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 n eiθ 0 o T = =∼ U(1) 0 e−iθ ∼ 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 1 s LsG (for s > 3/2) is defined as the set of maps from S to G of Sobolev class H , meaning that the Sobolev norm | f |s is finite (for example
2 2 2 (| f |2) = |f| + |df| 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. 1 The subset ΩsG of LsG (the based loop group) consists of those loops f : S → G for which f is the identity element at the basepoint ∗. There is a surjective map from 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 1 Z dY ωe(X,Y ) = < X(s), (θ) > dθ (1) 2π θ∈[0,2π] dθ 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 1 Z 2π E(f) = |f(θ)−1f 0(θ)|2dθ 4π 0 The moment map for the conjugation action of T (the “momentum” p) is Z 2π 1 −1 0 µ(f) = prLie(T ) f(s) f (s)ds 2π 0 (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 ΩG is the convex hull of the images of the fixed point set of T × S1 on ΩG, as for Hamiltonian torus actions on finite-dimensional symplectic manifolds. ∗ III. Motivation for computing KG(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 → G for which
1. dω = µ∗Λ where Λ is the generator of H3(G, Z) (this replaces dω = 0)
2. 1 ∗ i # ω = µ (θ + θ,¯ X) X 2 where θ (resp. θ¯) is the left-invariant (resp. right-invariant) Maurer-Cartan form θ = g−1dg, θ¯ = 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 component of 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 | | µ M −→ 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 flat connections 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 divided polynomial algebra Γ[s] is defined as a Z-algebra generated by sj (in degree 2j) where (j + k)! s · s = s . j k j!k! j+k Notice that a divided polynomial algebra can be described as the inverse limit of the symmetric polynomials in an exterior algebra. ∗ •HG(ΩG): Borel and others, late 1950s Tensor product of divided polynomial ∗ algebra Γ with HG(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) ⊗ KG(pt) •KT (ΩU(n)) (recursive calculation 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 complex dimension 2.
∗ 1 2r 1 2r Compute HG((P ) ) and KG((P ) ) via Bott periodicity. The answers as rings are as follows.
∗ 1 2r ∗ ¯ ¯ ¯2 ¯ • HG((P ) ) = HG(pt)[L1,..., L2r]/ < Lj − t > ¯ ∗ ∼ ¯ ¯ where t is an element of degree 4 which generates HG(pt) = Z[t] and Li are 2 1 2 1 elements of degree 2 corresponding to HG(P ) (isomorphic to H (P ), since G is simply connected.)
1 L2j corresponds to the canonical line bundle, over the 2jth copy of P . 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 HG corresponding to analogous elements in KG. 1 2r • KG((P ) ) = KG(pt)/I 2 where I is the ideal generated by Lj − vLj + 1 for j = 1,..., 2r Chern character
1. Chern homomorphism from K(X) to
∞ Y Hj(X) ⊗ Q j=0 (isomorphism with Q coefficients)
2. Chern homomorphism in KG: ∞ Y j KG(X) → HG(X; Q) j=0
(Reference: mimeographed notes by Atiyah and Segal, 1965)
3. Some properties of the Chern homomorphism: G (a) chG(x) = exp(c1 (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 := n r o X j f(z) ∈ G f(1) = I, f(z) = ajz , aj ∈ M2×2(C) j=−r i.e. the Fourier expansion of f is a finite Laurent polynomial expansion from −r to r. Set ∞ [ ΩpolySU(2) := Ωpoly,rSU(2) r=0
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.) •
∗ ∗ ∗ HG(ΩG) = HG(ΩpolyG) is the inverse limit of HG(Ωpoly,rG); K∗ (ΩG) = K∗ (Ω G) is the inverse limit of K∗ (Ω G). (Milnor lim1 G G poly G poly,r ←− sequence) Note that Bott-Tolman-Weitsman studied ΩG using an analogous filtration coming from the Morse theory of the energy functional 1 Z 2π dγ f(γ) := | |2 dt 4π 0 dt As R(G)-modules:
• ∞ ∞ even Y Y KG (ΩG) = KG(pt) = R(G) j=0 j=0 • odd KG (ΩG) = 0 • Theorem: (rephrasing of known result)
∗ ∗ 1 2r HG(Ωpoly,rG) is isomorphic to the subring of HG((P ) ) consisting of the symmetric polynomials in L¯1,..., L¯2r. ¯ ¯ 0 Lets ¯j be the j-th elementary symmetric polynomial in L1,..., L2r, and lets ¯j denote the corresponding polynomial in L¯1,..., L¯2r−2. The system maps of this inverse limit are determined by the following matrix, in 0 the basis of {s¯j} and {s¯j}.
1 0 −t¯ . . . 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 −t¯ 0 0 0 0 0 ... 0 1 0 −t¯ 0 0 0 0 ... 0 0 1 0 −t¯ 0 0 0 ... 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 • MAIN THEOREM:
1 2r KG(Ωpoly,rG) is isomorphic to the subring of KG((P ) ) consisting of the symmetric polynomials in L1,...,L2r. 0 Let sj be the j-th elementary symmetric polynomial in L1,...,L2r, and let sj denote the corresponding polynomial in L1,...,L2r−2. 2 Note that the relations Lj = vLj − 1 imply that any symmetric polynomial is actually 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 0 the basis of {sj} and {sj}. 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 a representation to its dimension.
1 2r 2 For (P ) , our relation Lj − vLj + 1 = 0 reduces (under setting v equal to 2) to
2 (Lj − 1) = 0, and we recover the familiar fact that K((P1)2r) is an exterior algebra.
1 2r K(Ωpoly,rG) = symmetric polynomials in K (P ) .
Note that the set of symmetric polynomials in the exterior algebra
Λ[y1, . . . , y2r] equals the truncated divided polynomial algebra 2r+1 Γ[y]/(y ) where y = y1 + ... + y2r. Taking the inverse limit tells us that K(ΩG) is a completed divided polynomial Q∞ 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
∼ 2r−1 • Ωpoly,r/Ωpoly,r−1 =G Thom(τ ) Here τ is the tangent bundle of P1. Hence
2r ∼ Y KG(Ωpoly,rG) = R(G) i=0 as R(G)-modules, using Thom isomorphism and induction. Taking the inverse limit gives
∞ ∼ Y KG(ΩG) = R(G) i=0 as R(G)-modules.
2r−1 ∼ 2r−1 2r−1 • Thom(τ ) =G P(τ ⊕ )/P(τ ) using Atiyah’s description of the Thom space. Here is the trivial bundle over P1. ∼ 1 1 • We show that P(τ ⊕ ) =G P × P where the subspace P(τ) gets mapped to the diagonal ∆ under this homeomorphism. Hence
∼ ∼ ∼ 1 1 Ωpoly,1G = Thom(τ) = P(τ ⊕ )/P(τ) = (P × P )/∆.
Therefore we have a quotient map
1 1 1 1 ∼ Φ2 : P × P → (P × P )/∆ = Ωpoly,1G. 1 2r We define Φ2r :(P ) → Ωpoly,rG as the composition
r 1 2r (Φ2) r Φ2r :(P ) → (Ωpoly,1G) → Ωpoly,rG where the last map is induced by pointwise matrix multiplication.
∗ ∗ ∗ 1 2r • Φ : KG(F2r) → KG((P ) ) is injective (this step uses Chern homomorphism). So multiplication in KG(Ωpoly,rG) is the restriction of multiplication on 1 2r KG((P ) ) ∗ ∗ • To finish the computation, we must compute the image of Φ on KG(Ωpoly,rG). To do this we first answer the corresponding question on cohomology. Use induction on r. The result is:
∗ ∗ 1 2r HG(Ωpoly,rG) = symmetric polynomials in HG((P ) ) • Key step:
1 2r KG(Ωpoly,rG) = symmetric polynomials in KG((P ) ).
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 1 2r conclude that KG(Ωpoly,rG) is the symmetric polynomials in KG((P ) ), as claimed in our theorem. ∼ W Note: It is not true in general that KG(X) = (KT (X)) (there are counterexamples 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)