\begin{itemize}%
\item William Cottrell, \emph{$p$-adic Strings and Tachyon Condensation}, 2002 (\href{http://jfi.uchicago.edu/~tten/teaching/Phys.291/Cottrell_Freund_2002.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Miriam Bocardo-Gaspar, H. García-Compeán, W. A. Zúñiga-Galindo, \emph{Regularization of p-adic String Amplitudes, and Multivariate Local Zeta Functions} (\href{https://arxiv.org/abs/1611.03807}{arXiv:1611.03807})

\end{itemize}
Relation to [[gravity]] and the zeros of the [[Riemann zeta function]] (hence the [[Riemann hypothesis]]):

\begin{itemize}%
\item [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], \emph{General relativity from $p$-adic strings} (\href{https://arxiv.org/abs/1901.02013}{arXiv:1901.02013})

\end{itemize}
[[!redirects p-adic string theories]]

[[!redirects adelic string theory]] [[!redirects adelic string theories]]

[[!redirects p-adic string]] [[!redirects p-adic strings]]



\end{document}