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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*{self-dual higher gauge theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology} [[!include differential cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Definition}{Definition (outline)}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{holographic_relation_to_higher_chernsimons_theory}{Holographic relation to higher Chern-Simons theory}\dotfill \pageref*{holographic_relation_to_higher_chernsimons_theory} \linebreak \noindent\hyperlink{idea_and_examples}{Idea and examples}\dotfill \pageref*{idea_and_examples} \linebreak \noindent\hyperlink{ConformalStructureFromPolarization}{Conformal structure from polarization}\dotfill \pageref*{ConformalStructureFromPolarization} \linebreak \noindent\hyperlink{PartitionFunction}{Partition function}\dotfill \pageref*{PartitionFunction} \linebreak \noindent\hyperlink{cohomological_description}{Cohomological description}\dotfill \pageref*{cohomological_description} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{ChiralBosonin2d}{Chiral boson in 2d (on the string)}\dotfill \pageref*{ChiralBosonin2d} \linebreak \noindent\hyperlink{chiral_2form_in_6d_on_the_m5brane}{Chiral 2-form in 6d (on the M5-brane)}\dotfill \pageref*{chiral_2form_in_6d_on_the_m5brane} \linebreak \noindent\hyperlink{RRFieldsin10d}{RR-Fields in 10d (on the ``9-brane'')}\dotfill \pageref*{RRFieldsin10d} \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{for_the_m5sigma_model}{For the M5-sigma model}\dotfill \pageref*{for_the_m5sigma_model} \linebreak \noindent\hyperlink{for_ordinary_higher_gauge_fields__ordinary_differential_cohomology}{For ordinary $U(1)$-higher gauge fields / ordinary differential cohomology}\dotfill \pageref*{for_ordinary_higher_gauge_fields__ordinary_differential_cohomology} \linebreak \noindent\hyperlink{for_rrfields__differential_ktheory}{For RR-fields / differential K-theory}\dotfill \pageref*{for_rrfields__differential_ktheory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The standard [[action functional]] for the [[higher U(1)-gauge field]] given by a [[circle n-bundle with connection]] $(P, \nabla)$ over a [[pseudo-Riemannian manifold|(pseudo)]] [[Riemannian manifold]] $(X,g)$ is \begin{displaymath} \nabla \mapsto \int_X F_\nabla \wedge \star_g F_{\nabla} \,, \end{displaymath} where $F_\nabla$ is the [[curvature]] $(n+1)$-form. If the [[dimension]] \begin{displaymath} dim X = 4 k + 2 \end{displaymath} then the [[Hodge star]] operator squares to $+1$ (Lorentzian signature) or $-1$ (Euclidean signature) on $\Omega^{k+1}(X)$. Therefore it makes sense in these dimensions to impose the \emph{self-duality} or \emph{chirality} constraint \begin{displaymath} \pm F_\nabla = \star F_\nabla \,. \end{displaymath} With this duality constraint imposed, one speaks of \textbf{self-dual higher gauge fields} or \textbf{chiral higher gauge fields} or \textbf{higher gauge fields with self-dual curvature}. (These are a higher degree/dimensional generalization of what in [[Yang-Mills theory]] are called [[Yang-Mills instanton]] field configurations.) Since imposing the self-duality constraint on the fields that enter the above [[action functional]] makes that functional vanish identically, self-dual higher gauge theory is notorious for being subtle in that either it does not have a [[Lagrangian]] field theory description, or else a somewhat intricate indirect one (e.g. \hyperlink{Witten96}{Witten 96, p. 7}, \hyperlink{BelovMooreI}{Belov-Moore 06a}). But instead one may regard the self-duality condition rather as part of the quantum theory (\hyperlink{Witten96}{Witten 96, p.8}, \hyperlink{Witten99}{Witten 99, section 3} \hyperlink{MW00}{DMW 00b, page 3}), namely as a choice of [[polarization]] of the [[phase space]] of an unconstrained theory in one dimension higher. By such as ``[[holographic principle]]'' the [[partition function]] of the self-dual theory on an $X$ of dimension $4 k +2$ is given by the [[state]] (wave function) of an abelian [[higher dimensional Chern-Simons theory]] in dimension $4 k + 3$. The way this works is understood in much mathematical detail for $k = 0$, see at \emph{[[AdS3-CFT2 and CS-WZW correspondence]]}. Motivated by this it has been proposed and studied in a fair bit of mathematical detail for $k = 1$ (\hyperlink{Witten96}{Witten 96}, [[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]]), see at \emph{[[M5-brane]]} and \emph{[[6d (2,0)-superconformal QFT]]} and \href{AdS-CFT#AdS7CFT6}{AdS7/CFT6}. In both these cases the [[higher gauge fields]] are [[cocycle]]s in [[ordinary differential cohomology]]. In (\hyperlink{MW00}{DMW 00}, \hyperlink{BelovMooreII}{Belov-Moore 06b}) it is suggested that similarly taking the self-dual fields to be cocycles in ([[differential K-theory|differential]]) [[complex K-theory]] produces the [[RR-fields]] of [[type II superstring theory]] in dimension 10. \hypertarget{Definition}{}\subsection*{{Definition (outline)}}\label{Definition} The [[holographic principle|holographic]] definition of self-dual abelian higher gauge field theory in dimension $4k+2$ given in (\hyperlink{Witten96}{Witten 96}, \hyperlink{Witten99}{Witten 99}) is in outline the following. Imagine the self-dual $2k$-form [[field (physics)|field]] on $\Sigma$ to be coupled to [[sources]] given by $(2k+1)$-form gauge fields, hence by [[cocycles]] in [[ordinary differential cohomology]] $[\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]$ of degree $(2k+2)$. One may try to write down a [[Lagrangian]] for this coupling (and that is what is discussed in (\hyperlink{BelovMooreI}{Belov-Moore 06I})) and the resulting [[partition function]] is then locally a function of the [[source fields]], hence a function on the [[intermediate Jacobian]] $[\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}]$ and globally (since the [[action functional]] will not be strictly [[gauge invariance|gauge invariant]]) a holomorphic [[section]] of a certain [[holomorphic line bundle]] on that space. However, no choice of [[Lagrangian]] will enforce a strictly self-dual gauge field (the anti-self dual component will remain in the space of fields, hence the partition function will depend on it). Therefore the idea is to turn this around, reject the idea that self-dual higher gauge theory is directly itself a [[Lagrangian quantum field theory]], and instead \emph{define} the partition function of the self-dual higher gauge theory (and thereby, indirectly, the self-dual higher gauge theory itself!) as being a certain section of a certain line bundle on the [[intermediate Jacobian]]. The question then is: which line bundle? Some plausibility arguments show that it must be a [[Theta characteristic]] bundle, hence a [[square root]] of the [[canonical bundle]] on the [[intermediate Jacobian]]. These have 1-dimensional spaces of holomorphic sections ([[theta functions]]) and hence any choice of that already uniquely fixes the would-be [[partition function]], too, up to a choice of global factor. (For that reason later authors often regard the partition function as a line bundle, somewhat abusing the terminology.) The claim then is that the correct choice of [[Theta characteristic]] to use is a [[quadratic refinement]] of the [[intersection pairing]] (the [[Beilinson-Deligne cup-product]]) \begin{displaymath} [\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}] \times [\Sigma, \mathbf{B}^{2k+1}\mathbb{G}^\times_{conn}] \stackrel{\cup}{\longrightarrow} [\Sigma, \mathbf{B}^{4k+3}\mathbb{G}^\times_{conn}] \stackrel{\int}{\longrightarrow} \mathbf{B}\mathbb{G}^\times_{conn} \,. \end{displaymath} Such [[quadratic refinement]] turns out to be given for $k = 0$ by a [[Spin structure]] (leading to ``[[Spin Chern-Simons theory]]'') and for $k =1$ by a [[Wu structure]] (and the latter case is the actual example of interest in (\hyperlink{Witten96}{Witten 96})). This is what the central theorem of ([[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]]) establishes rigorously. One notices now that while the self-dual $(4k+2)$-dimensional gauge field theory thus itself is not a [[Lagrangian quantum field theory]], it arises [[holographic principle|holographically]] form a field theory in dimension $4k+3$ that is, namely the [[intersection pairing]] above is the [[Lagrangian]] for [[higher dimensional Chern-Simons theory]] and the [[holomorphic line bundle]] which it gives rise to on the [[intermediate Jacobian]], is the [[prequantum line bundle]] of Chern-Simons theory, whose holomorphic sections are its [[quantum states]] (see at \emph{[[quantization of Chern-Simons theory]]}). From this perspective the [[square root]] [[quadratic refinement]] is the [[metaplectic correction]] in the [[geometric quantization]] of Chern-Simobs theory. This kind of relation \begin{tabular}{l|l|l} self-dual $(4k+2)$d gauge theory&&$(4k+3)$d Chern-Simons theory\\ \hline [[sources]]&$\leftrightarrow$&[[field (physics)\\ [[partition function]]/[[correlator]]&$\leftrightarrow$&[[wavefunction]]/[[quantum state]]\\ [[conformal blocks]]&$\leftrightarrow$&[[space of quantum states]]\\ \end{tabular} is the hallmark of the [[holographic principle]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{holographic_relation_to_higher_chernsimons_theory}{}\subsubsection*{{Holographic relation to higher Chern-Simons theory}}\label{holographic_relation_to_higher_chernsimons_theory} \hypertarget{idea_and_examples}{}\paragraph*{{Idea and examples}}\label{idea_and_examples} Generally, [[higher dimensional Chern-Simons theory]] in dimension $4k+3$ (for $k \in \mathbb{N}$) is [[holographic principle|holographically]] related to self-dual higher gauge theory in dimension $4k+2$ (at least in the abelian case). \begin{itemize}% \item $(k=0)$: ordinary 3-dimensional [[Chern-Simons theory]] is related to a [[string]] [[sigma-model]] on its boundary; \item $(k=1)$: [[7-dimensional Chern-Simons theory]] is related to a [[fivebrane]] model on its boundary; \item $(k=2)$: [[11-dimensional Chern-Simons theory]] (of [[field (physics)|fields]] which are [[cocycles]] in ([[twisted differential K-theory|twisted differential]]) [[complex K-theory]]) is related to a parts of a [[type II string theory]] on its boundary (or that of the space-filling [[D9-brane]], if one wishes) (\hyperlink{BelovMooreII}{Belov-Moore 06b}). \end{itemize} \hypertarget{ConformalStructureFromPolarization}{}\paragraph*{{Conformal structure from polarization}}\label{ConformalStructureFromPolarization} We indicate why [[higher dimensional Chern-Simons theory]] is -- if holographically related to anything -- holographically related to [[self-dual higher gauge theory]]. The [[phase space]] of [[higher dimensional Chern-Simons theory]] in [[dimension]] $4k+3$ on $\Sigma \times \mathbb{R}$ can be identified with the space of [[curvature|flat]] $2k+1$-forms on $\Sigma$. The [[presymplectic form]] on this space is given by the pairing \begin{displaymath} (\delta B_1, \delta B_2) \mapsto \int_\Sigma \delta B_1 \wedge \delta B_2 \end{displaymath} obtained as the [[integration of differential forms]] over $\Sigma$ of the [[wedge product]] of the two forms. The [[geometric quantization]] of the theory requires that we choose a [[polarization]] of the [[complexification]] of this space (split the space of forms into ``[[canonical coordinates]]'' and their ``[[canonical momenta]]''). One way to achieve this is to choose a [[conformal structure]] on $\Sigma$. The corresponding [[Hodge star operator]] \begin{displaymath} \star : \Omega^{2k+1}(\Sigma) \to \Omega^{2k+1}(\Sigma) \end{displaymath} provides the polarization by splitting into self-dual and anti-self-dual forms: notice that (by \href{Hodge+star+operator#eq:HodgeSquareOnRiemannian}{this Formula} formulas at [[Hodge star operator]]) we have on mid-dimensional forms \begin{displaymath} \star \star B = (-1)^{(2k+1)(4k+3)} B = - B \,. \end{displaymath} Therefore it provides a [[complex structure]] on $\Omega^{2k+1}(\Sigma) \otimes \mathbb{C}$. We see that the symplectic structure on the space of forms can equivalently be rewritten as \begin{displaymath} \begin{aligned} \int_X B_1 \wedge B_2 & = - \int_X B_1 \wedge \star \star B_2 \end{aligned} \,. \end{displaymath} Here on the right now the [[Hodge star operator|Hodge inner product]] of $B_1$ with $\star B_2$ appears, which is invariant under applying the Hodge star to both arguments. We then decompose $\Omega^{2k+1}(\Sigma)$ into the $\pm i$-[[eigenspace]]s of $\star$: say $B \in \Omega^{2k+1}(\Sigma)$ is \emph{imaginary self-dual} if \begin{displaymath} \star B = i B \end{displaymath} and \emph{imaginary anti-self-dual} if \begin{displaymath} \star B = - i B \,. \end{displaymath} Then for imaginary self-dual $B_1$ and $B_2$ we find that the symplectic pairing is \begin{displaymath} \begin{aligned} (B_1, B_2) &= -i \int_X B_1 \wedge \star B_2 \\ & = -i \int_X (\star B_1) \wedge \star (\star B_2) \\ & = +i \int_X B_1 \wedge \star B_2 \end{aligned} \,. \end{displaymath} Therefore indeed the symplectic pairing vanishes on the self-dual and on the anti-selfdual forms. Evidently these provide a decomposition into [[Lagrangian subspaces]]. (See also at \emph{[[Serre duality]]}.) Therefore a [[state]] of higher Chern-Simons theory on $\Sigma$ may locally be thought of as a function of the self-dual forms on $\Sigma$. Under holography this is (therefore) identified with the [[correlator]] of a [[self-dual higher gauge theory]] on $\Sigma$. \hypertarget{PartitionFunction}{}\subsubsection*{{Partition function}}\label{PartitionFunction} By the above discussion (\ldots{}) the [[partition function]] of self-dual higher gauge theory is given by (a multiple of) the unique [[holomorphic section]] of a [[square root]] of the [[line bundle]] classified by the [[secondary intersection pairing]]. (\hyperlink{Witten96}{Witten 96}, \hyperlink{HopkinsSinger}{Hopkins-Singer}). \hypertarget{cohomological_description}{}\subsubsection*{{Cohomological description}}\label{cohomological_description} [[!include moduli of higher lines -- table]] \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{ChiralBosonin2d}{}\subsubsection*{{Chiral boson in 2d (on the string)}}\label{ChiralBosonin2d} For $k = 0$ the self-dual theory abelian is that of a [[scalar field]] $\phi$ on a real-2-dimensional surface $\Sigma$ such that $\mathbf{d}\phi$ is self-dual. For any [[complex structure]] on $\Sigma$, making it a [[complex torus]], this makes $\phi$ a ``chiral'' boson, in the sense of a chiral half of the $U(1)$-[[WZW model]]. A quick review as a warm-up for the higher dimensional case is in (\hyperlink{Witten96}{Witten 96, section 2}). A detailed discussion is in (\hyperlink{GBMNV}{GBMNV}) and see the references at \emph{[[AdS3-CFT2 and CS-WZW correspondence]]}. \hypertarget{chiral_2form_in_6d_on_the_m5brane}{}\subsubsection*{{Chiral 2-form in 6d (on the M5-brane)}}\label{chiral_2form_in_6d_on_the_m5brane} The [[worldvolume]] theory of the [[M5-brane]], the [[6d (2,0)-superconformal QFT]], contains a self-dual 2-form field. Its \href{AdS-CFT#AdS7CFT6}{AdS7-CFT6 holographic} description by 7-dimensional Chern-Simons theory is due to (\hyperlink{Witten96}{Witten 96}). \hypertarget{RRFieldsin10d}{}\subsubsection*{{RR-Fields in 10d (on the ``9-brane'')}}\label{RRFieldsin10d} The [[RR-field]] in [[type II string theory]] are self-dual. Since the RR-fields are [[cocycles]] in ([[differential K-theory|differential]]) [[K-theory]], the proper discussion of this now involves generalizing from the above story [[ordinary cohomology]] and hence generalizing the concept of [[principally polarized variety|principally polarized]] [[intermediate Jacobians]] from ordinary cohomology to K-theory. (Of course one may, as a warmup, ``approximate'' K-theory classes on a 10-manifold by integral cohomology lifts of the degree-5 component of their [[Chern character]] and hence discuss that as abelian self-dual higher gauge theory in dimension 10. This is discussed in (\hyperlink{Witten99}{Witten 99, section 4})). The relevant analog of the [[intermediate Jacobian]] for K-tehory on a 10-manifold is naturally taken to be \begin{displaymath} K(X)\otimes_{\mathbb{Z}} \mathbb{R}/ K(X) \end{displaymath} (where the action is via the [[Chern character]]/realification $K \to K \otimes \mathbb{R}$). This is proposed and discussed in (\hyperlink{Witten99}{Witten 99, section 4.3}, \hyperlink{MooreWitten99}{Moore-Witten 99, section 3}, \hyperlink{MW00}{DMW 00, section 7.1}, \hyperlink{BelovMooreII}{Belov-Moore 06b, section 5}, \hyperlink{MSP11}{MPS 11}): where for ordinary cohomology the ([[quadratic refinement]] of the) [[intersection product]] (= [[Deligne-Beilinson cup product]] followed by [[fiber integration in ordinary differential cohomology]]) provided the [[symplectic structure]]/[[polarized variety|polarization]] of the intermediate Jacobian, here it is tensoring of [[Dirac operators]] followed by [[fiber integration in differential K-theory]], hence the [[index of a Dirac operator|index]] map in K-theory. See at \emph{\href{intermediate+Jacobian#ForComplexKTheory}{intermediate Jacobian -- For complex K-theory}}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include square roots of line bundles - table]] \begin{itemize}% \item [[Perry-Schwarz action]] \item [[Kervaire invariant]] \item [[bosonization]] \item [[higher dimensional Chern-Simons theory]] \item [[holographic principle]], [[AdS-CFT]] \item [[dual graviton]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Original articles on the general issue include \begin{itemize}% \item [[Marc Henneaux]], [[Claudio Teitelboim]], \emph{Dynamics of chiral (self-dual) $p$-forms}, Physics Letters B Volume 206, Issue 4, 2 June 1988, Pages 650-654 () \item [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], \emph{On Lorentz Invariant Actions for Chiral P-Forms}, Phys.Rev. D55 (1997) 6292-6298 (\href{https://arxiv.org/abs/hep-th/9611100}{arXiv:hep-th/9611100}) \item [[Edward Witten]], \emph{Duality Relations Among Topological Effects In String Theory}, JHEP 0005:031,2000 (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086}) \item Dmitriy Belov, [[Greg Moore]], \emph{Holographic Action for the Self-Dual Field} (\href{http://arxiv.org/abs/hep-th/0605038}{arXiv:hep-th/0605038}) \end{itemize} Surveys include \begin{itemize}% \item [[Greg Moore]], \emph{A Minicourse on Generalized Abelian Gauge Theory, Self-Dual Theories, and Differential Cohomology}, Lectures at Simons Center for Geometry and Physics (2011) (\href{http://www.physics.rutgers.edu/~gmoore/SCGP-Minicourse.pdf}{pdf}) \item [[Richard Szabo]], \emph{Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology} (\href{http://arxiv.org/abs/1209.2530}{arXiv:1209.2530}) \end{itemize} \hypertarget{for_the_m5sigma_model}{}\subsubsection*{{For the M5-sigma model}}\label{for_the_m5sigma_model} The [[sigma-model]] description of the (single) M5-brane of [[Green-Schwarz action functional]]-type was found in covariant form in \begin{itemize}% \item [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], \emph{Covariant Action for the Super-Five-Brane of M-Theory}, Phys. Rev. Lett. 78 (1997) 4332-4334 (\href{http://arxiv.org/abs/hep-th/9701149}{arXiv:hep-th/9701149}) \item [[Paolo Pasti]], [[Dmitri Sorokin]] and M. Tonin, \emph{Covariant Action for a D=11 Five-Brane with the Chiral Field}, Phys. Lett. B398 (1997) 41 (\href{https://arxiv.org/abs/hep-th/9701037}{arXiv:hep-th/9701037}) \end{itemize} following \begin{itemize}% \item [[Paul Townsend]], section 3.3 of \emph{D-branes from M-branes}, Phys. Lett. B373 (1996) 68-75 (\href{https://arxiv.org/abs/hep-th/9512062}{arXiv:hep-th/9512062}) (which did not yet have the [[Hopf-Wess-Zumino term]]) \end{itemize} and following the non-covariant form of the [[self-dual higher gauge field|self-duality mechanism]] ([[Perry-Schwarz action]]) of \begin{itemize}% \item [[Malcolm Perry]], [[John Schwarz]], \emph{Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory}, Nucl. Phys. B489 (1997) 47-64 (\href{http://arxiv.org/abs/hep-th/9611065}{arXiv:hep-th/9611065}) \item [[John Schwarz]], \emph{Coupling a Self-Dual Tensor to Gravity in Six Dimensions}, Phys.Lett.B395:191-195,1997 \item [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], \emph{World-Volume Action of the M Theory Five-Brane} (\href{http://arxiv.org/abs/hep-th/9701166}{arXiv:hep-th/9701166}) \end{itemize} Discussion in the [[superembedding approach]] is in \begin{itemize}% \item [[Dmitri Sorokin]], Section 5.2 of \emph{Superbranes and Superembeddings}, Phys.Rept.329:1-101, 2000 (\href{http://arxiv.org/abs/hep-th/9906142}{arXiv:hep-th/9906142}) \end{itemize} Discussion of the equivalence of these superficially different action functionals is in \begin{itemize}% \item [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], \emph{On the equivalence of different formulations of the M Theory five--brane}, Phys. Lett. B408 (1997) 135-141 (\href{http://arxiv.org/abs/hep-th/9703127}{arXiv:hep-th/9703127}) \end{itemize} \hypertarget{for_ordinary_higher_gauge_fields__ordinary_differential_cohomology}{}\subsubsection*{{For ordinary $U(1)$-higher gauge fields / ordinary differential cohomology}}\label{for_ordinary_higher_gauge_fields__ordinary_differential_cohomology} Self-duality for higher abelian gauge fields (in [[ordinary differential cohomology]]): Original reference on self-dual/chiral fields include \begin{itemize}% \item [[Xavier Bekaert]], [[Marc Henneaux]], \emph{Comments on Chiral $p$-Forms} (\href{http://arxiv.org/abs/hep-th/9806062}{arXiv:hep-th/9806062}) \item [[Mans Henningson]], [[Bengt Nilsson]], Per Salomonson, \emph{Holomorphic factorization of correlation functions in (4k+2)-dimensional (2k)-form gauge theory} (\href{http://arxiv.org/abs/hep-th/9908107}{arXiv:hep-th/9908107}) \item [[Mans Henningson]], \emph{The quantum Hilbert space of a chiral two-form in $d = 5 + 1$ dimensions} (\href{http://arxiv.org/abs/hep-th/0111150}{arxiv:hep-th/0111150}) \end{itemize} The chiral boson in 2d is discussed in detail in \begin{itemize}% \item [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Greg Moore]], P. Nelson, [[Cumrun Vafa]], \emph{Bosonization On Higher Genus Riemann Surfaces}, Commun. Math. Phys. 112 (1987) 503. \end{itemize} A quick exposition of the basic idea is in \begin{itemize}% \item [[Jacques Distler]], \emph{Actions for self-dual gauge fields} (\href{http://golem.ph.utexas.edu/~distler/blog/archives/000809.html}{blog}) \end{itemize} A precise formulation of the phenomenon in terms of [[ordinary differential cohomology]] is given in \begin{itemize}% \item [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], \emph{The Uncertainty of Fluxes} Commun. Math. Phys.271:247-274 (2007) (\href{http://arxiv.org/abs/hep-th/0605198}{arXiv:hep-th/0605198}) \item [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], \emph{Heisenberg Groups and Noncommutative Fluxes} , Annals Phys. 322:236-285 (2007) (\href{http://arxiv.org/abs/hep-th/0605200}{arXiv:hep-th/0605200}) \item Christian Becker, [[Marco Benini]], [[Alexander Schenkel]], [[Richard Szabo]], \emph{Abelian duality on globally hyperbolic spacetimes}, Commun. Math. Phys. 349, 361 (2017) (\href{https://arxiv.org/abs/1511.00316}{arXiv:1511.00316}) \end{itemize} The idea of describing self-dual higher gauge theory by [[holography]] with abelian [[higher dimensional Chern-Simons theory]] in one dimension higher originates in \begin{itemize}% \item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory} J. Geom. Phys.22:103-133,1997 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234}) \item [[Edward Witten]], \emph{Duality Relations Among Topological Effects In String Theory}, JHEP 0005:031,2000 (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086}) \end{itemize} Conceptual aspects of this are also discussed in section 6.2 of \begin{itemize}% \item S.L. Lyakhovich, [[A.A. Sharapov]], \emph{Quantizing non-Lagrangian gauge theories: an augmentation method}, JHEP 0701:047,2007 (\href{http://arxiv.org/abs/hep-th/0612086}{arXiv:hep-th/0612086}) \end{itemize} Motivated by this the [[ordinary differential cohomology]] of self-dual fields had been discussed in \begin{itemize}% \item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology, and M-Theory]]} \end{itemize} Discussion of [[S-duality]] in 6d self-dual higher gauge theory via [[non-commutative geometry|non-commutative]]-deformation: \begin{itemize}% \item [[Varghese Mathai]], [[Hisham Sati]], \emph{Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality}, J. Geom. Phys. 92:240-251, 2015 (\href{https://arxiv.org/abs/1404.2257}{arXiv:1404.2257}) \end{itemize} Discussion of the [[quantum anomaly]] of self-dual theories is in \begin{itemize}% \item Samuel Monnier, \emph{The anomaly line bundle of the self-dual field theory}, Comm. Math. Phys. 325 (2014) 41-72 (\href{http://arxiv.org/abs/1109.2904}{arXiv:1109.2904}) \item Samuel Monnier, \emph{The global gravitational anomaly of the self-dual field theory}, Comm. Math. Phys. 325 (2014) 73-104 (\href{http://arxiv.org/abs/1110.4639}{arXiv:1110.4639}, \href{http://www.physics.rutgers.edu/het/video/monnier11b.pdf}{pdf slides}) \end{itemize} (see also at [[M5-brane]] the \href{M5-brane#ReferencesAnomalyCancellation}{References on anomaly cancellation}). Discussion of the [[conformal blocks]] and [[geometric quantization]] of self-dual higher gauge theories is in \begin{itemize}% \item [[Kiyonori Gomi]], \emph{An analogue of the space of conformal blocks in $(4k+2)$-dimensions} (\href{http://www.math.kyoto-u.ac.jp/~kgomi/papers/acb.pdf}{pdf}) \item Samuel Monnier, \emph{Geometric quantization and the metric dependence of the self-dual field theory} (\href{http://arxiv.org/abs/1011.5890}{arXiv:1011.5890}) \end{itemize} For the case of nonabelian self-dual 1-form gauge fields see the references at \emph{[[Yang-Mills instanton]]}. \hypertarget{for_rrfields__differential_ktheory}{}\subsubsection*{{For RR-fields / differential K-theory}}\label{for_rrfields__differential_ktheory} The self-dual higher gauge field of the [[RR-field]] in terms of a [[quadratic form]] on [[differential K-theory]] is discussed originally around \begin{itemize}% \item [[Gregory Moore]], [[Edward Witten]], \emph{Self-Duality, Ramond-Ramond Fields, and K-Theory}, JHEP 0005:032,2000 (\href{http://arxiv.org/abs/hep-th/9912279}{arXiv:hep-th/9912279}) \end{itemize} and \begin{itemize}% \item [[Daniel Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]}. \end{itemize} for [[type I superstring theory]], and for [[type II superstring theory]] in \begin{itemize}% \item [[Edward Witten]], \emph{Duality Relations Among Topological Effects In String Theory}, JHEP 0005:031,2000 (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086}) \item [[Daniel Freed]], [[Michael Hopkins]], \emph{On Ramond-Ramond fields and K-theory}, JHEP 0005 (2000) 044 (\href{http://arxiv.org/abs/hep-th/0002027}{arXiv:hep-th/0002027}) \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}), summarised in \emph{A Derivation of K-Theory from M-Theory} (\href{http://arxiv.org/abs/hep-th/0005091}{arXiv:hep-th/0005091}) \item [[Stefan Müller-Stach]], Chris Peters, Vasudevan Srinivas, \emph{Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory},Journal de math\'e{}matiques pures et appliqu\'e{}es 98.5 (2012): 542-573 (\href{http://arxiv.org/abs/1105.4108}{arXiv:1105.4108}, \href{http://www-fourier.ujf-grenoble.fr/~peters/world.f/Torino.pdf}{pdf slides}) \end{itemize} with more refined discussion in [[twisted differential K-theory|twisted differential]] [[KR-theory]] 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:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides}) \end{itemize} See at \emph{[[orientifold]]} for more on this. The relation to [[11d Chern-Simons theory]] is made manifest in \begin{itemize}% \item Dmitriy Belov, [[Greg Moore]], \emph{Type II Actions from 11-Dimensional Chern-Simons Theories} (\href{http://arxiv.org/abs/hep-th/0611020}{arXiv:hep-th/0611020}) \end{itemize} Review is in (\hyperlink{Szabo12}{Szabo 12, section 3.6 and 4.6}). [[!redirects self-dual higher gauge theories]] [[!redirects higher self-dual gauge theory]] [[!redirects higher self-dual gauge theories]] [[!redirects self-dual higher gauge field]] [[!redirects self-dual higher gauge fields]] [[!redirects higher self-dual gauge field]] [[!redirects higher self-dual gauge field]] \end{document}