nLab
orbit method

Context

Representation theory

Geometric quantization

Contents

Idea

The Orbit method (or Kirillov’s method, or method of coadjoint orbits) is a method in geometric representation theory concerned with identifying unitary representations of Lie groups with the canonical GG-actions on spaces of sections of certain line bundles over coadjoint orbits of the Lie group. In terms of quantum physics this realizes GG-representations as actions of global gauge groups of quantum operators on spaces of quantum states under geometric quantization.

More in detail, the dual 𝔤 *\mathfrak{g}^* of a (say finite-dimensional real) Lie algebra has a canonical structure of a Poisson manifold – its Lie-Poisson structure –, namely for any a𝔤 *a\in \mathfrak{g}^*,

{f,g}(a)[df a,dg a],a. \{ f, g\}(a) \coloneqq \langle [d f_a, d g_a],a\rangle \,.

This Poisson manifold foliates into symplectic leaves which are the coadjoint orbits. The line bundles in question are the prequantum line bundles of these symplectic manifolds.

Hence, in the language of quantum physics, the orbit methods identifies unitary representations of Lie groups GG with the GG-action on spaces of states of the geometric quantization of a classical mechanical system with a global GG-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 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 (Graham-Vogan, Vogan 99).

Definitions and constructions

We list and discuss the basic notions, definitions and constructions in the context of the orbit method. A useful review is also in (Beasley, section 4).

The group and its Lie algebra

Throughout, let GG be a semisimple compact Lie group. For some considerations below we furthermore assume it to be simply connected.

Write 𝔤\mathfrak{g} for its Lie algebra. Its canonical (up to scale) binary invariant polynomial we write

,:𝔤𝔤. \langle -,-\rangle \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} \,.

Since this is non-degenerate, we may equivalently think of this as an isomorphism

𝔤𝔤 * \mathfrak{g} \simeq \mathfrak{g}^*

that identifies the vector space underlying the Lie algebra with its dual vector space 𝔤 *\mathfrak{g}^*.

The coadjoint orbit and the coset space/ flag manifold

We discuss the coadjoint orbits of GG and their relation to the coset space/flag manifolds of GG.

Write

  1. TGT \hookrightarrow G inclusion of the maximal torus of GG.

  2. 𝔱𝔤\mathfrak{t} \hookrightarrow \mathfrak{g} the corresponding Cartan subalgebra

In all of the following we consider an element λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^*.

Definition

For λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^* write

𝒪 λ𝔤 * \mathcal{O}_\lambda \hookrightarrow \mathfrak{g}^*

for its coadjoint orbit

𝒪 λ={Ad g *(λ,)𝔤 *|gG}. \mathcal{O}_{\lambda} = \{ Ad_g^*(\langle\lambda,-\rangle) \in \mathfrak{g}^* | g \in G \} \,.

Write G λGG_\lambda \hookrightarrow G for the stabilizer subgroup of λ,\langle \lambda,-\rangle under the coadjoint action.

Proposition

There is an equivalence

G/G λ𝒪 λ G/G_\lambda \stackrel{\simeq}{\to} \mathcal{O}_\lambda

given by

gG λAd g *λ,. g G_\lambda \mapsto Ad_g^* \langle\lambda,-\rangle \,.
Definition

An element λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^* is regular if its coadjoint action stabilizer subgroup coincides with the maximal torus: G λTG_\lambda \simeq T.

Example

For generic values of λ\lambda it is regular. The element in 𝔤 *\mathfrak{g}^* farthest from regularity is λ=0\lambda = 0 for which G λ=GG_\lambda = G instead.

The symplectic form

We describe a canonical symplectic form on the coadjoint orbit/coset 𝒪 λG/G λ\mathcal{O}_\lambda \simeq G/G_\lambda.

Write θΩ 1(G,𝔤)\theta \in \Omega^1(G, \mathfrak{g}) for the Maurer-Cartan form on GG.

Definition

Write

Θ λλ,θΩ 1(G) \Theta_\lambda \coloneqq \langle \lambda, \theta \rangle \in \Omega^1(G)

for the 1-form obtained by pairing the value of the Maurer-Cartan form at each point with the gixed element λ𝔤 *\lambda \in \mathfrak{g}^*.

Write

ν λd dRΘ λ \nu_\lambda \coloneqq d_{dR} \Theta_\lambda

for its de Rham differential.

Proposition

The 2-form ν λ\nu_\lambda from def. 3

  1. satisfies

    ν λ=12λ,[θθ]. \nu_\lambda = \frac{1}{2}\langle \lambda, [\theta\wedge \theta]\rangle \,.
  2. it descends to a closed GG-invariant 2-form on the coset space, to be denoted by the same symbol

    ν λΩ cl 2(G/G λ) G. \nu_\lambda \in \Omega^2_{cl}(G/G_\lambda)^G \,.
  3. this is non-degenerate and hence defines a symplectic form on G/G λG/G_\lambda.

The prequantum bundle

We discuss the geometric prequantization of the symplectic manifold given by the coadjoint orbit 𝒪 λ\mathcal{O}_\lambda equipped with its symplectic form ν λ\nu_\lambda of def. 2.

Assume now that GG is simply connected.

Proposition

The weight lattice Γ wt𝔱 *𝔱\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t} of the Lie group GG is isomorphic to the group of group characters

Γ wtHom LieGrp(T,U(1)) \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(T,U(1))

where the identification takes α,𝔱 *\langle \alpha , -\rangle \in \mathfrak{t}^* to ρ α:TU(1)\rho_\alpha \colon T \to U(1) given on t=exp(ξ)t = \exp(\xi) for ξ𝔱\xi \in \mathfrak{t} by

ρ α:exp(ξ)exp(iα,ξ). \rho_\alpha \colon \exp(\xi) \mapsto \exp(i \langle \alpha, \xi\rangle) \,.
Proposition

The symplectic form ν λΩ cl 2(G/T)\nu_\lambda \in \Omega^2_{cl}(G/T) of prop. 2 is integral precisely if λ,\langle \lambda, - \rangle is in the weight lattice.

The Hamiltonian GG-action / coadjoint moment map

The group GG canonically acts on the coset space G/G λG/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,ν λ)(G/T, \nu_\lambda) of prop. 2, equivalently a momentum map exhibiting this Hamiltonian action.

Wilson loops and 1d Chern-Simons σ\sigma-models with target the coadjoint orbit

Above (…) we discussed how an irreducible unitary representation of GG is encoded by the prequantization of a coadjoint orbit (𝒪 λ,ν λ)(\mathcal{O}_\lambda, \nu_\lambda). Here we discuss how to express Wilson loops/holonomy of GG-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.

Let A| S 1Ω 1(S 1,𝔤)A|_{S^1} \in \Omega^1(S^1, \mathfrak{g}) be a Lie algebra valued 1-form on the circle, equivalently a GG-principal connection on the circle.

For

ρ:GAut(V) \rho \colon G \to Aut(V)

a representation of GG, write

W S 1 R(A)hol S 1 R(A)Tr R(tra S 1(A)) W_{S^1}^R(A) \coloneqq hol^R_{S^1}(A) \coloneqq Tr_R( tra_{S^1}(A) )

for the holonomy of AA around the circle in this representation, which is the trace of its parallel transport around the circle (for any basepoint). If one thinks of AA as a background gauge field then this is alse called a Wilson loop.

Definition

Let the action functional

exp(iCS λ() A):[S 1,G/T]U(1) \exp(i CS_\lambda(-)^A) \;\colon\; [S^1, G/T] \to U(1)

be given by sending gT:S 1G/Tg T \colon S^1 \to G/T represented by g:S 1Gg \colon S^1 \to G to

exp(i S 1λ,A g), \exp(i \int_{S^1} \langle \lambda, A^g\rangle ) \,,

where

A gAd g(A)+g *θ A^g \coloneqq Ad_g(A) + g^* \theta

is the gauge transformation of AA under gg.

Proposition

The Wilson loop of AA over S 1S^1 in the unitarry irreducible representation RR is proportional to the path integral of the 1-dimensional sigma-model with

  1. target space the coadjoint orbit 𝒪 λG/T\mathcal{O}_\lambda \simeq G/T for λ,\langle \lambda, - \rangle the weight corresponding to RR under the Borel-Weil-Bott theorem

  2. action functional the functional of def. 4:

W S 1 R(A) [S 1,𝒪 λ]D(gT)exp(i S 1λ,A g). W_{S^1}^R(A) \propto \int_{[S^1, \mathcal{O}_\lambda]} D(g T) \exp(i \int_{S^1} \langle \lambda, A^g\rangle) \,.

See for instance (Beasley, (4.55)).

Remark

Notice that since 𝒪 λ\mathcal{O}_\lambda is a manifold of finite dimension, the path integral for a point particle with this target space can be and has been defined rigorously, see at path integral.

Formulation in higher geometry

We discuss here a natural equivalent reformulation of the above ingredients of the orbit method in terms of the higher geometry of smooth ∞-groupoids, and specifically in terms of the extended prequantum field theory of Chern-Simons theory with Wilson line defects (FSS).

  1. Survey

  2. Definitions and constructions

  3. Nonabelian charged particle trajectories – Wilson loops

  4. 3d Chern-Simons theory with Wilson loops.

Survey

We discuss how for λ𝔤\lambda \in \mathfrak{g} a regular element, there is a canonical diagram of smooth moduli stacks of the form

𝒪 λ G/T θ Ω 1(,𝔤)//T λ, BU(1) conn J * BG conn c B 3U(1) conn, \array{ \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} } \,,

where

  1. J\mathbf{J} is the canonical 2-monomorphism;

  2. the left square is a homotopy pullback square, hence θ\mathbf{\theta} is the homotopy fiber of J\mathbf{J};

  3. the bottom map is the extended Lagrangian for GG-Chern-Simons theory, equivalently the universal Chern-Simons circle 3-bundle with connection;

  4. the top map denoted λ,\langle \lambda,- \rangle is an extended Lagrangian for a 1-dimensional Chern-Simons theory;

  5. the total top composite modulates a prequantum circle bundle which is a prequantization of the canonical symplectic manifold structure on the coadjoint orbit Ω λG/T\Omega_\lambda \simeq G/T.

Definitions and constructions

Write H=\mathbf{H} = Smooth∞Grpd for the cohesive (∞,1)-topos of smooth \infty-groupoids.

For the following, let λ,𝔤 *\langle \lambda, - \rangle \in \mathfrak{g}^* be a regular element, def. 2, so that the stabilizer subgroup is identified with a maximal torus: G λTG_\lambda \simeq T.

As usual, write

BG connΩ 1(,𝔤)//GH \mathbf{B}G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G \in \mathbf{H}

for the moduli stack of GG-principal connections.

Definition

Write

J(Ω 1(,𝔤)//TΩ 1(,𝔤)//GBG conn)H (Δ 1) \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)}

for the canonical map, as indicated.

Remark

The map J\mathbf{J} is the differential refinement of the delooping BTBG\mathbf{B}T \to \mathbf{B}G of the defining inclusion. By the general discussion at coset space we have a homotopy fiber sequence

𝒪 λG/T BT *//T BG *//G. \array{ \mathcal{O}_\lambda \simeq G/T &\to& \mathbf{B}T & \simeq *//T \\ && \downarrow \\ && \mathbf{B}G & \simeq *//G } \,.

By the discussion at ∞-action this exhibits the canonical action ρ\rho of GG on its coset space: it is the universal rho-associated bundle.

The following proposition says what happens to this statement under differential refinement

Proposition

The homotopy fiber of J\mathbf{J} in def. 5 is

θ:G/TΩ 1(,𝔤)//T \mathbf{\theta} \colon G/T \stackrel{}{\to} \Omega^1(-,\mathfrak{g})//T

given over a test manifold UU \in CartSp by the map

θ U:C (U,G/T)Ω 1(U,𝔤) \mathbf{\theta}_U \colon C^\infty(U,G/T) \to \Omega^1(U,\mathfrak{g})

which sends gg *θg \mapsto g^* \theta, where θ\theta is the Maurer-Cartan form on GG.

Proof

We compute the homotopy pullback of J\mathbf{J} along the point inclusion by the factorization lemma as discussed at homotopy pullback – Constructions.

This says that with J\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(J)hofib(\mathbf{J}) which is the limit cone in

hofib(J) Ω 1(,𝔤) (BG conn) I BG conn * BG conn. \array{ hofib(\mathbf{J}) &\to& &\to& \Omega^1(-, \mathfrak{g}) \\ \downarrow && \downarrow && \downarrow \\ && (\mathbf{B}G_{conn})^I &\to& \mathbf{B}G_{conn} \\ \downarrow && \downarrow \\ * &\stackrel{}{\to}& \mathbf{B}G_{conn} } \,.

Unwinding the definitions shows that hofib(J)hofib(\mathbf{J}) has

  1. objects over a UU \in CartSp are equivalently morphisms 0gg *θ0 \stackrel{g}{\to} g^* \theta in Ω 1(U,𝔤)//C (U,G)\Omega^1(U,\mathfrak{g})//C^\infty(U,G), hence equivalently elements gC (U,G)g \in C^\infty(U,G);

  2. morphisms are over UU commuting triangles

    g 1 *θ t g 2 *θ g 1 g 2 0 \array{ g_1^* \theta &&\stackrel{t}{\to}&& g_2^* \theta \\ & {}_{\mathllap{g_1}}\nwarrow && \nearrow_{\mathrlap{g_2}} \\ && 0 }

    in Ω 1(U,𝔤)//C (U,G)\Omega^1(U,\mathfrak{g})//C^\infty(U,G) with tC (U,T)t \in C^\infty(U,T), hence equivalently morphisms

    g 1tg 2 g_1 \stackrel{t}{\to} g_2

    in C (U,G)//C (U,T)C^\infty(U,G)//C^\infty(U,T).

  3. The canonical map hofib(J)Ω 1(,𝔤)//Thofib(\mathbf{J}) \to \Omega^1(-,\mathfrak{g})//T picks the top horizontal part of these commuting triangles hence equivalently sends gg to g *θg^* \theta.

Remark

There is yet one more fiber sequence of similar structure. If we let LG[S 1,G]L G \coloneqq [S^1, G] denote the free loop group, then there is a fiber sequence

G/T LG/T LG/G ΩG. \array{ G/T &\to& L G / T \\ && \downarrow \\ && L G / G & \simeq \Omega G } \,.

The geometric quantization of LG/TL G / T yields the positive energy representations of the loop group LG 𝒞L G_\mathcal{C}. See at loop group – Properties – Representations for more on this.

Proposition

If λ,Γ wt𝔤 *\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^* is in the weight lattice, then there is a morphism of moduli stacks

λ,:Ω 1(,𝔤)//TBU(1) conn \langle \lambda, - \rangle \;\colon\; \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}

in H\mathbf{H} given over a test manifold UU \in CartSp by the functor

λ, U:Ω 1(U,𝔤)//C (U,G)Ω 1(U)//C (U,U(1)) \langle \lambda, - \rangle_U \;:\; \Omega^1(U,\mathfrak{g})//C^\infty(U,G) \to \Omega^1(U)//C^\infty(U,U(1))

which is given on objects by

Aλ,A A \mapsto \langle \lambda, A\rangle

and which maps morphisms labeled by exp(ξ)T\exp(\xi) \in T, ξC (,𝔱)\xi \in C^\infty(-,\mathfrak{t}) as

exp(ξ)exp(iλ,ξ). \exp(\xi) \mapsto \exp( i \langle \lambda, \xi \rangle ) \,.
Proof

That this construction defines a map *//T*//U(1)*//T \to *//U(1) is the statement of prop. 3. It remains to check that the differential 1-forms gauge-transform accordingly.

For this the key point is that since TG λT \simeq G_\lambda stabilizes λ,\langle \lambda , - \rangle under the coadjoint action, the gauge transformation law for points A:UBG connA \colon U \to \mathbf{B}G_{conn}, which for gC (U,G)g \in C^\infty(U,G) is

AAd gA+g *θ, A \mapsto Ad_g A + g^* \theta \,,

maps for g=exp(ξ)C (U,T)C (U,G)g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G) to the gauge transformation law in BU(1) conn\mathbf{B}U(1)_{conn}:

λ,A λ,Ad gA+λ,g *θ =λ,A+dλ,ξ \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}
Remark

The composite of the canonical maps of prop. 6 and prop. 7 modulates a canonical circle bundle with connection on the coset space/coadjoint orbit:

λ,θ:G/TθΩ 1(,𝔤)//Tλ,BU(1) conn. \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} \,.
Proposition

The curvature 2-form of the circle bundle λ,θ\langle \lambda, \mathbf{\theta}\rangle from remark 4 is the symplectic form of prop. 2. Therefore λ,θ\langle \lambda, \mathbf{\theta}\rangle is a prequantization of the coadjoint orbit (𝒪 λG/T,ν λ)(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda).

Proof

The curvature 2-form is modulated by the composite

ω:G/TθΩ 1(,𝔤)//Tλ,BU(1) connF ()Ω cl 2. \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} \,.

Unwinding the above definitions and propositions, one finds that this is given over a test manifold UU \in CartSp by the map

ω U:C (G/T)Ω cl 2(U) \omega_U \colon C^\infty(G/T) \to \Omega^2_{cl}(U)

which sends

[g]dλ,g *θ. [g] \mapsto d \langle \lambda, g^* \theta \rangle \,.

Nonabelian charged particle trajectories – Wilson loops

Let Σ\Sigma be an oriented closed smooth manifold of dimension 3 and let

C:S 1Σ C \;\colon\; S^1 \hookrightarrow \Sigma

be a submanifold inclusion of the circle: a knot in Σ\Sigma.

Let RR be an irreducible unitary representation of GG and let λ,\langle \lambda,-\rangle be a weight corresponding to it by the Borel-Weil-Bott theorem.

Regarding the inclusion CC as an object in the arrow category H Δ 1\mathbf{H}^{\Delta^1}, say that a gauge field configuration for GG-Chern-Simons theory on Σ\Sigma with Wilson loop CC and labeled by the representation RR is a map

ϕ:CJ \phi \;\colon\; C \to \mathbf{J}

in the arrow category H (Δ 1)\mathbf{H}^{(\Delta^1)} of the ambient cohesive (∞,1)-topos. Such a map is equivalently by a square

S 1 (A| S 1) g Ω 1(,𝔤)//T C g J Σ A BG conn \array{ 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} }

in H\mathbf{H}. In components this is

which fixes the field on the circle defect to be (A| S 1) g(A|_{S^1})^g, as indicated.

Moreover, a gauge transformation between two such fields κ:ϕϕ\kappa \colon \phi \Rightarrow \phi' is a GG-gauge transformation of AA and a TT-gauge transformation of A| S 1A|_{S^1} such that these intertwine the component maps gg and gg'. If we keep the bulk gauge field AA fixed, then his means that two fields ϕ\phi and ϕ\phi' as above are gauge equivalent precisely if there is a function t:S 1Tt \;\colon\; S^1 \to T such that g=gtg = g' t, hence gauge equivalence classes of fields for fixed bulk gauge field AA are parameterized by their components [g]=[g][S 1,G/T][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:

  1. that of 3d Chern-Simons theory, whose extended Lagrangian is c:BG connB 3U(1) conn\mathbf{c} \colon \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn};

  2. that of the 1-dimensional Chern-Simons theory discussed above whose extended Lagrangian is λ,:Ω 1(,𝔤)//TBU(1) conn\langle \lambda, -\rangle \colon \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}, by prop. 7.

These are obtained by postcomposing the above square on the right by these extended Lagrangians

S 1 (A| S 1) g Ω 1(,𝔤)//T λ, BU(1) conn C g J Σ A BG conn c BU(1) conn \array{ 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} }

and then preforming the fiber integration in ordinary differential cohomology over S 1S^1 and over Σ\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. 7 shows that this gives the 1d Chern-Simons action whose partition function is the Wilson loop observable by prop. 5 above.

3d Chern-Simons theory with Wilson loops

We discuss how an extended Lagrangian for GG-Chern-Simons theory with Wilson loop defects is naturally obtained from the above 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 above is naturally obtained this way.

More formally, we have an extended Chern-Simons theory as follows.

The moduli stack of fields ϕ:CJ\phi \colon C \to \mathbf{J} in H (Δ 1)\mathbf{H}^{(\Delta^1)} as above is the homotopy pullback

Fields(S 1Σ) [S 1,Ω 1(,𝔤)//T] [Σ,BG conn] [S 1,BG conn] \array{ \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}] }

in H\mathbf{H}, where square brackets indicate the internal hom in H\mathbf{H}.

Postcomposing the two projections with the two transgressions of the extended Lagrangians

exp(2πi Σ[Σ,c]):[Σ,BG conn][Σ,c][Σ,B 3U(1) conn]exp(2πi Σ())U(1) \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)

and

exp(2πi Σ[S 1,λ,]):[S 1,Ω 1(,𝔤)//T][Σ,λ,][S 1,BU(1) conn]exp(2πi S 1())U(1) \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)

to yield

Fields(S 1Σ) [S 1,Ω 1(,𝔤)//T] exp(2πi S 1[S 1,λ,]) U(1) [Σ,BG conn] [S 1,BG conn] exp(2πi Σ 2[Σ 3,c]) U(1) \array{ \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) }

and then forming the product yields the action functional

exp(2πi S 1[S 1,])exp(2πi Σ[Σ,c]):Fields(S 1Σ)U(1). \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) \,.

This is the action functional of 3d GG-Chern-Simons theory on Σ\Sigma with Wilson loop CC in the representation determined by λ\lambda.

Similarly, in codimension 1 let Σ 2\Sigma_2 now be a 2-dimensional closed manifold, thought of as a slice of Σ\Sigma above, and let i*Σ 2\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

exp(2πi Σ 2[Σ,c]):[Σ 2,BG conn][Σ 2,c][Σ 2,B 3U(1) conn]exp(2πi Σ 2())BU(1) conn \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}

and

exp(2πi i*[ i*,λ,]):[ i*,Ω 1(,𝔤)//T][ i*,λ,][ i*,BU(1) conn]exp(2πi i*())BU(1) conn \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}

and hence a total prequantum bundle

exp(2πi i*[ i*,β,])exp(2πi Σ 2[Σ 2,c]):Fields( i*Σ)BU(1) conn. \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} \,.

One checks that this is indeed the correct prequantization as considered in (Witten 98, p. 22).

Formulation in equivariant K-theory (Dirac induction)

Proposition

For GG a compact Lie group with Lie algebra 𝔤 *\mathfrak{g}^\ast, the push-forward in compactly supported twisted GG-equivariant K-theory to the point (the GG-equivariant index/Dirac induction) produces the Thom isomorphism

ind 𝔤 *:K G σ+dimG(𝔤 *) cptK G 0(*)Rep(G). ind_{\mathfrak{g}^\ast} \;\colon\; K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt} \stackrel{\simeq}{\to} K_G^0(\ast) \simeq Rep(G) \,.

Moreover, for i:𝒪𝔤 *i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast a regular coadjoint orbit, push-forward involves a twist σ\sigma of the form

Rep(G)K G 0(*)ind 𝒪K G σ(𝒪)+dim(𝒪)(𝒪)i !K G σ+dimG(𝔤 *) cpt 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}

and

  1. i !i_! is surjective

  2. ind 𝒪=ind 𝔤 *i !ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!.

This is (FHT II, (1.27), theorem 1.28).

Formulation in equivariant elliptic cohomology

The above formulation of the orbit method in equivariant K-theory has a higher order generalization where one replaces equivariant K-theory with equivariant elliptic cohomology. Here the “elliptic” orbit method directly knows about the representation theory of the loop group. (Ganter 12).

Theorems

Examples

References

Introductions and surveys include

  • Alexandre Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, 64, American Mathematical Society, (2004)

    David Vogan, Review of: Lectures on the orbit method (pdf)

  • David Vogan, Geometry and representations of reductive groups (2007) (pdf)

  • J. Maes, An introduction to the orbit method, Master thesis (2011) (pdf, pdf slides, web)

  • Craig Jackson, Symplectic manifolds, geometric quantization, and unitary representations of Lie groups (pdf)

  • Reyer Sjamaar, Notes on the orbit method and quantization (1997) (pdf)

Original references include

  • В. А. Гинзбург, Метод орбит в теории представлений комплексных групп Ли, Функц. анализ и его прил., 1981, том 15, в. 1, стр. 23–37, pdf; transl. V. A. Ginzburg, Method of orbits in the representation theory of complex Lie groups, Funct. Analysis and Its Appl. 1981, 15:1, 18–28, doi

  • Bertram Kostant, Orbits and quantization theory, Proc. ICM Nice 1970, 395-406, djvu:597 K, pdf:1.1 M

  • Bertram Kostant, Quantization and unitary representations. I. Prequantization, in: Lectures in Modern Analysis and Applications III, Lec. Notes in Math. 170, 87–208, MR294568; Russ. transl. by A. Kirillov: Uspehi Mat. Nauk 28 (1973), no. 1(169), 163–225, pdf

  • Alexandre Kirillov, Унитарные представления нильпотентных групп Ли, , Uspehi. Mat. Nauk. 17 (1962), 57-110, Rus. pdf; transl. Unitary representations of nilpotent Lie groups, Russian Math. Surveys, 1962, 17:4, 53–104, doi, MR142001

  • L. Auslander, Bertram Kostant, Quantization and representations of solvable Lie groups, Bull. Amer. Math. Soc. 73, 1967, 692–695, pdf; Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255–354, MR293012, doi

  • W. Graham, David Vogan, Geometric quantization for nilpotent coadjoint orbits, in Geometry and Representation Theory of real and p-adic groups. Birkhäuser, Boston-Basel-Berlin (1998)

  • David Vogan, The method of coadjoint orbits for real reductive groups, in Representation Theory of Lie Groups. IAS/Park City Mathematics Series 8 (1999), 179–238

Discussion with an eye towards application in gauge theory and in particular for Wilson loop observables in Chern-Simons theory (hinted at on p. 22, 23 of Edward Witten’s QFT and the Jones polynomial) is in section 4 of

  • Chris Beasley, Localization for Wilson Loops in Chern-Simons Theory, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) Chern-Simons Gauge Theory: 20 Years After, , AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. (arXiv:0911.2687)

referring to

  • S. Elitzur, Greg Moore, A. Schwimmer, and Nathan Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108–134.

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

  • Ranee Brylinski, Geometric Quantization of Real Minimal Nilpotent Orbits, DGA, vol. 9 (1998), 5-58 (arXiv:math/9811033)

    • Quantization of the 4-dimensional nilpotent orbit of SL(3,)SL(3,\mathbb{R}), Canad. J. Math. 49(1997), 916-943 (web)

    • Instantons and Kähler Geometry of Nilpotent Orbits (arXiv:math/9811032)

Discussion of the orbit method in terms of equivariant K-theory and Dirac induction is in

The generalization of this to elliptic cohomology is discussed in

Generalization to supergeometry is discussed in:

  • Gijs M. Tuynman, Geometric Quantization of Superorbits: a Case Study (arXiv:0901.1811)

A generalization to higher geometry and 2-group 2-representations is proposed in

The above discussion of the interpretation of the orbit method in terms of higher moduli stacks for differential cohomology appears in

See also

Revised on May 16, 2014 22:23:17 by Tim Porter (2.26.24.117)