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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*{orbit method} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory} [[!include representation theory - contents]] \hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization} [[!include geometric quantization - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{DefinitionsAndConstructions}{Definitions and constructions}\dotfill \pageref*{DefinitionsAndConstructions} \linebreak \noindent\hyperlink{the_group_and_its_lie_algebra}{The group and its Lie algebra}\dotfill \pageref*{the_group_and_its_lie_algebra} \linebreak \noindent\hyperlink{TheCoadjointOrbitAndTheCosetSpaceAndFlagManifold}{The coadjoint orbit and the coset space/ flag manifold}\dotfill \pageref*{TheCoadjointOrbitAndTheCosetSpaceAndFlagManifold} \linebreak \noindent\hyperlink{the_symplectic_form}{The symplectic form}\dotfill \pageref*{the_symplectic_form} \linebreak \noindent\hyperlink{the_prequantum_bundle}{The prequantum bundle}\dotfill \pageref*{the_prequantum_bundle} \linebreak \noindent\hyperlink{the_hamiltonian_action__coadjoint_moment_map}{The Hamiltonian $G$-action / coadjoint moment map}\dotfill \pageref*{the_hamiltonian_action__coadjoint_moment_map} \linebreak \noindent\hyperlink{WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit}{Wilson loops and 1d Chern-Simons $\sigma$-models with target the coadjoint orbit}\dotfill \pageref*{WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit} \linebreak \noindent\hyperlink{FormulationInHigherGeometry}{Formulation in higher geometry}\dotfill \pageref*{FormulationInHigherGeometry} \linebreak \noindent\hyperlink{FormulationInHigherGeometrySurvey}{Survey}\dotfill \pageref*{FormulationInHigherGeometrySurvey} \linebreak \noindent\hyperlink{FormulationInHigherGeometryDefinitions}{Definitions and constructions}\dotfill \pageref*{FormulationInHigherGeometryDefinitions} \linebreak \noindent\hyperlink{GaugeAndGravityWilsonLoops}{Nonabelian charged particle trajectories -- Wilson loops}\dotfill \pageref*{GaugeAndGravityWilsonLoops} \linebreak \noindent\hyperlink{ExtendedChern-SimonsTheoryAndWilsonLoops}{3d Chern-Simons theory with Wilson loops}\dotfill \pageref*{ExtendedChern-SimonsTheoryAndWilsonLoops} \linebreak \noindent\hyperlink{DiracInduction}{Formulation in equivariant K-theory (Dirac induction)}\dotfill \pageref*{DiracInduction} \linebreak \noindent\hyperlink{formulation_in_equivariant_elliptic_cohomology}{Formulation in equivariant elliptic cohomology}\dotfill \pageref*{formulation_in_equivariant_elliptic_cohomology} \linebreak \noindent\hyperlink{theorems}{Theorems}\dotfill \pageref*{theorems} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{Orbit method} (or \emph{Kirillov's method}, or \emph{method of coadjoint orbits}) is a method in [[geometric representation theory]] concerned with identifying [[unitary representations]] of [[Lie groups]] with the canonical $G$-[[actions]] on spaces of [[sections]] of certain [[line bundles]] over [[coadjoint orbits]] of the Lie group. In terms of [[quantum physics]] this realizes $G$-[[representations]] as actions of [[global gauge groups]] of [[quantum operators]] on [[spaces of quantum states]] under \emph{[[geometric quantization]]}. More in detail, the dual $\mathfrak{g}^*$ of a (say finite-dimensional real) [[Lie algebra]] has a canonical structure of a [[Poisson manifold]] -- its \emph{[[Lie-Poisson structure]]} --, namely for any $a\in \mathfrak{g}^*$, \begin{displaymath} \{ f, g\}(a) \coloneqq \langle [d f_a, d g_a],a\rangle \,. \end{displaymath} This Poisson manifold [[foliation|foliates]] into [[symplectic leaves]] which are the [[coadjoint orbits]]. The [[line bundles]] in question are the \emph{[[prequantum line bundles]]} of these [[symplectic manifolds]]. Hence, in the language of [[quantum physics]], the orbit methods identifies unitary representations of Lie groups $G$ with the $G$-action on [[spaces of states]] of the [[geometric quantization]] of a [[classical mechanical system]] with a global $G$-[[symmetry]]. Many important classes of unitary representations are obtained by that method. Notably in the case of [[compact Lie groups]], co-adjoint orbits are [[flag manifolds]] and the \emph{[[Borel-Weil theorem]]} says that under certain further conditions the expected unitary representations are obtained. The case of non-compact Lie groups is much less well understood, see for instance (\hyperlink{GrahamVogan}{Graham-Vogan}, \hyperlink{Vogan99}{Vogan 99}). \hypertarget{DefinitionsAndConstructions}{}\subsection*{{Definitions and constructions}}\label{DefinitionsAndConstructions} We list and discuss the basic notions, definitions and constructions in the context of the orbit method. A useful review is also in (\hyperlink{Beasley}{Beasley, section 4}). \hypertarget{the_group_and_its_lie_algebra}{}\subsubsection*{{The group and its Lie algebra}}\label{the_group_and_its_lie_algebra} Throughout, let $G$ be a [[semisimple Lie group|semisimple]] [[compact topological group|compact]] [[Lie group]]. For some considerations below we furthermore assume it to be [[simply connected topological space|simply connected]]. Write $\mathfrak{g}$ for its [[Lie algebra]]. Its canonical (up to scale) binary [[invariant polynomial]] we write \begin{displaymath} \langle -,-\rangle \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} \,. \end{displaymath} Since this is non-degenerate, we may equivalently think of this as an [[isomorphism]] \begin{displaymath} \mathfrak{g} \simeq \mathfrak{g}^* \end{displaymath} that identifies the [[vector space]] underlying the Lie algebra with its [[dual vector space]] $\mathfrak{g}^*$. \hypertarget{TheCoadjointOrbitAndTheCosetSpaceAndFlagManifold}{}\subsubsection*{{The coadjoint orbit and the coset space/ flag manifold}}\label{TheCoadjointOrbitAndTheCosetSpaceAndFlagManifold} We discuss the [[coadjoint orbits]] of $G$ and their relation to the [[coset space]]/[[flag manifolds]] of $G$. Write \begin{enumerate}% \item $T \hookrightarrow G$ inclusion of the [[maximal torus]] of $G$. \item $\mathfrak{t} \hookrightarrow \mathfrak{g}$ the corresponding [[Cartan subalgebra]] \end{enumerate} In all of the following we consider an element $\langle\lambda,-\rangle \in \mathfrak{g}^*$. \begin{defn} \label{}\hypertarget{}{} For $\langle\lambda,-\rangle \in \mathfrak{g}^*$ write \begin{displaymath} \mathcal{O}_\lambda \hookrightarrow \mathfrak{g}^* \end{displaymath} for its [[coadjoint orbit]] \begin{displaymath} \mathcal{O}_{\lambda} = \{ Ad_g^*(\langle\lambda,-\rangle) \in \mathfrak{g}^* | g \in G \} \,. \end{displaymath} Write $G_\lambda \hookrightarrow G$ for the [[stabilizer subgroup]] of $\langle \lambda,-\rangle$ under the coadjoint action. \end{defn} \begin{prop} \label{}\hypertarget{}{} There is an equivalence \begin{displaymath} G/G_\lambda \stackrel{\simeq}{\to} \mathcal{O}_\lambda \end{displaymath} given by \begin{displaymath} g G_\lambda \mapsto Ad_g^* \langle\lambda,-\rangle \,. \end{displaymath} \end{prop} \begin{defn} \label{RegularElement}\hypertarget{RegularElement}{} An element $\langle\lambda,-\rangle \in \mathfrak{g}^*$ is \textbf{regular} if its [[coadjoint action]] [[stabilizer subgroup]] coincides with the [[maximal torus]]: $G_\lambda \simeq T$. \end{defn} \begin{example} \label{}\hypertarget{}{} For generic values of $\lambda$ it is regular. The element in $\mathfrak{g}^*$ farthest from regularity is $\lambda = 0$ for which $G_\lambda = G$ instead. \end{example} \hypertarget{the_symplectic_form}{}\subsubsection*{{The symplectic form}}\label{the_symplectic_form} We describe a canonical [[symplectic form]] on the [[coadjoint orbit]]/[[coset]] $\mathcal{O}_\lambda \simeq G/G_\lambda$. Write $\theta \in \Omega^1(G, \mathfrak{g})$ for the [[Maurer-Cartan form]] on $G$. \begin{defn} \label{The2FormOnG}\hypertarget{The2FormOnG}{} Write \begin{displaymath} \Theta_\lambda \coloneqq \langle \lambda, \theta \rangle \in \Omega^1(G) \end{displaymath} for the [[differential 1-form]] obtained by pairing the value of the [[Maurer-Cartan form]] at each point with the fixed element $\lambda \in \mathfrak{g}^*$. Write \begin{displaymath} \nu_\lambda \coloneqq d_{dR} \Theta_\lambda \end{displaymath} for its [[de Rham differential]]. \end{defn} \begin{prop} \label{TheSymplecticFormOnTheCoset}\hypertarget{TheSymplecticFormOnTheCoset}{} The 2-form $\nu_\lambda$ from def. \ref{The2FormOnG} \begin{enumerate}% \item satisfies \begin{displaymath} \nu_\lambda = \frac{1}{2}\langle \lambda, [\theta\wedge \theta]\rangle \,. \end{displaymath} \item it descends to a closed $G$-invariant 2-form on the [[coset space]], to be denoted by the same symbol \begin{displaymath} \nu_\lambda \in \Omega^2_{cl}(G/G_\lambda)^G \,. \end{displaymath} \item this is non-degenerate and hence defines a [[symplectic form]] on $G/G_\lambda$. \end{enumerate} \end{prop} \hypertarget{the_prequantum_bundle}{}\subsubsection*{{The prequantum bundle}}\label{the_prequantum_bundle} We discuss the [[geometric prequantization]] of the [[symplectic manifold]] given by the [[coadjoint orbit]] $\mathcal{O}_\lambda$ equipped with its [[symplectic form]] $\nu_\lambda$ of def. \ref{TheSymplecticFormOnTheCoset}. Assume now that $G$ is [[simply connected topological space|simply connected]]. \begin{prop} \label{WeightsAndCharacters}\hypertarget{WeightsAndCharacters}{} The [[weight lattice]] $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is [[isomorphism|isomorphic]] to the group of [[group characters]] \begin{displaymath} \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(T,U(1)) \end{displaymath} where the identification takes $\langle \alpha , -\rangle \in \mathfrak{t}^*$ to $\rho_\alpha \colon T \to U(1)$ given on $t = \exp(\xi)$ for $\xi \in \mathfrak{t}$ by \begin{displaymath} \rho_\alpha \colon \exp(\xi) \mapsto \exp(i \langle \alpha, \xi\rangle) \,. \end{displaymath} \end{prop} \begin{prop} \label{}\hypertarget{}{} The [[symplectic form]] $\nu_\lambda \in \Omega^2_{cl}(G/T)$ of prop. \ref{TheSymplecticFormOnTheCoset} is integral precisely if $\langle \lambda, - \rangle$ is in the [[weight lattice]]. \end{prop} \hypertarget{the_hamiltonian_action__coadjoint_moment_map}{}\subsubsection*{{The Hamiltonian $G$-action / coadjoint moment map}}\label{the_hamiltonian_action__coadjoint_moment_map} The group $G$ canonically [[action|acts]] on the [[coset space]] $G/G_{\lambda}$ (by multiplication from the left). We discuss a lift of this action to a [[Hamiltonian action]] with respect to the [[symplectic manifold]] structure $(G/T, \nu_\lambda)$ of prop. \ref{TheSymplecticFormOnTheCoset}, equivalently a [[momentum map]] exhibiting this Hamiltonian action. \hypertarget{WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit}{}\subsubsection*{{Wilson loops and 1d Chern-Simons $\sigma$-models with target the coadjoint orbit}}\label{WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit} \hyperlink{DefinitionsAndConstructions}{Above} we discussed how an [[irreducible representation|irreducible]] [[unitary representation]] of $G$ is encoded by the [[prequantization]] of a [[coadjoint orbit]] $(\mathcal{O}_\lambda, \nu_\lambda)$. Here we discuss how to express [[Wilson loops]]/[[holonomy]] of $G$-[[principal connections]] in this representation as the [[path integral]] of a topological particle charged under this background field, whose [[action functional]] is that of a [[1-dimensional Chern-Simons theory]]. This was hinted at in \hyperlink{Witten89}{Witten 89, p. 22, 23}, details are in (\hyperlink{Beasley}{Beasley, section 4}). Let $A|_{S^1} \in \Omega^1(S^1, \mathfrak{g})$ be a [[Lie algebra valued 1-form]] on the circle, equivalently a $G$-[[principal connection]] on the circle. For \begin{displaymath} \rho \colon G \to Aut(V) \end{displaymath} a [[representation]] of $G$, write \begin{displaymath} W_{S^1}^R(A) \coloneqq hol^R_{S^1}(A) \coloneqq Tr_R( tra_{S^1}(A) ) \end{displaymath} for the [[holonomy]] of $A$ around the circle in this representation, which is the [[trace]] of its [[parallel transport]] around the circle (for any basepoint). If one thinks of $A$ as a [[background gauge field]] then this is alse called a [[Wilson loop]]. \begin{defn} \label{ActionFunctionalForTopologicalChargedParticle}\hypertarget{ActionFunctionalForTopologicalChargedParticle}{} Let the [[action functional]] \begin{displaymath} \exp(i CS_\lambda(-)^A) \;\colon\; [S^1, G/T] \to U(1) \end{displaymath} be given by sending $g T \colon S^1 \to G/T$ represented by $g \colon S^1 \to G$ to \begin{displaymath} \exp(i \int_{S^1} \langle \lambda, A^g\rangle ) \,, \end{displaymath} where \begin{displaymath} A^g \coloneqq Ad_g(A) + g^* \theta \end{displaymath} is the [[gauge transformation]] of $A$ under $g$. \end{defn} \begin{prop} \label{WilsonLoopIsPartitionFunctionOf1dCSSigmaModel}\hypertarget{WilsonLoopIsPartitionFunctionOf1dCSSigmaModel}{} The [[Wilson loop]] of $A$ over $S^1$ in the unitary irreducible representation $R$ is proportional to the [[path integral]] of the 1-dimensional [[sigma-model]] with \begin{enumerate}% \item [[target space]] the [[coadjoint orbit]] $\mathcal{O}_\lambda \simeq G/T$ for $\langle \lambda, - \rangle$ the [[weight (in representation theory)|weight]] corresponding to $R$ under the [[Borel-Weil-Bott theorem]] \item [[action functional]] the functional of def. \ref{ActionFunctionalForTopologicalChargedParticle}: \end{enumerate} \begin{displaymath} W_{S^1}^R(A) \propto \underset{g T \in [S^1, \mathcal{O}_\lambda]}{\int} \exp(i \int_{S^1} \langle \lambda, A^g\rangle) \; D(g T) \,. \end{displaymath} \end{prop} See for instance (\hyperlink{Beasley}{Beasley, (4.55)}). \begin{remark} \label{}\hypertarget{}{} Notice that since $\mathcal{O}_\lambda$ is a [[manifold]] of [[finite number|finite]] [[dimension]], the [[path integral]] for a point particle with this target space can be and has been defined rigorously, see at \emph{[[path integral]]}. \end{remark} \hypertarget{FormulationInHigherGeometry}{}\subsection*{{Formulation in higher geometry}}\label{FormulationInHigherGeometry} We discuss here a natural equivalent reformulation of the \hyperlink{DefinitionsAndConstructions}{above} ingredients of the orbit method in terms of the [[higher differential geometry]] of [[smooth ∞-groupoids]], and specifically in terms of the [[extended prequantum field theory]] of [[Chern-Simons theory]] with [[Wilson line]] [[QFT with defects|defects]] (\hyperlink{FSS}{FSS}). \begin{enumerate}% \item \hyperlink{FormulationInHigherGeometrySurvey}{Survey} \item \hyperlink{FormulationInHigherGeometryDefinitions}{Definitions and constructions} \item \hyperlink{GaugeAndGravityWilsonLoops}{Nonabelian charged particle trajectories -- Wilson loops} \item \hyperlink{ExtendedChern-SimonsTheoryAndWilsonLoops}{3d Chern-Simons theory with Wilson loops}. \end{enumerate} \hypertarget{FormulationInHigherGeometrySurvey}{}\subsubsection*{{Survey}}\label{FormulationInHigherGeometrySurvey} We discuss how for $\lambda \in \mathfrak{g}$ a regular element, there is a canonical diagram of [[smooth infinity-groupoid|smooth]] [[moduli stacks]] of the form \begin{displaymath} \itexarray{ \mathcal{O}_\lambda &\stackrel{\simeq}{\to}& G/T &\stackrel{\mathbf{\theta}}{\to}& \Omega^1(-,\mathfrak{g})//T &\stackrel{\langle \lambda, - \rangle}{\to}& \mathbf{B} U(1)_{conn} \\ && \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\mathbf{J}}} \\ && * &\stackrel{}{\to}& \mathbf{B}G_{conn} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1)_{conn} } \,, \end{displaymath} where \begin{enumerate}% \item $\mathbf{J}$ is the canonical [[2-monomorphism]]; \item the left square is a [[homotopy pullback]] square, hence $\mathbf{\theta}$ is the [[homotopy fiber]] of $\mathbf{J}$; \item the bottom map is the [[extended Lagrangian]] for $G$-[[Chern-Simons theory]], equivalently the universal [[Chern-Simons circle 3-bundle with connection]]; \item the top map denoted $\langle \lambda,- \rangle$ is an [[extended Lagrangian]] for a [[1-dimensional Chern-Simons theory]]; \item the total top composite modulates a [[prequantum circle bundle]] which is a [[prequantization]] of the canonical [[symplectic manifold]] structure on the [[coadjoint orbit]] $\Omega_\lambda \simeq G/T$. \end{enumerate} \hypertarget{FormulationInHigherGeometryDefinitions}{}\subsubsection*{{Definitions and constructions}}\label{FormulationInHigherGeometryDefinitions} Write $\mathbf{H} =$ [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of smooth $\infty$-groupoids. For the following, let $\langle \lambda, - \rangle \in \mathfrak{g}^*$ be a \emph{regular} element, def. \ref{RegularElement}, so that the [[stabilizer subgroup]] is identified with a [[maximal torus]]: $G_\lambda \simeq T$. As usual, write \begin{displaymath} \mathbf{B}G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G \in \mathbf{H} \end{displaymath} for the [[moduli stack]] of $G$-[[principal connections]]. \begin{defn} \label{InclusionOfModuliStacks}\hypertarget{InclusionOfModuliStacks}{} Write \begin{displaymath} \mathbf{J} \coloneqq \left( \; \Omega^1(-,\mathfrak{g})//T \;\to\; \Omega^1(-,\mathfrak{g})//G \simeq \mathbf{B}G_{conn} \; \right) \in \mathbf{H}^{(\Delta^1)} \end{displaymath} for the canonical map, as indicated. \end{defn} \begin{remark} \label{}\hypertarget{}{} The map $\mathbf{J}$ is the differential refinement of the [[delooping]] $\mathbf{B}T \to \mathbf{B}G$ of the defining inclusion. By the general discussion at [[coset space]] we have a [[homotopy fiber sequence]] \begin{displaymath} \itexarray{ \mathcal{O}_\lambda \simeq G/T &\to& \mathbf{B}T & \simeq *//T \\ && \downarrow \\ && \mathbf{B}G & \simeq *//G } \,. \end{displaymath} By the discussion at \emph{[[∞-action]]} this exhibits the canonical [[action]] $\rho$ of $G$ on its [[coset space]]: it is the [[universal associated ∞-bundle|universal rho-associated bundle]]. \end{remark} The following proposition says what happens to this statement under differential refinement \begin{prop} \label{ThetaAsHomotopyFiberOfJ}\hypertarget{ThetaAsHomotopyFiberOfJ}{} The [[homotopy fiber]] of $\mathbf{J}$ in def. \ref{InclusionOfModuliStacks} is \begin{displaymath} \mathbf{\theta} \colon G/T \stackrel{}{\to} \Omega^1(-,\mathfrak{g})//T \end{displaymath} given over a test manifold $U \in$ [[CartSp]] by the map \begin{displaymath} \mathbf{\theta}_U \colon C^\infty(U,G/T) \to \Omega^1(U,\mathfrak{g}) \end{displaymath} which sends $g \mapsto g^* \theta$, where $\theta$ is the [[Maurer-Cartan form]] on $G$. \end{prop} \begin{remark} \label{}\hypertarget{}{} This is a general phenomenon in the context of [[Cartan connections]]. See there at \emph{\href{Cartan+connection#InTermsOfSmoothModuliStacks}{Definition -- In terms of smooth moduli stacks}}. \end{remark} \begin{proof} \textbf{of prop. \ref{ThetaAsHomotopyFiberOfJ}}. We compute the [[homotopy pullback]] of $\mathbf{J}$ along the point inclusion by the [[factorization lemma]] as discussed at \emph{\href{homotopy%20pullback#ConstructionsGeneral}{homotopy pullback -- Constructions}}. This says that with $\mathbf{J}$ presented canonically as a map of presheaves of groupoids via the above definitions, its homotopy fiber is presented by the presheaf of groupids $hofib(\mathbf{J})$ which is the [[limit]] [[cone]] in \begin{displaymath} \itexarray{ hofib(\mathbf{J}) &\to& &\to& \Omega^1(-, \mathfrak{g})//T \\ \downarrow && \downarrow && \downarrow \\ && (\mathbf{B}G_{conn})^I &\to& \mathbf{B}G_{conn} \\ \downarrow && \downarrow \\ * &\stackrel{}{\to}& \mathbf{B}G_{conn} } \,. \end{displaymath} Unwinding the definitions shows that $hofib(\mathbf{J})$ has \begin{enumerate}% \item [[objects]] over a $U \in$ [[CartSp]] are equivalently morphisms $0 \stackrel{g}{\to} g^* \theta$ in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$, hence equivalently elements $g \in C^\infty(U,G)$; \item [[morphisms]] are over $U$ [[commuting diagram|commuting triangles]] \begin{displaymath} \itexarray{ g_1^* \theta &&\stackrel{t}{\to}&& g_2^* \theta \\ & {}_{\mathllap{g_1}}\nwarrow && \nearrow_{\mathrlap{g_2}} \\ && 0 } \end{displaymath} in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$ with $t \in C^\infty(U,T)$, hence equivalently morphisms \begin{displaymath} g_1 \stackrel{t}{\to} g_2 \end{displaymath} in $C^\infty(U,G)//C^\infty(U,T)$. This shows that $hofib(\mathbf{J}) \simeq G/T$. \item The canonical map $hofib(\mathbf{J}) \to \Omega^1(-,\mathfrak{g})//T$ picks the top horizontal part of these commuting triangles hence equivalently sends $g$ to $g^* \theta$. \end{enumerate} \end{proof} \begin{remark} \label{}\hypertarget{}{} There is yet one more fiber sequence of similar structure. If we let $L G \coloneqq [S^1, G]$ denote the free [[loop group]], then there is a fiber sequence \begin{displaymath} \itexarray{ G/T &\to& L G / T \\ && \downarrow \\ && L G / G & \simeq \Omega G } \,. \end{displaymath} The [[geometric quantization]] of $L G / T$ yields the [[positive energy representations]] of the [[loop group]] $L G_\mathcal{C}$. See at \emph{\href{loop+group#Representations}{loop group -- Properties -- Representations}} for more on this. \end{remark} \begin{prop} \label{Extended1dCSLagrangianFromLambda}\hypertarget{Extended1dCSLagrangianFromLambda}{} If $\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^*$ is in the [[weight lattice]], then there is a morphism of [[moduli stacks]] \begin{displaymath} \langle \lambda, - \rangle \;\colon\; \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn} \end{displaymath} in $\mathbf{H}$ given over a test manifold $U \in$ [[CartSp]] by the [[functor]] \begin{displaymath} \langle \lambda, - \rangle_U \;:\; \Omega^1(U,\mathfrak{g})//C^\infty(U,T) \to \Omega^1(U)//C^\infty(U,U(1)) \end{displaymath} which is given on objects by \begin{displaymath} A \mapsto \langle \lambda, A\rangle \end{displaymath} and which maps morphisms labeled by $\exp(\xi) \in T$, $\xi \in C^\infty(-,\mathfrak{t})$ as \begin{displaymath} \exp(\xi) \mapsto \exp( i \langle \lambda, \xi \rangle ) \,. \end{displaymath} \end{prop} \begin{proof} That this construction defines a map $*//T \to *//U(1)$ is the statement of prop. \ref{WeightsAndCharacters}. It remains to check that the differential 1-forms gauge-transform accordingly. For this the key point is that since $T \simeq G_\lambda$ stabilizes $\langle \lambda , - \rangle$ under the [[coadjoint action]], the [[gauge transformation]] law for points $A \colon U \to \mathbf{B}G_{conn}$, which for $g \in C^\infty(U,G)$ is \begin{displaymath} A \mapsto Ad_g A + g^* \theta \,, \end{displaymath} maps for $g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G)$ to the gauge transformation law in $\mathbf{B}U(1)_{conn}$: \begin{displaymath} \begin{aligned} \langle \lambda, A \rangle & \mapsto \langle \lambda, Ad_g A\rangle + \langle \lambda, g^* \theta\rangle \\ & = \langle \lambda, A \rangle + d \langle\lambda, \xi \rangle \end{aligned} \end{displaymath} \end{proof} \begin{remark} \label{ThePrequantumBundleFromCanonicalMaps}\hypertarget{ThePrequantumBundleFromCanonicalMaps}{} The composite of the canonical maps of prop. \ref{ThetaAsHomotopyFiberOfJ} and prop. \ref{Extended1dCSLagrangianFromLambda} modulates a canonical [[circle bundle with connection]] on the [[coset space]]/[[coadjoint orbit]]: \begin{displaymath} \langle \lambda, \mathbf{\theta}\rangle \colon G/T \stackrel{\mathbf{\theta}}{\to} \Omega^1(-,\mathfrak{g})//T \stackrel{\langle \lambda, - \rangle}{\to} \mathbf{B}U(1)_{conn} \,. \end{displaymath} \end{remark} \begin{prop} \label{}\hypertarget{}{} The [[curvature]] 2-form of the circle bundle $\langle \lambda, \mathbf{\theta}\rangle$ from remark \ref{ThePrequantumBundleFromCanonicalMaps} is the [[symplectic form]] of prop. \ref{TheSymplecticFormOnTheCoset}. Therefore $\langle \lambda, \mathbf{\theta}\rangle$ is a [[prequantization]] of the [[coadjoint orbit]] $(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda)$. \end{prop} \begin{proof} The curvature 2-form is modulated by the composite \begin{displaymath} \omega \colon G/T \stackrel{\mathbf{\theta}}{\to} \Omega^1(-,\mathfrak{g})//T \stackrel{\langle \lambda, - \rangle}{\to} \mathbf{B}U(1)_{conn} \stackrel{F_{(-)}}{\to} \Omega^2_{cl} \,. \end{displaymath} Unwinding the above definitions and propositions, one finds that this is given over a test manifold $U \in$ [[CartSp]] by the map \begin{displaymath} \omega_U \colon C^\infty(G/T) \to \Omega^2_{cl}(U) \end{displaymath} which sends \begin{displaymath} [g] \mapsto d \langle \lambda, g^* \theta \rangle \,. \end{displaymath} \end{proof} \hypertarget{GaugeAndGravityWilsonLoops}{}\subsubsection*{{Nonabelian charged particle trajectories -- Wilson loops}}\label{GaugeAndGravityWilsonLoops} Let $\Sigma$ be an [[orientation|oriented]] [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] 3 and let \begin{displaymath} C \;\colon\; S^1 \hookrightarrow \Sigma \end{displaymath} be a [[submanifold]] inclusion of the [[circle]]: a [[knot]] in $\Sigma$. Let $R$ be an [[irreducible representation|irreducible]] [[unitary representation]] of $G$ and let $\langle \lambda,-\rangle$ be a [[weight (in representation theory)|weight]] corresponding to it by the [[Borel-Weil-Bott theorem]]. Regarding the inclusion $C$ as an object in the [[arrow category]] $\mathbf{H}^{\Delta^1}$, say that a [[gauge field]] configuration for $G$-[[Chern-Simons theory]] on $\Sigma$ with [[Wilson loop]] $C$ and labeled by the [[representation]] $R$ is a map \begin{displaymath} \phi \;\colon\; C \to \mathbf{J} \end{displaymath} in the [[arrow category]] $\mathbf{H}^{(\Delta^1)}$ of the ambient [[cohesive (∞,1)-topos]]. Such a map is equivalently by a square \begin{displaymath} \itexarray{ S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T \\ \downarrow^{\mathrlap{C}} &\swArrow_{g}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \Sigma &\stackrel{A}{\to}& \mathbf{B}G_{conn} } \end{displaymath} in $\mathbf{H}$. In components this is \begin{itemize}% \item a $G$-[[principal connection]] $A$ on $\Sigma$; \item a $G$-valued function $g$ on $S^1$ \end{itemize} which fixes the field on the circle defect to be $(A|_{S^1})^g$, as indicated. Moreover, a \emph{[[gauge transformation]]} between two such fields $\kappa \colon \phi \Rightarrow \phi'$ is a $G$-gauge transformation of $A$ and a $T$-gauge transformation of $A|_{S^1}$ such that these intertwine the component maps $g$ and $g'$. If we keep the bulk gauge field $A$ fixed, then his means that two fields $\phi$ and $\phi'$ as above are gauge equivalent precisely if there is a function $t \;\colon\; S^1 \to T$ such that $g = g' t$, hence gauge [[equivalence classes]] of fields for fixed bulk gauge field $A$ are parameterized by their components $[g] = [g'] \in [S^1, G/T]$ with values in the coset space, hence in the coadjoint orbit. For every such field configuration we can evaluate two [[action functionals]]: \begin{enumerate}% \item that of 3d [[Chern-Simons theory]], whose [[extended Lagrangian]] is $\mathbf{c} \colon \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$; \item that of the [[1-dimensional Chern-Simons theory]] discussed \hyperlink{WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit}{above} whose [[extended Lagrangian]] is $\langle \lambda, -\rangle \colon \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}$, by prop. \ref{Extended1dCSLagrangianFromLambda}. \end{enumerate} These are obtained by postcomposing the above square on the right by these [[extended Lagrangians]] \begin{displaymath} \itexarray{ S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T &\stackrel{\langle \lambda, -\rangle}{\to}& \mathbf{B}U(1)_{conn} \\ \downarrow^{\mathrlap{C}} &\swArrow_{g}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \Sigma &\stackrel{A}{\to}& \mathbf{B}G_{conn} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}U(1)_{conn} } \end{displaymath} and then preforming the [[fiber integration in ordinary differential cohomology]] over $S^1$ and over $\Sigma$, respectively. For the bottom map this gives the ordinary action functional of [[Chern-Simons theory]]. For the top map inspection of the proof of prop. \ref{Extended1dCSLagrangianFromLambda} shows that this gives the 1d Chern-Simons action whose [[partition function]] is the [[Wilson loop]] observable by prop. \ref{WilsonLoopIsPartitionFunctionOf1dCSSigmaModel} above. \hypertarget{ExtendedChern-SimonsTheoryAndWilsonLoops}{}\subsubsection*{{3d Chern-Simons theory with Wilson loops}}\label{ExtendedChern-SimonsTheoryAndWilsonLoops} We discuss how an [[extended Lagrangian]] for $G$-[[Chern-Simons theory]] with [[Wilson loop]] [[QFT with defects|defects]] is naturally obtained from the \hyperlink{FormulationInHigherGeometryDefinitions}{above} [[higher geometry|higher geometric]] formulation of the orbit method. In particular we discuss how the relation between Wilson loops and [[1-dimensional Chern-Simons theory]] [[sigma-models]] with [[target space]] the [[coadjoint orbit]], as discussed \hyperlink{WilsonLoopsAnd1DCSSigmaModelWithTargetTheCoadjointOrbit}{above} is naturally obtained this way. More formally, we have an extended Chern-Simons theory as follows. The [[moduli stack]] of fields $\phi \colon C \to \mathbf{J}$ in $\mathbf{H}^{(\Delta^1)}$ as above is the [[homotopy pullback]] \begin{displaymath} \itexarray{ \mathbf{Fields}(S^1 \hookrightarrow \Sigma) &\stackrel{}{\to}& [S^1, \Omega^1(-,\mathfrak{g})//T] \\ \downarrow &\swArrow_\simeq& \downarrow \\ [\Sigma, \mathbf{B}G_{conn}] &\to& [S^1, \mathbf{B}G_{conn}] } \end{displaymath} in $\mathbf{H}$, where square brackets indicate the [[internal hom]] in $\mathbf{H}$. Postcomposing the two projections with the two [[transgressions]] of the [[extended Lagrangians]] \begin{displaymath} \exp(2 \pi i \int_\Sigma[\Sigma, \mathbf{c}]) \;\colon\; [\Sigma, \mathbf{B}G_{conn}] \stackrel{[\Sigma, \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_\Sigma (-))}{\to} U(1) \end{displaymath} and \begin{displaymath} \exp(2 \pi i \int_\Sigma[S^1, \langle \lambda, -\rangle]) \;\colon\; [S^1, \Omega^1(-,\mathfrak{g})//T] \stackrel{[\Sigma, \langle \lambda , -\rangle]}{\to} [S^1, \mathbf{B} U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1} (-))}{\to} U(1) \end{displaymath} to yield \begin{displaymath} \itexarray{ \mathbf{Fields}(S^1 \hookrightarrow \Sigma) &\stackrel{}{\to}& [S^1, \Omega^1(-,\mathfrak{g})//T] &\stackrel{\exp(2 \pi i \int_{S^1} [S^1, \langle \lambda, -\rangle] ) }{\to}& U(1) \\ \downarrow &\swArrow_\simeq& \downarrow \\ [\Sigma, \mathbf{B}G_{conn}] &\to& [S^1, \mathbf{B}G_{conn}] \\ \downarrow^{\mathrlap{\exp(2\pi i \int_{\Sigma_2} [\Sigma_3, \mathbf{c}])}} \\ U(1) } \end{displaymath} and then forming the product yields the action functional \begin{displaymath} \exp(2 \pi i \int_{S^1}[S^1, \langle -\rangle]) \cdot \exp(2 \pi i \int_{\Sigma}[\Sigma, \mathbf{c}]) \;:\; \mathbf{Fields}(S^1 \hookrightarrow \Sigma) \to U(1) \,. \end{displaymath} This is the action functional of 3d $G$-[[Chern-Simons theory]] on $\Sigma$ with Wilson loop $C$ in the representation determined by $\lambda$. Similarly, in [[codimension]] 1 let $\Sigma_2$ now be a 2-dimensional closed manifold, thought of as a slice of $\Sigma$ above, and let $\coprod_i {*} \to \Sigma_2$ be the inclusion of points, thought of as the punctures of the Wilson line above through this slice. Then we have [[prequantum bundles]] given by [[transgression]] of the extended Lagrangians to codimension 1 \begin{displaymath} \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma, \mathbf{c}\right]\right) \;\colon\; \left[\Sigma_2, \mathbf{B}G_{conn}\right] \stackrel{\left[\Sigma_2, \mathbf{c}\right]}{\to} \left[\Sigma_2, \mathbf{B}^3 U(1)_{conn}\right] \stackrel{\exp\left(2 \pi i \int_{\Sigma_2} \left(-\right)\right)}{\to} \mathbf{B}U\left(1\right)_{conn} \end{displaymath} and \begin{displaymath} \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \left\langle \lambda, -\right\rangle\right]\right) \;\colon\; \left[\coprod_i {*}, \Omega^1\left(-,\mathfrak{g}\right)//T\right] \stackrel{[\coprod_i {*}, \langle \lambda , -\rangle]}{\to} \left[\coprod_i {*}, \mathbf{B} U(1)_{conn}\right] \stackrel{\exp\left(2 \pi i \int_{\coprod_i {*}} \left(-\right)\right)}{\to} \mathbf{B}U(1)_{conn} \end{displaymath} and hence a total prequantum bundle \begin{displaymath} \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \langle \beta, -\rangle\right]\right) \otimes \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma_2, \mathbf{c}\right]\right) \;:\; \mathbf{Fields}\left(\coprod_i {*} \hookrightarrow \Sigma\right) \to \mathbf{B}U\left(1\right)_{conn} \,. \end{displaymath} One checks that this is indeed the correct prequantization as considered in (\hyperlink{Witten89}{Witten 89, p. 22}). \hypertarget{DiracInduction}{}\subsection*{{Formulation in equivariant K-theory (Dirac induction)}}\label{DiracInduction} \begin{prop} \label{}\hypertarget{}{} For $G$ a [[compact Lie group]] with [[Lie algebra]] $\mathfrak{g}^\ast$, the [[push-forward in generalized cohomology|push-forward]] in compactly supported [[twisted K-theory|twisted]] $G$-[[equivariant K-theory]] to the point (the $G$-equivariant [[index]]/[[Dirac induction]]) produces the [[Thom isomorphism]] \begin{displaymath} ind_{\mathfrak{g}^\ast} \;\colon\; K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt} \stackrel{\simeq}{\to} K_G^0(\ast) \simeq Rep(G) \,. \end{displaymath} Moreover, for $i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast$ a regular [[coadjoint orbit]], [[push-forward in generalized cohomology|push-forward]] involves a [[twisted K-theory|twist]] $\sigma$ of the form \begin{displaymath} Rep(G) \simeq K_G^0(\ast) \stackrel{ind_{\mathcal{O}}}{\leftarrow} K_G^{\sigma(\mathcal{O}) + dim(\mathcal{O})}(\mathcal{O}) \stackrel{i_!}{\to} K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt} \end{displaymath} and \begin{enumerate}% \item $i_!$ is surjective \item $ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!$. \end{enumerate} \end{prop} This is (\hyperlink{FHT}{FHT II, (1.27), theorem 1.28}). \hypertarget{formulation_in_equivariant_elliptic_cohomology}{}\subsection*{{Formulation in equivariant elliptic cohomology}}\label{formulation_in_equivariant_elliptic_cohomology} The \hyperlink{DiracInduction}{above} formulation of the orbit method in [[equivariant K-theory]] has a higher order generalization where one replaces [[equivariant K-theory]] with [[equivariant cohomology|equivariant]] [[elliptic cohomology]]. Here the ``elliptic'' orbit method directly knows about the [[representation theory]] of the [[loop group]]. (\hyperlink{Ganter12}{Ganter 12}). \hypertarget{theorems}{}\subsection*{{Theorems}}\label{theorems} \begin{itemize}% \item [[Borel-Weil-Bott theorem]] \end{itemize} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item [[geometric quantization of the 2-sphere]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} Introductions and surveys include \begin{itemize}% \item [[Alexandre Kirillov]], \emph{[[Lectures on the Orbit Method]]}, Graduate Studies in Mathematics, 64, American Mathematical Society, (2004) [[David Vogan]], \emph{Review of: Lectures on the orbit method} (\href{http://www.ams.org/journals/bull/2005-42-04/S0273-0979-05-01065-7/S0273-0979-05-01065-7.pdf}{pdf}) \item [[David Vogan]], \emph{Geometry and representations of reductive groups} (2007) (\href{http://www-math.mit.edu/~dav/rittC.pdf}{pdf}) \item J. Maes, \emph{A introduction to the orbit method}, Master thesis (2011) (\href{http://testweb.science.uu.nl/ITF/teaching/2011/Jeroen%20Maes.pdf}{pdf}, \href{http://www.imus.us.es/FSMYT12/Talk_Jeroen_Maes.pdf}{pdf slides}, \href{http://igitur-archive.library.uu.nl/student-theses/2011-0622-200341/UUindex.html}{web}) \item Craig Jackson, \emph{Symplectic manifolds, geometric quantization, and unitary representations of Lie groups} (\href{http://go.owu.edu/~chjackso/Papers/topic.pdf}{pdf}) \item Reyer Sjamaar, \emph{Notes on the orbit method and quantization} (1997) ([[SjamaarOrbitMethod.pdf:file]]) \end{itemize} Original references include \begin{itemize}% \item . . , \emph{ }, . ., 1981, 15, . 1, . 23--37, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1690&what=fullt&option_lang=rus}{pdf}; transl. V. A. Ginzburg, \emph{Method of orbits in the representation theory of complex Lie groups}, Funct. Analysis and Its Appl. \textbf{1981}, 15:1, 18--28, \href{http://dx.doi.org/10.1007/BF01082375}{doi} \item [[Bertram Kostant]], \emph{Orbits and quantization theory}, Proc. ICM Nice 1970, 395-406, \href{http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0395.0406.ocr.djvu}{djvu:597 K}, \href{http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0395.0406.ocr.pdf}{pdf:1.1 M} \item [[Bertram Kostant]], \emph{Quantization and unitary representations. I. Prequantization}, in: Lectures in Modern Analysis and Applications III, Lec. Notes in Math. \textbf{170}, 87--208, \href{http://www.ams.org/mathscinet-getitem?mr=294568}{MR294568}; Russ. transl. by A. Kirillov: Uspehi Mat. Nauk \textbf{28} (1973), no. 1(169), 163--225, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=4837&volume=28&year=1973&issue=1&fpage=163&what=fullt&option_lang=eng}{pdf} \item [[Alexandre Kirillov]], \emph{ }, , Uspehi. Mat. Nauk. 17 (1962), 57-110, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=6513&what=fullt&option_lang=rus}{Rus. pdf}; transl. \emph{Unitary representations of nilpotent Lie groups}, Russian Math. Surveys, 1962, 17:4, 53--104, \href{http://dx.doi.org/10.1070%2FRM1962v017n04ABEH004118}{doi}, \href{http://www.ams.org/mathscinet-getitem?mr=142001}{MR142001} \item L. Auslander, [[Bertram Kostant]], \emph{Quantization and representations of solvable Lie groups}, Bull. Amer. Math. Soc. \textbf{73}, 1967, 692--695, \href{http://www.ams.org/journals/bull/1967-73-05/S0002-9904-1967-11829-9/S0002-9904-1967-11829-9.pdf}{pdf}; \emph{Polarization and unitary representations of solvable Lie groups}, Invent. Math. \textbf{14} (1971), 255--354, \href{http://www.ams.org/mathscinet-getitem?mr=293012}{MR293012}, \href{http://dx.doi.org/10.1007/BF01389744}{doi} \item W. Graham, [[David Vogan]], \emph{Geometric quantization for nilpotent coadjoint orbits}, in Geometry and Representation Theory of real and p-adic groups. Birkh\"a{}user, Boston-Basel-Berlin (1998) \item [[David Vogan]], \emph{The method of coadjoint orbits for real reductive groups}, in Representation Theory of Lie Groups. IAS/Park City Mathematics Series 8 (1999), 179--238 \end{itemize} Discussion with an eye towards application in [[gauge theory]] and in particular for [[Wilson loop]] observables in [[Chern-Simons theory]], hinted at on \begin{itemize}% \item [[Edward Witten]], p. 22, 23 of \emph{Quantum Field Theory and the Jones Polynomial} Commun. Math. Phys. 121 (3) (1989) 351--399. MR0990772 (\href{http://projecteuclid.org/euclid.cmp/1104178138}{project EUCLID}) \end{itemize} is in section 4 of \begin{itemize}% \item [[Chris Beasley]], \emph{Localization for Wilson Loops in Chern-Simons Theory}, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) \emph{Chern-Simons Gauge Theory: 20 Years After}, , AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (\href{http://arxiv.org/abs/0911.2687}{arXiv:0911.2687}) \end{itemize} referring to \begin{itemize}% \item S. Elitzur, [[Greg Moore]], A. Schwimmer, and [[Nathan Seiberg]], \emph{Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory}, Nucl. Phys. B 326 (1989) 108--134. \end{itemize} A program of applying the orbit method to real nilpotent orbits of real semisimple Lie groups (closely related to [[quantization via the A-model]]) is in \begin{itemize}% \item Ranee Brylinski, \emph{Geometric Quantization of Real Minimal Nilpotent Orbits}, DGA, vol. 9 (1998), 5-58 (\href{http://arxiv.org/abs/math/9811033}{arXiv:math/9811033}) \begin{itemize}% \item \emph{Quantization of the 4-dimensional nilpotent orbit of $SL(3,\mathbb{R})$}, Canad. J. Math. 49(1997), 916-943 (\href{http://cms.math.ca/cjm/a148867}{web}) \item \emph{Instantons and K\"a{}hler Geometry of Nilpotent Orbits} (\href{http://arxiv.org/abs/math/9811032}{arXiv:math/9811032}) \end{itemize} \end{itemize} Discussion of the orbit method in terms of [[equivariant K-theory]] and [[Dirac induction]] is in \begin{itemize}% \item [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], part II, section 1 of \emph{[[Loop Groups and Twisted K-Theory]]} \item [[Peter Hochs]], section 2.2 of \emph{Quantisation of presymplectic manifolds, K-theory and group representations} (\href{http://arxiv.org/abs/1211.0107}{arXiv:1211.0107}) \end{itemize} The generalization of this to [[elliptic cohomology]] is discussed in \begin{itemize}% \item [[Nora Ganter]], \emph{The elliptic Weyl character formula} (\href{http://arxiv.org/abs/1206.0528}{arXiv:1206.0528}) \end{itemize} Generalization to [[supergeometry]] is discussed in: \begin{itemize}% \item Gijs M. Tuynman, \emph{Geometric Quantization of Superorbits: a Case Study} (\href{http://arxiv.org/abs/0901.1811}{arXiv:0901.1811}) \item Alexander Alldridge, Joachim Hilgert, Tilmann Wurzbacher, \emph{Superorbits} (\href{https://arxiv.org/abs/1502.04375}{arXiv:1502.04375}) \end{itemize} A generalization to [[higher geometry]] and [[2-group]] [[2-representations]] is proposed in \begin{itemize}% \item [[Bruce Bartlett]], \emph{On unitary 2-representations of finite groups and topological quantum field theory} (\href{http://arxiv.org/abs/0901.3975}{arXiv:0901.3975}) \end{itemize} The above discussion of the interpretation of the orbit method in terms of higher [[moduli stacks]] for [[differential cohomology]] appears in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], section 3.4.5 of \emph{[[schreiber:A higher stacky perspective on Chern-Simons theory]]}, in Damien Calaque et al. (eds.) \emph{Mathematical Aspects of Quantum Field Theories}, Mathematical Physics Studies, Springer 2014 (\href{http://arxiv.org/abs/1301.2580}{arXiv:1301.2580}) \end{itemize} See also \begin{itemize}% \item wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Orbit_method}{orbit method}} \end{itemize} [[!redirects method of coadjoint orbits]] \end{document}