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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*{S-duality} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory} [[!include duality in string theory -- contents]] \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{langlands_correspondence}{}\paragraph*{{Langlands correspondence}}\label{langlands_correspondence} [[!include Langlands correspondence -- contents ]] The term \emph{S-duality} can mean two different things: \begin{itemize}% \item in [[mathematics]] it is short for \emph{[[Spanier-Whitehead duality]]} : [[dual object|monoidal duality]] in the [[stable homotopy category]]; \item in [[physics]] it denotes a certain equivalence between [[quantum field theories]], this is what we discuss below \end{itemize} \vspace{.5em} \hrule \vspace{.5em} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{in_super_yangmills_theory}{In (super) Yang-Mills theory}\dotfill \pageref*{in_super_yangmills_theory} \linebreak \noindent\hyperlink{general_idea}{General idea}\dotfill \pageref*{general_idea} \linebreak \noindent\hyperlink{ForSYMFromCompactification}{From compactification of the 6d (2,0)-SCFT and AGT correspondence}\dotfill \pageref*{ForSYMFromCompactification} \linebreak \noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak \noindent\hyperlink{InTypeIIB}{Type IIB S-duality}\dotfill \pageref*{InTypeIIB} \linebreak \noindent\hyperlink{general_idea_2}{General idea}\dotfill \pageref*{general_idea_2} \linebreak \noindent\hyperlink{CohomologicalNatureOfTypeIIFieldsUnderSDuality}{Cohomological nature of type II fields under S-duality}\dotfill \pageref*{CohomologicalNatureOfTypeIIFieldsUnderSDuality} \linebreak \noindent\hyperlink{heterotictype_i_duality}{Heterotic/type I duality}\dotfill \pageref*{heterotictype_i_duality} \linebreak \noindent\hyperlink{for_type_iia}{For type IIA}\dotfill \pageref*{for_type_iia} \linebreak \noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{ReferencesInSYM}{In (super-)Yang-Mills theory}\dotfill \pageref*{ReferencesInSYM} \linebreak \noindent\hyperlink{in_type_ii_superstring_theory}{In type II superstring theory}\dotfill \pageref*{in_type_ii_superstring_theory} \linebreak \noindent\hyperlink{relation_to_geometric_langlands_duality}{Relation to geometric Langlands duality}\dotfill \pageref*{relation_to_geometric_langlands_duality} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In the original and restricted sense, \emph{S-duality} refers to the conjectured [[Montonen-Olive duality]] auto-[[equivalence]] of ([[super Yang-Mills theory|super]]) [[Yang-Mills theory]] in 4 dimensions under which the [[coupling constant]] is inverted, and more generally under which the combined [[coupling constant]] and [[theta angle]] tranform under an action of the [[modular group]]. At least for [[super Yang-Mills theory]] this conjecture can be argued for in detail. There is also a [[duality in string theory]] called \emph{S-duality}. Specifically in [[type IIB superstring theory]]/[[F-theory]] this is given by an action of the [[modular group]] on the [[axio-dilaton]], hence is, via the proportionality of the dilaton to the [[string coupling constant]], again a weak-strong coupling duality. Indeed, at least for [[super Yang-Mills theory]] Montonen-Olive S-duality may be understood as a special case of the [[duality in string theory|string duality]] (\hyperlink{Witten95a}{Witten 95a}, \hyperlink{Witten95b}{Witten 95b}): one may understand [[N=2 D=4 super Yang-Mills theory]] as the [[KK-compactification]] of the [[M5-brane]] [[6d (2,0)-superconformal QFT]] on the [[F-theory]] torus (\hyperlink{Johnson97}{Johnson 97}) to get the [[D3-brane]] worldvolume theory, and the remnant [[modular group]] action on the compactified torus is supposed to be the 4d Montonen-Olive S-duality (\hyperlink{Witten07}{Witten 07}). \hypertarget{in_super_yangmills_theory}{}\subsubsection*{{In (super) Yang-Mills theory}}\label{in_super_yangmills_theory} \hypertarget{general_idea}{}\paragraph*{{General idea}}\label{general_idea} In its original form, S-duality refers to \textbf{Montonen-Olive duality} , which is about the following phenomenon: The [[Lagrangian]] of [[Yang-Mills theory]] has two summands, \begin{displaymath} S_{YM} : \nabla \mapsto \int_X \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla\rangle + \int_{X} i \theta \langle F_\nabla \wedge F_\nabla \rangle \,, \end{displaymath} each pairing the [[curvature]] 2-form with itself in an [[invariant polynomial]], but the first involving the [[Hodge star operator]] dual, and the second not. One can combine the coefficients $\frac{1}{e^2}$ and $i \theta$ into a single \emph{[[complex number|complex]] [[coupling constant]]} \begin{displaymath} \tau = \frac{\theta}{2 \pi} + \frac{4 \pi i}{e^2} \,. \end{displaymath} Montonen-Olive duality asserts that the quantum field theories induced from one such parameter value and another one obtained from it by an [[action]] of [[SL(2,Z)]] on the [[upper half plane]] are equivalent. This is actually not quite true for ordinary Yang-Mills theory, but seems to be true for [[N=2 D=4 super Yang-Mills theory]]. \hypertarget{ForSYMFromCompactification}{}\paragraph*{{From compactification of the 6d (2,0)-SCFT and AGT correspondence}}\label{ForSYMFromCompactification} In (\hyperlink{Witten95a}{Witten 95a}, \hyperlink{Witten95b}{Witten 95b}, \hyperlink{Witten07}{Witten 07}) it was suggested that the above S-duality of [[N=2 D=4 super Yang-Mills theory]] may be understood geometrically by regarding the super Yang-Mills theory as the [[Kaluza-Klein compactification]] of the [[6d (2,0)-superconformal QFT]] -- that instead of a [[gauge field]] given by a [[principal bundle]] with [[connection on a bundle|connection]] involves a [[principal 2-bundle]] with [[connection on a 2-bundle|2-connection]] -- on a [[complex torus]]. The $SL(2,\mathbb{Z})$-invariance of the resulting 4-dimensional theory is then the [[modular group]] remnant of the conformal invariance of the 6-dimensional theory under conformal transformations of that torus. Moreover, Witten has suggested that this S-duality secretly drives a host of other subtle phenomena, notably that the [[geometric Langlands duality]] (see there for more) is just an aspect of a special case of this. The [[AGT correspondence]] refines this further and regards the [[6d (2,0)-superconformal QFT]] as something like a ``[[2d SCFT]] with values in 4d super-Yang-Mills theories''. This way the whole [[mapping class group]] of general 2d [[Riemann surfaces]] acts as a generalized S-duality on 4d super-Yang-Mills theory \hypertarget{in_string_theory}{}\subsubsection*{{In string theory}}\label{in_string_theory} In [[string theory]], S-duality is supposed to apply to whole string theories and make [[type II string theory]] be S-dual to itself and make [[heterotic string theory]] be S-dual to [[type I string theory]]. \hypertarget{InTypeIIB}{}\paragraph*{{Type IIB S-duality}}\label{InTypeIIB} \hypertarget{general_idea_2}{}\paragraph*{{General idea}}\label{general_idea_2} [[Type IIB string theory]] is obtained by [[KK-compactification]] of [[M-theory]] on a [[torus]] bundle followed by [[T-duality|T-dualizing]] one of the torus cycles. This perspective -- referred to as [[F-theory]] -- exhibits the [[axio-dilaton]] of type IIB string theory as the fiber of an [[elliptic fibration]] (essentially the torus bundle that M-theory was compactified on (\hyperlink{Johnson97}{Johnson 97})). The [[modular group]] acts on this [[elliptic fibration]], and this is S-duality for type IIB-strings. In particular the transformation $\tau \mapsto - \frac{1}{\tau}$ inverts the type II coupling constant. See at \emph{[[F-theory]]} for more. The type IIB [[F1-string]] and the [[D1-brane]] appear this way by [[double dimensional reduction]] from the [[M2-brane]] wrapping (either) one of the two cycles of the compactifying torus. S-duality mixes these strings by the evident [[modular group]] action on the $(p,q)\in \mathbb{Z}^2$ labels of the [[(p,q)-strings]]. Here at least part of the S-duality action on $(p,q)$-strings may be seen as a system of autoequivalences of the [[super L-infinity algebras]] which defines the [[extended super spacetime]] constituted by the type II superstring (\href{Bandos00}{Bandos 00}, \hyperlink{FSS13}{FSS 13, section 4.3}). Similarly the [[D5-brane]] and the [[NS5-brane]] are the [[double dimensional reduction]] of the [[M5-brane]] wrapping one of the two cycles of the compactifying torus, and hence the S-duality modular group also acts on $(p,q)$-5-branes, exchanging them. Finally, the [[D3-brane]] is instead the [[double dimensional reduction]] of the [[M5-brane]], wrapping \emph{both} compactifying dimensions. Accordingly the worldvolume theory of the D3, which is [[super Yang-Mills theory]] in $d = 4$ has an S-self-duality. That is supposed to be the [[Montonen-Olive duality]] discussed above, which is thereby unified with type IIB S-duality. \hypertarget{CohomologicalNatureOfTypeIIFieldsUnderSDuality}{}\paragraph*{{Cohomological nature of type II fields under S-duality}}\label{CohomologicalNatureOfTypeIIFieldsUnderSDuality} While [[F-theory]] does capture much of this [[non-perturbative effect|non-perturbative]] S-duality, there currently remains a puzzle as to the correct [[differential cohomology]] nature of all the fields under S-duality: by the above S-duality mixes the [[Kalb-Ramond field]] $\hat B_{NS}$ with the degree-3 component $\hat B_{RR}$ of the [[RR-field]]. But the best available description of the fine-structure of these fields is (see also at \emph{[[orientifold]]}) that $\hat B_{NS}$ is a [[cocycle]] in ([[twisted cohomology|twisted]]) [[ordinary differential cohomology]] while $\hat B_{RR}$ is (only) one component of a cocycle in ([[twisted cohomology|twisted]]) [[KU]] (or really: [[KR-theory]]). This issue was first highlighted in (\hyperlink{DMW00}{DMW 00, section 11}). In (\href{supergravity+C-field#DFM}{DFM 03, section 9}) it was observed that taking into account the [[cubical structure in M-theory]] on the [[higher dimensional Chern-Simons theory|11-dimensional Chern-Simons term]] of the [[supergravity C-field]] the conceptual mismatch is alleviated, but not quite resolved. See also (\hyperlink{BEJVS05}{BEJVS 05}) On the other hand, as discussed at \emph{[[cubical structure in M-theory]]}, this structure plausibly relates to a [[generalized cohomology theory]] beyond [[ordinary cohomology]] and beyond [[K-theory]], namely to [[elliptic cohomology]]/[[tmf]]. Hints like this led in (\hyperlink{KrizSati05}{KrizSati 05}) to the [[conjecture]] that the right [[cohomology theory]] to capture the S-duality of [[type II superstring theory|type IIB]]/[[F-theory]] is [[modular equivariant elliptic cohomology]]. \hypertarget{heterotictype_i_duality}{}\paragraph*{{Heterotic/type I duality}}\label{heterotictype_i_duality} \begin{quote}% Something substantial should go here, for the moment the following is copied from a discussion forum comment by some Olof \href{http://physics.stackexchange.com/a/65546/5603}{here}: \end{quote} For the Het/I relation, the first observation is that the massless spectra of the two models agree. Moreover, if we make the identification \begin{displaymath} G^I_{\mu\nu} = e^{-\Phi_h} G^h_{\mu\nu} , \qquad \Phi^I = - \Phi^h , \qquad \tilde{F}^I_3 = \tilde{H}^h_3 , \qquad A^I_1 = A^h_1 \end{displaymath} the low energy effective supergravity actions of the two models match. Since the [[string coupling constants]] $g_s^I$ and $g_s^h$ are given as the expectation values of the exponentials of the dilatons $\exp(\Phi^I)$ and $\exp(\Phi^h)$, respectively, the above equations relates the type-I theory at strong coupling to the heterotic theory at weak coupling: \begin{displaymath} g^I_s = \frac{1}{g^h_s} . \end{displaymath} From the relative scaling of the metric in (1) we also see that the string length in the two theories are related by \begin{displaymath} l^I_s = l^h_s \sqrt{g^h_s}. \end{displaymath} As a non-perturbative check we can consider the tension of the type-I D1 brane. The brane is a BPS object, so for all values of the coupling $g_s^I$ the tension is given by the same formula \begin{displaymath} T^I_{D1} = \frac{1}{g_s^I} \frac{1}{2\pi\left(l^I_s\right)^2} = \frac{g^h_s}{2\pi\left(l^h_s\sqrt{g^h_s}\right)^2} = \frac{1}{2\pi\left(l^h_s\right)^2} \end{displaymath} where I've used relations (2) and (3). But this is equal to the tension of the fundamental heterotic string \begin{displaymath} T^h_{F1} = \frac{1}{2\pi\left(l^h_s\right)^2}. \end{displaymath} This indicates that it is sensible to identify the strong coupling limit of the type-I D1 brane with the heterotic string. \hypertarget{for_type_iia}{}\paragraph*{{For type IIA}}\label{for_type_iia} A priori [[type IIA superstring theory]] does not have S-duality, but by compactifying [[M-theory]] on a torus one can sort of read off what the non-perturbative additions to type IIA should be that make it have S-duality after all, see \begin{itemize}% \item Gottfried Curio, Boris Kors, [[Dieter Lüst]], \emph{Fluxes and Branes in Type II Vacua and M-theory Geometry with G(2) and Spin(7) Holonomy}, Nucl.Phys.B636:197-224,2002 (\href{http://arxiv.org/abs/hep-th/0111165}{arXiv:hep-th/0111165}) \end{itemize} \hypertarget{overview}{}\subsection*{{Overview}}\label{overview} [[!include heterotic S-duality -- table]] [[!include gauge theory from AdS-CFT -- table]] \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Yang-Mills theory]] \item [[topologically twisted D=4 super Yang-Mills theory]] \end{itemize} [[duality in physics]], [[duality in string theory]] \begin{itemize}% \item \textbf{S-duality} \begin{itemize}% \item [[electro-magnetic duality]] \begin{itemize}% \item [[Montonen-Olive duality]] \end{itemize} \item [[Seiberg duality]] \item [[geometric Langlands correspondence]] \item [[quantum geometric Langlands correspondence]] \end{itemize} \item [[duality in string theory]] \begin{itemize}% \item [[T-duality]], [[U-duality]] \item [[AdS/CFT correspondence]] \end{itemize} \item [[type II string theory]], [[F-theory]] \begin{itemize}% \item [[(p,q)-string]] \item [[modular equivariant elliptic cohomology]] \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Mike Duff]], chapter 6 of \emph{[[The World in Eleven Dimensions]]: Supgergravity, Supermembranes and M-theory} IoP 1999 \end{itemize} \hypertarget{ReferencesInSYM}{}\subsubsection*{{In (super-)Yang-Mills theory}}\label{ReferencesInSYM} It was originally noticed in \begin{itemize}% \item P. Goddard, J. Nuyts, and [[David Olive]], \emph{Gauge Theories And Magnetic Charge}, Nucl. Phys. B125 (1977) 1-28. \end{itemize} that where [[electric charge]] in [[Yang-Mills theory]] takes values in the [[weight lattice]] of the [[gauge group]], then [[magnetic charge]] takes values in the lattice of what is now called the [[Langlands dual group]]. This led to the electric/magnetic duality conjecture formulation in \begin{itemize}% \item [[Claus Montonen]], [[David Olive]], \emph{Magnetic Monopoles As Gauge Particles?} Phys. Lett. B72 (1977) 117-120. \end{itemize} According to (\hyperlink{KapustinWitten06}{Kapustin-Witten 06, pages 3-4}) the observaton that the Montonen-Olive dual charge group coincides with the [[Langlands dual group]] is due to \begin{itemize}% \item [[Michael Atiyah]], private communication to [[Edward Witten]], 1977 \end{itemize} Discussion of the duality for abelian gauge theory ([[electromagnetism]]) is is \begin{itemize}% \item [[Edward Witten]], \emph{On S-duality in abelian gauge theory} Selecta Mathematica, (2):383-410, 1995 (\href{http://arxiv.org/abs/hep-th/9505186}{arXiv:hep-th/9505186}) \item Jose Barbon, \emph{Generalized abelian S-duality and coset constructions}, Nuclear Physics B, 452(1):313-330, 1995 (\href{http://arxiv.org/abs/hep-th/9506137}{arXiv:hep-th/9506137}) \item Gerald Kelnhofer, \emph{Functional integration and gauge ambiguities in generalized abelian gauge theories} J. Geom. Physics, 59:1017-1035, 200 \end{itemize} See also the references at \emph{[[electro-magnetic duality]]}. The insight that the Montonen-Olive duality works more naturally in [[super Yang-Mills theory]] is due to \begin{itemize}% \item [[David Olive]], [[Edward Witten]], \emph{Supersymmetry Algebras That Include Topological Charges}, Phys. Lett. B78 (1978) 97-101. \end{itemize} and that it works particularly for [[N=4 D=4 super Yang-Mills theory]] is due to \begin{itemize}% \item H. Osborn, \emph{Topological Charges For $N = 4$ Supersymmetric Gauge Theories And Monopoles Of Spin 1}, Phys. Lett. B83 (1979) 321-326. \end{itemize} The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-action in this case is due to \begin{itemize}% \item [[John Cardy]], E. Rabinovici, \emph{Phase Structure Of Zp Models In The Presence Of A Theta Parameter}, Nucl. Phys. B205 (1982) 1-16; \item [[John Cardy]], \emph{Duality And The Theta Parameter In Abelian Lattice Models}, Nucl. Phys. B205 (1982) 17-26. \item A. Shapere and [[Frank Wilczek]], \emph{Selfdual Models With Theta Terms}, Nucl. Phys. B320 (1989) 669-695. \end{itemize} The understanding of this $SL(2,\mathbb{Z})$-symmetry as a remnant conformal transformation on a 6-dimensional [[principal 2-bundle]]-theory -- the [[6d (2,0)-superconformal QFT]] -- compactified on a torus is described in \begin{itemize}% \item [[Edward Witten]], pages 4-5 of \emph{Some Comments On String Dynamics}, Proceedings of \emph{String95} (\href{http://arxiv.org/abs/hepth/9507121}{arXiv:hepth/9507121}) \item [[Edward Witten]], \emph{On S-Duality in Abelian Gauge Theory} (\href{http://arxiv.org/abs/hep-th/9505186}{arXiv:hep-th/9505186}) \item [[Edward Witten]], \emph{[[Conformal field theory in four and six dimensions]]} (\href{http://arxiv.org/abs/0712.0157}{arXiv:0712.0157}) \end{itemize} \hypertarget{in_type_ii_superstring_theory}{}\subsubsection*{{In type II superstring theory}}\label{in_type_ii_superstring_theory} The suggestion of an $SL(2,\mathbb{Z})$-duality action in [[type II superstring theory]] goes back to \begin{itemize}% \item [[John Schwarz]], [[Ashoke Sen]], \emph{Duality Symmetries Of $4D$ Heterotic Strings}, Phys. Lett. 312B (1993) 105-114, \item [[Ashoke Sen]], \emph{Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and $SL(2,\mathbb{Z})$ Invariance in String Theory} (\href{http://arxiv.org/abs/hep-th/9402032}{arXiv:hep-th/9402032}) \end{itemize} \emph{Duality Symmetric Actions}, Nucl. Phys. B411 (1994) 35-63 (\href{http://arxiv.org/abs/hep-th/9304154}{arXiv:hep-th/9304154}) \begin{itemize}% \item [[Chris Hull]], [[Paul Townsend]], \emph{Unity of Superstring Dualities}, Nucl.Phys.B438:109-137,1995 (\href{http://arxiv.org/abs/hep-th/9410167}{arXiv:hep-th/9410167}) \item [[John Schwarz]], \emph{An $SL(2,\mathbb{Z})$ Multiplet of Type IIB Superstrings}, Phys.Lett. B360 (1995) 13-18; Erratum-ibid. B364 (1995) 252 (\href{http://arxiv.org/abs/hep-th/9508143}{arXiv:hep-th/9508143}) \end{itemize} The geometric understanding of S-duality in [[type II superstring theory]] via [[M-theory]]/[[F-theory]] goes maybe back to \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 textbook account is in \begin{itemize}% \item [[Peter West]], section 17.3 of \emph{[[Introduction to Strings and Branes]]}, Cambridge University Press 2012 \end{itemize} A 2-loop test is in \begin{itemize}% \item [[Eric D'Hoker]], Michael Gutperle, [[Duong Phong]], \emph{Two-loop superstrings and S-duality}, Nucl.Phys. B722 (2005) 81-118 (\href{http://arxiv.org/abs/hep-th/0503180}{arXiv:hep-th/0503180}) \end{itemize} S-duality acting on the [[worldsheet]] theory if [[(p,q)-strings]] is discussed for instance in \begin{itemize}% \item [[Igor Bandos]], \emph{Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane}, Nucl.Phys.B599:197-227,2001 (\href{http://arxiv.org/abs/hep-th/0008249}{arXiv:hep-th/0008249}) \end{itemize} Closely related to this, S-duality in [[type II string theory]] as an operation on the [[extended super spacetime]] [[super L-infinity algebra]] is \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], section 4.3 of \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]} (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264}) \end{itemize} The cohomological problem of the type II S-duality action on the 3-form flux was originally highlighted in \begin{itemize}% \item D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], \emph{$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory}, Adv.Theor.Math.Phys.6:1031-1134 2003 (\href{http://arxiv.org/abs/hep-th/0005090}{arXiv:hep-th/0005090}) \end{itemize} The conjecture that with combined targetspace/worldsheet modular transformations the type IIB S-duality is reflected in [[modular equivariant elliptic cohomology]] is due to \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} See also \begin{itemize}% \item [[Peter Bouwknegt]], [[Jarah Evslin]], [[Branislav Jurco]], [[Mathai Varghese]], [[Hisham Sati]], \emph{Flux Compactifications on Projective Spaces and The S-Duality Puzzle}, Adv.Theor.Math.Phys. 10 (2006) 345-394 (\href{http://arxiv.org/abs/hep-th/0501110}{arXiv:hep-th/0501110}) \end{itemize} Discussion of [[Montonen-Olive duality]] in [[D=4 super Yang-Mills theory]] via [[ABJM-model]] as [[D3-brane]] model after [[double dimensional reduction]] followed by [[T-duality]]: \begin{itemize}% \item [[Koji Hashimoto]], Ta-Sheng Tai, Seiji Terashima, \emph{Toward a Proof of Montonen-Olive Duality via Multiple M2-branes}, JHEP 0904:025, 2009 (\href{https://arxiv.org/abs/0809.2137}{arxiv:0809.2137}) \end{itemize} See also: \begin{itemize}% \item Surya Raghavendran, Philsang Yoo, \emph{Twisted S-Duality} (\href{https://arxiv.org/abs/1910.13653}{arxiv:1910.13653}) \end{itemize} \hypertarget{relation_to_geometric_langlands_duality}{}\subsubsection*{{Relation to geometric Langlands duality}}\label{relation_to_geometric_langlands_duality} The relation of S-duality to [[geometric Langlands duality]] was understood in \begin{itemize}% \item [[Anton Kapustin]], [[Edward Witten]], \emph{Electric-Magnetic Duality And The Geometric Langlands Program} , Communications in number theory and physics, Volume 1, Number 1, 1--236 (2007) (\href{http://arxiv.org/abs/hep-th/0604151}{arXiv:hep-th/0604151}) \end{itemize} Exposition of this is in \begin{itemize}% \item [[Edward Frenkel]], \emph{What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?} KITP talk 2011 (\href{http://online.kitp.ucsb.edu/online/bblunch/frenkel/}{web}) \end{itemize} [[!redirects Montonen-Olive duality]] [[!redirects complexified coupling constant]] \end{document}