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projective unitary group

Contents

Definition

Definition

For \mathcal{H} a Hilbert space, the projective unitary group PU()P U(\mathcal{H}) is the quotient of the unitary group U()U(\mathcal{H}) by its center ZU()U(1)Z U(\mathcal{H}) \simeq U(1), the circle group

PU():=U()/U(1). P U(\mathcal{H}) := U(\mathcal{H})/U(1) \,.

This is naturally a topological group. For \mathcal{H} of finite-dimension nn PU(n):=PU()P U(n) := P U(\mathcal{H}) is also naturally a Lie group.

Properties

Homotopy type

Prop

If \mathcal{H} is an infinite-dimensional separable Hilbert space the underlying topological space of its projective unitary group has the homotopy type of an Eilenberg-MacLane space K(,2)K(\mathbb{Z}, 2).

Proof

The unitary group U()U(\mathcal{H}) in this case is contractible (by Kuiper's theorem) and the circle group U(1)U(1) acts free and faithfully on it. Therefore the quotient map U()PU()U(\mathcal{H}) \to P U(\mathcal{H}) is a model for the circle group-universal principal bundle and in particular the topological space underlying PU()P U(\mathcal{H}) is equivalent to the classifying space BU(1)B 2K(,2)B U(1) \simeq B^2 \mathbb{Z} \simeq K(\mathbb{Z},2).

Prop

For \mathcal{H} an infinite-dimensional separable Hilbert space PU()P U(\mathcal{H})-principal bundles over a topological space XX are classidied by third integral cohomology of XX

PU()Bund(X)H 3(X,). P U(\mathcal{H}) Bund(X) \simeq H^3(X, \mathbb{Z}) \,.
Proof

By prop. 1 we have that the classifying space of PU()P U(\mathcal{H}) itself is an Eilenberg-MacLane space

BPU()BBU(1)B 3K(,3). B P U(\mathcal{H}) \simeq B B U(1) \simeq B^3 \mathbb{Z} \simeq K(\mathbb{Z}, 3) \,.

This is the classifying space for degree-3 integral cohomology (see Eilenberg-MacLane spectrum for more on this).

Prop

Every circle 2-bundle/bundle gerbe on XX is equivalent to the lifting gerbe of some PU()P U(\mathcal{H})-principal bundle to a U()U(\mathcal{H})-bundle, and the equivalence classes of these structures correspond uniquely.

Proof

The twisted bundles of a given bundle gerbe are given by the twisted cohomology relative to the morphism BPU()B 2U(1)\mathbf{B} P U(\mathcal{H}) \to \mathbf{B}^2 U(1) that is part of the long fiber sequence

BU(1)BU()BPU()B 2U(1). \mathbf{B} U(1) \to \mathbf{B} U(\mathcal{H}) \to B \mathbf{P} U(\mathcal{H}) \to \mathbf{B}^2 U(1) \,.

Since the topological space underlying U()U(\mathcal{H}) is contractible, on the underlying topological spaces this is

K(,2)*K(,3)K(,3). K(\mathbb{Z},2) \to * \to K(\mathbb{Z},3) \stackrel{\simeq}{\to} K(\mathbb{Z}, 3) \,.

This means that the morphism that sends PU()P U(\mathcal{H})-bundles to the twist that they induce is an isomorphism.

(Somebody should force me to say this in more detail).

For more on this see also twisted K-theory.

Action on Fredholm operators

Let \mathcal{H} be an infinite-dimensional separable Hilbert space.

Since by the above PU()BU(1)PU(\mathcal{H}) \simeq B U(1) and since there is a canonical action of line bundles on complex vector bundles, hence on the topological K-theory of a manifold XX, there must also be a natural action of PU()×FredFredPU(\mathcal{H}) \times Fred \to Fred of PU()PU(\mathcal{H}) on the space of Fredholm operators (on connected components).

This is given by letting a projective unitary act by conjugation on a Fredholm operator: (g,F)gFg 1(g, F) \mapsto g F g^{-1}.

Revised on March 22, 2014 08:35:11 by Urs Schreiber (82.113.121.50)