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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*{n-plectic geometry} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry} [[!include symplectic geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{PoissonLInfinityAlgebras}{Poisson $L_\infty$-algebras}\dotfill \pageref*{PoissonLInfinityAlgebras} \linebreak \noindent\hyperlink{prequantization}{Prequantization}\dotfill \pageref*{prequantization} \linebreak \noindent\hyperlink{review_of_the_symplectic_situation}{Review of the symplectic situation}\dotfill \pageref*{review_of_the_symplectic_situation} \linebreak \noindent\hyperlink{2plectic_geometry_and_courant_algebroids}{2-plectic geometry and Courant algebroids}\dotfill \pageref*{2plectic_geometry_and_courant_algebroids} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{central_extensions_under_geometric_quantization}{Central extensions under geometric quantization}\dotfill \pageref*{central_extensions_under_geometric_quantization} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{ReferencesGeneral}{General}\dotfill \pageref*{ReferencesGeneral} \linebreak \noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} $n$-plectic geometry is a generalization of [[symplectic geometry]] to [[higher category theory]]. It is closely related to [[multisymplectic geometry]] and [[n-symplectic manifold]]s. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{}\hypertarget{}{} For $n \in \mathbb{N}$, an \textbf{[[n-plectic vector space]]} is a [[vector space]] $V$ (over the [[real numbers]]) equipped with an $(n+1)$-linear skew function \begin{displaymath} \omega : \wedge^{n+1} V \to \mathbb{R} \end{displaymath} such that regarded as a function \begin{displaymath} V \to \wedge^n V^* \end{displaymath} is has trivial [[kernel]]. \end{defn} Let $X$ be a [[smooth manifold]], $\omega \in \Omega^{n+1}(X)$ a [[differential form]]. \begin{defn} \label{}\hypertarget{}{} We say $(X,\omega)$ is a \textbf{$n$-plectic manifold} if \begin{itemize}% \item $\omega$ is closed: $d \omega = 0$; \item for all $x \in X$ the map \begin{displaymath} \hat \omega : T_x X \to \Lambda^n T_x^n X \end{displaymath} given by contraction of vectors with forms \begin{displaymath} v \mapsto \iota_v \omega \end{displaymath} is [[injective]]. \end{itemize} \end{defn} See also the definition at \emph{[[multisymplectic geometry]]}. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item $n = 1$ -- this is the case of an ordinary [[symplectic manifold]] this appears in [[Hamiltonian mechanics]]; \item $n = 2$, this appears naturally in 1+1 dimensional [[quantum field theory]]. \end{itemize} \begin{enumerate}% \item For $X$ orientable, take $\omega$ the [[volume form]]. This is $(dim(X)-1)$-plectic. \item $\wedge^n T^* X \to X$ \item $G$ a [[compact space|compact]] [[simple group|simple]] [[Lie group]], let $\nu : (x,y,z) \mapsto \langle x, [y,z]\rangle$ be the canonical [[Lie algebra cohomology|Lie algebra 3-cocycle]] and extend it left-invariantly to a 3-form $\omega_\nu$ on $G$. Then $(G,\omega_\nu)$ is 2-plectic. \end{enumerate} \hypertarget{PoissonLInfinityAlgebras}{}\subsection*{{Poisson $L_\infty$-algebras}}\label{PoissonLInfinityAlgebras} To an ordinary [[symplectic manifold]] is associated its [[Poisson algebra]]. Underlying this is a [[Lie algebra]], whose Lie bracket is the \emph{Poisson bracket}. We discuss here how to an $n$-plectic manifold for $n \gt 1$ there is correspondingly assoociated not a Lie algebra, but a [[Lie n-algebra]]: the \textbf{[[Poisson bracket Lie n-algebra]]}. It is natural to call this a \textbf{Poisson Lie $n$-algebra} (see for instance at \emph{[[Poisson Lie 2-algebra]]}). (Not to be confused with the Lie algebra of a [[Poisson Lie group]], which is a Lie group that itself is equipped with a compatible [[Poisson manifold]] structure. It is a bit unfortunate that there is no better established term for the Lie algebra underlying a Poisson algebra apart from ``Poisson bracket''.) Consider the ordinary case in $n=1$ for how a [[Poisson algebra]] is obtained from a [[symplectic manifold]] $(X, \omega)$. Here \begin{displaymath} \hat \omega : T_x X \to T^*_x X \end{displaymath} is an [[isomorphism]]. Given $f \in C^\infty(X)$, $\exists ! \nu_f \in \Gamma(T X)$ such that $d f = - \omega(v_f, -)$ Define $\{f,g\} := \omega(v_f, v_g)$. Then $(C^\infty(X,), \{-,-\})$ is a [[Poisson algebra]]. We can generalize this to $n$-plectic geometry. Let $(X,\omega)$ be $n$-plectic for $n \gt 1$. Observe that then $\hat \omega : T_x X \to \wedge^n T_x X$ is no longer an [[isomorphism]] in general. \textbf{Definition} Say \begin{displaymath} \alpha \in \Omega^{n-1}(X) \end{displaymath} is \textbf{Hamiltonian} precisely if \begin{displaymath} \exists v_\alpha \in \Gamma(T X) \end{displaymath} such that \begin{displaymath} d \alpha = - \omega(v_\alpha, -) \,. \end{displaymath} This makes $v_\alpha$ uniquely defined. Denote the collection of [[Hamiltonian forms]] by $\Omega^{n-1}_{Hamilt}(X)$. Define a bracket \begin{displaymath} \{-,-\} : \Omega^{n-1}_{Hamilt}(X)^{\otimes_2} \to \Omega^{n-1}_{Hamilt}(X) \end{displaymath} by \begin{displaymath} \{\alpha, \beta\} = - \omega(v_\alpha, v_\beta, -, \cdots, -) \,. \end{displaymath} This satisfies \begin{enumerate}% \item k \begin{displaymath} d \{\alpha, \beta\} = - \omega([v_\alpha, v_\beta], -, \cdots, -) \,. \end{displaymath} \item $\{-,-\}$ is skew-symmetric; \item $\{\alpha_1, \{\alpha_2, \alpha_3\}\}$ + cyclic permutations\newline $d \omega(v_{\alpha_1}, v_{\alpha_2}, v_{\alpha_3}, -, \cdots)$. \end{enumerate} So the Jacobi dientity fails up to an exact term. This will yield the structure of an [[L-infinity algebra]]. \textbf{Proposition} Given an $n$-plectic manifold $(X,\omega)$ we get a [[Lie n-algebra]] structure on the complex \begin{displaymath} C^\infty(X) \stackrel{d_{dR}}{\to} \Omega^1(X) \stackrel{d_{dR}}{\to} \to \cdots \to \Omega^{n-1}_{Hamilt}(X) \end{displaymath} (where the rightmost term is taken to be in degree 0). where \begin{itemize}% \item the unary bracket is $d_{dR}$; \item the $k$-ary bracket is \begin{displaymath} [\alpha_1, \cdots, \alpha_k] = \left\{ \itexarray{ \pm \omega(v_{\alpha_1}, \cdots, v_{\alpha_k}) & if \forall i : \alpha_i \in \Omega^{n-1}_{Hamilt}(X) \\ 0 & otherwise } \right. \end{displaymath} \end{itemize} This is the \textbf{[[Poisson bracket Lie n-algebra]]}. This appears as (\hyperlink{Rogers11}{Rogers 11, theorem 3.14}). For $n = 1$ this recovers the definition of the [[Lie algebra]] underlying a [[Poisson algebra]]. \hypertarget{prequantization}{}\subsection*{{Prequantization}}\label{prequantization} \hypertarget{review_of_the_symplectic_situation}{}\subsubsection*{{Review of the symplectic situation}}\label{review_of_the_symplectic_situation} Recall for $n=1$ the mechanism of [[geometric quantization]] of a [[symplectic manifold]]. Given a 2-form $\omega$ and the corresponding complex [[line bundle]] $P$, consider the [[Atiyah Lie algebroid]] sequence \begin{displaymath} ad P \to T P/U(1) \to T X \end{displaymath} The smooth [[section]]s of $T P/U(1) \to X$ are the $U(1)$ invariant [[vector field]]s on the total space of $P$. Using a [[connection on a bundle|connection]] $\nabla$ on $P$ we may write such a section as \begin{displaymath} s(v) + f \partial_t \end{displaymath} for $v \in \Gamma(T X)$ a vector field downstairs, $s(v)$ a horizontal lift with respect to the given connection and $f \in C^\infty(X)$. Locally on a suitable patch $U \subset X$ we have that $s(V)|_U = v|_U + \iota_v \theta_i|_U$ . We say that $\tilde v = s(v) + f \partial_t$ \textbf{preserves the splitting} iff $\forall u \in \Gamma(X)$ we have \begin{displaymath} [\tilde v, s(u)] = s([v,u]) \,. \end{displaymath} One finds that this is the case precisely if \begin{displaymath} d f = - \iota_v \omega \,. \end{displaymath} This gives a [[homomorphism]] of [[Lie algebra]]s \begin{displaymath} C^\infty(X) \to \Gamma(T P / U(1)) \end{displaymath} \begin{displaymath} f \mapsto s(v_f) + f \partial_t \,. \end{displaymath} \hypertarget{2plectic_geometry_and_courant_algebroids}{}\subsubsection*{{2-plectic geometry and Courant algebroids}}\label{2plectic_geometry_and_courant_algebroids} We consder now [[geometric quantization|prequantization]] of [[2-plectic manifolds]]. Let $(X,\omega)$ be a 2-plectic manifold such that the [[de Rham cohomology]] class $[\omega]/2 \pi i$ is in the image of [[integral cohomology]] (Has integral periods.) We can form a cocycle in [[Deligne cohomology]] from this, encoding a [[bundle gerbe]] with connection. On a [[cover]] $\{U_i \to X\}$ of $X$ this is given in terms of [[Cech cohomology]] by data \begin{itemize}% \item $(g_{i j k} : U_{i j k} \to U(1)) \in C^\infty(U_{i j k}, U(1))$ \item $A_{i j} \in \Omega^1(U_{i j})$; \item $B_i \in \Omega^2(U_i)$ \end{itemize} satisfying a cocycle condition. Now recall that an exact [[Courant algebroid]] is given by the following data: \begin{itemize}% \item a [[vector bundle]] $E \to X$; \item an \emph{anchor} morphism $\rho : E \to T X$ to the [[tangent bundle]]; \item an \emph{inner product} $\langle -,-\rangle$ on the fibers of $E$; \item a bracket $[-,-]$ on the sections of $E$. \end{itemize} Satisfying some conditions. The fact that the [[Courant algebroid]] is \emph{exact} means that \begin{displaymath} 0 \to T^* X \to E \to T X \to 0 \end{displaymath} is an [[exact sequence]]. The \textbf{standard Courant algebroid} is the example where \begin{itemize}% \item $E = T X \oplus T^* X$; \item $\langle v_1 + \alpha_1, v_2 + \alpha_2\rangle = \alpha_2(v_1) + \alpha_1(v_2)$; \item the bracket is the skew-symmetrization of the Dorfman bracket \begin{displaymath} (v_1 + \alpha_1, v_2 + \alpha_2) = [v_1, v_2] - \mathbb{L}_{v_1}\alpha_2 - (d \alpha_1)(v_2,-) \end{displaymath} \end{itemize} Now with respect to the above Deligne cocycle, build a Courant algebroid as follows: \begin{itemize}% \item on each patch $U_i$ is is the standard Courant algebroid $E_i := T U_i \oplus T^* U_i$; \item glued together on double intersections using the $d A_{i j}$ \end{itemize} This gives an exact Courant algebroid $E \to X$ as well as a splitting $s : T X \to E$ given by the $\{B_i\}$. The bracket on this $E$ is given by the skew-symmetrization of \begin{displaymath} [ [ s(v_1) \alpha_1, s(v_2) + \alpha_2 ] ] = s([v_1, v_2]) + \mathcal{L}_{v_1} \alpha_2 - (d \alpha_2)(v_2, -) - \omega(v_1, v_2, \cdots) \,. \end{displaymath} Here a section $e = s(v) + ...$ preserves the splitting precisely if for all $u \in \Gamma(T X)$ we have \begin{displaymath} [ [ e, s(u)] ]_D = s([v,u]) \end{displaymath} exactly if $\alpha$ is Hamiltonian and $v = v_\alpha$. \textbf{Theorem} Recall that to every [[Courant algebroid]] $E$ is associated a [[Lie 2-algebra]] $L_\infty(E)$. Then: we have an embedding of [[L-infinity algebra]]s \begin{displaymath} \phi : L_\infty(X,\omega) \to L_\infty(E) \end{displaymath} given by $\phi(\alpha) = s(v_\alpha) + \alpha$. \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[geometry of physics -- prequantum geometry]] \end{itemize} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{central_extensions_under_geometric_quantization}{}\subsubsection*{{Central extensions under geometric quantization}}\label{central_extensions_under_geometric_quantization} [[!include geometric quantization extensions - table]] \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[2-plectic geometry]] \item [[symplectic infinity-groupoid]] \item [[higher geometric quantization]] \end{itemize} [[!include Isbell duality - table]] [[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]] \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{ReferencesGeneral}{}\subsubsection*{{General}}\label{ReferencesGeneral} The observation that the would-be Poisson bracket induced by a higher degree closed form extends to the [[Poisson bracket Lie n-algebra]] is due to \begin{itemize}% \item [[Chris Rogers]], \emph{$L_\infty$ algebras from multisymplectic geometry} , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (\href{http://arxiv.org/abs/1005.2230}{arXiv:1005.2230}, \href{http://link.springer.com/article/10.1007%2Fs11005-011-0493-x}{journal}). \item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (2011) (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068}) \end{itemize} with first discussion of application to [[prequantization]] in \begin{itemize}% \item [[Chris Rogers]], \emph{2-plectic geometry, Courant algebroids, and categorified prequantization} , \href{http://arxiv.org/abs/1009.2975}{arXiv:1009.2975}. \item [[Chris Rogers]], \emph{Higher geometric quantization}, talk at \emph{Higher Structures 2011} in G\"o{}ttingen (\href{http://www.crcg.de/wiki/Higher_geometric_quantization}{pdf slides}) \end{itemize} Discussion in the more general context of [[higher differential geometry]]/[[extended prequantum field theory]] is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]}, \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]} \end{itemize} See also the references at [[multisymplectic geometry]] and [[n-symplectic manifold]]. A [[higher differential geometry]]-generalization of [[contact geometry]] in line with [[multisymplectic geometry]]/$n$-plectic geometry is discussed in \begin{itemize}% \item Luca Vitagliano, \emph{L-infinity Algebras From Multicontact Geometry} (\href{http://arxiv.org/abs/1311.2751}{arXiv.1311.2751}) \end{itemize} \hypertarget{applications}{}\subsubsection*{{Applications}}\label{applications} Some more references on application, on top of those mentioned in the articles \hyperlink{ReferencesGeneral}{above}. A survey of some (potential) applications of 2-plectic geometry in [[string theory]] and [[M2-brane]] models is in section 2 of \begin{itemize}% \item [[Christian Saemann]], [[Richard Szabo]], \emph{Groupoid quantization of loop spaces} (\href{http://arxiv.org/abs/1203.5921}{arXiv:1203.5921}) \end{itemize} and in \begin{itemize}% \item [[Christian Saemann]], [[Richard Szabo]], \emph{Quantization of 2-Plectic Manifolds} (\href{http://arxiv.org/abs/1106.1890}{arXiv:1106.1890}) \end{itemize} [[!redirects n-plectic manifold]] [[!redirects n-plectic manifolds]] [[!redirects Hamiltonian 1-form]] [[!redirects Hamiltonian 1-forms]] [[!redirects Hamiltonian n-form]] [[!redirects Hamiltonian n-forms]] [[!redirects n-plectic structure]] [[!redirects n-plectic structures]] [[!redirects 3-plectic geometry]] [[!redirects pre-n-plectic manifold]] [[!redirects pre-n-plectic manifolds]] \end{document}