geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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 $G$-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 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 Lie-Poisson structure –, namely for any $a\in \mathfrak{g}^*$,
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 $G$ 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 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).
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).
Throughout, let $G$ be a semisimple compact Lie group. For some considerations below we furthermore assume it to be simply connected.
Write $\mathfrak{g}$ for its Lie algebra. Its canonical (up to scale) binary invariant polynomial we write
Since this is non-degenerate, we may equivalently think of this as an isomorphism
that identifies the vector space underlying the Lie algebra with its dual vector space $\mathfrak{g}^*$.
We discuss the coadjoint orbits of $G$ and their relation to the coset space/flag manifolds of $G$.
Write
$T \hookrightarrow G$ inclusion of the maximal torus of $G$.
$\mathfrak{t} \hookrightarrow \mathfrak{g}$ the corresponding Cartan subalgebra
In all of the following we consider an element $\langle\lambda,-\rangle \in \mathfrak{g}^*$.
For $\langle\lambda,-\rangle \in \mathfrak{g}^*$ write
for its coadjoint orbit
Write $G_\lambda \hookrightarrow G$ for the stabilizer subgroup of $\langle \lambda,-\rangle$ under the coadjoint action.
There is an equivalence
given by
An element $\langle\lambda,-\rangle \in \mathfrak{g}^*$ is regular if its coadjoint action stabilizer subgroup coincides with the maximal torus: $G_\lambda \simeq T$.
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.
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 $G$.
Write
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
for its de Rham differential.
The 2-form $\nu_\lambda$ from def. 3
satisfies
it descends to a closed $G$-invariant 2-form on the coset space, to be denoted by the same symbol
this is non-degenerate and hence defines a symplectic form on $G/G_\lambda$.
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. 2.
Assume now that $G$ is simply connected.
The weight lattice $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is isomorphic to the group of group characters
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
The symplectic form $\nu_\lambda \in \Omega^2_{cl}(G/T)$ of prop. 2 is integral precisely if $\langle \lambda, - \rangle$ is in the weight lattice.
The group $G$ canonically 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. 2, equivalently a momentum map exhibiting this Hamiltonian action.
Above we discussed how an 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 Witten 89, p. 22, 23, details are in (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
a representation of $G$, write
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.
Let the action functional
be given by sending $g T \colon S^1 \to G/T$ represented by $g \colon S^1 \to G$ to
where
is the gauge transformation of $A$ under $g$.
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
target space the coadjoint orbit $\mathcal{O}_\lambda \simeq G/T$ for $\langle \lambda, - \rangle$ the weight corresponding to $R$ under the Borel-Weil-Bott theorem
action functional the functional of def. 4:
See for instance (Beasley, (4.55)).
Notice that since $\mathcal{O}_\lambda$ 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.
We discuss here a natural equivalent reformulation of the 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 defects (FSS).
We discuss how for $\lambda \in \mathfrak{g}$ a regular element, there is a canonical diagram of smooth moduli stacks of the form
where
$\mathbf{J}$ is the canonical 2-monomorphism;
the left square is a homotopy pullback square, hence $\mathbf{\theta}$ is the homotopy fiber of $\mathbf{J}$;
the bottom map is the extended Lagrangian for $G$-Chern-Simons theory, equivalently the universal Chern-Simons circle 3-bundle with connection;
the top map denoted $\langle \lambda,- \rangle$ is an extended Lagrangian for a 1-dimensional Chern-Simons theory;
the total top composite modulates a prequantum circle bundle which is a prequantization of the canonical symplectic manifold structure on the coadjoint orbit $\Omega_\lambda \simeq G/T$.
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 regular element, def. 2, so that the stabilizer subgroup is identified with a maximal torus: $G_\lambda \simeq T$.
As usual, write
for the moduli stack of $G$-principal connections.
Write
for the canonical map, as indicated.
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
By the discussion at ∞-action this exhibits the canonical action $\rho$ of $G$ on its coset space: it is the universal rho-associated bundle.
The following proposition says what happens to this statement under differential refinement
The homotopy fiber of $\mathbf{J}$ in def. 5 is
given over a test manifold $U \in$ CartSp by the map
which sends $g \mapsto g^* \theta$, where $\theta$ is the Maurer-Cartan form on $G$.
This is a general phenomenon in the context of Cartan connections. See there at Definition – In terms of smooth moduli stacks.
of prop. 6.
We compute the homotopy pullback of $\mathbf{J}$ along the point inclusion by the factorization lemma as discussed at 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
Unwinding the definitions shows that $hofib(\mathbf{J})$ has
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 $g \in C^\infty(U,G)$;
morphisms are over $U$ commuting triangles
in $\Omega^1(U,\mathfrak{g})//C^\infty(U,G)$ with $t \in C^\infty(U,T)$, hence equivalently morphisms
in $C^\infty(U,G)//C^\infty(U,T)$. This shows that $hofib(\mathbf{J}) \simeq G/T$.
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 $g^* \theta$.
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
The geometric quantization of $L G / T$ yields the positive energy representations of the loop group $L G_\mathcal{C}$. See at loop group – Properties – Representations for more on this.
If $\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^*$ is in the weight lattice, then there is a morphism of moduli stacks
in $\mathbf{H}$ given over a test manifold $U \in$ CartSp by the functor
which is given on objects by
and which maps morphisms labeled by $\exp(\xi) \in T$, $\xi \in C^\infty(-,\mathfrak{t})$ as
That this construction defines a map $*//T \to *//U(1)$ 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 $T \simeq G_\lambda$ 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
maps for $g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G)$ to the gauge transformation law in $\mathbf{B}U(1)_{conn}$:
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:
The curvature 2-form of the circle bundle $\langle \lambda, \mathbf{\theta}\rangle$ from remark 5 is the symplectic form of prop. 2. Therefore $\langle \lambda, \mathbf{\theta}\rangle$ is a prequantization of the coadjoint orbit $(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda)$.
The curvature 2-form is modulated by the composite
Unwinding the above definitions and propositions, one finds that this is given over a test manifold $U \in$ CartSp by the map
which sends
Let $\Sigma$ be an oriented closed smooth manifold of dimension 3 and let
be a submanifold inclusion of the circle: a knot in $\Sigma$.
Let $R$ be an irreducible unitary representation of $G$ and let $\langle \lambda,-\rangle$ be a 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
in the arrow category $\mathbf{H}^{(\Delta^1)}$ of the ambient cohesive (∞,1)-topos. Such a map is equivalently by a square
in $\mathbf{H}$. In components this is
a $G$-principal connection $A$ on $\Sigma$;
a $G$-valued function $g$ on $S^1$
which fixes the field on the circle defect to be $(A|_{S^1})^g$, as indicated.
Moreover, a 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:
that of 3d Chern-Simons theory, whose extended Lagrangian is $\mathbf{c} \colon \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$;
that of the 1-dimensional Chern-Simons theory discussed above whose extended Lagrangian is $\langle \lambda, -\rangle \colon \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}$, by prop. 7.
These are obtained by postcomposing the above square on the right by these extended Lagrangians
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. 7 shows that this gives the 1d Chern-Simons action whose partition function is the Wilson loop observable by prop. 5 above.
We discuss how an extended Lagrangian for $G$-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 $\phi \colon C \to \mathbf{J}$ in $\mathbf{H}^{(\Delta^1)}$ as above is the homotopy pullback
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
and
to yield
and then forming the product yields the action functional
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
and
and hence a total prequantum bundle
One checks that this is indeed the correct prequantization as considered in (Witten 89, p. 22).
For $G$ a compact Lie group with Lie algebra $\mathfrak{g}^\ast$, the push-forward in compactly supported twisted $G$-equivariant K-theory to the point (the $G$-equivariant index/Dirac induction) produces the Thom isomorphism
Moreover, for $i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast$ a regular coadjoint orbit, push-forward involves a twist $\sigma$ of the form
and
$i_!$ is surjective
$ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!$.
This is (FHT II, (1.27), theorem 1.28).
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).
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, A 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
is in section 4 of
referring to
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 $SL(3,\mathbb{R})$, 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
Daniel Freed, Michael Hopkins, Constantin Teleman, part II, section 1 of Loop Groups and Twisted K-Theory
Peter Hochs, section 2.2 of Quantisation of presymplectic manifolds, K-theory and group representations (arXiv:1211.0107)
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)
Alexander Alldridge, Joachim Hilgert, Tilmann Wurzbacher, Superorbits (arXiv:1502.04375)
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