The Based Loop Group of SU(2) Lisa Jeffrey. Department of Mathematics University of Toronto. Joint work with Megumi Harada and Paul Selick
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1 The Based Loop Group of SU(2) Lisa Jeffrey Department of Mathematics University of Toronto Joint work with Megumi Harada and Paul Selick
2 I. The based loop group ΩG Let G = SU(2) and let T be its maximal torus { T = eiθ 0 } = U(1) 0 e iθ The Weyl group is W = Z 2. The based loop group ΩG is defined as ΩG = {γ : S 1 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 L s G (for s > 3/2) is defined as the set of maps from S 1 to G of Sobolev class H s, meaning that the Sobolev norm f s is finite (for example ( f 2 ) 2 = f 2 + df 2 using the L 2 norm). Strictly speaking the Sobolev norm is defined on the Lie algebra of L s G and the exponential map is used pointwise to transfer this definition to L s G. L s G is an infinite dimensional Hilbert manifold. It contains the set C (G) of C maps from S 1 to G, which however is not a Hilbert manifold.
3 The subset Ω s G of L s G (the based loop group) consists of those loops f : S 1 G for which f is the identity element at the basepoint. There is a surjective map from L s G to Ω s G defined as follows: F : h h( ) 1 h This map sends the submanifold of constant loops (which may be identified with the group G) to the identity element in Ω 1 G, so it identifies Ω s G with the homogeneous space L s G/G. The space Ω s G is symplectic. The symplectic form at the identity element e is ω e (X, Y ) = 1 < X(s), dy (θ) > dθ (1) 2π dθ for X, Y L s (g). θ [0,2π]
4 II. Torus actions on the based loop group The rotation group S 1 acts on Ω 1 G as follows: (e iθ f)(s) = f(θ) 1 f(s + θ) (2) for s, θ [0, 2π]. The maximal torus T acts on Ω 1 G 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 2π 0 f(θ) 1 f (θ) 2 dθ The moment map for the conjugation action of T (the momentum p) is µ(f) = 1 2π pr Lie(T ) 2π 0 f(s) 1 f (s)ds (where the projection is onto the Lie algebra of the maximal torus T ).
5 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 S 1 on ΩG is the convex hull of the images of the fixed point set of T S 1 on ΩG, as for Hamiltonian torus actions on finite-dimensional symplectic manifolds.
6 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 H 3 (G, Z) (this replaces dω = 0) 2. i X #ω = 1 2 µ (θ + θ, X) where θ (resp. θ) is the left-invariant (resp. right-invariant) Maurer-Cartan form θ = g 1 dg, θ = dgg 1 Ω 1 (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.
7 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 M µ G
8 The vertical map from Lg to G is the holonomy. In case M = G sg, 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
9 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 G 2N of an even-dimensional number of copies of G; conjugacy classes in G.
10 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 s j (in degree 2j) where s j s k = (j + k)! s 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.
11 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) K G (G): Brylinski-Zhang, 2000: H (G) K G (pt) K T (ΩU(n)) (recursive calculation using Kac-Moody algebras): Kostant-Kumar, 1990
12 Homotopy filtrations of ΩSU(2) James, 1955 Pressley-Segal, 1986 Bott-Tolman-Weitsman, 2004
13 V. Background about cohomology and K-theory of products of copies of P 1 Note that P 1 = G/T and G acts on it by left multiplication. K G (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. Compute H G ((P1 ) 2r ) and K G ((P 1 ) 2r ) via Bott periodicity.
14 The answers as rings are as follows. HG((P 1 ) 2r ) = HG(pt)[ L 1,..., L 2r ]/ < L 2 j t > where t is an element of degree 4 which generates HG (pt) = Z[ t] and L i are elements of degree 2 corresponding to HG 2 (P1 ) (isomorphic to H 2 (P 1 ), since G is simply connected.) L 2j corresponds to the canonical line bundle, over the 2jth copy of P 1. L 2j 1 is the hyperplane line bundle (the dual of the canonical line bundle) over the (2j 1)-th copy of P 1. Here we have written bars for elements in HG elements in K G. corresponding to analogous
15 K G ((P 1 ) 2r ) = K G (pt)/i where I is the ideal generated by L 2 j vl j + 1 for j = 1,..., 2r
16 Chern character 1. Chern homomorphism from K(X) to H j (X) Q j=0 (isomorphism with Q coefficients) 2. Chern homomorphism in K G : K G (X) j=0 H j G (X; Q) (Reference: mimeographed notes by Atiyah and Segal, 1965) 3. Some properties of the Chern homomorphism: (a) ch G (x) = exp(c G 1 (x)) if x is a line bundle (b) ch G Q is an isomorphism on the spaces we will discuss
17 VI. Statement of Results Ω poly,r G := { f(z) G f(1) = I, f(z) = r j= r } a j z j, a j M 2 2 (C) i.e. the Fourier expansion of f is a finite Laurent polynomial expansion from r to r. Set Ω poly SU(2) := Ω poly,r SU(2) We refer to Ω poly SU(2) as the the space of polynomial (based) loops in SU(2). The inclusion Ω poly G 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.) r=0
18 H G (ΩG) = H G (Ω polyg) is the inverse limit of H G (Ω poly,rg); KG (ΩG) = K G (Ω polyg) is the inverse limit of KG (Ω 1 poly,rg). (Milnor lim 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 2π 0 dγ dt 2 dt
19 As R(G)-modules: K even G (ΩG) = K G (pt) = j=0 K odd G (ΩG) = 0 R(G) j=0
20 Theorem: (rephrasing of known result) H G (Ω poly,rg) is isomorphic to the subring of H G ((P1 ) 2r ) consisting of the symmetric polynomials in L 1,..., L 2r. Let s j be the j-th elementary symmetric polynomial in L 1,..., L 2r, and let s j denote the corresponding polynomial in L 1,..., L 2r 2.
21 The system maps of this inverse limit are determined by the following matrix, in the basis of { s j } and { s j }. 1 0 t t t t
22 MAIN THEOREM: K G (Ω poly,r G) is isomorphic to the subring of K G ((P 1 ) 2r ) consisting of the symmetric polynomials in L 1,..., L 2r. Let s j be the j-th elementary symmetric polynomial in L 1,..., L 2r, and let s j denote the corresponding polynomial in L 1,..., L 2r 2. Note that the relations L 2 j = vl j 1 imply that any symmetric polynomial is actually a linear combination of s 0, s 1,..., s 2r. The s j are the R(G)-module generators. The system maps of this inverse limit are determined by the following matrix, in the basis of {s j } and {s j }.
23 1 v v v v v
24 VII. Comparison with known results The reduction K G (X) K(X) is induced by ɛ : R(G) Z where ɛ takes a representation to its dimension. For (P 1 ) 2r, our relation L 2 j vl j + 1 = 0 reduces (under setting v equal to 2) to (L j 1) 2 = 0, and we recover the familiar fact that K((P 1 ) 2r ) is an exterior algebra. K(Ω poly,r G) = symmetric polynomials in K ( (P 1 ) 2r). Note that the set of symmetric polynomials in the exterior algebra Λ[y 1,..., y 2r ] equals the truncated divided polynomial algebra Γ[y]/(y 2r+1 ) where y = y y 2r.
25 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=0 Z.
26 VIII. Outline of proof Ω poly,r /Ω poly,r 1 =G Thom(τ 2r 1 ) Here τ is the tangent bundle of P 1. Hence K G (Ω poly,r G) = 2r i=0 R(G) as R(G)-modules, using Thom isomorphism and induction. Taking the inverse limit gives as R(G)-modules. K G (ΩG) = R(G) i=0 Thom(τ 2r 1 ) = G P(τ 2r 1 ɛ)/p(τ 2r 1 ) using Atiyah s description of the Thom space. Here ɛ is the trivial bundle over P 1. We show that P(τ ɛ) = G P 1 P 1 where the subspace P(τ) gets mapped to
27 the diagonal under this homeomorphism. Hence Ω poly,1 G = Thom(τ) = P(τ ɛ)/p(τ) = (P 1 P 1 )/. Therefore we have a quotient map Φ 2 : P 1 P 1 (P 1 P 1 )/ = Ω poly,1 G.
28 We define Φ 2r : (P 1 ) 2r Ω poly,r G as the composition Φ 2r : (P 1 ) 2r (Φ 2) r (Ω poly,1 G) r Ω poly,r G where the last map is induced by pointwise matrix multiplication. Φ : KG (F 2r) KG ((P1 ) 2r ) is injective (this step uses Chern homomorphism). So multiplication in K G (Ω poly,r G) is the restriction of multiplication on K G ((P 1 ) 2r )
29 To finish the computation, we must compute the image of Φ on KG (Ω poly,rg). To do this we first answer the corresponding question on cohomology. Use induction on r. The result is: HG (Ω poly,rg) = symmetric polynomials in HG ((P1 ) 2r )
30 Key step: K G (Ω poly,r G) = symmetric polynomials in K G ((P 1 ) 2r ). The argument requires first proving the analogous statement for K T. 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(Φ 2r ). Having obtained the computation of K T (Ω poly,r G), we take Weyl invariants to conclude that K G (Ω poly,r G) is the symmetric polynomials in K G ((P 1 ) 2r ), as claimed in our theorem. Note: It is not true in general that K G (X) = (K T (X)) W (there are counterexamples due to Reyer Sjamaar). However, our module calculations (see Section V) show us that this equality holds in our case.
31 Finally, take the inverse limit to get K G (ΩG)
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