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\begin{document}

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\section*{string theory}

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

[[!include gravity contents]]

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

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

\hypertarget{higher_spin_geometry}{}\paragraph*{{Higher spin geometry}}\label{higher_spin_geometry}

[[!include higher spin geometry - contents]]

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{conceptually_from_qft_worldline_formalism}{Conceptually: From QFT worldline formalism}\dotfill \pageref*{conceptually_from_qft_worldline_formalism} \linebreak
\noindent\hyperlink{IdeaPhenomenology}{Phenomenologically}\dotfill \pageref*{IdeaPhenomenology} \linebreak
\noindent\hyperlink{Scales}{Scales and (no) parameters}\dotfill \pageref*{Scales} \linebreak
\noindent\hyperlink{CriticalStringsAndQuantumAnomalies}{Critical string theories and quantum anomalies}\dotfill \pageref*{CriticalStringsAndQuantumAnomalies} \linebreak
\noindent\hyperlink{subtopics}{Subtopics}\dotfill \pageref*{subtopics} \linebreak
\noindent\hyperlink{ingredients}{Ingredients}\dotfill \pageref*{ingredients} \linebreak
\noindent\hyperlink{critical_string_models}{Critical string models}\dotfill \pageref*{critical_string_models} \linebreak
\noindent\hyperlink{extended_objects}{Extended objects}\dotfill \pageref*{extended_objects} \linebreak
\noindent\hyperlink{scattering_amplitudes}{Scattering amplitudes}\dotfill \pageref*{scattering_amplitudes} \linebreak
\noindent\hyperlink{EllipticGenera}{Elliptic genera, elliptic cohomology and $tmf$}\dotfill \pageref*{EllipticGenera} \linebreak
\noindent\hyperlink{topological_strings}{Topological strings}\dotfill \pageref*{topological_strings} \linebreak
\noindent\hyperlink{string_phenomenology}{String phenomenology}\dotfill \pageref*{string_phenomenology} \linebreak
\noindent\hyperlink{string_theory_results_applied_elsewhere}{String theory results applied elsewhere}\dotfill \pageref*{string_theory_results_applied_elsewhere} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{higher_structures}{Higher structures}\dotfill \pageref*{higher_structures} \linebreak
\noindent\hyperlink{more_technical_details}{More technical details}\dotfill \pageref*{more_technical_details} \linebreak
\noindent\hyperlink{gut}{GUT}\dotfill \pageref*{gut} \linebreak
\noindent\hyperlink{elliptic_genera_elliptic_homology_and_tmf}{Elliptic genera, elliptic homology and tmf}\dotfill \pageref*{elliptic_genera_elliptic_homology_and_tmf} \linebreak
\noindent\hyperlink{quantum_anomalies}{Quantum anomalies}\dotfill \pageref*{quantum_anomalies} \linebreak
\noindent\hyperlink{FieldMedalWork}{Fields medal (and other) work related to string theory}\dotfill \pageref*{FieldMedalWork} \linebreak
\noindent\hyperlink{ReferencesHistory}{History}\dotfill \pageref*{ReferencesHistory} \linebreak


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

String theory is a [[theory (physics)|theory]] in fundamental [[physics]]. Certainly it is, mathematically, a structure that contains in various limits a plethora of [[nLab:quantum field theories]]. Its interest for experimental high energy physics lies in the hypothesis that it provides a [[theory of everything]] in the sense of fundamental physics, but the jury on that is still out.

Below we indicate the basic idea and provide pointers to further details. See also the \emph{[[string theory FAQ]]}.

\hypertarget{conceptually_from_qft_worldline_formalism}{}\subsubsection*{{Conceptually: From QFT worldline formalism}}\label{conceptually_from_qft_worldline_formalism}

\emph{Perturbative string theory} is something at least close to a [[vertical categorification|categorification]] of the following description of \emph{[[perturbation theory|perturbative]] [[quantum field theory]]} in terms of sums over [[Feynman diagrams]].

Recall that in [[quantum field theory]] one approach to make sense of the [[path integral]] is the \emph{[[perturbation series]] expansion}, which interprets the path integral for the [[scattering amplitudes]] (the \emph{[[S-matrix]]}) as a certain sum over [[graphs]] of certain numbers assigned to each [[graph]].

The graphs are called \emph{[[Feynman diagrams]]}, the numbers assigned to them are called \emph{([[renormalization|renormalized]]) [[scattering amplitudes]]} and the sum over graphs of (renormalized) amplitudes is the \emph{[[perturbation series]]} or [[S-matrix]].

The amplitude assigned to a single graph with $n$ external edges is interpreted as the amplitude for $n$ ``quanta'' or ``[[particles]]'' of the [[field (physics)|fields]] in question to interact in the way indicated by the graph.

Crucial for the motivation of the idea of string theory is the observation that this (renormalized) amplitude assigned to a graph is itself the \emph{correlator} of a 1-dimensional quantum field theory on that graph: the ``worldline quantum field theory'' describing the (relativistic) quantum mechanics of these particles. This is usually a [[sigma-model]] with parameter space the given graph and target space the spacetime on which the fields live for which the perturbation series computes the path integral.

When made explicit this is called the \emph{[[worldline formalism]]} for computing the quantum field perturbation series. (See there for more details.)

The premise of perturbative string theory is to replace the [[perturbation series]] over correlators of a 1-d QFT over graphs by a sum of correlators of a 2-dQFT over 2-dimensional surfaces -- called \emph{[[worldsheets]]} and hence produce an [[S-matrix]] this way. Again in simple cases this 2d QFT is a [[sigma-model]] whose target is the spacetime in which one computes interactions.



\begin{quote}%
graphics grabbed from \href{https://benjaminjurke.com/content/articles/2010/string-theory/}{Jurke 10}


\end{quote}
In analogy to the previous case, one thinks of the amplitude assigned this way to a surface as the amplitude for the boundary arcs -- the \emph{string}s -- to interact in the way given by the surface.

Some of the motivations for considering this replacement of graphs by surfaces have been the following:

\begin{itemize}%
\item the 2-d correlators are better behaved in that they don't have to be renormalized. The ``counterterms'' appearing in renormalization of ordinary QFT can be identified with contributions to the correlators that come from the linear extension of the strings (see the above reference for more on this);


\item there are fewer choices involved: a Feynman graph is really a decorated graph with the decoration from some more or less arbitrary index set, describing the nature of the particles associated with a given edge and the nature of the interaction associated with a given vertex. In the sum over surfaces there is no extra decoration (except on the boundary of the surfaces) and one finds that instead a single string diagram (a 2d QFT correlator for a given surface) encodes already a sum over (infinitely) many particle species decorations and all possible interaction decorations for them;


\item while there are fewer choices to be made by hand, it turns out that the effective particle content that does arise automatically from this prescription happens to be structurally of the kind one would hope for: the massless effective particles described by the string perturbation series happen to be \emph{gauge bosons}, \emph{fermions} charged under them, and, notably, \emph{graviton}s. This is structurally exactly the [[Yang-Mills theory]] input of the [[standard model of particle physics]] combined with perturbative [[quantum gravity]] that one would hope to see.



\end{itemize}
These aspects have motivated the impression that the string perturbation series might be considerably closer to the true formalism of fundamental physics than ordinary perturbative quantum field theory. This impression is however offset by the following problems:

\begin{itemize}%
\item while the worldsheet 2d QFT whose correlators are summed over surfaces are themselves much easier to handle than the full target space quantum physics they are used to encode, a fully complete and rigorous theory of 2d QFT is available only in simple special cases.


\item In particular, even though there are fewer arbitrary choices involved in the string perturbation series as compared to the ordinary Feynman perturbation series, one crucial choice still present is that of this worldsheet 2d QFT. By the above, every choice of worldsheet QFT (called a choice of ``vacuum'') corresponds to a choice of effective target space geometry (to be thought of as the one that the perturbation series computes the quantum perturbations about) and particle content (see [[2-spectral triple]] for more on that). One would therefore like to understand the space of all worldsheet QFTs whose effective target space geometry and particle content is close to the one experimentally observed. After many years of rather na\"i{}ve approaches to handle or not to handle this, it has more recently at least come to the general attention that there is something to be better understood here.


\item More fundamentally, already the role of the original perturbation series in quantum field theory is actually not fully understood. Its main success is the observation that truncating or resumming the perturbation series in a more or less ad hoc way, it does yield values that very well describe a plethora of real world measurements. One imagines that there is a \emph{non-perturbative} definition of [[quantum field theory]] such that in certain well-defined circumstances the perturbation series does yield an approximation to it and is \emph{a posteriori} justified. If so, there should be an analogous nonperturbative definition of string theory. There is a large ratio of speculations as to what that might be over solid results about it.



\end{itemize}
\hypertarget{IdeaPhenomenology}{}\subsubsection*{{Phenomenologically}}\label{IdeaPhenomenology}

While therefore the premise of perturbative string theory is conceptually suggestive for various reasons, there is to date no connection to experimental [[phenomenology]] (apart from the fact that conceptual insights into string theory have helped analyze quantum field theoretic data, see at \emph{[[string theory results applied elsewhere]]}). As a result much of the substantial outcome of string theory research is more in [[mathematical physics]] (if done well, at least), exploring the general theory space of [[quantum field theories]] and their [[UV-completions]], than in realistic [[model (in theoretical physics)|model building]] (though there is no lack of trying), where it remains very speculative. This has led to public or semi-public debates about the value of string theory for actual physics. See at \emph{[[criticism of string theory]]} for pointers.

\hypertarget{Scales}{}\subsubsection*{{Scales and (no) parameters}}\label{Scales}

\begin{itemize}%
\item [[string tension]] $T = 1/(2\pi \alpha^\prime)$


\item [[string length scale]] $l_s = \sqrt{\alpha'}$


\item [[string coupling constant]] $g_s = e^\lambda$

(= radius of [[M-theory]] compactification circle (\href{http://ncatlab.org/nlab/show/M-theory#Witten95}{Witten 95}))


\item [[coupling constant]] of [[gravity]]: $\kappa \propto g_s (\alpha')^6 = g_s l_s^{12}$


\item [[Newton's constant]] (before compactification) $G_N = \kappa^2/8\pi$


\item [[Planck length]] $l_P = (8 \pi G_N)^{1/24} = \kappa^{1/12} \propto l_s g_s^{1/12}$


\item [[Planck mass]] $M_p = l_p^{-1}$



\end{itemize}
(see e.g. \href{http://arxiv.org/abs/0908.0333}{arXiv:0908.0333})

see also at \emph{[[non-perturbative effect]]} the section \emph{\href{non-perturbative+effect#WorldsheetAndBraneInstantons}{Worldsheet and brane instantons}}

\hypertarget{CriticalStringsAndQuantumAnomalies}{}\subsection*{{Critical string theories and quantum anomalies}}\label{CriticalStringsAndQuantumAnomalies}

The [[action functional]] for the [[string]]-[[sigma model]] in general has a [[quantum anomaly]] of both kinds:

\begin{enumerate}%
\item For both the bosonic string and the superstring the corresponding [[Polyakov action]] has a \href{}{gauge anomaly} for the [[conformal field theory|conformal symmetry]], depending on the [[dimension]] $d$ of [[target space]], and on the strength of the [[dilaton]] background field. For vanishing dilaton field this anomaly vanishes exactly for $d = 26$ for the bosonic model, and in $d = 10$ for the [[superstring]].

For target spaces of these dimensions one speaks of \textbf{critical string theory}. In as far as string theory is expected to have relevance for physics at all, it is usually expected to be in this critical dimension. But also noncritical string models can and have been considered.


\item Apart from the gauge anomaly, the [[action functional]] of the [[string]]-[[sigma-model]] also in general has an \emph{\href{http://ncatlab.org/nlab/show/quantum+anomaly#AnomalousActionFunctional}{anomalous action functional}} , for two reasons:

\begin{enumerate}%
\item The [[holonomy|higher holonomy]] of the higher [[background gauge field]]s is in general not a function, but a [[section]] of a [[line bundle]];


\item The fermionic [[path integral]] over the [[worldsheet]]-[[spinor]]s of the [[superstring]] produces as section of a [[Pfaffian line bundle]].



\end{enumerate}
In order for the action functional to be well-defined, the [[tensor product]] of these different anomaly [[line bundle]]s over the bosonic [[configuration space]] must have trivial class (as [[connection on a bundle|bundles with connection]], even). This gives rise to various further anomaly cancellation conditions:

\begin{enumerate}%
\item For the [[heterotic string]] (necessarily closed) the anomaly cancellation condition is known as the \emph{[[Green-Schwarz mechanism]]} : it says that the background fields of [[gravity]] and [[B-field]] must organize to a [[twisted differential string structure]] whose twist is given by the background [[Yang-Mills field]].


\item For the open [[type II string]] the condition is known as the [[Freed-Witten anomaly cancellation]] condition: it says that the restriction of the [[B-field]] to any [[D-brane]] must consistute the twist of a [[twisted spin{\tt \symbol{94}}c structure]] on the brane.

A more detailed analysis of these type II anomalies is in (\hyperlink{DFMI}{DFMI}) and (\hyperlink{DFMII}{DFMII}).

See also \emph{[[Diaconescu-Moore-Witten anomaly]]}.



\end{enumerate}


\end{enumerate}
\hypertarget{subtopics}{}\subsection*{{Subtopics}}\label{subtopics}

\hypertarget{ingredients}{}\subsubsection*{{Ingredients}}\label{ingredients}

\begin{itemize}%
\item [[quantum field theory]]


\item [[sigma-model]],

\begin{itemize}%
\item [[2d CFT]], [[2d SCFT]], [[2-spectral triple]]

\begin{itemize}%
\item [[Dirac-Ramond operator]]


\item [[Witten genus]]


\item [[Gepner model]], [[flop transition]]



\end{itemize}

\item [[world sheets for world sheets]]



\end{itemize}

\item [[Goddard-Thorn no-ghost theorem]], [[GSO projection]]


\item [[perturbation theory]]

\begin{itemize}%
\item [[string scattering amplitude]], [[S-matrix]]

\end{itemize}

\item [[effective QFT|effective background QFT]]

\begin{itemize}%
\item [[gravity]], [[supergravity]], [[Yang-Mills theory]], [[quantum gravity]]


\item [[quantum anomaly]]


\item [[twisted smooth cohomology in string theory]]



\end{itemize}


\end{itemize}
\hypertarget{critical_string_models}{}\subsubsection*{{Critical string models}}\label{critical_string_models}

\begin{itemize}%
\item [[heterotic string theory]]

\begin{itemize}%
\item [[Green-Schwarz mechanism]], [[differential string structure]],


\item [[dual heterotic string theory]]

\begin{itemize}%
\item [[differential fivebrane structure]]

\end{itemize}

\item [[Witten genus]], [[string orientation of tmf]]



\end{itemize}

\item [[type II string theory]]

\begin{itemize}%
\item [[type II geometry]]


\item [[elliptic genus]]


\item [[Diaconescu-Moore-Witten anomaly]]



\end{itemize}

\item [[type I string theory]]


\item [[type 0 string theory]]


\item [[little string theory]]


\item [[Kaluza-Klein mechanism]]

\begin{itemize}%
\item [[double dimensional reduction]]


\item [[moduli stabilization]]


\item [[flux compactification]]


\item [[landscape of string theory vacua]]



\end{itemize}

\item [[string field theory]]


\item [[duality in string theory]]

\begin{itemize}%
\item [[T-duality]], [[mirror symmetry]]


\item [[S-duality]], [[electric-magnetic duality]]


\item [[U-duality]]


\item [[open/closed string duality]]

\begin{itemize}%
\item [[AdS/CFT]], [[holographic principle]]


\item [[KLT relations]]



\end{itemize}


\end{itemize}

\item [[11-dimensional supergravity]], [[M-theory]]

\begin{itemize}%
\item [[Hořava-Witten theory]]

\end{itemize}


\end{itemize}
\hypertarget{extended_objects}{}\subsubsection*{{Extended objects}}\label{extended_objects}

\begin{itemize}%
\item [[brane]]


\item \textbf{[[D-brane]]}

\begin{itemize}%
\item [[D0-brane]], [[D2-brane]], [[D4-brane]]


\item [[D1-brane]], [[D3-brane]], [[D5-brane]]


\item [[RR-field]], [[differential K-theory]]



\end{itemize}

\item [[black brane]]

\begin{itemize}%
\item [[black holes in string theory]]

\end{itemize}

\item \textbf{[[NS-brane]]}

\begin{itemize}%
\item [[string]], [[sigma-model]]

\begin{itemize}%
\item [[spinning string]], [[superstring]]


\item [[B2-field]]



\end{itemize}

\item [[NS5-brane]]

\begin{itemize}%
\item [[B6-field]]

\end{itemize}


\end{itemize}

\item \textbf{[[M-brane]]}

\begin{itemize}%
\item [[M2-brane]]

\begin{itemize}%
\item [[C3-field]]


\item [[ABJM theory]], [[BLG model]]



\end{itemize}

\item [[M5-brane]]

\begin{itemize}%
\item [[C6-field]]


\item [[6d (2,0)-supersymmetric QFT]]



\end{itemize}


\end{itemize}


\end{itemize}
[[!include table of branes]]

\hypertarget{scattering_amplitudes}{}\subsubsection*{{Scattering amplitudes}}\label{scattering_amplitudes}

\begin{itemize}%
\item [[S-matrix]]


\item [[Veneziano amplitude]]


\item [[string scattering amplitude]]


\item [[p-adic string theory]]



\end{itemize}
\hypertarget{EllipticGenera}{}\subsubsection*{{Elliptic genera, elliptic cohomology and $tmf$}}\label{EllipticGenera}

\begin{quote}%
A properly developed theory of [[elliptic cohomology]] is likely to shed some light on what string theory really means. (\hyperlink{Witten87}{Witten 87, very last sentence})


\end{quote}
The [[large volume limit]] of the [[partition function]] of the [[superstring]] on a given [[target space|target]] [[spacetime]] is an [[elliptic genus]] of that manifold (\hyperlink{Witten87}{Witten 87}), the \emph{[[Witten genus]]} (see there for more).

Since the [[Witten genus]] in turn is the [[decategorification]] of the [[string orientation of tmf]], this suggests that [[tmf]]-[[generalized (Eilenberg-Steenrod) cohomology]] classifies full string theories, in refinement of how the classification of [[D-brane charge]] (just the [[boundary conditions]] for [[open strings]]) is given by [[K-theory]].

A non-trivial conistency check of this idea is announced in (\hyperlink{Nikolaus14}{Nikolaus 14}).

\hypertarget{topological_strings}{}\subsubsection*{{Topological strings}}\label{topological_strings}

\begin{itemize}%
\item [[A-model]], [[B-model]]

\end{itemize}
\hypertarget{string_phenomenology}{}\subsubsection*{{String phenomenology}}\label{string_phenomenology}

\begin{itemize}%
\item [[string phenomenology]]

\begin{itemize}%
\item [[G2-MSSM]]

\end{itemize}


\end{itemize}
\hypertarget{string_theory_results_applied_elsewhere}{}\subsubsection*{{String theory results applied elsewhere}}\label{string_theory_results_applied_elsewhere}

Beyond the speculative hypothetized role of string theory as a theory behind observed [[particle physics]], the theory has shed light on many aspects of [[quantum field theory]], both on the conceptual structure of quantum field theory as such as well as on concrete theories and their concrete properties. Some of these string theory results enter crucially in computations that are used to interpret particle physics experiments such as the [[LHC]].

For more see

\begin{itemize}%
\item [[string theory results applied elsewhere]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Textbooks on [[string theory]] and [[M-theory]] include the following (for more see at \emph{[[books about string theory]]}):

\begin{itemize}%
\item [[Mike Duff]]

\emph{[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and $M$-theory},

IoP 1999 (\href{https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720}{publisher})


\item [[Joseph Polchinski]], \emph{[[String theory]]}, Cambridge Monographs on Mathematical Physics (2001)


\item [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds.

\emph{[[Quantum Fields and Strings]], A course for mathematicians}, 2 vols. Amer. Math. Soc. Providence 1999. (\href{http://www.math.ias.edu/qft}{web version})


\item [[Michael Douglas]], [[Elias Kiritsis]] et. al. (eds.),

\emph{[[String theory and the real world]]},

Les Houches Session LXXXVII 2007


\item [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} ,Proceedings of Symposia in Pure Mathematics, \href{http://www.ams.org/bookstore?fn=20&arg1=pspumseries&ikey=PSPUM-83}{volume 83} AMS (2011)


\item [[Luis Ibáñez]], [[Angel Uranga]],

\emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]},

Cambridge University Press 2012


\item [[Peter West]],

\emph{[[Introduction to Strings and Branes]]},

Cambridge University Press 2012


\item [[Gregory Moore]],

\emph{[[The Impact of D-Branes on Mathematics]]},

talk at \href{http://online.kitp.ucsb.edu/online/joefest-c14/}{PolchinskiFest 2014} (\href{http://www.physics.rutgers.edu/~gmoore/JOEFEST-THOUGHTS.pdf}{pdf})


\item [[Gregory Moore]],

\emph{[[Physical Mathematics and the Future]]}

talk at \href{http://physics.princeton.edu/strings2014/}{Strings 2014} (\href{http://physics.princeton.edu/strings2014/slides/Moore.pdf}{talk slides}, \href{http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf}{companion text pdf}, [[MooreVisionTalk2014.pdf:file]])


\item [[Gordon Kane]],

\emph{String theory and the real world},

Morgan \& Claypool, 2017 ()

(on [[M-theory on G2-manifolds]] and the [[G2-MSSM]])



\end{itemize}
A large body of references is organized at the

\begin{itemize}%
\item \href{http://www.stringwiki.org/w/index.php?title=String_Theory_Wiki}{String Theory Wiki}

\end{itemize}
A quick survey of the big picture as of 2016 is in

\begin{itemize}%
\item [[Ashoke Sen]], \emph{What is string theory?}, talk at YITP50, Stony Brook 2016 (\href{http://media.scgp.stonybrook.edu/presentations/2016/20161010_Sen.pdf}{pdf})

\end{itemize}
A useful survey of the status of string theory as a theory of [[quantum gravity]] is in

\begin{itemize}%
\item [[Matthias Blau]], \emph{String theory as a theory of quantum gravity -- A status report} Talk at \emph{\href{http://www.conferences.itp.phys.ethz.ch/doku.php?id=qg11:programme}{Quantum theory and Gravitation}} Z\"u{}rich (2011) (\href{http://www.conferences.itp.phys.ethz.ch/lib/exe/fetch.php?media=qg11:zurichblau.pdf}{pdf})

\end{itemize}
An article summarizing information about [[cohomology|cohomological]] models for aspects of string theory and listing plenty of useful further references is

\begin{itemize}%
\item [[Hisham Sati]], \emph{[[Geometric and topological structures related to M-branes]]}

\end{itemize}
\hypertarget{higher_structures}{}\subsubsection*{{Higher structures}}\label{higher_structures}

Discussion from the [[nPOV]]/[[schreiber:Higher Structures]]:

\begin{itemize}%
\item [[Branislav Jurco]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]],

\emph{Higher Structures in M-Theory},

Introduction to \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory, LMS-EPSRC Durham Symposium, August 2018}}, Fortsch. d. Phys. 2019 (\href{https://arxiv.org/abs/1903.02807}{arXiv:1903.02807})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]],

\emph{[[schreiber:The rational higher structure of M-theory]]}

in \emph{Proceedings of the \href{http://www.maths.dur.ac.uk/lms/}{LMS-EPSRC Durham Symposium}}\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory}, August 2018, Fortschritte der Physik, 2019 (\href{https://arxiv.org/abs/1903.02834}{arXiv:1903.02834})


\item [[Urs Schreiber]],

\emph{[[schreiber:Introduction to Higher Supergeometry]]}

Lectures at \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory, LMS-EPSRC Durham Symposium, August 2018}}



\end{itemize}
\hypertarget{more_technical_details}{}\subsubsection*{{More technical details}}\label{more_technical_details}

\hypertarget{gut}{}\paragraph*{{GUT}}\label{gut}

A key article for the idea that string theory provides a framework for realistic (chiral) [[grand unified theories]] combined with [[quantum gravity]] was

\begin{itemize}%
\item [[Edward Witten]], \emph{Some properties of $O(32)$ superstrings}, Phys. Lett. B, volume 149, December 1984 ([[Witten84.pdf:file]])

\end{itemize}
based on the insights of [[Green-Schwarz anomaly cancellation]].

\hypertarget{elliptic_genera_elliptic_homology_and_tmf}{}\paragraph*{{Elliptic genera, elliptic homology and tmf}}\label{elliptic_genera_elliptic_homology_and_tmf}

The [[partition function]] of the [[superstring]] was understood to be an [[elliptic genus]] (the [[Witten genus]]) in

\begin{itemize}%
\item [[Edward Witten]], \emph{Elliptic Genera And Quantum Field Theory} , Commun.Math.Phys. 109 525 (1987) (\href{http://projecteuclid.org/euclid.cmp/1104117076}{Euclid})

\end{itemize}
A nontrivial consistency check on the suggestion that this means that string backgrounds are classified in [[tmf]] is given in

\begin{itemize}%
\item [[Thomas Nikolaus]], \emph{[[T-Duality in K-theory and Elliptic Cohomology]]}, talk at \emph{String Geometry Network Meeting}, Feb 2014, ESI Vienna (\href{http://www.ingvet.kau.se/juerfuch/conf/esi14/esi14_34.html}{website})

\end{itemize}
\hypertarget{quantum_anomalies}{}\paragraph*{{Quantum anomalies}}\label{quantum_anomalies}

Discussion of type II [[quantum anomalies]] is in

\begin{itemize}%
\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides})


\item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Spin structures and superstrings} (\href{http://arxiv.org/abs/1007.4581}{arXiv:1007.4581})



\end{itemize}
Discussion of [[superstring]] [[perturbation theory]] is in

\begin{itemize}%
\item [[Edward Witten]], \emph{Superstring Perturbation Theory Revisited} (\href{http://arxiv.org/abs/1209.5461}{arXiv:1209.5461})

\end{itemize}
\hypertarget{FieldMedalWork}{}\subsubsection*{{Fields medal (and other) work related to string theory}}\label{FieldMedalWork}

Pure [[mathematics]] work which is closely related string theory and was awarded with a [[Fields medal]] includes the following.

[[Richard Borcherds]], 1998

\begin{itemize}%
\item [[vertex algebras]]

\begin{itemize}%
\item [[monster vertex algebra]], [[Goddard-Thorn theorem]]

\end{itemize}


\end{itemize}
[[Maxim Kontsevich]], 1998

\begin{itemize}%
\item [[homological mirror symmetry]],


\item [[Gromov-Witten theory]]


\item [[Kontsevich formality|formality theorem]] and [[formal deformation quantization]] via [[holographic principle|holography]] of [[Poisson sigma-model]] string.



\end{itemize}
[[Edward Witten]], 1990

\begin{itemize}%
\item [[knot invariants]] via [[WZW model]]-string/[[Chern-Simons theory]] [[AdS3-CFT2 and CS-WZW correspondence|holography]];


\item [[elliptic genus]], [[Witten genus]] and rigidity via [[superstring]] [[partition functions]];


\item [[mirror symmetry]] via [[A-model]]/[[B-model]] [[topological string]])



\end{itemize}
[[Grigori Perelman]], 2006

\begin{itemize}%
\item [[Ricci flow]] for string [[sigma-model]] in [[dilaton gravity]] background

\end{itemize}
[[Maryam Mirzakhani]], 2014

\begin{itemize}%
\item The dynamics and geometry of Riemann surfaces and their moduli spaces, including a new proof of the [[Witten conjecture]]

\end{itemize}
In (\href{Mumford+conjecture#Madsen07}{Madsen 07}) it says with respect to the [[proof]] (\href{Mumford+conjecture#MadsenWeiss02}{Madsen-Weiss 02}) of the [[Mumford conjecture]]:

\begin{quote}%
These tools $[$ used in the proof $]$ are all rather old, known for at least twenty years, and one may wonder why they have not before been put to use in connection with the Riemann moduli space. Maybe we lacked the inspiration that comes from the renewed interaction with physics, exemplified in conformal field theories.


\end{quote}
\hypertarget{ReferencesHistory}{}\subsubsection*{{History}}\label{ReferencesHistory}

\begin{itemize}%
\item [[Murray Gell-Mann]], introductory talk at \emph{\href{https://en.wikipedia.org/wiki/Shelter_Island_Conference}{Shelter Island II}}, 1983 ([[Gell-Mann\_ShelterIslandII\_1983.pdf:file]])

in: \emph{Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics}. MIT Press. pp. 301--343. ISBN 0-262-10031-2.


\item [[John Schwarz]], \emph{The Early Years of String Theory: A Personal Perspective} (\href{http://arxiv.org/abs/0708.1917}{arXiv:0708.1917}), published as \emph{Gravity, unification, and the superstring}, in Filippo Colomo, [[Paolo Di Vecchia]] (eds.) \emph{The birth of string theory}, Cambridge University Press, (2011) (\href{https://doi.org/10.1017/CBO9780511977725.005}{doi:10.1017/CBO9780511977725.005})

(on the early history of [[string theory]] up to the ``first superstring revolution'', the construction of the [[Green-Schwarz anomaly cancellation|Green-Schwarz-anomaly]]-free $SO(32)$ [[type I string theory]] and [[heterotic string]])


\item [[John Schwarz]], \emph{The Second Superstring Revolution}, Colloquium-level lecture presented at the Sakharov Conference (Moscow, May 1996) (\href{https://arxiv.org/abs/hep-th/9607067}{arXiv:hep-th/9607067})

(on the ``second superstring revolution'': the realization of [[D-branes]], [[dualities in string theory]] and [[M-theory]])


\item [[John Schwarz]], \emph{String Theory in the Twentieth Century}, talk at \href{http://ymsc.tsinghua.edu.cn:8090/strings/}{Strings 2016} (\href{https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2016/main/Schwarz.pdf}{pdf}, [[SchwarzString2016.pdf:file]])



\end{itemize}
[[!redirects string theories]]

[[!redirects superstring theory]] [[!redirects superstring theories]]

[[!redirects perturbative string theory]]



\end{document}