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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{F-theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{From11dSupergravity}{Relation to M-theory}\dotfill \pageref*{From11dSupergravity} \linebreak \noindent\hyperlink{relation_to_mtheory_on_manifolds}{Relation to M-theory on $G_2$-manifolds}\dotfill \pageref*{relation_to_mtheory_on_manifolds} \linebreak \noindent\hyperlink{relation_to_heterotic_mtheory_on_adesingularities}{Relation to heterotic M-theory on ADE-singularities}\dotfill \pageref*{relation_to_heterotic_mtheory_on_adesingularities} \linebreak \noindent\hyperlink{RelationToOrientifolds}{Relation to orientifold type II backgrounds}\dotfill \pageref*{RelationToOrientifolds} \linebreak \noindent\hyperlink{relation_to_heterotic_string_theory}{Relation to heterotic string theory}\dotfill \pageref*{relation_to_heterotic_string_theory} \linebreak \noindent\hyperlink{SingularLocusAndD7Branes}{The singular locus of the elliptic fibration and the D7-branes}\dotfill \pageref*{SingularLocusAndD7Branes} \linebreak \noindent\hyperlink{Fbranescan}{F-brane scan}\dotfill \pageref*{Fbranescan} \linebreak \noindent\hyperlink{sduality_operation_on_branes}{S-duality operation on $(p,q)$-branes}\dotfill \pageref*{sduality_operation_on_branes} \linebreak \noindent\hyperlink{ModelBuildingAndPhenomenology}{Model building and phenomenology}\dotfill \pageref*{ModelBuildingAndPhenomenology} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{relation_to_orientifolds}{Relation to orientifolds}\dotfill \pageref*{relation_to_orientifolds} \linebreak \noindent\hyperlink{relation_to_elliptic_cohomology}{Relation to elliptic cohomology}\dotfill \pageref*{relation_to_elliptic_cohomology} \linebreak \noindent\hyperlink{relation_to_heterotic_string_string_theory}{Relation to heterotic string string theory}\dotfill \pageref*{relation_to_heterotic_string_string_theory} \linebreak \noindent\hyperlink{relation_to_mtheory_on_manifolds_2}{Relation to M-theory on $G_2$-manifolds}\dotfill \pageref*{relation_to_mtheory_on_manifolds_2} \linebreak \noindent\hyperlink{relation_to_the_6d_superconformal_theory}{Relation to the 6d superconformal theory}\dotfill \pageref*{relation_to_the_6d_superconformal_theory} \linebreak \noindent\hyperlink{ReferencesPhenomnenology}{Phenomenology and model building}\dotfill \pageref*{ReferencesPhenomnenology} \linebreak \noindent\hyperlink{cosmological_constant}{Cosmological constant}\dotfill \pageref*{cosmological_constant} \linebreak \noindent\hyperlink{FluxAndInstantons}{4-Form flux and instantons}\dotfill \pageref*{FluxAndInstantons} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} \emph{F-theory} is a toolbox for describing [[type IIB string theory]] backgrounds -- \emph{including} [[non-perturbative effects]] induced from the presence of [[D7-branes]] and [[(p,q)-strings]] -- in terms of [[complex numbers|complex]] [[elliptic fibrations]] whose fiber modulus $\tau$ encodes the [[axio-dilaton]] (the [[string coupling constant]] and the degree-0 [[RR-field]]) tranforming under the $SL(2, \mathbb{Z})$ [[S-duality]]/[[U-duality]] [[group]]. See also at \emph{[[duality in string theory]]}. More technically, F-theory is what results when [[KK-compactification|KK-compactifying]] [[M-theory]] on an [[elliptic fibration]] (which yields [[type IIA superstring theory]] compactified on a [[circle]]-[[fiber bundle]]) followed by [[T-duality]] with respect to one of the two cycles of the elliptic fiber. The result is (uncompactified) [[type IIB superstring theory]] with [[axio-dilaton]] given by the moduli of the original elliptic fibration, see \hyperlink{From11dSupergravity}{below}. Or rather, this is [[type IIB string theory]] with some [[non-perturbative effects]] included, reducing to [[perturbative string theory]] in the [[Sen limit]]. With a full description of [[M-theory]] available also F-theory should be a full non-perturbative description of type IIB string theory, but absent that it is some kind of approximation. For instance while the [[modular group|modular]] [[structure group]] of the [[elliptic fibration]] in principle encodes (necessarily non-perturbative) [[S-duality]] effects, it is presently not actually known in full detail how this affects the full theory, notably the proper charge quantization law of the 3-form fluxes, see at \emph{\href{S-duality#CohomologicalNatureOfTypeIIFieldsUnderSDuality}{S-duality -- Cohomological nature of the fields under S-duality}} for more on that. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{From11dSupergravity}{}\subsubsection*{{Relation to M-theory}}\label{From11dSupergravity} The [[duality between M-theory and F-theory]]: The following line of argument shows why first compactifying M-theory on a torus $S_1^A \times S_1^B$ to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape $\frac{R_A}{R_B}$ of the torus, but not on its other geometry. By the [[dualities in string theory]], 10-dimensional [[type II string theory]] is supposed to be obtained from the [[UV-completion]] of [[11-dimensional supergravity]] by first [[Kaluza-Klein mechanism|dimensionally reducing]] over a circle $S^1_A$ -- to obtain [[type IIA supergravity]] -- and then applying [[T-duality]] along another circle $S^1_B$ to obtain [[type IIB supergravity]]. To obtain type IIB sugra in noncompact 10 dimensions this way, also $S^1_B$ is to be compactified (since [[T-duality]] sends the radius $r_A$ of $S^1_A$ to the inverse radius $r_B = \ell_s^2 / R_A$ of $S^1_B$). Therefore type IIB sugra in $d = 10$ is obtained from 11d sugra compactified on the [[torus]] $S^1_A \times S^1_B$. More generally, this torus may be taken to be an [[elliptic curve]] and this may vary over the 9d base space as an [[elliptic fibration]]. Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity [[vacuum]] is the input data that F-theory is concerned with. A schematic depiction of this story is the following: \newline In the simple case where the elliptic fiber is indeed just $S^1_A \times S^1_B$, the [[imaginary part]] of its complex modulus is \begin{displaymath} Im(\tau) = \frac{R_A}{R_B} \,. \end{displaymath} By following through the above diagram, one finds how this determines the [[coupling constant]] in the [[type II string theory]]: First, the [[KK-compactification]] of [[M-theory]] on $S^1_A$ yields a type IIA [[string coupling]] \begin{displaymath} g_{IIA} = \frac{R_A}{\ell_s} \,. \end{displaymath} Then the T-duality operation along $S^1_B$ divides this by $R_B$: \begin{displaymath} \begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,. \end{displaymath} \hypertarget{relation_to_mtheory_on_manifolds}{}\paragraph*{{Relation to M-theory on $G_2$-manifolds}}\label{relation_to_mtheory_on_manifolds} In order to get minimal [[N=1 d=4 supergravity]] after [[KK-compactification]], one needs [[M-theory on G2-manifolds]] and [[F-theory on CY4-manifolds]]. Discussion of the relation between then [[G2-manifold]] fibers for [[M-theory on G2-manifolds]] and the corresponding [[Calabi-Yau manifold|Calabi-Yau 4-manifold]] fibers in F-theory includes (\hyperlink{GukovYauZaslow02}{Gukov-Yau-Zaslow 02}, \hyperlink{Belhaj02}{Belhaj 02}). \hypertarget{relation_to_heterotic_mtheory_on_adesingularities}{}\paragraph*{{Relation to heterotic M-theory on ADE-singularities}}\label{relation_to_heterotic_mtheory_on_adesingularities} Relation to [[heterotic M-theory on ADE-singularities]]: \begin{itemize}% \item Monika Marquart, [[Daniel Waldram]], \emph{F-theory duals of M-theory on $S^1/\mathbb{Z}_2 \times T^4 / \mathbb{Z}_N$} (\href{https://arxiv.org/abs/hep-th/0204228}{arXiv:hep-th/0204228}) \item Christoph Lüdeling, Fabian Ruehle, \emph{F-theory duals of singular heterotic K3 models}, Phys. Rev. D 91, 026010 (2015) (\href{https://arxiv.org/abs/1405.2928}{arXiv:1405.2928}) \end{itemize} \hypertarget{RelationToOrientifolds}{}\subsubsection*{{Relation to orientifold type II backgrounds}}\label{RelationToOrientifolds} The general [[vacuum]] of [[type II superstring theory]] (including [[type I superstring theory]]) is an [[orientifold]]. The [[target space]] data of an [[orientifold]] is a $\mathbb{Z}_2$-[[principal bundle]]/[[local system]], possibly singular (hence possibly on a [[smooth stack]]). On the other hand, the non-singular part of the [[elliptic fibration]] that defines the F-theory is a $SL_2(\mathbb{Z})$-[[local system]] (being the ``homological invariant'' of the [[elliptic fibration]]). An argument due to (\hyperlink{Sen96}{Sen 96}, \hyperlink{Sen97a}{Sen 97a}) says that the F-theory data does induce the [[orientifold]] data along the [[subgroup]] inclusion $\mathbb{Z}_2 \hookrightarrow SL_2(\mathbb{Z})$. See at \emph{[[Sen limit]]}. The degeneration locus of the elliptic fibration -- where the [[discriminant]] $\Delta$ vanishes and its fibers are the [[nodal curve]] -- is interpreted as that of [[D7-branes]] and [[O-planes|O7-planes]] (\hyperlink{Sen97a}{Sen 97a, (3)}, \hyperlink{Blumenhagen10}{Blumenhagen 10, (11)}), exhibiting [[gauge enhancement]] \hyperlink{Sen97b}{Sen 97b} see \hyperlink{SingularLocusAndD7Branes}{below}. Reasoning like this might suggest that in generalization to how type II [[orientifolds]] involve $\mathbb{Z}_2$-[[equivariant]] [[K-theory]] (namely [[KR-theory]]), so F-theory should involve $SL_2(\mathbb{Z})$-equivariant [[elliptic cohomology]]. This was conjectured in (\hyperlink{KrizSati05}{Kriz-Sati 05, p. 3, p.17, 18}). For more on this see at \emph{[[modular equivariant elliptic cohomology]]}. \hypertarget{relation_to_heterotic_string_theory}{}\subsubsection*{{Relation to heterotic string theory}}\label{relation_to_heterotic_string_theory} The [[duality between F-theory and heterotic string theory]]:A subclass of K3 manifolds elliptically fibered. F-theory on an [[elliptic fibration|elliptically fibered]] [[K3]] is supposed to be equivalent to [[heterotic string theory]] [[KK-compactification|compactified]] on a 2-[[torus]]. An early argument for this is due to (\hyperlink{Sen96}{Sen 96}). More generally, heterotic string theory on an elliptically fibered Calabi-Yau $Z \to B$ of complex dimension $(n-1)$ is supposed to be equivalent $F$-theory on an $n$-dimensional $X\to B$ with elliptic K3-fibers. A detailed discussion of the [[equivalence]] of the respective [[moduli spaces]] is originally due to (\hyperlink{FriedmanMorganWitten97}{Friedman-Morgan-Witten 97}). A review of this is in (\hyperlink{Donagi98}{Donagi 98}). \hypertarget{SingularLocusAndD7Branes}{}\subsubsection*{{The singular locus of the elliptic fibration and the D7-branes}}\label{SingularLocusAndD7Branes} In passing from [[M-theory]] to [[type IIA string theory]], the locus of any [[Kaluza-Klein monopole]] in 11d becomes the locus of [[D6-branes]] in 10d. The locus of the [[Kaluza-Klein monopole]] in turn (as discussed there) is the locus where the $S^1_A$-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the $S^1_A \times S^1_B$-[[elliptic fibration]] degenerates to the [[nodal curve]]. Since the [[T-duality|T-dual]] of [[D6-branes]] are [[D7-branes]], it follows that [[D7-branes]] in F-theory ``are'' the singular locus of the elliptic fibration. Now an [[elliptic fibration|elliptically fibered]] complex [[K3-surface]] \begin{displaymath} \itexarray{ T &\longrightarrow& K3 \\ && \downarrow \\ && \mathbb{C}\mathbb{P}^1 } \end{displaymath} may be parameterized via the [[Weierstrass elliptic function]] as the solution locus of the equation \begin{displaymath} y^2 = x^3 + f(z) x + g(z) \end{displaymath} for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a [[polynomial]] of degree 8 and $g$ of degree twelve. The [[j-invariant]] of the complex [[elliptic curve]] which this parameterizes for given $z$ is \begin{displaymath} j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3} \,. \end{displaymath} The [[poles]] $j\to \infty$ of the [[j-invariant]] correspond to the [[nodal curve]], and hence it is at these poles that the [[D7-branes]] are located. Since the order of the poles is 24 (the polynomial degree of the [[discriminant]] $\Delta = 27 g^2 + 4 f^3$) there are necessarily \emph{24 D7-branes}. (\hyperlink{Sen96}{Sen 96, page 5} \hyperlink{Sen97b}{Sen 97b}, see also \hyperlink{Morrison04}{Morrison 04, sections 8 and 17}, \hyperlink{Denef08}{Denef 08, around (3.41)}). Notice that the \emph{net charge} of these 24 D7-branes is supposed to vanish, due to [[S-duality]] effects (e.g. \hyperlink{Denef08}{Denef 08, below (3.41)}). \hypertarget{Fbranescan}{}\subsubsection*{{F-brane scan}}\label{Fbranescan} [[!include F-branes -- table]] \hypertarget{sduality_operation_on_branes}{}\subsubsection*{{S-duality operation on $(p,q)$-branes}}\label{sduality_operation_on_branes} The F-theory picture gives a geometric interpretation of the [[S-duality]] expected in [[type II string theory]], by which all branes carry \emph{two} integer charges $(p,q)$ acted on by $SL(2,\mathbb{Z})$. For instance the fundamental string ([[F1-brane]]) and the [[D1-brane]] combine to the $(p,q)$-string, and similarly the [[NS5-brane]] and the [[D5-brane]] combine to a $(p,q)$-5-brane. Namely in the F-theory picture this comes from [[wrapped brane|wrapping]] the [[M2-brane]] and the [[M5-brane]], respectively, on either of the two cycles of the elliptic fibration (and the [[T-duality|T-dualizing]]). (e.g. \hyperlink{Johnson97}{Johnson 97, p. 4}) \hypertarget{ModelBuildingAndPhenomenology}{}\subsubsection*{{Model building and phenomenology}}\label{ModelBuildingAndPhenomenology} For F-theory a fairly advanced [[model (physics)|model]] building and [[string phenomenology]] has been developed. A detailed review is in (\hyperlink{Denef08}{Denef 08}). Via the relation between [[supersymmetry and Calabi-Yau manifolds]] there is particular interest in F-theory compactified on [[Calabi-Yau variety|Calabi-Yau spaces]] of ([[complex manifold|complex]]) [[dimension]] 4. For more on this see at \emph{[[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]}. A large number of realizations of the exact field content of the [[standard model of particle physics]] (or rather the [[MSSM]]) is claimed to be realized in in F-theory in \hyperlink{CveticHalversonLinLiuTian19}{Cvetic-Halverson-Lin-Liu-Tian 19}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[F'-theory]] \item [[12-dimensional supergravity]] \end{itemize} [[!include F-theory compactifications -- table]] \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Discussion of [[T-duality]] in the strong coupling limit is due to \begin{itemize}% \item [[John Schwarz]], \emph{M Theory Extensions of T Duality} (\href{https://arxiv.org/abs/hep-th/9601077}{arXiv:hep-th/9601077}) \item [[Jorge Russo]], \emph{T-duality in M-theory and supermembranes}, Phys.Lett. B400 (1997) 37-42 (\href{https://arxiv.org/abs/hep-th/9701188}{arXiv:hep-th/9701188}) \end{itemize} That [[S-duality]] of [[type II string theory]] may be interpreted in terms of [[conformal transformations]] on the [[fiber]] for [[M-theory]] compactified on a [[torus]] was originally observed in \begin{itemize}% \item [[John Schwarz]], \emph{An $SL(2,\mathbb{Z})$-Multiplet of Type IIB Superstrings}, Phys.Lett.B 360:13-18, 1995; ERRATUM-ibid.B364:252, 1995 (\href{https://arxiv.org/abs/hep-th/9508143}{arXiv:hep-th/9508143}) \item [[Paul Aspinwall]], \emph{Some Relationships Between Dualities in String Theory}, Nucl.Phys.Proc.Suppl. 46 (1996) 30-38 (\href{https://arxiv.org/abs/hep-th/9508154}{arXiv:hep-th/9508154}) \end{itemize} The original article on F-theory as such is \begin{itemize}% \item [[Cumrun Vafa]], \emph{Evidence for F-theory}, Nucl.Phys.B469:403-418,1996, (\href{http://arxiv.org/abs/hep-th/9602022}{arXiv:hep-th/9602022}) \end{itemize} An early survey of its relation to [[M-theory]] with [[M5-branes]] is in \begin{itemize}% \item [[Clifford Johnson]], \emph{From M-theory to F-theory, with Branes}, Nucl.Phys. B507 (1997) 227-244 (\href{http://arxiv.org/abs/hep-th/9706155}{arXiv:hep-th/9706155}) \end{itemize} A more recent survey is \begin{itemize}% \item [[Ralph Blumenhagen]], \emph{Basics of F-theory from the Type IIB Perspective} (\href{http://arxiv.org/abs/1002.2836}{arXiv:1002.2836}) \end{itemize} Lecture notes include \begin{itemize}% \item [[David Morrison]], \emph{TASI Lectures on Compactification and Duality} (\href{http://arxiv.org/abs/hep-th/0411120}{arXiv:hep-th/0411120}) \item [[Timo Weigand]], \emph{Lectures on F-theory compactifications and model building} Class. Quantum Grav. 27 214004 (\href{http://arxiv.org/abs/1009.3497}{arXiv:1009.3497}) \item [[Timo Weigand]], \emph{TASI Lectures on F-theory} (\href{https://arxiv.org/abs/1806.01854}{arXiv:1806.01854}) \end{itemize} Textbook accounts include \begin{itemize}% \item [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], section 8.4 of \emph{String Theory and M-Theory: A Modern Introduction}, 2007 \end{itemize} Further survey includes \begin{itemize}% \item [[Timo Weigand]], \emph{F-theory: Progress and Prospects}, 2014 (\href{https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/string_pheno_ringberg/slides_frontiers/weigand.pdf}{pdf}) \item [[Cumrun Vafa]], \emph{Reflections on F-theory}, 2015 (\href{http://wwwth.mpp.mpg.de/conf/f-theory15/talks/Vafa.pdf}{pdf}) \end{itemize} Related conferences include \begin{itemize}% \item \emph{\href{http://www.match.uni-heidelberg.de/GPF/}{Physics and Geometry of F-theory 2014}} \item \emph{\href{http://f-theory15.mpp.mpg.de}{Physics and Geometry of Ftheory 2015}} \end{itemize} \hypertarget{relation_to_orientifolds}{}\subsubsection*{{Relation to orientifolds}}\label{relation_to_orientifolds} F-theory lifts of [[orientifold]] backgrounds were first identified in \begin{itemize}% \item [[Ashoke Sen]], \emph{F-theory and Orientifolds}, Nucl.Phys.B475:562-578,1996 (\href{http://arxiv.org/abs/hep-th/9605150}{arXiv:hep-th/9605150}) \item [[Ashoke Sen]], \emph{Orientifold Limit of F-theory Vacua}, Nucl. Phys. Proc. Suppl. 68 (1998) 92 (Nucl. Phys. Proc. Suppl. 67 (1998) 81) (\href{http://arxiv.org/abs/hep-th/9702165}{arXiv:hep-th/9702165}) \end{itemize} and the corresponding [[gauge enhancement]] in \begin{itemize}% \item [[Ashoke Sen]], \emph{A Note on Enhanced Gauge Symmetries in M- and String Theory}, JHEP 9709:001,1997 (\href{http://arxiv.org/abs/hep-th/9707123}{arXiv:hep-th/9707123}) \end{itemize} with more details including \begin{itemize}% \item Zurab Kakushadze, [[Gary Shiu]], S.-H. Henry Tye, \emph{Type IIB Orientifolds, F-theory, Type I Strings on Orbifolds and Type I - Heterotic Duality}, Nucl.Phys. B533 (1998) 25-87 (\href{http://arxiv.org/abs/hep-th/9804092}{arXiv:hep-th/9804092}) \end{itemize} This is further expanded on in \begin{itemize}% \item [[Hisham Sati]], \emph{The Elliptic curves in gauge theory, string theory, and cohomology}, JHEP 0603 (2006) 096 (\href{http://arxiv.org/abs/hep-th/0511087}{arXiv:hep-th/0511087}) \end{itemize} \hypertarget{relation_to_elliptic_cohomology}{}\subsubsection*{{Relation to elliptic cohomology}}\label{relation_to_elliptic_cohomology} A series of articles arguing for a relation between the [[elliptic fibration]] of F-theory and [[elliptic cohomology]] (see also at [[modular equivariant elliptic cohomology]]) \begin{itemize}% \item [[Igor Kriz]], [[Hisham Sati]], \emph{Type II string theory and modularity}, JHEP 0508 (2005) 038 (\href{http://arxiv.org/abs/hep-th/0501060}{arXiv:hep-th/0501060}) \end{itemize} \hypertarget{relation_to_heterotic_string_string_theory}{}\subsubsection*{{Relation to heterotic string string theory}}\label{relation_to_heterotic_string_string_theory} \begin{itemize}% \item Robert Friedman, [[John Morgan]], [[Edward Witten]], \emph{Vector Bundles And F Theory} (\href{http://arxiv.org/abs/hep-th/9701162}{arXiv:hep-th/9701162}) \item [[Ron Donagi]], \emph{ICMP lecture on heterotic/F-theory duality} (\href{http://arxiv.org/abs/hep-th/9802093}{arXiv:hep-th/9802093}) \end{itemize} \hypertarget{relation_to_mtheory_on_manifolds_2}{}\subsubsection*{{Relation to M-theory on $G_2$-manifolds}}\label{relation_to_mtheory_on_manifolds_2} \begin{itemize}% \item [[Sergei Gukov]], [[Shing-Tung Yau]], [[Eric Zaslow]], \emph{Duality and Fibrations on $G_2$ Manifolds} (\href{http://arxiv.org/abs/hep-th/0203217}{arXiv:hep-th/0203217}) \item Adil Belhaj, \emph{F-theory Duals of M-theory on $G_2$ Manifolds from Mirror Symmetry} (\href{http://arxiv.org/abs/hep-th/0207208}{arXiv:hep-th/0207208}) \item [[Mariana Graña]], C. S. Shahbazi, [[Marco Zambon]], \emph{$Spin(7)$-manifolds in compactifications to four dimensions}, JHEP11(2014)046 (\href{http://arxiv.org/abs/1405.3698}{arXiv:1405.3698}) \end{itemize} \hypertarget{relation_to_the_6d_superconformal_theory}{}\subsubsection*{{Relation to the 6d superconformal theory}}\label{relation_to_the_6d_superconformal_theory} Realization to the [[6d (2,0)-supersymmetric QFT]] is discussed in \begin{itemize}% \item Jonathan Heckman, [[David Morrison]], [[Cumrun Vafa]], \emph{On the Classification of 6D SCFTs and Generalized ADE Orbifolds} (\href{http://arxiv.org/abs/1312.5746}{arXiv:1312.5746}) \end{itemize} \hypertarget{ReferencesPhenomnenology}{}\subsubsection*{{Phenomenology and model building}}\label{ReferencesPhenomnenology} A large body of literature is concerned with particle physics [[string phenomenology]] modeled in the context of F-theory, in particular [[GUTs]]: \begin{itemize}% \item [[Frederik Denef]], \emph{Les Houches Lectures on Constructing String Vacua}, in \emph{[[String theory and the real world]]} (\href{http://arxiv.org/abs/0803.1194}{arXiv:0803.1194}) \item [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], \emph{GUTs and Exceptional Branes in F-theory - I}, JHEP 0901:058,2009 (\href{http://arxiv.org/abs/0802.3391}{arXiv:0802.3391}) \item [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], \emph{GUTs and Exceptional Branes in F-theory - I} (\href{http://arxiv.org/abs/0802.3391}{arxiv:0802.3391}), \emph{II: Experimental Predictions} (\href{http://arxiv.org/abs/0806.0102}{arxiv:0806.0102}) \item [[Martin Wijnholt]], \emph{String compactification}, \href{https://pitp2014.ias.edu}{PITP 2014} lecture notes ([[WijnholtCompactification14.pdf:file]], \href{https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P1_wijnholt.pdf}{slides for lecture 1}, \href{https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P2_wijnholt.pdf}{slides for lecture 2}, \href{https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P3_wijnholt.pdf}{slides for lecture 3}) \item Gianluca Zoccarato, \emph{Yukawa couplings at the point of $E_8$ in F-theory}, 2014 (\href{http://stringpheno2014.ictp.it/parallels/tuesday/F-theory(B}{pdf}/zoccarato.pdf)) \end{itemize} Discussion of the \emph{exact} [[gauge group]] of the [[standard model of particle physics]], $G = \big( SU(3) \times SU(2) \times U(1)\big)/\mathbb{Z}_6$ including its $\mathbb{Z}_6$-[[quotient group|quotient]] (see \href{standard+model+of+particle+physics#GaugeGroup}{there}) and the exact [[fermion]] field content, realized in F-theory is in \begin{itemize}% \item Denis Klevers, Damian Kaloni Mayorga Pena, Paul-Konstantin Oehlmann, Hernan Piragua, Jonas Reuter, \emph{F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches}, JHEP01(2015)142 (\href{https://arxiv.org/abs/1408.4808}{arXiv:1408.4808}) \item [[Mirjam Cvetic]], Ling Lin, section 3.3 of \emph{The global gauge group structure of F-theory compactifications with $U(1)$s} (\href{https://arxiv.org/abs/1706.08521}{arXiv:1706.08521}) \end{itemize} Based on this large number of realizations of the exact field content of the [[standard model of particle physics]] (or rather [[MSSM]]) in [[F-theory]] is claimed to be realized in \begin{itemize}% \item [[Mirjam Cvetic]], [[James Halverson]], Ling Lin, Muyang Liu, Jiahua Tian, \emph{A Quadrillion Standard Models from F-theory} (\href{https://arxiv.org/abs/1903.00009}{arXiv:1903.00009}) \item [[Washington Taylor]], Andrew P. Turner, \emph{Generic construction of the Standard Model gauge group and matter representations in F-theory} (\href{https://arxiv.org/abs/1906.11092}{arXiv:1906.11092}) \end{itemize} \hypertarget{cosmological_constant}{}\paragraph*{{Cosmological constant}}\label{cosmological_constant} An argument for [[non-perturbative effect|non-perturbative]] non-[[supersymmetry|supersymmetric]] 4d [[string phenomenology]] with fundamentally vanishing [[cosmological constant]], based on 3d [[M-theory on 8-manifolds]] decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]): \begin{itemize}% \item [[Edward Witten]], \emph{The Cosmological Constant From The Viewpoint Of String Theory}, lecture at \href{http://inspirehep.net/record/972507}{DM2000} (\href{https://arxiv.org/abs/hep-ph/0002297}{arXiv:hep-ph/0002297}) (see p. 7) \item [[Edward Witten]], \emph{Strong coupling and the cosmological constant}, Mod. Phys. Lett. A 10:2153-2156, 1995 (\href{https://arxiv.org/abs/hep-th/9506101}{arXiv:hep-th/9506101}) \item [[Edward Witten]], Section 3 of \emph{Some Comments On String Dynamics}, talk at \href{https://cds.cern.ch/record/305869}{Strings95} (\href{http://arxiv.org/abs/hep-th/9507121}{arXiv:hep-th/9507121}) \end{itemize} The realization of this scenario in [[F-theory]]: \begin{itemize}% \item [[Cumrun Vafa]], Section 4.3 of: \emph{Evidence for F-Theory}, Nucl. Phys. B469:403-418, 1996 (\href{https://arxiv.org/abs/hep-th/9602022}{arxiv:hep-th/9602022}) \item [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Gianluca Zoccarato, \emph{F-theory and Dark Energy}, Fortschritte der Physik (\href{https://arxiv.org/abs/1811.01959}{arXiv:1811.01959}, \href{https://doi.org/10.1002/prop.201900057}{doi:10.1002/prop.201900057}) \item [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, Gianluca Zoccarato, \emph{Pixelated Dark Energy} (\href{https://arxiv.org/abs/1901.10489}{arXiv:1901.10489}) \end{itemize} \hypertarget{FluxAndInstantons}{}\subsubsection*{{4-Form flux and instantons}}\label{FluxAndInstantons} The image of the [[supergravity C-field]] from [[11-dimensional supergravity]] to F-theory yields the \emph{$G_4$-[[flux compactification|flux]]}. \begin{itemize}% \item Andres Collinucci, Raffaele Savelli, \emph{On Flux Quantization in F-Theory} (2010) (\href{http://arxiv.org/abs/1011.6388}{arXiv:1011.6388}) \item Sven Krause, Christoph Mayrhofer, [[Timo Weigand]], \emph{$G_4$ flux, chiral matter and singularity resolution in F-theory compactifications} (\href{http://arxiv.org/abs/1109.3454}{arXiv:1109.3454}) \item [[Thomas Grimm]], Denis Klevers, Maximilian Poretschkin, \emph{Fluxes and Warping for Gauge Couplings in F-theory} (\href{http://arxiv.org/abs/1202.0285}{arXiv:1202.0285}) \item Sven Krause, Christoph Mayrhofer, [[Timo Weigand]], \emph{Gauge Fluxes in F-theory and Type IIB Orientifolds} (2012) (\href{http://arxiv.org/abs/1202.3138}{arXiv:1202.3138}) \end{itemize} and with [[M5-brane]] [[instanton]] contributions: \begin{itemize}% \item Max Kerstan, [[Timo Weigand]], \emph{Fluxed M5-instantons in F-theory} (\href{http://arxiv.org/abs/1205.4720}{arXiv:1205.4720}) \end{itemize} Reviewed in \begin{itemize}% \item [[Timo Weigand]], \emph{Fluxes and M5-instantons in F-theory}, 2012 (\href{http://people.physik.hu-berlin.de/~ahoop/weigand.pdf}{pdf slides}) \end{itemize} For more on this see also at \emph{[[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]}. \end{document}