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quantization of loop groups

Contents

Contents

Definition

Let GG be a compact Lie group and write LGL G for its loop group. See there for details and notation.

We discuss the quantization of loop groups in the sense of geometric quantization of their canonical prequantum bundle.

Let GG be a compact Lie group. Let TGT \hookrightarrow G be the inclusion of a maximal torus. 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 } \,.
Remark

By the discussion at orbit method, if GG is a semisimple Lie group, then G/TG/T is isomorphic to the coadjoint orbit of an element λ,𝔤 *\langle \lambda , -\rangle \in \mathfrak{g}^* for which TG λT \simeq G_\lambda is the stabilizer subgroup.

If moreover GG is simply connected, then 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(G,U(1)). \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1)) \,.
Proposition

The irreducible projective positive energy representations of LGL G correspond precisley to the possible geometric quantizations of LG/TL G / T (as in the orbit method).

More in detail:

The degree-2 integral cohomology of LG/TL G / T is

H 2(LG/T)H 2(G/T,)T^. H^2(L G / T) \simeq \mathbb{Z} \oplus H^2(G / T, \mathbb{Z}) \simeq \mathbb{Z} \oplus \hat T \,.

Writing L n,λL_{n,\lambda} for the corresponding complex line bundle with level nn \in \mathbb{Z} and weight λT^\lambda \in \hat T we have that

  1. the space of holomorphic sections of L n,λL_{n,\lambda} is either zero or is an irreducible positive energy representation;

  2. every such arises this way;

  3. and is non-zero precisely if (n,λ)(n,\lambda) is positive in the sense that for each positive coroot? h αh_\alpha of GG

    0λ(h α)nh α,h α. 0 \leq \lambda(h_\alpha) \leq n \langle h_\alpha, h_\alpha\rangle \,.

This appears for instance as (Segal, prop. 4.2).

References

The standard textbook on loop groups is

  • Andrew Pressley, Graeme Segal, Loop groups Oxford University Press (1988)

A review talk is

Created on November 6, 2013 at 03:05:42. See the history of this page for a list of all contributions to it.