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\section*{string theory results applied elsewhere}

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

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

[[!include string theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{worldline_formalism}{Worldline formalism}\dotfill \pageref*{worldline_formalism} \linebreak
\noindent\hyperlink{general_relation_between_yangmills_theory_and_gravity}{General relation between Yang-Mills theory and gravity}\dotfill \pageref*{general_relation_between_yangmills_theory_and_gravity} \linebreak
\noindent\hyperlink{ApplicationsToYangMillsTheory}{Applications to Yang-Mills theory}\dotfill \pageref*{ApplicationsToYangMillsTheory} \linebreak
\noindent\hyperlink{worldline_formalism_2}{Worldline formalism}\dotfill \pageref*{worldline_formalism_2} \linebreak
\noindent\hyperlink{qft_duality_and_specifically_montonenolive_electricmagnetic_duality}{QFT Duality and specifically Montonen-Olive electric/magnetic duality}\dotfill \pageref*{qft_duality_and_specifically_montonenolive_electricmagnetic_duality} \linebreak
\noindent\hyperlink{application_to_qcd_and_experimental_particle_physics}{Application to QCD and experimental particle physics}\dotfill \pageref*{application_to_qcd_and_experimental_particle_physics} \linebreak
\noindent\hyperlink{QCDScatteringAmplitudes}{Scattering amplitudes}\dotfill \pageref*{QCDScatteringAmplitudes} \linebreak
\noindent\hyperlink{QuarkGluonPlasma}{Quark-gluon plasma}\dotfill \pageref*{QuarkGluonPlasma} \linebreak
\noindent\hyperlink{to_the_confinement_problem}{To the confinement problem}\dotfill \pageref*{to_the_confinement_problem} \linebreak
\noindent\hyperlink{to_super_yangmills_theory}{To super Yang-Mills theory}\dotfill \pageref*{to_super_yangmills_theory} \linebreak
\noindent\hyperlink{ApplicationsToGravity}{Applications to gravity}\dotfill \pageref*{ApplicationsToGravity} \linebreak
\noindent\hyperlink{GravityScattetingAmplitudes}{Scattering amplitudes}\dotfill \pageref*{GravityScattetingAmplitudes} \linebreak
\noindent\hyperlink{GravitationalWaveSignatures}{Gravitational wave signatures}\dotfill \pageref*{GravitationalWaveSignatures} \linebreak
\noindent\hyperlink{black_hole_entropy}{Black hole entropy}\dotfill \pageref*{black_hole_entropy} \linebreak
\noindent\hyperlink{application_to_pure_mathematics}{Application to pure mathematics}\dotfill \pageref*{application_to_pure_mathematics} \linebreak


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

Beyond the speculative hypothetized role of [[string theory]] as a physical theory of [[fundamental strings]] that constitute the observed [[fundamental particles]] in the [[standard model of particle physics]], the theory has shed light on many aspects of [[quantum field theory]] as such, both on the conceptual structure of QFT as well as on concrete theories and their concrete properties such as of. This entry lists such instances of string theory results having lead to insights in non-stringy physics and in particular into experimentally confirmed physics, such as [[QCD]] in the [[standard model of particle physics]].

\begin{quote}%
Le plus court chemin entre deux v\'e{}rit\'e{}s dans le domaine r\'e{}el passe par le domaine complexe.

[[Jacques Hadamard]] (\href{http://homepage.math.uiowa.edu/~jorgen/hadamardquotesource.html}{source})


\end{quote}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The two basic theories that underlie observed fundamental physics -- and which string theory unifies at least qualitatively and in [[perturbation theory]] -- are [[Yang-Mills theory]] and [[gravity|Einstein gravity]]/[[general relativity]].

\begin{enumerate}%
\item \hyperlink{ApplicationsToYangMillsTheory}{Applications to Yang-Mills theory}


\item \hyperlink{ApplicationsToGravity}{Applications to Gravity}



\end{enumerate}
Many of the insights are based on the gauge/gravity duality in string theory:

\begin{enumerate}%
\item \href{}{General relation between Yang-Mills theory and gravity}

\end{enumerate}
\hypertarget{worldline_formalism}{}\subsubsection*{{Worldline formalism}}\label{worldline_formalism}

The [[worldline formalism]] for expressing QFT [[scattering amplitudes]] in an effective [[gauge invariance|gauge invariant]] way (different from but equivalent to the [[Feynman rules]]) was originally found by taking the point-particle limit of the expressions for [[string scattering amplitudes]]. See at \emph{[[worldline formalism]]} for more.

Example:

The first calculuation along these lines was actually done earlier in (\hyperlink{MetsaevTseytlin88}{Metsaev-Tseytlin 88}), where the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] was obtained as the point-particle limit of the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in (\hyperlink{Nepomechie83}{Nepomechie 83}) that when computed via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].

\begin{itemize}%
\item R.I. Nepomechie, \emph{Remarks on quantized Yang-Mills theory in 26 dimensions}, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 ()


\item [[Ruslan Metsaev]], [[Arkady Tseytlin]], \emph{On loop corrections to string theory effective actions}, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 ()



\end{itemize}
\hypertarget{general_relation_between_yangmills_theory_and_gravity}{}\subsubsection*{{General relation between Yang-Mills theory and gravity}}\label{general_relation_between_yangmills_theory_and_gravity}

[[AdS/CFT correspondence]] [[open/closed string duality]]

\begin{itemize}%
\item Spenta R. Wadia, \emph{Gauge/Gravity Duality and Some Applications} (\href{http://arxiv.org/abs/1009.0212}{arXiv:1009.0212})

\end{itemize}
talks at

\begin{itemize}%
\item [[Zvi Bern]], Thomas Gehrmann, Frank Petriello, Anastasia Volovich, \emph{The Harmony of Scattering Amplitudes} KITP Program (2011) (\href{http://online.kitp.ucsb.edu/online/qcdscat11/}{web})

\end{itemize}
\hypertarget{ApplicationsToYangMillsTheory}{}\subsubsection*{{Applications to Yang-Mills theory}}\label{ApplicationsToYangMillsTheory}

\hypertarget{worldline_formalism_2}{}\paragraph*{{Worldline formalism}}\label{worldline_formalism_2}

The [[worldline formalism]] for expressing QFT [[scattering amplitudes]] in an effective [[gauge invariance|gauge invariant]] way (different from but equivalent to the [[Feynman rules]]) was originally found by taking the point-particle limit of the expressions for [[string scattering amplitudes]]. See at \emph{[[worldline formalism]]} for more.

Example:

The first calculuation along these lines was actually done earlier in (\hyperlink{MetsaevTseytlin88}{Metsaev-Tseytlin 88}), where the [[1-loop]] [[beta function]] for pure [[Yang-Mills theory]] from the [[partition function]] of a [[bosonic string|bosonic]] [[open string]] in a Yang-Mills [[background field]]. This provided a theoretical explanation for the observation, made earlier in (\hyperlink{Nepomechie83}{Nepomechie 83}) that when computed in via [[dimensional regularization]] then this [[beta function]] coefficient of [[Yang-Mills theory]] vanishes in [[spacetime]] [[dimension]] 26. This of course is the critical dimension of the [[bosonic string]].

\begin{itemize}%
\item R.I. Nepomechie, Phys. Lett. B128 (1983) 177


\item \{MetsaevTseytlin88\} [[Ruslan Metsaev]], [[Arkady Tseytlin]], \emph{On loop corrections to string theory effective actions}, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (\href{https://doi.org/10.1016/0550-3213(88}{doi:10.1016/0550-3213(88)90306-9}90306-9))



\end{itemize}
\hypertarget{qft_duality_and_specifically_montonenolive_electricmagnetic_duality}{}\paragraph*{{QFT Duality and specifically Montonen-Olive electric/magnetic duality}}\label{qft_duality_and_specifically_montonenolive_electricmagnetic_duality}

By embedding quantum field theories in string theory (typically as the [[worldvolume]] theories of various [[branes]]) the various [[duality in string theory|dualities of string theory]] will relate different QFTs in ways that are typically far from obvious from just looking at these QFTs themselves.

The investigation specifically of [[N=1 D=4 super Yang-Mills theory]] and [[N=2 D=4 super Yang-Mills theory]] in this fashion has come to be known as \emph{[[geometric engineering of quantum field theory]]}.

[[Montonen-Olive duality]] of ([[super Yang-Mills theory|super]]) [[Yang-Mills theory]] derives from conformal invariance of the [[6d (2,0)-supersymmetric QFT]] (see there) [[Kaluza-Klein mechanism|compactified]] on a [[torus]].

[[!include gauge theory from AdS-CFT -- table]]

See also

\begin{itemize}%
\item [[Matthew Strassler]], \emph{Millenial Messages for QCD from the Superworld and from the String} (\href{http://arxiv.org/abs/hep-th/0309140}{arXiv:hep-th/0309140})

\end{itemize}
\hypertarget{application_to_qcd_and_experimental_particle_physics}{}\paragraph*{{Application to QCD and experimental particle physics}}\label{application_to_qcd_and_experimental_particle_physics}

The realization of [[Yang-Mills theory]] that describes [[quarks]] and their interaction by the [[strong nuclear force]] carried by [[gluons]] is \emph{[[quantum chromodynamics]]} ([[QCD]]).

\begin{enumerate}%
\item \hyperlink{QCDScatteringAmplitudes}{QCD Scattering amplitudes}


\item \hyperlink{QuarkGluonPlasma}{Quark-gluon plasma}



\end{enumerate}
\hypertarget{QCDScatteringAmplitudes}{}\paragraph*{{Scattering amplitudes}}\label{QCDScatteringAmplitudes}

The [[string scattering amplitudes]] exhibit certain relations due to the extended nature of the string which are retained in the point particle limit and hence explain and serve to discover subtle relations in [[QFT]] [[scattering amplitudes]].

\begin{itemize}%
\item [[twistor string theory]] explains some ([[N=4 D=4 super Yang-Mills theory|super]]) [[Yang-Mills theory]] [[scattering amplitudes]]


\item Precision scattering amplitudes in [[QCD]] use [[twistor string theory]] [[on-shell recursion]] methods,



\end{itemize}
This also goes by the term ``on-shell methods''. See also at \emph{[[amplituhedron]]}.

Reviews include

\begin{itemize}%
\item [[Matthew Strassler]], \emph{From string theory to the large hadron collider} (\href{http://profmattstrassler.com/2012/08/15/from-string-theory-to-the-large-hadron-collider/}{blog post})


\item [[Lance Dixon]], \emph{Calculating Amplitudes}, December 2013 (\href{http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/}{web})


\item Rutger Boels, \emph{On-shell recursion for string theory amplitudes on the disk and the sphere} (\href{http://www.itp.uni-hannover.de/nordic/slides/Boels.pdf}{pdf})



\end{itemize}
Original articles include

\begin{itemize}%
\item [[Zvi Bern]], [[Lance Dixon]], [[David Kosower]], \emph{On-Shell Methods in Perturbative QCD} (\href{http://arxiv.org/abs/0704.2798}{arXiv:0704.2798})


\item [[Joseph Polchinski]], [[Matthew Strassler]], \emph{Hard Scattering and Gauge/String Duality}, Phys.Rev.Lett.88:031601,2002, (\href{http://lanl.arxiv.org/abs/hep-th/0109174}{arXiv:hep-th/0109174})



\end{itemize}
See also

\begin{itemize}%
\item Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi, \emph{Elliptic polylogarithms and Feynman parameter integrals} (\href{https://arxiv.org/abs/1902.09971}{arXiv:1902.09971})

reviewed in

Lorenzo Tancredi, \emph{Feynman integrals and higher genus surfaces}, talk at \emph{\href{https://indico.cern.ch/event/750565/}{Amplitudes 2019}} (\href{https://indico.cern.ch/event/750565/contributions/3438642/attachments/1871838/3080501/Tancredi.pdf}{pdf})



\end{itemize}
See also below \emph{\hyperlink{GravityScattetingAmplitudes}{Application to gravity -- Scattering amplitudes}}.

\hypertarget{QuarkGluonPlasma}{}\paragraph*{{Quark-gluon plasma}}\label{QuarkGluonPlasma}

Properties of [[quark-gluon plasma]] from [[AdS/CFT|AdS/CFT-dual]] [[type II string theory]]

\begin{itemize}%
\item Pavel Kovtun, \emph{Quark-Gluon Plasma and String Theory}, RHIC news (2009) (\href{http://www.bnl.gov/rhic/news/091107/story2.asp}{blog entry})


\item Makoto Natsuume, \emph{String theory and quark-gluon plasma} (\href{http://arxiv.org/abs/hep-ph/0701201}{arXiv:hep-ph/0701201})


\item [[Steven Gubser]], \emph{Using string theory to study the quark-gluon plasma: progress and perils} (\href{http://arxiv.org/abs/0907.4808}{arXiv:0907.4808})



\end{itemize}
\hypertarget{to_the_confinement_problem}{}\paragraph*{{To the confinement problem}}\label{to_the_confinement_problem}

Discussion of [[confinement]] in the context of the [[AdS-CFT correspondence]] is in

\begin{itemize}%
\item [[David Berman]], Maulik K. Parikh, \emph{Confinement and the AdS/CFT Correspondence}, Phys.Lett. B483 (2000) 271-276 (\href{http://es.arxiv.org/abs/hep-th/0002031}{arXiv:hep-th/0002031})


\item Henrique Boschi Filho, \emph{AdS/QCD and confinement}, Seminar at the \emph{Workshop on Strongly Coupled QCD: The confinement problem}, November 2011 (\href{http://www.if.ufrj.br/~boschi/pesquisa/seminarios/AdS_QCD_Confinement_UERJ_2011.pdf}{pdf})



\end{itemize}
\hypertarget{to_super_yangmills_theory}{}\paragraph*{{To super Yang-Mills theory}}\label{to_super_yangmills_theory}

[[Seiberg duality]] in [[super Yang-Mills theory]] is conceptually explained by [[type II string theory]] on certain [[D-brane]] configurations (\ldots{})

\hypertarget{ApplicationsToGravity}{}\subsubsection*{{Applications to gravity}}\label{ApplicationsToGravity}

\hypertarget{GravityScattetingAmplitudes}{}\paragraph*{{Scattering amplitudes}}\label{GravityScattetingAmplitudes}

\begin{itemize}%
\item [[open/closed string duality]] in [[string scattering amplitudes]] allows to compute [[gravity]] [[scattering amplitudes]] in terms of [[Yang-Mills theory]] scattering amplitudes: the \emph{[[KLT relations]]}


\item [[Zvi Bern]], \emph{Perturbative Quantum Gravity and its Relation to Gauge Theory}, Living Rev Relativ. 2002; 5(1): 5. (\href{https://arxiv.org/abs/gr-qc/0206071}{arXiv:gr-qc/0206071}, \href{https://dx.doi.org/10.12942%2Flrr-2002-5}{doi:10.12942/lrr-2002-5})


\item more on this string-organizatioon of [[graviton]] [[scattering]] amplitudes is in

David C. Dunbar, Paul S. Norridge, \emph{Calculation of Graviton Scattering Amplitudes using String-Based Methods}, Nucl.Phys. B433 (1995) 181-208 (\href{http://arxiv.org/abs/hep-th/9408014}{arXiv:hep-th/9408014})


\item [[KLT relations]] used for instance to demonstrate:

\begin{itemize}%
\item multi-loop finiteness of [[N=8 d=4 supergravity]]

\end{itemize}


\end{itemize}
\hypertarget{GravitationalWaveSignatures}{}\paragraph*{{Gravitational wave signatures}}\label{GravitationalWaveSignatures}

Application of the [[KLT relation]]/[[double copy]]-technique to computation of [[gravitational wave]]-signature of [[relativistic binary]] mergers:

\begin{itemize}%
\item [[Zvi Bern]], Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng, \emph{Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order}, Phys. Rev. Lett. 122, 201603 (2019) (\href{https://arxiv.org/abs/1901.04424}{arXiv:1901.04424})

\end{itemize}
\hypertarget{black_hole_entropy}{}\paragraph*{{Black hole entropy}}\label{black_hole_entropy}

Semi-classical QFT computations suggest that there should be [[entropy]] associated with [[black holes]], the \emph{[[Bekenstein-Hawking entropy]]}, without however providing microscopic degrees of freedom of which this would be an entropy in the ordinary sense.

Since the quantum dynamics of general black holes is outside the reach of perturbative methods in string theory, certain [[BPS state|supersymmetric black hole]] solutions in [[supergravity]] have properties independent of the coupling and are known to be the strong-coupling limit of what at weak coupling is a certain configuration of [[branes]] in flat space. Therefore the ordinary entropy of these brane configurations should match the [[Bekenstein-Hawking entropy]] of the corresponding black holes, and this has been confirmed to good precision.

While this argument does not give direct information about the origin of the BH-entropy of physically observed black holes, it does show conceptually, in the general context of black holes in theories of gravity, BH-entropy can be accounted for by microscopic degrees of freedom in a theory of [[quantum gravity]].

Reviews include

chapter 5 of

\begin{itemize}%
\item [[Ofer Aharony]], S.S. Gubser, [[Juan Maldacena]], H. Ooguri, Y. Oz, \emph{Large N Field Theories, String Theory and Gravity} (\href{http://arxiv.org/abs/hep-th/9905111}{arXiv:hep-th/9905111})

\end{itemize}
\hypertarget{application_to_pure_mathematics}{}\subsubsection*{{Application to pure mathematics}}\label{application_to_pure_mathematics}

Reviews include

\begin{itemize}%
\item [[Gregory Moore]], \emph{[[The Impact of D-Branes on Mathematics]]} (2014), \emph{[[Physical Mathematics and the Future]]} (2014)


\item [[Mina Aganagic]], \emph{String Theory and Math: Why This Marriage May Last}, (\href{https://arxiv.org/abs/1508.06642}{arXiv:1508.06642})



\end{itemize}
See also \href{string+theory#FieldMedalWork}{Fields medal (and other) work related to string theory}

\begin{itemize}%
\item [[twisted K-theory]]


\item [[monstrous moonshine]]


\item The [[string orientation of tmf]] was directly motivated from the index of the heterotic string (``[[Witten genus]]'').


\item The relation between [[Bruhat-Tits buildings]] and zeros of the [[Riemann zeta function]], as described in \href{https://ncatlab.org/nlab/show/p-adic+string+theory#HuangStoicaYau19}{Huang-Stoica-Yau}.


\item [[mirror symmetry]] took complex geometers by complete surprise, but is easily seen from the [[2d (2,0)-superconformal QFT|2d (2,0)-sigma model]] with [[target space]] the respective CY manifolds.


\item [[Morse theory]] of loop spaces via loop space [[supersymmetric quantum mechanics]] describing superstring propagation.



\end{itemize}
(\ldots{})



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