The Based Loop Group of SU(2)

Lisa Jeffrey

Department of Mathematics

University of Toronto

Joint work with Megumi Harada and Paul Selick

I. The based loop group ΩG

Let G = SU(2) and let T be its maximal torus

T ={ eiθ 0

0 e−iθ

} ∼= U(1)The Weyl group is W ∼= Z2.

The based loop group ΩG is defined as

ΩG = {γ : S1 → G | γ(∗) = e}

where ∗ is the basepoint. We require that γ be continuous.

Let G be a compact, simple and simply connected Lie group. The loop group

LsG (for s > 3/2) is defined as the set of maps from S1 to G of Sobolev class Hs,

meaning that the Sobolev norm | f |s is finite (for example

(| f |2)2 = |f |2 + |df |2

using the L2 norm). Strictly speaking the Sobolev norm is defined on the Lie

algebra of LsG and the exponential map is used pointwise to transfer this

definition to LsG. LsG is an infinite dimensional Hilbert manifold. It contains the

set C∞(G) of C∞ maps from S1 to G, which however is not a Hilbert manifold.

The subset ΩsG of LsG (the based loop group) consists of those loops f : S1 → G

for which f is the identity element at the basepoint ∗. There is a surjective mapfrom LsG to ΩsG defined as follows:

F : h→ h(∗)−1h

This map sends the submanifold of constant loops (which may be identified with

the group G) to the identity element in Ω1G, so it identifies ΩsG with the

homogeneous space LsG/G.

The space ΩsG is symplectic. The symplectic form at the identity element e is

ωe(X,Y ) =1

2π

∫θ∈[0,2π]

< X(s),dY

dθ(θ) > dθ (1)

for X,Y ∈ Ls(g).

II. Torus actions on the based loop group

The rotation group S1 acts on Ω1G as follows:

(eiθf)(s) = f(θ)−1f(s+ θ) (2)

for s, θ ∈ [0, 2π]. The maximal torus T acts on Ω1G by conjugation: for t ∈ T ,

(tf)(s) = tf(s)t−1 (3)

These actions commute. It can be shown that these actions are Hamiltonian and

the moment maps are as follows. The moment map for the rotation action (the

“energy” E) is

E(f) =1

4π

∫ 2π0|f(θ)−1f ′(θ)|2dθ

The moment map for the conjugation action of T (the “momentum” p) is

µ(f) =1

2πprLie(T )

∫ 2π0

f(s)−1f ′(s)ds

(where the projection is onto the Lie algebra of the maximal torus T ).

The torus action on the loop group was studied by Pressley-Segal (1988),

Atiyah-Pressley (1983) and other authors such as Harada-Holm-Jeffrey-Mare

(2006).

Atiyah and Pressley proved that the image of the moment map for T × S1 on ΩGis the convex hull of the images of the fixed point set of T × S1 on ΩG, as forHamiltonian torus actions on finite-dimensional symplectic manifolds.

III. Motivation for computing K∗G(ΩG)

Alekseev-Malkin-Meinrenken (1998):

Quasi-Hamiltonian G-spaces are G-spaces with a 2-form with a structure

analogous to a Hamiltonian G-spaces. A symplectic form is closed and

nondegenerate.

A quasi-Hamiltonian G-space is equipped with a 2-form ω and a map µ : M → Gfor which

1. dω = µ∗Λ where Λ is the generator of H3(G,Z) (this replaces dω = 0)

2.

iX#ω =1

2µ∗(θ + θ̄, X)

where θ (resp. θ̄) is the left-invariant (resp. right-invariant) Maurer-Cartan

form

θ = g−1dg, θ̄ = dgg−1 ∈ Ω1(G,g).

(This replaces the condition that the fundamental vector field X# associated

to X ∈ Lie(G) is the Hamiltonian vector field associated to the X componentof a Lie algebra valued moment map.

3. The kernel of ω is X# for X satisfying

Ad(µ(x))X

)= −X

(this replaces the condition that the moment map is nondegenerate)

Alekseev, Malkin and Meinrenken established a bijective correspondence between

Hamiltonian LG-spaces M and quasi-Hamiltonian G-spaces M .

ΩG = ΩG

| |

v v

| |

M Φ−→ Lg∗

| |

v v

| |

Mµ−→ G

The vertical map from Lg∗ to G is the holonomy. In case M = Gsg, he moment

map µ is the product of commutators. For this example M is the space of flatconnections on a surface of genus g with one boundary compomemt. amd the

map Φ is restriction to the boundary

The prototype examples of Hamiltonian LG spaces are:

• moduli spaces of flat G connections on 2-manifolds;

• coadjoint orbits in the dual of Lie(G).

The prototype examples of quasi-Hamiltonian G spaces are:

• products G2N of an even-dimensional number of copies of G;

• conjugacy classes in G.

IV. History

Computations:

•H∗(ΩG): Bott, 1950s: divided polynomial algebra Recall that the dividedpolynomial algebra Γ[s] is defined as a Z-algebra generated by sj (in degree 2j)

where

sj · sk =(j + k)!

j!k!sj+k.

Notice that a divided polynomial algebra can be described as the inverse limit of

the symmetric polynomials in an exterior algebra.

•H∗G(ΩG): Borel and others, late 1950s Tensor product of divided polynomialalgebra Γ with H∗G(pt) = Z[t] (polynomial ring on one generator of degree 4)

•K(ΩG): known to many people (e.g. Adams, Atiyah, Hirzebruch, Segal, Serre),1960s

Completed divided polynomial algebra over the representation ring R(G)

•KG(G): Brylinski-Zhang, 2000: H∗(G)⊗K∗G(pt) •KT (ΩU(n)) (recursivecalculation using Kac-Moody algebras): Kostant-Kumar, 1990

Homotopy filtrations of ΩSU(2)

James, 1955

Pressley-Segal, 1986

Bott-Tolman-Weitsman, 2004

V. Background about cohomology and K-theory of products of copies

of P1

Note that P1 ∼= G/T and G acts on it by left multiplication.

KG(pt) = R(G) where R(G) is the representation ring of G.

R(G) ∼= Z[v] where v is the fundamental representation of G in complexdimension 2.

Compute H∗G((P1)2r) and KG((P

1)2r) via Bott periodicity.

The answers as rings are as follows.

• H∗G((P1)2r) = H∗G(pt)[L̄1, . . . , L̄2r]/ < L̄2j − t̄ >

where t̄ is an element of degree 4 which generates H∗G(pt)∼= Z[̄t] and L̄i are

elements of degree 2 corresponding to H2G(P1) (isomorphic to H2(P1), since G is

simply connected.)

L2j corresponds to the canonical line bundle, over the 2jth copy of P1. L2j−1 is

the hyperplane line bundle (the dual of the canonical line bundle) over the

(2j − 1)-th copy of P1.

Here we have written bars for elements in H∗G corresponding to analogous

elements in KG.

• KG((P1)2r) = KG(pt)/I

where I is the ideal generated by L2j − vLj + 1 for j = 1, . . . , 2r

Chern character

1. Chern homomorphism from K(X) to

∞∏j=0

Hj(X)⊗Q

(isomorphism with Q coefficients)

2. Chern homomorphism in KG:

KG(X)→∞∏j=0

HjG(X; Q)

(Reference: mimeographed notes by Atiyah and Segal, 1965)

3. Some properties of the Chern homomorphism:

(a) chG(x) = exp(cG1 (x)) if x is a line bundle

(b) chG ⊗Q is an isomorphism on the spaces we will discuss

VI. Statement of Results

Ωpoly,rG := {f(z) ∈ G

∣∣∣ f(1) = I, f(z) = r∑j=−r

ajzj , aj ∈M2×2(C)

}i.e. the Fourier expansion of f is a finite Laurent polynomial expansion from −rto r. Set

ΩpolySU(2) :=∞⋃r=0

Ωpoly,rSU(2)

We refer to ΩpolySU(2) as the the space of polynomial (based) loops in

SU(2).

•The inclusion ΩpolyG→ ΩG is a G-homotopy equivalence

(A non-equivariant version of the proof of this appears in Pressley-Segal. Our

proof uses the ideas in Pressley-Segal along with some ideas from Milnor, Morse

Theory.)

•

H∗G(ΩG) = H∗G(ΩpolyG) is the inverse limit of H

∗G(Ωpoly,rG);

K∗G(ΩG) = K∗G(ΩpolyG) is the inverse limit of K

∗G(Ωpoly,rG). (Milnor lim←−

1

sequence)

Note that Bott-Tolman-Weitsman studied ΩG using an analogous filtration

coming from the Morse theory of the energy functional

f(γ) :=1

4π

∫ 2π0| dγdt|2 dt

As R(G)-modules:

•

KevenG (ΩG) =∞∏j=0

KG(pt) =∞∏j=0

R(G)

•KoddG (ΩG) = 0

• Theorem: (rephrasing of known result)

H∗G(Ωpoly,rG) is isomorphic to the subring of H∗G((P

1)2r) consisting of the

symmetric polynomials in L̄1, . . . , L̄2r.

Let s̄j be the j-th elementary symmetric polynomial in L̄1, . . . , L̄2r, and let s̄′j

denote the corresponding polynomial in L̄1, . . . , L̄2r−2.

The system maps of this inverse limit are determined by the following matrix, in

the basis of {s̄j} and {s̄′j}.

1 0 −t̄ . . . 0 0 0 0 0

0 1 0 . . . 0 0 0 0 0

0 0 1 . . . 0 0 0 0 0...

......

. . ....

......

......

0 0 0 . . . 1 0 −t̄ 0 0

0 0 0 . . . 0 1 0 −t̄ 0

0 0 0 . . . 0 0 1 0 −t̄

0 0 0 . . . 0 0 0 0 0

0 0 0 . . . 0 0 0 0 0

• MAIN THEOREM:

KG(Ωpoly,rG) is isomorphic to the subring of KG((P1)2r) consisting of the

symmetric polynomials in L1, . . . , L2r.

Let sj be the j-th elementary symmetric polynomial in L1, . . . , L2r, and let s′j

denote the corresponding polynomial in L1, . . . , L2r−2.

Note that the relations L2j = vLj − 1 imply that any symmetric polynomial isactually a linear combination of s0, s1, . . . , s2r. The sj are the R(G)-module

generators.

The system maps of this inverse limit are determined by the following matrix, in

the basis of {sj} and {s′j}.

1 v 1 . . . 0 0 0 0 0

0 1 v . . . 0 0 0 0 0

0 0 1 . . . 0 0 0 0 0...

......

. . ....

......

......

0 0 0 . . . 1 v 1 0 0

0 0 0 . . . 0 1 v 1 0

0 0 0 . . . 0 0 1 v 1

0 0 0 . . . 0 0 0 0 0

0 0 0 . . . 0 0 0 0 0

VII. Comparison with known results

The reduction KG(X)→ K(X) is induced by � : R(G)→ Z where � takes arepresentation to its dimension.

For (P1)2r, our relation L2j − vLj + 1 = 0 reduces (under setting v equal to 2) to

(Lj − 1)2 = 0,

and we recover the familiar fact that K((P1)2r) is an exterior algebra.

K(Ωpoly,rG) = symmetric polynomials in K((P1)2r

).

Note that the set of symmetric polynomials in the exterior algebra

Λ[y1, . . . , y2r] equals the truncated divided polynomial algebra

Γ[y]/(y2r+1) where y = y1 + . . .+ y2r.

Taking the inverse limit tells us that K(ΩG) is a completed divided polynomial

algebra. If we ignore the ring structure we have K(ΩG) =∏∞i=0 Z.

Similarly H∗(ΩG) is a divided polynomial algebra, as originally computed by

Bott. If we ignore the ring structure we have H∗(ΩG) = ⊕∞i=0Z.

VIII. Outline of proof

• Ωpoly,r/Ωpoly,r−1 ∼=G Thom(τ2r−1)

Here τ is the tangent bundle of P1. Hence

KG(Ωpoly,rG) ∼=2r∏i=0

R(G)

as R(G)-modules, using Thom isomorphism and induction. Taking the inverse

limit gives

KG(ΩG) ∼=∞∏i=0

R(G)

as R(G)-modules.

• Thom(τ2r−1) ∼=G P(τ2r−1 ⊕ �)/P(τ2r−1)

using Atiyah’s description of the Thom space. Here � is the trivial bundle over P1.

• We show that P(τ ⊕ �) ∼=G P1 ×P1 where the subspace P(τ) gets mapped to

the diagonal ∆ under this homeomorphism. Hence

Ωpoly,1G ∼= Thom(τ) ∼= P(τ ⊕ �)/P(τ) ∼= (P1 ×P1)/∆.

Therefore we have a quotient map

Φ2 : P1 ×P1 → (P1 ×P1)/∆ ∼= Ωpoly,1G.

We define Φ2r : (P1)2r → Ωpoly,rG as the composition

Φ2r : (P1)2r →(Φ2)r (Ωpoly,1G)r → Ωpoly,rG

where the last map is induced by pointwise matrix multiplication.

• Φ∗ : K∗G(F2r)→ K∗G((P1)2r) is injective (this step uses Chern homomorphism).So multiplication in KG(Ωpoly,rG) is the restriction of multiplication on

KG((P1)2r)

• To finish the computation, we must compute the image of Φ∗ on K∗G(Ωpoly,rG).To do this we first answer the corresponding question on cohomology. Use

induction on r.

The result is:

H∗G(Ωpoly,rG) = symmetric polynomials in H∗G((P

1)2r)

• Key step:

KG(Ωpoly,rG) = symmetric polynomials in KG((P1)2r).

The argument requires first proving the analogous statement for KT .

The reason this indirect route is necessary is because some of the relevant

diagrams in the induction are only T -equivariant and not G-equivariant.

The difficult part is to use Chern and Thom to show that every symmetric

polynomial is in Im(Φ2r).

Having obtained the computation of KT (Ωpoly,rG), we take Weyl invariants to

conclude that KG(Ωpoly,rG) is the symmetric polynomials in KG((P1)2r), as

claimed in our theorem.

Note: It is not true in general that KG(X) ∼= (KT (X))W (there arecounterexamples due to Reyer Sjamaar). However, our module calculations (see

Section V) show us that this equality holds in our case.

• Finally, take the inverse limit to get KG(ΩG)