Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the * Banach-Steinhaus theorem* shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H).

If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {U_{i}}_{i∈I} of X, the transition functions between trivializations P|_{Ui} can be lifted to bundle isomorphisms g_{ij} on double intersections U_{ij} = U_{i} ∩ U_{j} which are projectively coherent, i.e., over each of the triple intersections U_{ijk} = U_{i} ∩ U_{j} ∩ U_{k} the composition g_{ki} g_{jk} g_{ij} is given as multiplication by a U(1)-valued function f_{ijk} : U_{ijk} → U(1). The collection {(U_{ij}, f_{ijk})} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class B_{P} in the sheaf cohomology group H^{2}(X, U(1)_{X}). On the other hand, the sheaf cohomology H^{1}(X, PU(H)_{X}) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H^{1}(X,PU(H)_{X}).

There is an isomorphism

H^{1}(X, PU(H)_{X}) →^{≈} H^{2}(X, U(1)_{X}) provided by the

boundary map [P] ↦ B_{P}. There is also an isomorphism

H^{2}(X, U(1)_{X}) →^{≈} H^{3}(X, Z_{X}) ≅ H^{3}(X, Z)

The image δ(P) ∈ H^{3}(X, Z) of B_{P} is called the * Dixmier-Douady* invariant of P. When δ(P) = [H] is represented in H

^{3}(X, R) by a closed three-form H on X, called the H-flux of the given B-field B

_{P}, we will write P = P

_{H}. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By

*every torsion element of H*

**Serre’s theorem**^{3}(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by

where we identify the cyclic group Z_{n} with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(C^{n}). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element B_{P} ∈ H^{2}(X, U(1)_{X}) comes from H^{2}(X, (Z_{n})_{X}), and therefore its order divides n.

One also has δ(P_{1} ⊗ P_{2}) = δ(P_{1}) + δ(P_{2}) and δ(P^{∨}) = −δ(P). This follows from the commutative diagram

and the fact that P^{∨} ⊗ P = P(E) where E is the vector bundle of * Hilbert-Schmidt endomorphisms *of P . Putting everything together, it follows that the cohomology group H

^{3}(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product.

We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that P_{x} = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z_{2}-graded Hilbert space. We define Fred^{0}(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T^{2} − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred^{0}(H) ֒→ B(H) × K(H) given by T → (T, T^{2} − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology.

Let P = P_{H} → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred^{0}(P) whose fibres are Fred^{0}(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps

K^{0}(X, H) = [X, Fred^{0}(P_{H})]

The group K^{0}(X, H) depends functorially on the pair (X, P_{H}), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K^{0}(X, H) → K^{0}(X, H′). Addition in K^{0}(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K^{0}(X, H_{2}) with [H_{2}] = δ(P ⊗ P(C^{2})) = δ(P) = [H]. Under the isomorphism H ⊗ C^{2} ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C^{2}) for any projective bundle P and so K^{0}(X, H_{2}) is canonically isomorphic to K^{0}(X, H). When [H] is a non-torsion element of H^{3}(X, Z), so that P = P_{H} is an infinite-dimensional bundle of projective spaces, then the index map K^{0}(X, H) → Z is zero, i.e., any section of Fred^{0}(P) takes values in the index zero component of Fred^{0}(H).

Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and

End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H)

as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle P_{H} we can associate a bundle of compact operators E_{H} → X given by

E_{H} = P_{H} ×_{PU} K

with δ(E_{H}) = [H]. The Banach algebra A_{H} := C_{0}(X, E_{H}) of continuous sections of E_{H} vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K^{•}(X, H) of X is canonically isomorphic to the algebraic K-theory group K_{•}(A_{H}).

We will also need a smooth version of this definition. Let A^{∞}_{H} be the smooth subalgebra of A_{H} given by the algebra CT^{∞}(X, H) = C^{∞}(X, L^{1}_{PH}),

where L^{1}_{PH} = P_{H} ×_{PU}L^{1}. Then the inclusion CT^{∞}(X, H) → CT(X, H) induces an isomorphism K_{•}CT(X, H) →^{≈} K_{•}CT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism K_{•}CT^{∞}(X, H) ≅ K^{•}(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = P_{H} over X.

Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories

E(B, X) ≅ LF (C(X, B))

Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups

K(LF (C(X, B))) ≅ K(E(B, X))

This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group

K(X, A) := K LF(A)

For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have

K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K^{0}(X)

The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, M_{m}(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of * Azumaya algebras* over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the

*of X and denoted Br(X). By Serre’s theorem there is an isomorphism*

**Brauer group**δ : Br(X) →^{≈} tor(H^{3}(X, Z))

where tor(H^{3}(X, Z)) is the torsion subgroup of H^{3}(X, Z).

If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K_{0}(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism

K(X, C(X, A)) ≅ K Proj (C(X, A))

where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H^{3}(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K^{0}(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K^{0}(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K_{0}(C(X)) = K^{0}(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre C^{nm} ≅ (C^{m})^{⊕n}, which is naturally an M_{m}(C)-module.

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