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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*{Perry-Schwarz action} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak \noindent\hyperlink{for_the_selfdual_field_and_trivial_target_space_metric}{For the self-dual field and trivial target space metric}\dotfill \pageref*{for_the_selfdual_field_and_trivial_target_space_metric} \linebreak \noindent\hyperlink{worldvolume_and_selfduality}{Worldvolume and self-duality}\dotfill \pageref*{worldvolume_and_selfduality} \linebreak \noindent\hyperlink{compactification}{$S^1$-compactification}\dotfill \pageref*{compactification} \linebreak \noindent\hyperlink{selfduality_after_compactification}{Self-duality after $S^1$-compactification}\dotfill \pageref*{selfduality_after_compactification} \linebreak \noindent\hyperlink{the_gauge_field}{The gauge field}\dotfill \pageref*{the_gauge_field} \linebreak \noindent\hyperlink{weak_selfduality_and_psequations_of_motion}{Weak self-duality and PS-equations of motion}\dotfill \pageref*{weak_selfduality_and_psequations_of_motion} \linebreak \noindent\hyperlink{lagrangian_density}{Lagrangian density}\dotfill \pageref*{lagrangian_density} \linebreak \noindent\hyperlink{example_reduction_to_5d_maxwell_theory}{Example: Reduction to 5d Maxwell theory}\dotfill \pageref*{example_reduction_to_5d_maxwell_theory} \linebreak \noindent\hyperlink{for_the_full_m5brane_sigma_model}{For the full M5-brane sigma model}\dotfill \pageref*{for_the_full_m5brane_sigma_model} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A [[Lagrangian density]]/[[action functional]] for the [[self-dual higher gauge field]] in 6d and/or the [[M5-brane]] [[Green-Schwarz sigma model]], after [[KK-compactification]] to 5 [[worldvolume]] [[dimensions]]. \hypertarget{details}{}\subsection*{{Details}}\label{details} \hypertarget{for_the_selfdual_field_and_trivial_target_space_metric}{}\subsubsection*{{For the self-dual field and trivial target space metric}}\label{for_the_selfdual_field_and_trivial_target_space_metric} We review the definitions from \hyperlink{PerrySchwarz96}{Perry-Schwarz 96}, for the [[worldvolume]] [[Lagrangian density]] of just the [[self-dual higher gauge field]] on a [[circle principal bundle]]-[[worldvolume]] for would-be [[target space]] being [[Minkowski spacetime]]. In doing so, we translate to [[coordinate chart|coordinate]]-invariant [[Cartan calculus]]-formalism and generalized to [[KK-compactification]] on possibly non-[[trivial bundle|trivial]] [[circle principal bundle]]: \hypertarget{worldvolume_and_selfduality}{}\paragraph*{{Worldvolume and self-duality}}\label{worldvolume_and_selfduality} Let \begin{displaymath} (\Sigma^6, g) \end{displaymath} be a [[pseudo-Riemannian manifold]] of [[dimension]] 6 and of [[signature of a quadratic form|signature]] $(-,+,+,+,+,+)$, to be called the \emph{[[worldvolume]]}. In this dimension and with this signature, the [[Hodge star operator]] squares to $+1$. This allows to consider for a [[differential 3-form]] \begin{displaymath} H \;\in\; \Omega^3\big(\Sigma^6\big) \end{displaymath} the condition that it be self-dual (\hyperlink{PerrySchwarz96}{PS 96 (2)}) \begin{equation} H \;=\; \star H \,. \label{SelfDualityIn6d}\end{equation} We will assume in the following that $H$ is [[exact differential form]], hence that there exists a [[differential 2-form]] \begin{displaymath} B \in \Omega^2\big( \Sigma^6 \big) \end{displaymath} such that (\hyperlink{PerrySchwarz96}{PS 96 (4)}) \begin{displaymath} H = d B \,. \end{displaymath} \hypertarget{compactification}{}\paragraph*{{$S^1$-compactification}}\label{compactification} Consider then on $\Sigma^6$ the structure of an $S^1 = U(1)$-[[circle principal bundle|principal bundle]] \begin{equation} \, \label{FibrationWorldvolume}\end{equation} $\backslash$begin\{xymatrix\} S{\tt \symbol{94}}1 $\backslash$arr \& $\backslash$Sigma{\tt \symbol{94}}6 $\backslash$ard $\backslash$ \& $\backslash$Sigma{\tt \symbol{94}}5 $\backslash$end\{xymatrix\} Write \begin{equation} v^5 \in \Gamma( T \Sigma^6 ) \label{FiberVectorField}\end{equation} for the [[vector field]] which reflects the infinitesimal [[circle group]]-[[action]] on \eqref{FibrationWorldvolume}. We will write \begin{displaymath} \mathcal{L}_{v^5} \;=\; \big[d, \iota_{v^5} \big] \;\colon\; \Omega^\bullet\big( \Sigma^6 \big) \longrightarrow \Omega^\bullet\big( \Sigma^6 \big) \end{displaymath} for the [[Lie derivative]] of [[differential forms]] along $v^5$, and make use of [[Cartan's magic formula]] expressing it as an [[anti-commutator]], as shown. Next consider an [[Ehresmann connection]] on the $S^1$-bundle \eqref{FibrationWorldvolume}, hence a [[differential 1-form]] \begin{displaymath} \theta^5 \;\in\; \Omega^1\big( \Sigma^6 \big) \end{displaymath} such that \begin{equation} \iota_{v^5} \theta^5 = 1 \phantom{AA} \text{and} \phantom{AA} \mathcal{L}_{v^5} \theta^5 = 0 \label{EhresmannConditions}\end{equation} So in particular \begin{displaymath} \theta^5 \wedge \iota_{v^5} \;:\; \Omega^\bullet\big( \Sigma^6\big) \longrightarrow \Omega^\bullet\big( \Sigma^6\big) \end{displaymath} is a [[projection operator]]: \begin{displaymath} \theta^5 \wedge \iota_{v^5} \circ \theta^5 \wedge \iota_{v^5} \;=\; \theta^5 \wedge \iota_{v^5} \end{displaymath} The complementary projection is that onto [[horizontal differential forms]] \begin{displaymath} (-)^{\mathrm{hor}} := \big(\mathrm{id} - \theta^5 \iota_{v^5}) \;:\; \Omega^\bullet\big( \Sigma^6\big) \longrightarrow \Omega^\bullet\big( \Sigma^6\big) \end{displaymath} We require $v^5$ \eqref{FiberVectorField} to be a [[spacelike]] [[isometry]]. This means that \begin{equation} \star \circ \iota_{v^5} = - \theta^5 \wedge \circ \star \;:\; \Omega^3\big( \Sigma^6\big) \longrightarrow \Omega^4\big( \Sigma^6 \big) \label{HodgeStarCommutingWithIsometryContraction}\end{equation} \hypertarget{selfduality_after_compactification}{}\paragraph*{{Self-duality after $S^1$-compactification}}\label{selfduality_after_compactification} Set (\hyperlink{PerrySchwarz96}{PS 96 (5)}) \begin{equation} \mathcal{F} \;\coloneqq\; \iota_{v^5} H \label{DefOfCalF}\end{equation} and (\hyperlink{PerrySchwarz96}{PS 96 (6)}) \begin{equation} \tilde H \;\coloneqq\; \iota_{v^5} \star H \label{DefOfTildeH}\end{equation} With this notation the self-duality condition \eqref{SelfDualityIn6d} is equivalently (\hyperlink{PerrySchwarz96}{PS 96 (9)}, see \eqref{EquivalentIncarnationsOfSelfDuality} below): \begin{equation} \mathcal{F} \;=\; \tilde H \label{SelfDualityIntermsOfcalF}\end{equation} To make this fully explicit, notice that we have the following chain of logical equivalences: \begin{equation} \begin{aligned} \big( H = \star H \big) & \Leftrightarrow \left( \itexarray{ & \phantom{\text{and}\;} \iota_{v_5} H = \iota_{v^5} \star H \\ & \text{and}\; \theta^5 \wedge H = \theta^5 \wedge \star H } \right) \\ & \Leftrightarrow \big( \iota_{v^5} H = \iota_{v^5} \star H \big) \\ &\Leftrightarrow \big( \mathcal{F} \;=\; \widetilde H \big) \end{aligned} \label{EquivalentIncarnationsOfSelfDuality}\end{equation} Here the first step is decomposition of the self-duality equation into components, the second step follows by \eqref{HodgeStarCommutingWithIsometryContraction} and the third step invokes the definitions \eqref{DefOfCalF} and \eqref{DefOfTildeH} and the fourth step the equality \eqref{DecompositionOfCalF}. \hypertarget{the_gauge_field}{}\paragraph*{{The gauge field}}\label{the_gauge_field} Define the [[vector potential]] (\hyperlink{PerrySchwarz96}{PS 96 above (4)}) \begin{equation} A \;\coloneqq\; - \iota_{v^5} B \label{AField}\end{equation} With this we have \begin{displaymath} B \;=\; A \wedge \theta^5 + B^{\mathrm{hor}} \,. \end{displaymath} Set also (\hyperlink{PerrySchwarz96}{PS 96 above (4)}) \begin{displaymath} F \;\coloneqq\; \big( d A \big)^{\mathrm{hor}} \end{displaymath} then (\hyperlink{PerrySchwarz96}{PS 96 (5)}) \begin{equation} \begin{aligned} \mathcal{F} &\coloneqq \iota_{v^5} H \\ & = \iota_{v^5} d B \\ & = - d \iota_{v^5} B + [\iota_{v^5}, d] B \\ & = d A + \mathcal{L}_5 B \\ & = F + \theta^5 \wedge \iota_{v^5} d A + \mathcal{L}_5 B^{\mathrm{hor}} + \underset{ = -\theta^5 \wedge \mathcal{L}_{v^5} A }{ \underbrace{ \mathcal{L}_5 \theta^5 \wedge \iota_{v^5} B } } \\ & = F + \mathcal{L}_{v^5} B^{\mathrm{hor}} \end{aligned} \label{DecompositionOfCalF}\end{equation} where in the last step under the brace we used \eqref{EhresmannConditions} and \eqref{AField}. Hence in terms of $F$ and $B^{\mathrm{hor}}$ the self-duality condition \eqref{SelfDualityIn6d}, \eqref{SelfDualityIntermsOfcalF} is equivalently expressed as on the right of the following \begin{equation} \big( H = \star H \big) \;\Leftrightarrow\; \big( \widetilde H \;=\; F + \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \label{SelfDualityDecomposition}\end{equation} \hypertarget{weak_selfduality_and_psequations_of_motion}{}\paragraph*{{Weak self-duality and PS-equations of motion}}\label{weak_selfduality_and_psequations_of_motion} Notice that \begin{displaymath} \begin{aligned} \theta^5 \wedge d \big( (d A)^{hor} \big) & = \theta^5 \wedge d \big( d A - \theta^5 \wedge \iota_{v_5} d A\big) \\ & = \theta^5 \wedge (d \theta^5) \wedge \iota_{v_5} d A \end{aligned} \,. \end{displaymath} Hence assume now hat the Ehresmann connection is [[flat connection|flat]], hence $d \theta^5 = 0$. Then the self-duality condition in the form \eqref{SelfDualityDecomposition} \begin{displaymath} \widetilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \;=\; (d A)^{\mathrm{hor}} \end{displaymath} implies, after applying $\theta^5 \wedge d$ to both sides, the second-order equation (\hyperlink{PerrySchwarz96}{PS 96 (16)}) \begin{equation} (H = \star H) \;\;\;\Rightarrow\;\;\; \theta^5 \wedge d \big( \widetilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \;=\; 0 \label{TheEquationOfMotion}\end{equation} This equation by itself is hence a weakened form of the self-duality condition, a kind of ``self-duality up to horizontally closed terms''. The proposal of \hyperlink{PerrySchwarz96}{Perry-Schwarz 96, Sec. 2} is to take this as the relevant [[equation of motion]] for the theory on $S^1$. \hypertarget{lagrangian_density}{}\paragraph*{{Lagrangian density}}\label{lagrangian_density} Therefore one islooking now for a [[Lagrangian density]] whose [[Euler-Lagrange equations]] are \eqref{TheEquationOfMotion}: The \emph{Perry-Schwarz-Lagrangian} is (\hyperlink{PerrySchwarz96}{PS 96 (17)}) \begin{equation} L \;\coloneqq\; - \tfrac{1}{2} \big( \tilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge \star \tilde H \label{CartanCalculusPerrySchwarzLagrangian}\end{equation} With \eqref{HodgeStarCommutingWithIsometryContraction} the Lagrangian \eqref{CartanCalculusPerrySchwarzLagrangian} becomes \begin{equation} \begin{aligned} L & = - \tfrac{1}{2} \big( \tilde H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge H \wedge \theta^5 \\ & = - \tfrac{1}{2} \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge H \wedge \theta^5 \end{aligned} \label{LagrangianUsingIsometry}\end{equation} where in the second line we inserted the definition \eqref{DefOfTildeH}. Notice that \eqref{LagrangianUsingIsometry} is the quadratic part of the following form-valued [[bilinear form]] on 2-form fields: \begin{displaymath} (B, B^\prime) \;\mapsto\; - \tfrac{1}{2} \big( \iota_{v^5} \star (d B) - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge (d B^\prime) \wedge \theta^5 \end{displaymath} Moreover, this bilinear form is \emph{symmetric} up to a total derivative. For the first summand this is manifest from its incarnation in \eqref{CartanCalculusPerrySchwarzLagrangian}, since the Hodge pairing is symmetric, and for the second term this follows by ``local integration by parts''. As a consequence, the [[Euler-Lagrange equations]] of the Perry-Schwarz Lagrangian density \eqref{LagrangianUsingIsometry} may be computed from twice the variation of just the second factor \begin{displaymath} \begin{aligned} \delta L_{\mathrm{sd}} & = 2 \Big( - \tfrac{1}{2} \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge d(\delta B) \wedge \theta^5 \Big) \\ & = \Big( d \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \Big) \wedge (\delta B) \wedge \theta^5 + d(\cdots) \end{aligned} \end{displaymath} to indeed be \eqref{TheEquationOfMotion}: \begin{equation} \theta^5 \wedge d \big( \iota_{v^5} \star H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \;=\; 0 \,. \label{EquationsOfMotion}\end{equation} Notice that if we do use the self-duality condition \eqref{SelfDualityIn6d} on the Perry-Schwarz Lagrangian \eqref{LagrangianUsingIsometry} it becomes \begin{equation} L_{\mathrm{sd}} \;=\; - \tfrac{1}{2} \big( \iota_{v^5} H - \mathcal{L}_{v^5} B^{\mathrm{hor}} \big) \wedge H \wedge \theta^5 \phantom{AAAAA} \text{if} \;\; H = \star H \label{NonCovariantLagrangianInFlatSpacetime}\end{equation} \hypertarget{example_reduction_to_5d_maxwell_theory}{}\paragraph*{{Example: Reduction to 5d Maxwell theory}}\label{example_reduction_to_5d_maxwell_theory} Consider the special case that \begin{displaymath} \mathcal{L}_{v^5} B = 0 \,, \end{displaymath} which corresponds to keeping only the 0-mode under [[KK-compactification]] along the circle fiber. Then \eqref{DecompositionOfCalF} becomes \begin{displaymath} \mathcal{F} = F \end{displaymath} and so the self-duality condition \eqref{SelfDualityDecomposition} now becomes \begin{displaymath} \iota_{v^5} \star H \;=\; F \,. \end{displaymath} which means that \begin{displaymath} H = F \wedge \theta^5 + \star_5 F \end{displaymath} \begin{quote}% (check relative sign) \end{quote} Since $d H = d \circ d B = 0$, this implies \begin{displaymath} \begin{aligned} \big( d H = 0 \big) & \Leftrightarrow \Big( d \big( F \wedge \theta^5 + \star_5 F \big) = 0 \Big) \\ & \Leftrightarrow \left( \left\{ \itexarray{ d_5 F & = 0 \\ d_5 \star_5 F & = 0 } \right. \right) \end{aligned} \end{displaymath} These are of course [[Maxwell's equations]] on $\Sigma^5$. (\hyperlink{PerrySchwarz96}{PS 96 above (16)}) \hypertarget{for_the_full_m5brane_sigma_model}{}\subsubsection*{{For the full M5-brane sigma model}}\label{for_the_full_m5brane_sigma_model} (\ldots{}) (\hyperlink{Schwarz97}{Schwarz 97}, \hyperlink{APPS97}{APPS 97}) (\ldots{}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[D=5 super Yang-Mills theory]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The Perry-Schwarz action is due to \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 (\href{http://cds.cern.ch/record/317663}{cds:317663}, ) \item [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], \emph{World-Volume Action of the M Theory Five-Brane}, Nucl.Phys. B496 (1997) 191-214 (\href{http://arxiv.org/abs/hep-th/9701166}{arXiv:hep-th/9701166}) \end{itemize} A similar construction but with compactification along the [[timelike]] direction is due to \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 () \end{itemize} The covariant version via a [[scalar field|scalar]] [[auxiliary field]] is due to \begin{itemize}% \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 [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario 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}) \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}) \end{itemize} Speculations about non-abelian generalizations (for several coincident M5-branes): \begin{itemize}% \item [[Chong-Sun Chu]], \emph{A proposal for the worldvolume action of multiple M5-branes}, 2013 (\href{http://hep.phy.ntnu.edu.tw/old_version/talks/2013/2013-05-09Chong-SunChu.pdf}{pdf}) \end{itemize} The above text follows \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 2 of: \emph{[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]} (\href{https://arxiv.org/abs/1908.00042}{arXiv:1908.00042}) \end{itemize} [[!redirects Perry-Schwarz actions]] [[!redirects Perry-Schwarz action functional]] [[!redirects Perry-Schwarz action functionals]] [[!redirects Perry-Schwarz Lagrangian]] [[!redirects Perry-Schwarz Lagrangians]] [[!redirects Perry-Schwarz Lagrangian density]] [[!redirects Perry-Schwarz Lagrangian densities]] \end{document}