Twisted differential K-characters and D-branes

We analyse in detail the language of partially non-abelian Deligne cohomology and of twisted differential K-theory, in order to describe the geometry of type II superstring backgrounds with D-branes. This description will also provide the opportunity to show some mathematical results of independent interest. In particular, we begin classifying the possible gauge theories on a D-brane or on a stack of D-branes using the intrinsic tool of long exact sequences. Afterwards, we recall how to construct two relevant models of differential twisted K-theory, paying particular attention to the dependence on the twisting cocycle within its cohomology class. In this way we will be able to define twisted K-homology and twisted Cheeger-Simons K-characters in the category of simply-connected manifolds, eliminating any unnatural dependence on the cocycle. The ambiguity left for non simply-connected manifolds will naturally correspond to the ambiguity in the gauge theory, following the previous classification. This picture will allow for a complete characterization of D-brane world-volumes, the Wess-Zumino action and topological D-brane charges within the K-theoretical framework, that can be compared step by step to the old cohomological classification. This has already been done for backgrounds with vanishing B-field; here we remove this hypothesis.


Introduction
In [5] and [17] we showed how to classify the possible gauge theories on a D-brane or on a stack of D-branes in type II superstring theory. In particular, the Freed-Witten anomaly cancellation imposes a suitable interaction between the A-field and the B-field on the worldvolume, so that the A-field does not always represent an ordinary gauge theory. For this reason, a classification of the possible natures of such a field was necessary, and we realized it using the language of partially non-abelian Deligne cohomology. Nevertheless, we achieved this aim "by hand", through the local representation of the fields in a suitable (good) cover, since we were not able to use a more intrinsic machinery when certain twistings were present. In the first part of this paper we reproduce that classification in a completely intrinsic way, only using the tool of long exact sequences in Deligne cohomology.
Afterwards, we link the previous description of the gauge theories to the following more general discussion, concerning the rigorous geometrical and topological foundations of type II superstring theory. As we already summarized in [15], there are two fundamental pictures that describe and classify D-brane charges and the Ramond-Ramond fields in this framework. The first one relies on classical cohomology. In particular, a D-brane world-volume is a submanifold, which becomes a singular cycle via a suitable triangulation, and the Poincaré dual of the underlying homology class is the topological charge. The Ramond-Ramond fields are classified by ordinary differential cohomology, for which the Deligne cohomology provides a concrete model [8], and the Wess-Zumino action turns out to be the holonomy of a differential cohomology class along the world-volume. The other fundamental classification scheme relies on K-theory (see [14,30,31,32,19,34] among many others; see also [13] for a more refined proposal, that we do not consider here, and the recent preprint [24] for a detailed analysis of the Ramond-Ramond fields). In particular, a D-brane world volume defines a K-homology class in the space-time through the Chan-Patton bundle, and its Poincaré dual is by definition the topological charge. The Ramond-Ramond fields are classified by a differential K-theory class, that is supposed to couple with the world-volume, so that the Wess-Zumino action is well-defined. In [15] we tried to clarify how to correctly define the world-volume through the notion of differential K-character [4,16], in order to get such a coupling. In this way we managed to draw a complete parallel between the two classification schemes. Nevertheless, since we considered ordinary K-theory, we supposed that the Bfield was vanishing. Here we remove this hypothesis, therefore we develop an analogous construction for twisted K-theory and its differential extension.
In this setting, we pay particular attention to the unnatural dependence on the twisting cocycle, developing a model of twisted K-homology and of Cheeger-Simons characters that only depends on the twisting cohomology class as far as it is possible. Coherently, we show that the ambiguity left in such definitions, when certain hypotheses are not satisfied, in spite of being a limit of this model from a purely mathematical point of view, has an interesting physical meaning, since it naturally fits in the classification of the gauge theories we started from. Moreover, we show that the Freed-Witten anomaly cancellation, that has been introduced originally in order for the world-sheet action to be well-defined, is also the condition for the Wess-Zumino action to be well-defined, since it is essentially the hypothesis imposed on the twisting classes in the definition of Cheeger-Simons K-character.
The paper is organized as follows. In chapter 2 we deal with partially non-abelian Deligne cohomology and we use it to classify the gauge theories on a D-brane. In chapter 3 we review and analyse two relevant models of differential twisted K-theory, also completing some constructions that we did not find explicitly in literature. In chapter 4 we provide the definition of twisted Cheeger-Simons characters and of twisted K-homology from the viewpoint sketched above. We conclude with chapter 5, in which we apply the previous material to complete the rigorous gemetrical description of type II supestring background with D-branes.

Partially non-abelian Deligne cohomology
We are going to recall the notion of twisted vector bundle with connection and to insert it within the framework of partially non-abelian Deligne cohomology. In this way we will be able to classify the possible gauge theories on a D-brane or on a stack of D-branes, using the intrinsic language of long exact sequences.

Brief review on the Deligne cohomology
Given a smooth manifold X, we consider the complex of sheaves: where U(1) is the sheaf of smooth U(1)-valued functions, Ω k R is the sheaf of real k-forms, d is the exterior differential anddf := 1 2πi f −1 df . The Deligne cohomology group of degree p on X is by definition the sheaf hypercohomology group of the complex (1), i.e.,Ȟ p (X, S p X ). It can be concretely described through a good cover U = {U i } i∈I of X as follows: we consider the double complex whose rows are theČech complexes of the sheaves involved in (1), and we consider the cohomology of the associated total complex. This means that a p-cocycle consists in a sequence ({g i 0 ···ip }, {(C 1 where C k is a kform, satisfying the conditions: 2πi g −1 i 0 ...ip dg i 0 ...ip δ p g i 0 ...ip = 1. (2) We call G := [g, C 1 , . . . , C p ] the corresponding cohomology class. The local forms C p correspond to the Ramond-Ramond potentials in string theory (if we consider this model) and their differentials dC p glue to a global gauge-invariant closed form G p+1 (the Ramond-Ramond field strength), which is called curvature. Moreover, from the underlying class [{g i 0 ···ip }] ∈Ȟ p (U, U(1)), applying the isomorphismȞ p (U, U(1)) ≃ H p+1 (X; Z), we get the first Chern class c 1 (G) ∈ H p+1 (X; Z). The de-Rham cohomology class, represented by the curvature, is the real image of the first Chern class (Dirac quantization condition), therefore the curvature has integral periods. Of course we are free to add any coboundary to the cocycle (g, C 1 , . . . , C p ), the meaningful datum being the corresponding cohomology class, since it is determined by the two real physical observables: the field strength G p+1 (corresponding to the field F in electromagnetism) and the holonomy or Wess-Zumino action (providing the additional piece of information that, in electromagnetism, is measured by the phase difference in the Aranhov-Bohm effect). The latter can be computed on a singular p-cycle of X, integrating the local k-forms C k on the k-simplicies and summing the results in a suitable way [22]. It is related to the curvature by a Stokes-type formula, hence, if the class is flat (i.e. the curvature vanishes), then its holonomy on a cycle only depends on the underlying homology class. (5) where ρ !,q (ω) = ρ * ω for any q ≥ 1 and ρ !,0 (f ) = f • ρ. The corresponding cone complex is the following one: The cohomology groups of (Č C C • (ρ),Ď D D • ) are by definition the relative Deligne cohomology groups of ρ.
Remark 2.1 The case p = 1 is quite clear geometrically. In fact, a relative Deligne class is represented by a cocyle of the form (α, β), where α = ({g ij }, {A i }) ∈Ž 1 (S 1 X ) represents a line bundle with connection (L, ∇) and β = {h i } ∈Č 0 (U(1) Y ) represents a trivialization (i.e. a global non-vanishing section) of ρ * L. Let us see why such a construction is natural. Fixing a piece-wise smooth curve γ : I → X, such that γ(∂I) ⊂ Y , the parallel transport of ∇ along γ is not well-defined as a complex number, since γ is not a closed curve. Nevertheless, since γ(∂I) ⊂ Y , the fixed trivialization of L| Y provides a canonical way to identify the fibres L γ(0) and L γ( 1) with C, hence the parallel transport, as a unitary liner map from L γ(0) to L γ( 1) , becomes a well-defined number belonging to U(1). Therefore, the 1-Deligne cohomology group of ρ naturally leads to the notion of relative holonomy.
In general, if (α, β) represents a relative cocycle of degree p, then α represents a Deligne p-cohomology class on X such that ρ * α has trivial first Chern class. This means that ρ * α is topologically trivial, but not necessarily trivial as a Deligne class. It follows that ρ * α is cohomologous to a cochain of the form (1, 0, . . . , 0, G), where G is a global potential. The cochain β provides a suitable reparametrization, i.e. ρ * α −Ďβ = (1, 0, . . . , 0, G). By definition, the curvature of [(α, β)] is the relative form (F, G) ∈ Ω p (ρ), where F is the curvature of α in X and G is the global potential on Y . Now we can understand why, in (5), the complex on Y has been truncated at degree p − 1, not p. That's because, if we reach p even in the lower row, then the cocycle condition also imposes G = 0, hence ρ * α must be trivial (not only topologically). Therefore, the morphism ρ ! : S p X → ρ * S p Y (with p on both sides) leads to a proper subgroup of relative p-Deligne cohomology, whose elements are called parallel classes: We set:Ĥ We get the following diagram, generalizing (3) to the relative framework: Moreover,Ĥ p+1 par (ρ) is the subgroup ofĤ p+1 (ρ) formed by classes with curvature of the form (F, 0). It is possible to define the holonomy of a class (α, β) ∈Ĥ p+1 (ρ) on relative p-cycles, such cycles being defined through the homological version of the cone complex.
Remark 2.2 In the case p = 1, using the notations of remark 2.1, a class is parallel when the fixed trivialization of L| Y is a global parallel section with respect to ∇. It follows that ∇| Y = 0.

Twisted vector bundles
We fix a paracompact topological space X and a good cover U = {U i } i∈I , whose existence we assume by hypothesis. Every smooth manifold admits a good cover [6]. We denote by U(r) the sheaf of U(r)-valued continuous functions on X and, when r = 1, we denote byČ • (U, U(1)),Ž • (U, U(1)) andȞ • (U, U(1)) the corresponding Cech cochains, cocyles and cohomology classes, with respect to the fixed good cover U.
Definition 2.3 Given a cochain ζ := {ζ ijk } ∈Č 2 (U, U(1)), a ζ-twisted vector bundle of rank r on X is a collection of trivial Hermitian vector bundles π i : E i → U i of rank r and of unitary vector bundle isomorphisms ϕ ij : E i | U ij → E j | U ij , such that ϕ ki ϕ jk ϕ ij = ζ ijk · id.
Non-integral vector bundles. The following notion will naturally appear in the classification of the gauge theories on a D-brane, and it is the natural generalization of the "line bundles with non-integral first Chern class" defined in [5].
In this case the image of [ζ] in the cohomology of U(1) is always a torsion class, but the corresponding U(1)-cohomology class is not necessarily torsion, unless the transition functions can be chosen constant too. We denote by NIVB ζ (X) the set of non-integral vector bundles with twisting class ζ. In this case we have isomorphisms analogous to (10) and ( 11), but with respect to a U(1)-cochain η. It follows that the set of isomorphisms of the form Φ η is a torsor over the image of the natural map H 1 (U, U(1)) → H 1 (U, U(1)), where H 1 (U, U(1)) ≃ H 1 (X; R/Z) ≃ Hom(H 1 (X; Z); R/Z) and of course H 1 (U, U(1)) ≃ H 2 (X; Z). The image is canonically isomorphic to Tor H 2 (X; Z). Therefore, if Tor H 2 (X; Z) = 0 (in particular, if X is simply connected), then NIVB [ζ] (X) is canonically defined, with [ζ] ∈ H 2 (X; R/Z).
For any k ≥ 1, we call Γ k the subgroup of U(1) formed by k-th roots of unity and we set Γ := k∈N Γ k . It follows that Γ k is the image of the group embedding Z k ֒→ U(1), a → e 2πi a k , and Γ is the image of Q ֒→ U(1), q → e 2πiq . It is possible to define non-integral vector bundle with twisting cocycle in Γ or Γ k . This general picture can be useful is some contexts (e.g. dealing with fractionary branes), but for our purposes it will be enough to consider Z 2 (or Γ 2 ).

Twisted connections
In section 2.1 we considered the sheaf complex S p X , defined in (1), whose Cech p-hypercohomology group is by definition the Deligne p-cohomology group of the smooth manifold X. Now we are interested in S 2 X as a source of twistings. In fact, as a bundle can be twisted with respect to a 2-cocycle of the sheaf U(1), similarly a bundle with connection can be twisted with respect to a 2-cocycle of the complex S 2 X [17].
Definition 2.10 Given a Cech 2-cocycle (ζ, Λ, B) of the complex S 2 X and a ζ-twisted vector bundle E := ({E i }, {ϕ ij }), a (ζ, Λ, B)-twisted connection on E is defined by a connection ∇ i on E i for each i ∈ I, compatible with the Hermitian metric and with curvature ∇ 2 i , in such a way that: where r = rk(E).
With this definition, the main features of topological twisting hold for connections too. In particular, if we suppose that (ζ, Λ, B) is only a cochain, the existence of a corresponding twisted connection implies that it is a cocycle, as the reader can prove by direct computation. It follows that the twisting class [(ζ, Λ, B)] ∈Ȟ 2 (S 2 X ) is well-defined, and it must be a torsion class, as we are going to show in remark 2.11.
If, for each i ∈ I, we fix a set of r pointwise-independent local sections s 1,i , . . . , s r,i : U i → E i of unit norm, we get the transition functions g ij : U ij → U(r) and the local poten- In this case, if we choose local sections s 1,i , . . . , s r,i : . . , f i (s i,r (x))}, that satisfy the following relations (as for ordinary vector bundles): Remark 2.11 The class [(ζ, Λ, B)] is necessarily torsion (in particular, flat). In fact, computing determinants and traces in (13), we getĎ 1 (det g, TrA) = (ζ r , rΛ, rB) = r(ζ, Λ, B), whereĎ 1 is the coboundary of the total complex associated to theČech double complex induced by S 2 X . It follows that r[(ζ, Λ, B)] = 0. In particular, the order of [(ζ, Λ, B)] divides r. One can prove that, for any cocycle representing a torsion class, there exist some corresponding twisted bundles with connection [17]. This confirms the naturality of definition 2.10, since it extends to the differential setting all of the fundamental features of topological twisting.
It follows from the previous remark that, as topological twisting classes are classified by Tor H 3 (X; Z), similarly differential twisting classes are classified by Tor H 2 (X; R/Z). If H 2 (X; Z) is finitely generated (in particular, if X is compact), then Tor H 2 (X; R/Z) ≃ H 2 (X; Q/Z). Moreover, if E is an ordinary vector bundle, then an ordinary connection ∇ on E is a (1, 0, 1 r TrF i )-twisted connection, TrF i being a global integral form. Coherently, the class [(1, 0, 1 r TrF i )] is torsion and its order divides r.
Remark 2.12 If we consider the q-cohomology of (1), for q = p, we get the following picture. If q < p, thenȞ q (S p X ) ≃ H q (X; R/Z). This is due to the fact that the q-cochains and the q-coboundaries of S p X coincides with the ones of S q X , but the cocycle condition is stronger, since it also imposes flatness. If q > p, thenȞ q (S p X ) ≃ H q+1 (X; Z), since, the sheaves Ω • R being acyclic, the reader can verify that the cohomology class [(ζ, Λ 1 , . . . , We now weaken the twisting condition as follows [33]. We observe that, if (ζ, Λ, B) is a 2-cocycle of S 2 X , then (ζ, Λ) is a 2-cocycle of S 1 X . Conversely, any 2-cocycle (ζ, Λ) of S 1 X can be completed to a 2-cocycle (ζ, Λ, B) of S 2 X , the B-component being unique up to a global formB ∈ Ω 2 (X). It follows from remark 2.12 that [(ζ, Definition 2.13 Given a Cech 2-cocycle (ζ, Λ) of the complex S 1 X and a ζ-twisted vector bundle E := ({E i }, {ϕ ij }), a (ζ, Λ)-twisted connection on E is defined by a connection ∇ i on E i for each i ∈ I, compatible with the Hermitian metric, in such a way that: Again, if we suppose that (ζ, Λ) is only a cochain, the existence of a corresponding connection implies that it is a cocycle. It follows that the twisting (torsion) class [(ζ, Λ)] ∈Ȟ 2 (S 1 X ) ≃ H 3 (X; Z) is well-defined. For any cocycle representing a torsion class, there exists a twisted bundle with connection. With the same notations of equations (13), fixing a set of pointwiseindependent local sections of each E i we get: while condition (14) remains unchanged. If E is an ordinary vector bundle, then an ordinary connection ∇ on E is a (1, 0)-twisted connection.
For any (ζ, Λ, B) fixed, we have the following commutative diagram: Moreover: Any two families of local potentials B and B ′ in the disjoint union differ by a global closed formB ∈ Ω 2 cl (X) such that a finite multiple ofB has integral periods. If H 2 (X; Z) is finitely generated (in particular, if X is compact), it means thatB has rational periods.
Non-integral vector bundles with connection. The following definition is the differential extension of 2.8.
Definition 2. 15 We call non-integral vector bundle with connection a (ζ, 0)-twisted vector bundle with connection, for any fixed ζ.
Since (ζ, 0) must be a cocycle, it follows that ζ is constant, therefore we are endowing a nonintegral (topological) vector bundle, as defined in 2.8, with a connection, the latter satisfying by definition ∇ i = ϕ * ij ∇ j . It follows from (16) that the corresponding isomorphism class can be represented in the form [{g ij , A i }], with the latter condition being identical to the one holding for ordinary connections. We denote by NIVB∇ ζ (X) the set of non-integral vector bundles with connection twisted by (ζ, 0). Similarly we define NIVB∇ [2] ζ (X), considering definition 2.9. For this kind of bundles we can define the Chern classes in the usual way. In fact, the gauge transformations of the curvature are given by F j = g −1 ij F i g ij , as for ordinary vector bundles (cfr. formula (94) with B global). Therefore, one can define the Chern classes via the elementary symmetric polynomials P k , which are invariant by conjugation: The cohomology class of P k (F ) only depends on the (topological) non-integral bundle, not on the connection. The main difference with respect to ordinary vector bundles is that, when the twisting class is not trivial, a Chern class is not the real image of an integral class in general (i.e. it is represented by a form whose periods are not necessarily integral). 1 As a particular case, the Chern classes are defined for Z 2 -non-integral bundles. In this case they are half-integral. A similar consideration holds more generally for Z n -bundles or Q-bundles.

Partially non-abelian Deligne cohomology
Let us show how relative Deligne cohomology can be suitably generalized in order to describe twisted vector bundles with connection in a quite intrinsic way. We start from the topological setting, then we consider the differential extension, analysing the absolute and the relative versions in each case. Moreover, we show the corresponding long exact sequences.
Topological framework -Absolute version. We define the partially non-abelian Cech complex of the sheaf U(r), for a fixed rank r, as follows (we leave the cover U implicit): whereδ 0 is not a morphism but a group action, defined by {h i } · {g ij } := {h i g ij h −1 j }. As usual we setδ 1 {g ij } := {g ki g jk g ij }. Moreover,δ 2 {ζ ijk } :=δ 2 {det ζ ijk }, whereδ 2 on the r.h.s. is the usual one of the (abelian) complexČ • (U(1)). All of the other groups and boundaries are the ones ofČ • (U(1)), but they will be irrelevant for the present discussion. The 1-degree cohomology, defined as the quotient of the kernel ofδ 1 up to the actionδ 0 , is canonically isomorphic to the pointed set of rank-r vector bundles on X up to isomorphism. The (irrelevant) higher-degree cohomology sets and groups are defined as usual, while the 0-degree cohomology of (21) is the kernel ofδ 0 , i.e. the set of 0-cochains such that {h i }·1 = 1. We get the global sections of U(r), as it has to be.
Topological framework -Relative version. Given a continuous closed embedding ρ : Y ֒→ X, we consider the complexes of sheaves U(1) X and ρ * U(r) Y on X. We have the natural morphism In order to define the relative Cech cohomology sets of ρ ! , we generalize the cone complex to this partially non-abelian framework as follows: Let us analyse the corresponding 2-degree cohomology. The kernel ofδ δ δ 2 is formed by classes ({ζ ijk }, {g ij }) such that {ζ ijk } is a cocycle and g ki g jk g ij = ρ ! ζ ijk , therefore we get transition functions of ρ ! ζ ijk -twisted vector bundles. The action of (0, {h −1 i }) throughδ δ δ 1 replaces g ij by h i g ij h −1 j , therefore it changes the representative within the same isomorphism class, and the action of ({η ij }, 0) corresponds to the isomorphism (10). This means that the groupȞ 2 (ρ ! ) naturally encodes twisted vector bundles on Y with respect to any class induced by pull-back from X, up to isomorphism and change of representative in the twisting class. We observe that, choosing Y = X and ρ = id X , we get twisted vector bundles on X.
Remark 2. 16 It is important to observe thatȞ 2 (ρ ! ) is not twisted, hence it encodes twisted bundles in ordinary cohomology. The same remark holds about the differential extension described below.
Differential extension -Absolute version I. We have seen that, in the topological framework, partially non-abelian Cech cohomology describes in a unitary way twisted vector bundles. We are going to do the same in the differential framework, i.e. we introduce partially non-abelian Deligne cohomology in order to describe twisted vector bundles with connection. First of all, we start from (21), that is the Cech complex of the sheaf U(r) on X, and we extend it to the Cech double-complex of the sheaf complex whered is the action f · ω := f ωf −1 + 1 2πi f −1 df . We define such a double complex as: where the first coboundaryĎ 0 is the action and all the others are the ones of the complex U(1) → Ω 1 R . The 1-degree cohomology of ( 25), defined as the quotient ofĎ 1 up to the actionĎ 0 , is canonically isomorphic to the set of rank r vector bundles with connection on X. The (irrelevant) higher-degree cohomology sets are defined as usual, while the 0-degree cohomology is the kernel ofĎ 0 , i.e. the set of 0-cochains such that {h i } · 1 = 1. We get the global constant sections of U(r), as in the abelian case. Moreover, we callȞ the set of 1-cochains of (25) whose 1-coboundary is (ζ, Λ), up to the action ofĎ 0 . We get the canonical bijection VB∇ (ζ,Λ) (X) ≃ r∈NȞ 1 (ζ,Λ) (U, S 1,r X ), obtained fixing local point-wise independent sections and computing the corresponding transition functions and potentials. Moreover, if (ξ, Θ) = (ζ, Λ) ·δ 1 (η, λ) in S 1 X , we get the following isomorphism, extending (10) to the differential setting: Equivalently, using the language of twisted bundles with connection (and without fixing the rank r): Fixing (ζ, Λ) and (ξ, Θ), two choices (η, λ) and (η ′ , λ ′ ) differ by a cocycle. Moreover, Φ (η,λ) = Φ (η ′ ,λ ′ ) if and only if (η, λ) − (η ′ , λ ′ ) is a coboundary, hence the set of isomorphisms of the form (27) or (28) is a torsor overȞ 1 (S 1 X ).
Differential extension -Absolute version II. If, instead of (24), we consider the degree-2 complex then the double complex (25) is replaced by the following one: We callȞ the set of 1-cochains of (30) whose 1-coboundary is (ζ, Λ, B), up to the action ofĎ 0 . We get the canonical bijection VB∇ (ζ,Λ,B) (X) ≃ r∈NȞ (ζ, Λ, B) ·δ 1 (η, λ) in S 2 X , we get the following isomorphism: Equivalently, using the language of twisted bundles with connection, we get ( 28) adding B and C to the twisting cocycles. The set of isomorphisms of the form (32) is a torsor over the flat part ofȞ 1 (S 1 X ), i.e. over H 1 (X; R/Z) ≃ Hom(H 1 (X; Z); R/Z). It follows that, if Differential extension -Relative version I. Given a smooth closed embedding ρ : Y ֒→ X, we compute the relative Deligne cohomology of degree 2, but in the partially non-abelian setting. We have the natural morphism ρ ! : S 2 X → ρ * S 1,r Y defined as follows: where ρ !,0 (f ) := (f • ρ) · I r and ρ !,1 (ω) := ρ * ω · I r . The relative Cech hypercohomology groups of ρ ! are by definition the cohomology groups of the corresponding cone complex, that is the total complex associated to the following double complex: We denote it as (ζ, Λ, B, g, A) for simplicity. The 2-coboundary is by definitionĎ D D 2 (ζ, Λ, B, g, , therefore a 2-cocycle is formed by a Deligne 2-cocycle (ζ, Λ, B) on X and a representative of a ρ * (ζ, Λ)-twisted line bundle on Y .
Differential extension -Relative version II. We consider the natural morphism ρ ! : S 2 X → ρ * S 2,r Y defined as follows: The relative Cech hypercohomology groups of ρ ! are by definition the cohomology groups of the corresponding cone complex, that is the total complex associated to the following double complex: The only difference with respect to (34) is that, in the cocycle condition, we also have the constraint rB i − Tr(F i ) = 0, i.e. the second component of the curvature must vanish. It follows that a 2-cocycle is formed by a Deligne 2-cocycle (ζ, Λ, B) on X and a representative of a ρ * (ζ, Λ, B)-twisted vector bundle on Y . The action of (0, 0, h) is a gauge transformation of a ρ * (ζ, Λ, B)-twisted vector bundle, as in (14), and the action of (η, λ, 0) corresponds to the isomorphism (32).
Summary. Generalizing (8) to this partially non-abelian framework, we set: It follows thatĤ 3,r (ρ) naturally encodes (ζ, Λ)-twisted bundles andĤ 3,r par (ρ) naturally encodes (ζ, Λ, B)-twisted bundles. We observe that B is fixed even in a non-parallel cocycle, since it allows to define the gauge-invariant field-strength rB − Tr F , but the twisted bundle itself is not linked to B. In particular, the relative curvature of the class [(ζ, Λ, B, g, A)] ∈Ĥ 3,r (ρ) is the relative form (H, rB − Tr F ), where H = dB. Moreover, in the absolute setting, we can extend diagram (17) to the following one: Therefore, formula (18) is equivalent to the following one: Non-integral vector bundles. We denote byȞ 1 ζ (U, S 1,r X ) the group (26) when Λ = 0, canonically isomorphic to NIVB∇ ζ (X) (see definition 2. 15). If ξ = ζ ·δ 1 η as U(1)-cochains, we get the following isomorphism, analogous to (27): Equivalently, we get the isomorphism analogous to (28). It follows that the set of isomorphisms of the form (40) is a torsor overȞ 1 (U, U(1)) ≃ H 1 (X; R/Z) ≃ Hom(H 1 (X; Z); R/Z). Therefore, if H 1 (X; Z) = 0 (in particular, if X is simply connected), then the semi-group (NIVB∇ [ζ] (X), ⊕) is canonically defined for any [ζ] ∈ H 2 (X; R/Z). We observe that the Chern classes (20) are invariant under the tensor product with a flat twisted line bundle, since the latter has vanishing real first Chern class. Since this is exactly the ambiguity in the definition of NIVB∇ [ζ] (X) when H 1 (X; Z) is not vanishing, it follows that the Chern classes only depend on [ζ] in any case. The cohomological description of non-integral vector bundles can be realized as follows. Topologically (i.e. without connection) we replace (22) by defined in the same way, but considering only constant cochains on X. We get the cone complex analogous to (23). It follows thatȞ 1 (ρ ! fl ) naturally encodes non-integral vector bundles on Y , twisted with respect to any constant class induced by pull-back from X. If we introduce a (ζ, 0)-connection, then we replace (33) by ρ ! fl : U(1) X → ρ * S 1,r Y , defined by: The setȞ 2 (ρ ! fl ) naturally encodes non-integral vector bundles with connection on Y , with respect to any class induced by pull-back from X. We can also consider parallel classes, i.e. we introduce a (ζ, 0, B)-connection, where B is necessarily global. In this case we replace (35) AgainȞ 2 (ρ ! fl ) is the required set. We remark that the first line of (42) and the first line of (43) are not equivalent. In fact, the 2-cohomology of the former is H 2 (X; R/Z), which is the flat part ofĤ 2 (X), while the 2-cohomology of the latter is H 2 (X; R/Z) ⊕ Ω 2 X,R , that is not a subgroup ofĤ 2 (X). 2 This is due to the fact that a U(1)-class already provides the complete information about the corresponding Deligne class, therefore the choice of B is a piece of information more.
Similar considerations hold replacing U(1) X by Z 2 (more generally, by Z n or Q). In particular, when Λ = 0 and ζ and ξ are cohomologous as Γ 2 -cochains, the set of isomorphisms of the form (40) is a torsor overȞ 1 (U, Remarks on canonicity. We have seen that the set VB∇ (ζ,Λ) (X) is independent of the twisting cocycle (within a fixed cohomology class) up to the action ofĤ 2 (X), while the sets VB∇ (ζ,Λ,B) (X) and NIVB∇ ζ (X) are independent of the cocycle up to the action of H 1 (X; R/Z), i.e. the flat part ofĤ 2 (X). Topologically, the sets VB ζ (X) and NIVB ζ (X) are independent of the cocycle up to the action of H 2 (X; Z) and Tor H 2 (X; Z) respectively. As we already pointed out, an interesting consequence is that, if X is simply connected, then the sets VB∇ [(ζ,Λ,B)] (X), NIVB∇ [ζ] (X) and NIVB [ζ] (X) are canonically defined. On the contrary, VB∇ [(ζ,Λ)] (X) is almost never well defined.

Twisted bundles vs non-twisted cohomology
The sets VB [ζ] (X), VB∇ [(ζ,Λ)] (X), VB∇ [(ζ,Λ,B)] (X) and NIVB∇ [ζ] (X), even when they are defined only up to the action of a non-trivial (abelian) cohomology group, can be described through non-twisted cohomology as well, and this is the information that will lead to a natural description of the possible gauge theories on a D-brane or on a stack of D-branes (more natural than in [17]).
Abelian framework -Absolute topological version. In order to make the exposition clearer, we start from the abelian setting, since cohomology is more natural in this context. In particular, we consider non-integral line bundles, i.e. line bundles twisted by a constant cocycle ζ, and we show that they are classified up to the action of H 1 (X; R/Z) by the 1cohomology of the sheaf U(1)/U(1). In fact, given a class [{g ij }] ∈Ȟ 1 (U(1)/U(1)), the cocycle condition imposes that ζ ijk := g ki g jk g ij is constant, since it must vanish up to the quotient by U(1). Moreover, each g ij is defined up to a constant transition function, therefore the cohomology class [{g ij }] represents a non-integral line bundle up to a flat (nontwisted) line bundle, 3 i.e. up to the action of H 1 (X; R/Z). Since, multiplying g ij by any constant transition function η ij , we get the twisting cocycle ζ ·δ 1 η, it follows that the only meaningful information on the twisting is the cohomology class [ζ ijk ] ∈Ȟ 2 (U(1)), thereforě H 1 (U(1)/U(1)) is canonically isomorphic to the rank-1 subset of NIVB [ζ] (X). Actually, the same argument can be applied to classes of any degree, i.e. to [ In order to describe this picture with the language of homological algebra, we consider the exact sequence of sheaves 0 → U(1) → U(1) → U(1)/U(1) → 0, inducing the corresponding long exact sequence The Bockstein map t associates to [{g i 0 ...ip }] its twisting class [{ζ i 0 ...i p+1 }], the latter coinciding with the image in U(1) of the first Chern class, as the reader can verify both by direct computation and through homological algebra. More precisely, from the long exact sequence , the latter isomorphism being precisely the first Chern class, whose image in H p+1 (X; U(1)) is the twisting class. We observe that the morphismȞ (1)) for any constant twisting cocycle ζ. In this case, the image is contained in t −1 [ζ], that, by exactness, is a coset of the image ofȞ p (U(1)). This is the reason whyȞ p (U(1)/U(1)) encodes any twisted class, up to its natural ambiguity.
Abelian framework -Absolute differential version. If we introduce a connection on a non-integral line bundle, we have to consider the complex and the corresponding Deligne cohomology groupȞ p (S p X ). An element of this group is a class [g, Λ 1 , . . . , Λ p ], satisfying the usual cocycle condition in Deligne cohomology, except for the fact thatδ p g = ζ, with ζ constant. We get the curvature F , locally described by F = dΛ p , which is not necessarily an integral form. Its de-Rham cohomology class corresponds to the real first Chern class of [g]. Composing the projection [g, Λ 1 , . . . , Λ p ] → [g] with the map t in sequence (44), we get the mapt : (1)). Since we have the natural projection S p X →S p X , that quotients out g up to constant functions, we get the following exact sequence, whose blue segment refines (44) to the differential setting: Exactness inȞ p (S p X ) is due to the fact that a non-integral class vanishes inȞ p (S p X ) if and only if it can be realized by constant transition functions and zero potentials, i.e. if it is flat. Exactness inȞ p (S p X ) is due to the fact that the twisting class vanishes if and only if the non-integral Deligne class is the projection (up to constant functions) of a non-twisted one. Exactness in H p+1 (X; U(1)) is due to the fact that the U(1)-cohomology class of ζ is trivialized by the transition functions g (in particular,t is not surjective in general). Again Actually, the sheaf cohomology of (45) has been introduced only as a preliminary step towards the partially non-abelian setting. In fact, since the quotient by U(1) cuts the flat part, the curvature is the only meaningful information, henceȞ p (S p X ) ≃ Ω p+1 cl (X).
Abelian framework -Relative topological version. Given a smooth closed embedding ρ : Y ֒→ X, we have seen that the corresponding relative Cech cohomology of the sheaf U(1) is by definition the one of the cone complex of ρ ! : U(1) X → ρ * U(1) Y . In the non-integral setting, it is natural to considerρ ! : both in X and Y , induce the following diagram: that induces the morphisms in cohomology From the short exact sequence 0 we obtain the following long exact sequence, that is the relative version of (44): The image of the mapt is a class [ζ] = [(ζ X , ζ Y )], whose meaning is the following one. Given a class [(g, h)] ∈Ȟ p (ρ ! ), we have thatδ p g = ζ X (i.e. ζ X is the twisting cocycle of g), and h is a class such that ρ * g =δ p−1 h · ζ Y , i.e. ζ Y is the constant correction of the relative cocycle condition in Y . The pair (ζ X , ζ Y ) is a relative cocycle. With respect toρ ! , from the short exact sequence 0 → ρ * (U(1) Y ) → ρ ! →ρ ! → 0, we obtain the following long exact sequence: The image of the mapt is a class [ζ Y ], whose meaning is the following one. Given a class [(g, h)] ∈Ȟ p (ρ ! ), we have thatδ p g = 0 (i.e. g is not twisted), and h is a class such that ζ Y is the constant correction of the relative cocycle condition in Y and it turns out to be a cocycle, as it is easy to verify by direct computation too.
We have a natural morphism of exact sequences from 0 inducing the following morphism of long exact sequences from (50) to (49): The central vertical arrow is the map appearing in (48) and the one on the right is the Bockstein map of the long exact sequence induced by ρ in singular cohomology with U(1)coefficients.

(53)
Partially non-abelian framework -Absolute topological version. In order to extend the previous picture to any non-integral vector bundle (not necessarily of rank 1), we consider the sheaf quotient U(r)/U(1). We get the following Cech complex, analogous to (21): The groupȞ 1 (U(r)/U(1)) encodes non-integral vector bundles up to constant functions. We have the natural projection U(r) → U(r)/U(1), inducing the corresponding push-forward in cohomology, and we define the map t : . We get the exact sequence, that generalizes (44) with p = 1: In this non-abelian framework, we can also consider twisted vector bundles, without imposing that the twisting cocycle is constant. In this case we have to consider the sheaf quotient U(r)/U(1) (that vanishes in the abelian context). We get the following Cech complex, analogous to (54): The groupȞ 1 (U(r)/U(1)) encodes twisted vector bundles up to U(1)-valued transition functions. We define the mapt : and we get the exact sequence: There is a natural morphism of exact sequences from (55) to (57), the right vertical arrow being the Bockstein map induced by the sequence 0 → Z → R → U(1) → 0 and the central one being induced by the projection U(r)/U(1) → U(r)/U(1). Moreover, as in the abelian framwework, the morphismȞ 1 (U(r)) →Ȟ 1 (U(r)/U(1)) in sequence (57) is the untwisted version ofȞ 1 ζ (U(r)) →Ȟ 1 (U(r)/U(1)) for any twisting cocycle ζ. In this case the image is contained int −1 [ζ], that, by exactness, is a coset of the image ofȞ 1 (U(r)).
Partially non-abelian framework -Absolute differential version. In order to endow the bundle with a connection, we consider the following complex, analogous to (45): We get the Cech double complex analogous to (25). The groupȞ 1 (S 1,r X ) corresponds to the set of non-integral rank-r vector bundles with connection, up to constant U(1)-valued transition functions. We have the natural projection S 1,r X →S 1,r X , inducing the corresponding push-forward in cohomology, and we define the mapt : . We get the following exact sequence, that generalizes (46) and refines (55): We briefly justify the exactness of this sequence. The map i is the embedding of H 1 (X; U(1)) inȞ 1 (S 1,r X ) that sends a flat line bundle with connection L to the direct sum L ⊕ · · · ⊕ L iterated r times, hence it is injective. The kernel of p consists of the set of rank-r vector bundles with connection that are trivial when applying the quotient by U(1) as the centre of U(r), hence it coincides with the image of i. The kernel oft is formed by vector bundles with connection that are twisted with respect to a U(1)-cocycle representing a trivial cohomology class, i.e. by twisted bundles that are equivalent to a non-twisted one up to the action of U(1).
If we consider twisted vector bundles, without imposing that the twisting cocycle is constant, we consider the following complex: We get the Cech double complex analogous to (25), such thatȞ 1 (S 1,r X ) corresponds to the set of twisted vector bundles with connection, up to (non-twisted) lined bundles with connection. We get the following exact sequence, that generalizes (46) and refines (57): We have a natural morphism of exact sequences from (59) to (61), the right vertical arrow being the Bockstein map induced by the sequence 0 → Z → R → U(1) → 0 and the central one being induced by the projectionS 1,r X →S 1,r X . If we include 2-forms in the codomain of t, we can refine Partially non-abelian framework -Relative topological version. The picture is similar to the abelian one, but considering also the quotient by U(1). We have seen that the non-twisted relative cohomology is the one of ρ ! : U(1) X → ρ * U(r) Y . The five twisted versions are: Diagram (47), which is essentially ρ ! →ρ ! →ρ ! , now can be enriched as follows: The corresponding t-maps provide the twisting class, that is a cohomology class of the complex formed by the denominators. Therefore, we get the following extension of diagram Partially non-abelian framework -Relative differential version. Again the picture is similar to the abelian one, but considering also the quotient by U(1). We have seen that the non-twisted relative cohomology is the one of ρ ! : S 2 X → ρ * S 1,r Y . The five twisted versions are: Y . Diagrams (62) and (63) holds in the differentials setting too, composing the t-maps in (63) with the projections to the underlying topological class.
Partially non-abelian framework -Parallel classes. The same constructions can be applied to parallel classes too. We leave the detailed to the interested reader, since they will not be necessary in the following.
Finer quotients. As we have seen in definition 2.9, we can consider the quotients U(r)/Γ k or U(r)/Γ, that are intermediate between U(r) and U(r)/U(1). All of the previous diagrams and sequences can be enriched with such sheaves, both in the topological and differential frameworks. We consider directly the differential non-abelian setting for Z 2 -twistings, in order not to make the exposition too long. We set: We remark that Z 2 = {0, 1} (with additive notation) and Γ 2 = {±1} (with multiplicative notation). We get the projections S p Of course we have other intermediate possibilities that we are neglecting. In diagram (62) the upper triangle can be enriched as follows: Diagram (63) gets enriched coherently, in the differential version as well. In particular, we have the maps:t

Long exact sequences
In the abelian setting, the long exact sequence in Deligne cohomology, associated to ρ : Y ֒→ X, is made by segments of the form Considering remark 2.12, that holds in the relative case as well, we get the sequence of the flat part until degree p − 1 in X, and we get the sequence of (topological) singular cohomology from degree p in Y on. Hence, the complete exact sequence is the following one: We can construct an analogous sequence with parallel classes, but we will not need it in the present paper. If we considerρ ! :S p X → ρ * S p−1 Y , then the flat part vanishes at the quotient, hence integral cohomology is replaced by the real one. We get the following sequence: The natural projection morphisms S p X →S p X and S p−1 Y →S p−1 Y , inducing ρ ! →ρ ! , induce a morphism of exact sequences from (67) to (68). The reader can construct the analogous sequence induced byρ ! : S p X → ρ * S p−1 Y . Considering the partially non-abelian generalization, from the five maps of diagram (62) we get the long exact sequence analogous to (67) and (68) (with p = 2), whose most relevant segments are the following ones: The morphisms of diagram (62) induce the corresponding morphisms of exact sequences. From the maps (64), we get:

Classification of gauge theories
Given a smooth embedding ρ : Y ֒→ X, we call id Y the identity map of Y and we consider the following natural morphisms of complexes of sheaves on X: This is equivalent to considering the following diagram: The first and the last vertical morphisms are the ones appearing in diagrams (62) and (65), with respect to ρ and id Y respectively. The central one is formed byρ ! : on the left and the natural projectionS 1,r [2] Y →S 1,r Y on the right. Considering the exact sequences (69), (70), (76) and (72) respectively, and considering the t-maps in (66), (59) and (61) respectively, from (77) we get the following diagram in cohomology, that is the key to classify the possible gauge theories on a stack of D-branes: Let us call B := [(ζ, Λ, B)] the B-field class on X, with curvature H = dB, and A := [(g, A)] the A-field class on Y , i.e. the (ξ, Θ)-twisted bundle with connection, that is the gauge theory to be classified. In order to choose a sign convention coherent with formula (158) in chapter 4, we consider the dual bundle A * := [(ḡ, −Ā)], which is −(ξ, Θ)-twisted. 4 Moreover, we call S the spin c -gerbe on Y , which is a flat gerbe classified by the second Stiefel-Whitney class w 2 (Y ) ∈ H 2 (Y ; Z 2 ). 5 In order for the open string action to be well-defined, A * has to trivialize topologically the tensor product ρ * B ⊗ S, hence ρ * (ζ, Λ) + (ǫ, 0) = −(ξ, Θ), where ǫ is a Γ 2 -cocycle representing w 2 (Y ). Equivalently: i.e. ρ * ζ · ξ = ǫ and Λ + Θ = 0. Considering the coboundaryĎ D D 2 of the complex (34), this Since ǫ vanishes quotienting out by ), the latter group being part of diagram (78). More precisely, , the latter being a coset of Ker(t [2] ). Such a coset is the most general classification of the possible A-field and B-field configurations in type II superstring theory (cfr. [5] and [17]). If w 2 (Y ) = 0, then we can choose ǫ = 1 and lift the class to [(ζ, Λ, B,ḡ, −Ā)] ∈Ȟ 2 (ρ ! ), the lift being part of the physical datum, since it is not unique in general.
Freed-Witten anomaly. Fixing X, Y and B, condition (80) can be realized by some configurations of A if and only if the Freed-Witten anomaly vanishes [21,26] . We denote such a condition by (FW). Let us show in detail the equivalence (80) ⇔ (FW). From (80) we easily get (FW) by taking the first Chern class on both sides. Conversely, assuming (FW), we choose any representatives (ζ, Λ) and ǫ and, up to multiplying any fixed ξ by a coboundary, we set (ξ, Θ) := (ǫ, 0) − ρ * (ζ, Λ). Since [ξ] is the twisting class of a vector bundle by construction, it has finite order, hence there exist some (ξ, Θ)-twisted bundles on Y independently of Θ, i.e. there exist some admissible A-field configurations.
Since [ξ] is part of A and the only constraint about it consists in being a finite-order class, it follows that, fixing X, Y and B, (FW) can be realized if and only if ρ * [ζ] has finite order (equivalently, ρ * H is exact). In general, the rank r of the A-filed depends on the configuration, since, given [ξ], not every rank is realizable. If we have a single D-brane, i.e. we impose r = 1, then [ξ] = 0 in H 3 (Y ; Z), hence the only possibility is ρ * [ζ] = W 3 (Y ), which is a much stronger condition than ρ * [ζ] being torsion. This is the original condition described in the seminal paper [21], that has been refined to the partially non-abelian setting (i.e. to stacks of D-branes) in [26].
The classification. Now we can classify the possible gauge theories on the world-volume, analysing diagram (78).
(162) in chapter 4, so that formula (158) would change coherently. We could also change the convention in the definition of twisting for a vector bundle, introducing a minus sign when necessary. All of such choices are equivalent. 5 As a (complex) gerbe, the flat holonomy of S is classified by the image of w 2 (Y ) through the inclusion Z 2 ֒→ U(1). It follows that its first Chern class is W 3 (Y ). Anyway, such a gerbe is the complexification of a real one, classified by w 2 itself, hence we can choose its transition functions in Γ 2 . In this way we can choose the more refined functionρ !
Case I: Vanishing B-field and spin world-volume. If the B-field vanishes on the whole space-time X, i.e. [(ζ, Λ, B)] = 0, and w 2 (Y ) = 0, then [(ζ, Λ, B,ḡ, −Ā)] ∈Ȟ 2 (ρ ! ) and, in particular, it belongs to the kernel of the mapȞ 2 (ρ ! ) →Ȟ 2 (S 2 X ). By exactness of the first line of (78), it can be lifted to a class [(g, up to the restriction to Y of a flat line bundle on X. Such an ambiguity is the "residual gauge freedom", whose physical meaning has been described in detail in [5,17,15].
Case II: Vanishing B-field. If the B-field vanishes on the whole space-time X, but we have no hypotheses on the world-volume, then ) and, in particular, it belongs to the kernel of the mapȞ 2 (ρ !
[2] ) →Ȟ 2 (S 2 X ). By exactness of the second line of (78), it can be lifted to a class [(g, . This means that we get a Z 2 -non-integral vector bundle with connection, twisted by w 2 (Y ), up to the restriction of a flat line bundle on X and up to a real line bundle on Y , the latter ambiguity being the action of H 1 (X; Z 2 ) on Z 2 -twisted bundles. This case has not been considered explicitly in [5,17,15]. (77), by exactness of the third line we get a unique lift [(g, we get a non-integral vector bundle on Y , that is canonically defined up to a flat line bundle on Y (i.e. up to the action of H 1 (Y ; R/Z); see comments after definition 2. 8). By construction and because of the chosen sign convention, we have that Case IV: No hypotheses on the B-field. In a completely generic configuration, we have to project [(ζ, Λ, B,ḡ, −Ā)] ∈Ȟ 2 (ρ ! [2] ) toȞ 2 (ĩd ! Y ) and, by exactness, we get a unique class [(g, A)] ∈Ȟ 1 (S 1,r Y ), i.e. a twisted vector bundle up to any line bundle with connection, because of the Freed-Witten anomaly.
The classification for simply-connected world-volumes. If the world-volume is simply connected, many of the ambiguities in the previous classification disappear, hence the classification becomes simpler and more natural. In particular, any non-integral vector bundle with connection is canonically defined from the U(1)-twisting class, hence ordinary and Z 2 -twisted vector bundles are just particular cases. With the cohomological language, the ) becomes an embedding, as the reader can verify by diagram chasing in (78). Therefore, diagram (78) becomes the following one: (81) Hence, case I of the previous classification is just a particular case of II, therefore we can ), without analysing w 2 (Y ). Moreover, case II becomes a particular case of III as well, since, in spite ofȞ 2 (ρ !
[2] ) →Ȟ 2 (ĩd ! Y ) not being an embedding in general (since it forgets the behaviour of B outside Y ), the eventual lift toȞ 1 (S 1,r [2] Y ) in the second line is a particular case of the eventual lift toȞ 1 (S 1,r Y ) in the third line. Therefore, the only meaningful information is whether ρ * H vanishes or not, leading to the following two possibilities: Case III (including I and II): (81), by exactness of the third line we get a unique lift [(g, A)] ∈Ȟ 1 (S 1,r Y ), i.e. we get a non-integral vector bundle on Y , that is canonically defined. By construction [(g, and, by exactness, we get a unique class [(g, A)] ∈Ȟ 1 (S 1,r Y ), i.e. a twisted vector bundle up to any line bundle with connection, belongingt Remark 2. 17 The two cases of the classification for simply-connected world-volumes can be unified through the following statement, that will be important in order to define the Wess-Zumino action: is the unique (up to isomorphism) non-integral line bundle with connection whose curvature isB ′ −B.
This is the general case IV, but, when ρ * H = 0, we have the canonical choiceB = 0, so that we obtain a well-defined twisted vector bundle with connection, as stated in case III. In order to prove (⋆), we observe that, since [ξ] is a torsion class, we can always choose ξ constant and Θ = 0, i.e. we can represent . In this way we get a (ξ, 0)-twisted bundle [(E, ∇)] (i.e. a non-integral vector bundle) and a fixed formB. Changing representative to (ξ ′ , 0,B ′ ), through any cochain (η, λ) such thať Since Y is simply-connected, the latter is completely determined by its curvatureB ′ −B. In particular, ifB ′ =B, then [(E, ∇)] does not change, hence it is completely determined byB.
Let us prove the statement (⋆) with the intrinsic language of homological algebra. We Y , described by the following complex, intermediate between (42) and (43), but with X = Y and quotienting out by Z 2 in the image: [2] fl ), the latter being formed precisely by a ξ-twisted non-integral vector bundle and a global 2-formB. The lift is unique up to the action ofȞ 1 where the latter isomorphism identifies a non-integral line bundle with its curvature F (since Y is simply connected, the curvature is the only meaningful information). Since a non-zero curvature F shiftsB toB + F , it follows that, fixingB, the lift is unique.
We conclude observing that, fixingB, we can write the (relative) curvature of [(ζ, Λ, B,ḡ, where F is the field strength of the corresponding bundle [(E, ∇)]. Therefore, fixingB is equivalent to fixing Tr F , and this choice determines the twisted vector bundle up to isomorphism, since, Y being simply-connected, it leaves no freedom for the action ofȞ 1 (S 1 Y ).
Final remarks. Up to know we considered the case of vanishing B-field in the whole X, corresponding to the kernel ofȞ 2 (ρ ! ) →Ȟ 2 (S 2 (77), and the case of flat B-field on Y , corresponding to the kernel ofȞ 2 (ĩd ! Y ) →Ȟ 2 (S 2 Y ). The reader may inquire why we did not consider the intermediate cases of flat B-field on X or of vanishing B-field on Y . The reason is that we get no new information with respect to case III, that include both. In fact, the case H = 0 corresponds to the mapρ ! , whose exact sequence is (70). By exactness, we get a unique lift [(g, A)] ∈Ȟ 1 (S 1,r Y ), exactly as in the more general case III. Similarly, the case of vanishing B-field on Y corresponds to the map id ! Y , whose exact sequence is (71). Again we get a unique lift [(g, A)] ∈Ȟ 1 (S 1,r Y ). Moreover, it is worth to mention the case in which the B-field is flat on Y and its holonomy is exactly . In this case we get a non-twisted vector bundle on Y , but anyway it is meaningful only up to a flat line bundle on Y , since we fit in case III, with a class belonging to Ker(t).
The last remark is about the world-volume being spin c , since W 3 (Y ) appears in Freed-Witten anomaly. Actually the most interesting cases of the previous classification deal with non-integral bundles, that's why w 2 (Y ) is more interesting than W 3 (Y ). The latter is partially meaningful only in case IV, since we are forced to deal with generic twisted bundles. In this case, the condition W 3 (Y ) = 0 is equivalent to the fact that [(g, A)] ∈ Ker(t), i.e. we get a non-twisted vector bundle, but anyway up to any twisted line bundle.

Twisted differential K-theory
It is well known that K-theory is a better tool than ordinary cohomology in order to classify D-brane charges [14,30,32,15,19]. When the B-field is vanishing on the whole spacetime and the world-volume is spin, we use ordinary K-theory, otherwise we must consider its twisted version [11,12,7,26]. Since there are various models in the literature, we summarize the ones that we are going to use in this paper, focusing on the features that will be important for our purposes.
This section is supposed to be mainly expository, but we think it is necessary in order to provide an organic presentation of differential twisted K-theory with the language of the present paper, paying particular attention to the dependence on the twisting cocycle. Subsection 3.1 is based mainly on [27], but giving more emphasis to the structure of twisted cohomology theory only through the language of twisted vector bundles. Subsection 3.2 is based mainly on [33], but we complete the construction to the differential cohomology groups of any degree, considering product and integration, and we link it more explicitly to relative Deligne cohomology. Subsection 3.3 is based on [12], except for 3.3.3 and 3.3.4. The former deals with the dependence on the cocycle in the differential setting. Surely its content is well known to the experts too, but we could not find a similar discussion in literature and it is very relevant for our purposes. In 3.3.4 we sketch the construction of the isomorphism between the two models of differential twisted K-theory we are dealing with, deferring the details to a future paper. We conclude this section with subsection 3.4, in which we introduce a simple but useful definition, that will be applied to define the topological charge of a D-brane.

Topological twisted K-theory -Finite order
Any general definition of twisted K-theory involves some infinite-dimensional geometric objects, like projective Hilbert bundles. Nevertheless, this is not necessary when the twisting class has finite order [27], as we are going to summarize.
The direct sum of ζ-twisted vector bundles is defined as The set VB ζ (X), endowed with this operation, is a commutative semi-group, hence we can define the corresponding Grothendieck group, that we call ζtwisted K-theory group of X and we denote by K ζ (X). This is the 0-degree group. In order to construct twisted K-theory of any degree, we consider the n-dimensional torus T n := (S 1 ) n and the natural embeddings ι j : T n−1 × X ֒→ T n × X, for j ∈ {1, . . . , n}, defined supposing that S 1 has a marked point (e.g. 1 ∈ S 1 ⊂ C). Fixing a good cover of S 1 , we easily get a good cover of T n × X by cartesian product; moreover, the cocycle ζ on X determines a cocycle on T n × X, that we also denote by ζ, by pull-back with respect to the projection T n × X → X. Then we set: This construction concerns non-positive degrees. Before extending it to positive degrees, we need to introduce the Bott periodicity, hence we need products. Given a ζ-twisted bundle The result is a (ζη)-twisted bundle. Therefore, we define the product where π X : X × Y → X and π Y : X ×Y → Y are the projections. Considering the corresponding K-theory classes, we get Composing with the pull-back via the diagonal map, we get . In particular, this shows that K • ζ (X) is a graded module over K • (X). Now we can define the Bott periodicity. We consider the dual of the tautological line bundle of P 1 (C) ≃ S 2 ; its pullback via the projection T 2 → S 2 is a line bundle η on the torus, such that η −1 is a generator of K −2 (pt) ≃K(T 2 ) ≃ Z. We get the morphism B : α, which is a group isomorphism. This shows that only the parity of the (non-positive) degree is meaningful up to canonical isomorphism, hence, for n ≥ 0, we set K n ζ (X) := K −n ζ (X).
Dependence on the cocycle. If ζ and ξ are cohomologous and we fix η such that ξ = ζ ·δ 1 η, the isomorphism (10) extends to the corresponding Grothendieck groups, defining . This shows that the isomorphism class of the group K ζ (X) only depends on [ζ] in a non-canonical way, the set of isomorphisms of the form Φ η being a torsor overȞ 1 (U, U(1)) ≃ H 2 (X, Z). In particular, if H 2 (X, Z) = 0, then K [ζ] (X) is canonically defined and does not depend on the cover. In general, only the quotient up to the action of H 2 (X, Z), that we denote by K [ζ] (X) anyway, depends on [ζ] in a canonical way.
Non-integral vector bundles and K-theory. Of course we are free to choose ζ constant, getting the Grothendieck group of non-integral vector bundles twisted by ζ ∈Ž 2 (U, U(1)). All of the previous considerations keep on holding, except for the fact that the set of isomorphisms of the form Φ η is a torsor over the image of the natural map H 1 (U, U(1)) → H 1 (U, U(1)), canonically isomorphic to Tor H 2 (X; Z). Therefore, if Tor H 2 (X; Z) = 0 (in particular, if X is simply connected), then K [ζ] (X) is canonically defined with [ζ] ∈ H 2 (X; R/Z).

Relative version
Given a pair of spaces (X, Y ), such that U| Y is a good cover, we define the relative ζ-twisted K-theory group K ζ (X, Y ) as follows. We call L ζ (X, Y ) the set of triples (E, F, α), where E and F are ζ-twisted vector bundles on X and α : E| Y → F | Y is an isomorphism. There exists a natural operation of direct sum in L ζ (X, Y ), defined by (E, F, α) Moreover, a triple is called elementary if it is of the form (E, E, id). We introduce the following equivalence relation in L ζ (X, Y ): the triple (E, F, α) is equivalent to (E ′ , F ′ , α ′ ) if there exist two elementary triples (G, G, id) and (H, H, H, H, id). We call K ζ (X, Y ) the quotient of L ζ (X, Y ) by this relation. We obtain an abelian group, whose zero-element is the class [(E, E, id)], for any E, and such that −[(E, F, The definition of the relative groups of any degree is analogous to (84). In particular, we consider the n-dimensional torus T n := (S 1 ) n and the natural embeddings ι j : (T n−1 × X, T n−1 × Y ) ֒→ (T n × X, T n × Y ), for j ∈ {1, . . . , n}, and we set: There is a natural product K In particular, the Bott periodicity morphism B : This shows that only the parity of the (non-positive) degree is meaningful up to canonical isomorphism, hence, for n ≥ 0, we set K n ζ (X, Y ) := K −n ζ (X, Y ). With these definitions we get a twisted cohomology theory on the category of pairs (X, Y ) such that X is compact and Y ⊂ X is closed. 6 Moreover, the isomorphism (11) easily extends to the relative setting, applying it to both E and F in the triple (E, F, α), hence the isomorphism class of K ∈ H 2 (X; R/Z), when Tor H 2 (X; Z) = 0.

Compact support
When X is locally compact, we are going to define the compactly-supported ζ-twisted Ktheory groups K • ζ,cpt (X). Given a compact subset K ⊂ X, we set X \\ K := X \ K, i.e. the complement of the interior of K in X. We say that K is U-compact if U| X\\K is a good cover. We denote by K X,U the directed set formed by the U-compact subsets of X, the partial ordering being given by set inclusion. We assume that U is refined enough so that the union of the U-compact subsets is the whole X. We think of K X,U as a category, whose objects are the U-compact subsets of X and whose morphisms are defined as follows: the set Hom K X,U (K, H) contains one element if K ⊂ H and it is empty otherwise. Calling Top 2 the category of pairs of topological spaces, there is a natural contravariant functor C X : K X,U → Top 2 , assigning to an object K the pair (X, X \\ K) and to a morphism K ֒→ H the natural inclusion (X, X \\ H) ֒→ (X, X \\ K). Calling A Z the category of graded abelian groups, we define K • ζ,cpt (X) as the colimit of the composition functor K • ζ • C X : K X,U → A Z : Since K • ζ and C X are both contravariant, the composition is covariant. Concretely, an element of K • ζ,cpt (X) is an equivalence class [α], represented by a class α ∈ K • ζ (X, X \\ K), K being a U-compact subset of X. The colimit is taken over the groups K • ζ (X, X \\K), where, if K ⊂ H, the corresponding morphism in the direct system is the pull-back i * KH : We have defined the compactly-supported groups associated to a space X. We can make K • ζ,cpt a covariant functor, defining its behaviour on open embeddings. In fact, let us fix an open embedding ι : Y ֒→ X, such that the good cover of X restricts to a good cover of Y . For any U-compact subset K ⊂ Y , from the embedding of pairs ι K : (Y, Y \\K) ֒→ (X, X \\ι(K)), we get the induced morphism ι * , which is an excision isomorphism. If K ⊂ H, the following diagram commutes: therefore we get an induced morphism between the colimits, i.e., ι * : K • ζ,cpt (Y ) → K • ζ,cpt (X), as desired. We also have the following natural product Finally, the following canonical isomorphism will be useful later on: where R := i * , for i : R × X ֒→ S 1 × X the open embedding induced by R ֒→ R + ≈ S 1 .
In order to show that (88) is an isomorphism, on the l.h.s. we take the direct limit of K • ζ ((R × X), (R × X) \\ ι(K)) over the cofinal subset of K R×X,U formed by the elements K := I n × K ′ , with I n := [−n, n] ⊂ R and K ′ ⊂ X. The direct limit is the group of compactly-supported classes in S 1 × X relative to {∞} × X. 7 Considering the embedding i ∞ : X ֒→ X ×S 1 , x → (x, ∞), such a group is the kernel of i * ∞ : K • ζ,cpt (S 1 ×X) → K • ζ,cpt (X), which is exactly K •−1 ζ,cpt (X) by definition (84). The isomorphism (11), extended to relative twisted K-theory, induces an isomorphism between the compactly-supported groups. It follows that the isomorphism class of K • ζ,cpt (X) only depends on [ζ]. In particular, if H 2 (X; Z) = 0, then K • [ζ],cpt (X) is canonically defined. If Tor H 2 (X; Z) = 0 and we choose ζ constant, then K • [ζ],cpt (X) is well defined.

S 1 -Integration
We are going to define the integration map calling ζ both the twisting cocycle on X and its pull-back on S 1 × X. Let us consider the embedding ι 1 : X ֒→ S 1 × X, defined through a marked point of S 1 , and the projection π 1 : S 1 × X → X. We set S 1 : The construction can be iterated and it can be extended to positive degrees by the Bott periodicity.

Thom isomorphism
We recall some basic facts on spin geometry in order to fix the notation within the framework of twisted bundles. Given a real orientable vector bundle π : E → X of rank 2r, with fixed metric and orientation, the good cover U = {U i } i∈I on X induces the good cover π * U := {E i := π −1 (U i )} i∈I on E. Let us consider the frame bundle p : SO(E) → X and the corresponding restrictions where ϕ ′′ ij := ϕ ′ ij × ρ 1 : Spin(E i ) × ρ S → Spin(E j ) × ρ S. We also have the global splitting of ǫ-twisted vector bundles S( . Let us consider the projection π : E → X and the pull-back π * S(E) = π * S + (E)⊕π * S − (E) on E. We define the twisted-bundle morphism µ : π * S + (E) → π * S − (E) as follows. For any fixed point e ∈ E x , with x ∈ U i , the morphism µ acts between the fibres (S + (E i )) x and (S − (E i )) x , both contained in (S(E i )) x , as the Clifford multiplication by e ∈ Cl(E x ) = Cl((E i ) x ). 8 It is easy to verify that µ is actually a morphism of twisted bundles and that it is an isomorphism on the closure of the complement of the disk bundle D E of E. Refining π * U on E in a suitable way, 9 we get a good cover V = {V j } j∈J such that D E is V-compact and the union of the V-compact sets is the whole E. In this way we get a classũ := [π * S + (E), π * S − (E), µ] ∈ K ǫ (E, E \\ D E ), representing a compactly-supported class u ∈ K ǫ,cpt (E), the latter being a (twisted) Thom class. Of course, when w 2 (E) = 0, we can choose ǫ = 1 and we get an ordinary Thom class.
Up to now the rank of E has been taken even by hypothesis. If rk(E) is odd, we consider a Thom class in E ⊕ 1 → X, where 1 = X × R is the trivial real line bundle. 10 We get u ∈ K ǫ,cpt (E ⊕ 1) = K ǫ,cpt (E × R) ≃ K −1 ǫ,cpt (E), the last isomorphism being (88). It follows that, if rk E = n (even or odd), then u ∈ K n ǫ,cpt (E). The Thom isomorphism is defined in the following way, choosing a refinement map φ : J → I from π * U to V and using the product (87): Of course we are identifying ζ and ǫ in X with π * ζ and π * ǫ in E as twisting cocycles. In general (91) depends on ζ, ǫ and φ. Nevertheless, it becomes canonical when H 2 (X; Z) = 0. In fact, in this case, H 2 (E; Z) = 0 as well, since E retracts by deformation on X. It follows that both K • ζ (X) and K •+n ζǫ,cpt (E) only depends on the cohomology class of their twisting cocycle, hence the isomorphism (91) can be written intrinsically as follows: Since ǫ is constant, if ζ is constant too we get the a canonical isomorphism similar to (92) on any manifold such that Tor H 2 (X; Z) = 0, replacing W 3 (E) by w 2 (E). The Thom isomorphism, stated with the present language, is coherent with the the one constructed in [11]. We defer the details of the comparison to a future paper.
Thom isomorphism and spin c structures. Let us suppose that W 3 (E) = 0, but w 2 (E) = 0. In this case we cannot choose ǫ = 1, but we should recover the ordinary Thom isomorphism anyway, trough a spin c -structure of E or E ⊕ 1. Let us show how. Again we summarize some basic facts about spin c geometry in order to fix the notation.
We get the principal bundle isomorphisms We can construct a global bundle Spin c (E) if and only if it is possible to choose these data in such a way that ǫ ijk θ ijk = 1, that is equivalent to θ ijk = ǫ ijk . Fixing a set of local unitary sections t i : U i → L| U i , we get the set of transition functions h ij : U ij → U(1) such that t i = h ij t j . We lift the sections t i to This is equivalent to the triviality of [ǫ] as a U(1)-cocyle, that is equivalent to W 3 (E) = 0, sinceȞ 2 (X, U(1)) ≃ H 3 (X; Z).
Let us suppose that W 3 (E) = 0 and let us show how to recover the Thom isomorphism in ordinary K-theory from (91). Choosing ζ = ǫ, we get the isomorphism T : K • ǫ (X) → K •+n cpt (E). Since W 3 (E) = 0 and since W 3 (E) is the twisting (integral) class represented by ǫ, it follows that K • ǫ (X) ≃ K • (X) in a non-canonical way. In order to find an isomorphism of the form (10), we must fix a trivialization of ǫ in U(1). If we call {h ′ ij } such a trivialization, we get (93). This means that the choice of a spin c structure is equivalent to the choice of (91), the latter in the form T : K • ǫ (X) → K •+n cpt (E), is the ordinary Thom isomorphism, with respect to the Thom class induced by the chosen spin c -structure. 11 The choice of ǫ as a representative of w 2 (E) is immaterial (even if Tor (2) H 2 (X; Z) does not vanish), since, choosing another representative ǫ ′ and a cochain υ such that ǫ ′ = ǫ ·δ 1 υ, the identifications ϕ ′ ij → ϕ ′ ij υ ij and ψ ′ ij → ψ ′ ij υ ij determine the same isomorphism ϕ c ij , hence the two Z 2 -ambiguities cut, inducing the same ordinary Thom isomorphism.

Freed-Lott model of differential twisted K-theory
We are going to describe differential twisted K-theory as the Grothendieck group of a suitable semi-group, as in the topological setting, following the construction shown in [20] about ordinary differential K-theory and in [33] in the twisted case. Moreover, we complete the construction of [33] to any degree, considering product and integration, and we link it more explicitly to relative Deligne cohomology.

Preliminary notions
We briefly review the notions of Chern character and Cheeger-Simons class in the twisted framework, since they will be applied to define differential K-theory in this context.
Chern character of a twisted connection. We have seen in section 2.1 that, given a classα ∈Ĥ p (ρ), its curvature is a relative form (ω, η) ∈ Ω p (ρ), where ω is the curvature of 11 If we consider the associated bundles L ′ i := U ′ (L i )× 1 C, where '1' denotes the fundamental representation of U(1), and the isomorphisms ψ ′′ ij := ψ ′ ij × 1 : With this language, the isomorphism Φ h ′ can be also written in the the class on X and η is the global potential representing the pull-back on Y . In the partially non-abelian setting, it is natural to define the curvature of [(ζ, Λ, B, g, A)] ∈Ĥ 3,r (ρ) as the relative 3-form (H, rB i −Tr F i ), the first component being the field strength on the space-time X and the second one being the field strength on the world-volume Y . Coherently, the class is parallel when the second component vanishes. Actually, as for ordinary connections, we can extract higher-degree gauge-invariant forms as well, starting from the following simple lemma.  by (g, A), we have that: where we recall that The proof can be realized by direct computation. We remark that formula (94) holds for any B completing the twisting cocycle (ζ, Λ), no matter if the connection is (ζ, Λ, B)-twisted or not. It easily follows that e F j −B j Ir = g −1 ij e F i −B i Ir g ij , therefore we get the globally-defined even form ch(∇) := Tr e F i −B i Ir .

Cheeger-Simons class. Let us consider two (ζ, Λ)-twisted connections ∇ = {∇ i } and
Fixing a completion (ζ, Λ, B), so that the Chern character (95) is well-defined, we get the Cheeger-Simons class CS(∇, ∇ ′ ) ∈ Ω odd (X)/Im(d H ), defined as follows. On the space I × X we have the good cover π * U and the twisted bundle π * I E, where π I : I × X → X is the natural projection. We endow each vector bundle π * I E i with the connection∇ i , that interpolates between ∇ i and ∇ ′ i . More precisely, for any sections s of π * I E i and (a, V ) of the tangent bundle T (I × X), we set The following relation holds: Chern character in twisted K-theory. Equations (96) and (100) imply that ch(E) ∈ H ev dR,H (X) is well-defined, since it does not depend on the choice of the connection. We get the natural transformation: where H is the curvature of any Deligne cocycle (ζ, Λ, B). From definition (84) we easily get: where ch α is computed thinking of α ∈ K 0 ζ (T n × X).
The direct sum between differential vector bundles is defined as: An isomorphism of differential vector bundles Φ : The isomorphism classes of differential vector bundles form an abelian semi-group, that we call (DiffVect (ζ,Λ,B) (X), ⊕). Moreover, the following map is well-defined and it is a surjective semi-group homomorphism: Since (DiffVect (ζ,Λ,B) (X), ⊕) is an abelian semi-group, it is natural to consider its Grothendieck group.

Definition 3.3
The (ζ, Λ, B)-twisted differential K-theory group of X is the group: By definition an element ofK( is the class of (E, ∇, ω) up to the stable equivalence relation. It follows that, if (E, ∇, ω) is equivalent to (E ′ , ∇ ′ , ω ′ ), then the two twisted bundles E and E ′ represent the same K-theory class, therefore the following map is well-defined and it is a surjective group homomorphism: Axioms of (twisted) differential cohomology. We are looking for a differential extension of twisted K-theory, therefore we need to define the following natural transformations: 12 satisfying the following axioms, analogous to R1-R3 of [10, p. 4]: R2. the following diagram is commutative: R3. the following sequence is exact: where Ω •−1 ch,H (X) denotes the forms representing a class belonging to the image of ch •−1 : We fulfil these axioms, only in degree 0 up to now, through the following definition. Extension to any degree. We begin definingK −n (ζ,Λ,B) (X) for every n ≥ 0. In particular, we use a general property of differential cohomology [18,Lemma 2.16] as the definition. We use the same notations of formula (84) about the torus T n and the embeddings i 1 , . . . , i n .
Definition 3.5 The groupK −n (ζ,Λ,B) (X) is the subgroup ofK (ζ,Λ,B) (T n × X) whose elements are the classesα such that: • for every j = 1, . . . , n: • there exists ρ ∈ Ω −n (X) such that, calling π T n : T n × X → X the projection, we have: It follows from (107) and (84) that the following natural transformation is well-defined: Moreover, we can define the curvature of a class inK −n (ζ,Λ,B) (X) as the form ρ appearing in (108), or, equivalently: Finally, we define the natural transformation: for any (E, ∇). It is quite easy to prove that these definitions fulfil the axioms R1-R3 in any negative degree.
Product. We have the following natural product: In particular, we get a module structure onK (ζ,Λ,B) (X) overK(X), such that I and R are multiplicative and a(ω) ·α = a(ω ∧ R(α)), as required by the axioms of multiplicativity in differential cohomology. We can also define in a similar way the exterior product applying the pull-backs through the projections π X : X × Y → X and π Y : X × Y → Y to the corresponding terms. Calling ∆ : X → X × X the diagonal embedding, one has (E, ∇, ω) ⊗ (F,∇, ρ) = ∆ * ((E, ∇, ω) ⊠ (F,∇, ρ)). The exterior product can be extended to any non-positive degree: In fact, givenα where ⊠ 0 is the exterior product in degree 0. Then we consider the natural diffeomorphism ϕ n,m : T n+m × X × Y → T n × X × T m × Y and we setα ⊠β := (−1) nm ϕ * n,m (α ⊠ 0β ). The factor (−1) nm is necessary to make the first Chern class and the curvature multiplicative. 13 Then, via the diagonal embedding ∆ : T n+m × X → T n+m × X × X, we get the interior product The reader can verify by direct computation that a −n (ω) ·α = a −n−m (ω ∧ R −m (α)). 13 For example, considering the curvature, we have )), therefore we need to multiply by (−1) nm to get R −n−m (α ⊠β) = π * X R −n (α) ∧ π * Y R −m (β).
S 1 -integration. We have to define the map: satisfying the following axioms: I1. Calling t : S 1 → S 1 the conjugation e iθ → e −iθ and considering the map t×id : I2. S 1 •π * 1 = 0, where π 1 : S 1 × X → X is the natural projection. I3. The integration map (114) commutes with I, R and a.
Bott periodicity and positive degrees. Bott periodicity can be extended to the differential framework as follows. Let us consider the generator η − 1 ∈K(T 2 ) ≃ Z leading to topological Bott periodicity. There exists a unique differential refinementη − 1 such that R(η − 1) = dt 1 ∧ dt 2 and i * 1 (η − 1) = i * 2 (η − 1) = 0, where i 1 , i 2 : T → T 2 are the natural embeddings: such a class is the unique element ofK −2 (pt) with curvature 1 and first Chern class 1. Then the mapB is an isomorphism too. Thus, for any n > 0, we setK n (ζ,Λ,B) (X) :=K −n (ζ,Λ,B) (X) and we easily extend to positive degrees the functors I, R and a, together with product and S 1 -integration.
Parallel and compactly-supported versions. We just need the definition of parallel relative classes, since it will be used in order to define the compactly-supported version. It is defined exactly as in the topological case (see section 3.1.1), replacing the set L ζ (X, Y ) by the setL (ζ,Λ,B) (X, Y ). The latter is formed by triples (E, ∇ E , ω E ), (F, ∇ F , ω F ), α , where (E, ∇ E , ω E ) and (F, ∇ F , ω F ) are (ζ, Λ)-twisted vector bundles on X, such that ω E and ω F vanish on Y , and α : E| Y → F | Y is an isomorphism such that α * ∇ F = ∇ E . We define in an analogous way the notions of isomorphism of triples and elementary triple, getting the corresponding setK (ζ,Λ,B) (X, Y ). With this definition, we can construct the compactlysupported version exactly as in topological framework (see section 3.1.2), since parallel classes satisfy excision as well [18,Lemma 2.15]. We get the groupK (ζ,Λ,B),cpt (X).

Generic twisting class
Up to now we considered a twisting class of finite order. Considering the B-field, this finiteorder hypothesis corresponds to the exactness of the H-flux. This is always true on a Dbrane world-volume, because of the Freed-Witten anomaly, but not on the whole space-time. That's why we need to briefly review the general case, in which some infinite-dimensional tools are necessary, and explicitly relate it to the language that we have used up to now, analysing in particular the dependence on the cocycle. We choose the model described in [12]. In a future paper, it would be interesting to compare it with the more recent model in [23], that could also be used as a reference here.

Topological twisted K-theory
We fix a separable Hilbert space H. We can easily generalize definition 2.3 as follows.
Definition 3.6 Given a cocycle ζ := {ζ ijk } ∈Ž 2 (U, U(1)), a ζ-twisted Hilbert bundle with fibre H on X is a collection of trivial Hilbert bundles π i : E i → U i with fibre H and of Hilbert bundle isomorphisms ϕ ij : The corresponding definition of (iso)morphism coincides with 2.4. For every ζ ∈Ž 2 (U, U(1)), not necessarily of finite order in cohomology, there exists a ζ-twisted Hilbert bundle [2], the main difference with respect to the finite-dimensional setting being that any two ζ-twisted Hilbert bundles (for a fixed ζ) are isomorphic [27].

Projective Hilbert bundles. Given a twisted bundle
to the corresponding projective space, we get a well-defined (non-twisted) projective bundle, that we denote by P(E). It follows from local triviality that every projective bundle can be obtained in this way up to isomorphism, therefore we get a surjective map from isomorphism classes of twisted bundles to isomorphism classes of projective bundles. In the finite-dimensional case such a map is not injective for a fixed ζ (for example, every line bundle projects to the trivial one). On the contrary, in the infinite-dimensional case, the unique isomorphism class of ζ-twisted Hilbert bundles induces a unique isomorphism class of projective bundles. Moreover, fixing ζ and ζ ′ := ζ ·δ 1 η, let us consider the bijection where VB ζ (X) denotes the set of ζ-twisted Hilbert bundles on X (not quotiented out up to isomorphism). Since P(E) = P(Φ η (E)), the isomorphism class of P(E) only depends on ). It follows that H 3 (X; Z) classifies projective Hilbert bundles on X. Ifδ 1 η = 1, then, since any two ζ-twisted bundles are isomorphic, there exists an isomor- This means that f i : E i → E i and ϕ ij η ij f i = f i ϕ ij , hence f induces an automorphismf : P(E) → P(E). Let us see that any automorphismf can be realized in this way from suitable η and f . In fact, by local triviality, we can liftf to f i : E i → E i for each i. Since the family {f i } glues tof , there exists η ij such f j ϕ ij = ϕ ij f i η ij . The latter condition necessarily impliesδ 1 η = 1. Moreover, the only freedom we had in constructing the cocycle η was the choice of the lifts f i . Any other choice is of the form f i ξ i , that replaces η by η ·δ 0 ξ. Therefore, the following map is well-defined: It is easy to prove that it is a group homomorphism. Moreover, it follows from the previous construction thatf ∈ Aut(P(E)) lifts to an automorphisms of E if and only if Φ(f ) = 0, therefore Φ(f ) can be thought of as the obstruction to the existence such a lift. This remark leads quite easily to the following lemma, that also follows from the fact that PU(H) is an Lemma 3.7 The morphism (122) is surjective. Moreover, its kernel is the connected component of the identity of Aut(P(E)), therefore Φ induces a canonical bijection between the connected components of Aut(P(E)) and H 2 (X, Z).
Definition of twisted K-theory. We fix a cocycle ζ ∈Ž 2 (U, U(1)) and a ζ-twisted We denote by Γ(F P(E) ) its set of global sections and byΓ(F P(E) ) the corresponding quotient with respect to homotopy of sections. The latter carries a natural abelian group structure, induced by composition of Fredholm operators.
Definition 3.8 The twisted K-theory group K ζ (X) is defined as the abelian groupΓ(F P(E) ) for any ζ-twisted Hilbert bundle E.
Since the space of bounded invertible operators in H is contractible (like U(H)), a section of F P(E) , which is point-wise invertible, is always homotopic to the identity. Therefore, if a section is point-wise invertible in a subset of X, we consider it trivial on such a subset. This fact justifies the following definition. Definition 3.9 A section of F P(E) is called compactly supported if it is point-wise invertible in the complement of a compact subset of X. We denote by Γ cpt (F P(E) ) andΓ cpt (F P(E) ) respectively the space of compactly-supported sections of F P(E) and its quotient up to compactlysupported homotopy. We define the compactly supported twisted K-theory group K ζ,cpt (X) as the abelian groupΓ cpt (F P(E) ) for any ζ-twisted Hilbert bundle E. 14 Dependence on the cocycle. Definitions 3.8 and 3.9 seem to depend on E, not only on ζ. Nevertheless, fixing two ζ-twisted bundles E and E ′ , an isomorphism f : E → E ′ is unique up to an automorphism of E. It follows from lemma 3.7 that the induced isomorphism f : P(E) → P(E ′ ) is unique up to an automorphism of P(E) connected to the identity, the latter inducing the identity onΓ(F P(E) ) andΓ cpt (F P(E) ). Hence, K ζ (X) and K ζ,cpt (X) are canonically defined. 15 On the contrary, the definition is not canonical if we only fix the cohomology class [ζ]. In fact, let us consider a ζ-twisted bundle E and a ζ ′ -twisted bundle E ′ , such that ζ ′ = ζ ·δ 1 η. We have the isomorphism analogous to the one induced by (10) in the finite-order setting, defined as follows. We fix an isomorphismf : P(E) → P(E ′ ), belonging to the inverse image of [η] through (122), and we apply the induced one between the corresponding K-theory groups. This is equivalent to inducing the identity betweenΓ(F P(E) ) andΓ(F P(Φη(E) ) ), that represent respectively K ζ (X) and K ζ ′ (X).
The isomorphism (123) depends on η up to coboundaries. Equivalently, the set of isomorphisms of the form (123) is a torsor over H 2 (X; Z), hence, if ζ = ζ ′ , we get an action of H 2 (X; Z) on K ζ (X). Only the quotient up to such an action is well-defined. Of course, if H 2 (X; Z) = 0, then we have the canonical group K [ζ] (X), as in the finite-order setting. Analogous considerations hold about compactly-supported K-theory.

Differential extension
We follow [12] to define differential twisted K-theory with no hypotheses on the torsion class. We keep on denoting by H a separable infinite-dimensional Hilbert space. Moreover, given two such Hilbert spaces H 1 and H 2 , for any p ∈ [1, +∞) we denote by L p (H 1 , H 2 ) the corresponding p-Schatten class, i.e. the space of boundend linear operators A : H 1 → H 2 such that We set L p (H) := L p (H, H).
Schatten Grassmannians. We setĤ := H⊕H and we denote by H + and H − respectively the first and the second component of the direct sum. It follows thatĤ is Z 2 -graded, the corresponding involution being ǫ :Ĥ →Ĥ such that H ± is the ±1-eigenspace. Given a closed subspace V ⊂Ĥ, we get the corresponding decompositionĤ = V ⊕ V ⊥ and the corresponding self-adjoint involution ǫ V . We define the space Gr p (Ĥ, ǫ) as the subspace of the Grassmannian ofĤ such that V ∈ Gr p (Ĥ, ǫ) if and only if ǫ V = ǫ + A, with A ∈ L p (Ĥ). The space Gr p (Ĥ, ǫ), identified with a subspace of bounded linear operators onĤ through ǫ V , is a Banach manifold and it is smoothly homotopically equivalent to Fred(H). In particular, its homotopy type does not depend on p.
In order to construct an explicit homotopy equivalence, we consider the group GL p (Ĥ, ǫ), formed by bounded invertible operators onĤ of the form: It follows from this definition that a ∈ Fred(H + ). From now on we denote Gr p (Ĥ, ǫ) and GL p (Ĥ, ǫ) respectively by Gr p and GL p . We have a natural transitive action GL p × Gr p → Gr p , (A, V ) → A(V ). In particular, a closed subspace V ⊂Ĥ belongs to Gr p if and only if there exists A ∈ GL p such that A(H + ) = V . We have the following natural maps, that turn out to be homotopy equivalences: The group PU(H) acts by conjugation on each of the previous spaces. We denote such actions respectively by ρ : PU(H) → C 0 (Fred(H)), ρ ′′ : PU(H) → C 0 (GL p ) and ρ ′ : PU(H) → C 0 (Gr p ). In ρ ′ and ρ ′′ we are applying the diagonal embedding PU(H) ֒→ PU(Ĥ). We get the three bundles It is straightforward to verify that the homotopy equivalences ϕ and ψ in (124) commute with the actions of PU(H), therefore we get the induced maps: As above, we denote by Γ(F P(E) ) the space of sections of F P(E) and byΓ(F P(E) ) the quotient up to homotopy. We use the same notation for F ′ P(E) and F ′′ P(E) . By definition K ζ (X) := Γ(F P(E) ) and we set K ′ ζ (X) :=Γ(F ′ P(E) ). The maps (125) induce the corresponding ones between sections, that are isomorphisms up to homotopy, since the maps (124) are homotopy equivalences. We get the following canonical isomorphism: The group K ′ ζ (X) turns out to be more suitable than K ζ (X) in order to define the corresponding differential extension.
Natural connection on U(H). Since PU(H) := U(H)/U(1), the projection π : U(H) → PU(H) naturally induces a principal U(1)-bundle structure, which can be endowed with the universal connection for line bundles θ : T U(H) → R. We recall some basic properties of θ, that we are going to use in the following sections. First of all, given two functions f, g : Z → U(H), for any smooth manifold Z, we have that where f g denotes the point-wise product. Moreover, if ζ : Z → U(1), then, thinking of U(1) ⊂ U(H) as the centre, we have: Such a formula easily follows from the fact that θ, restricted to the centre, coincides with the Maurier-Cartan 1-form of U(1). Moreover, for any 1-form Λ : In fact, since θ is the universal connection for line bundles and since Λ represents any connection ∇ Λ on Z×C, we can find a homotopically-trivial functionh : Z → PU(H), covered by h ′ :h * U(H) → U(H), such that the connection ∇ Λ corresponds to (h ′ ) * θ. Choosing a trivialization s : Z →h * U(H), inducing Λ as the potential representing ∇ Λ , we set h := h ′ • s and we get that h * θ = Λ.
Realizing a Deligne cocycle. Given a ζ-twisted Hilbert bundle E = ({E i }, {ϕ ij }), let us fix a system of local sections {s n i : U i → E i } n∈N , that locally trivializes E (i.e. that trivializes each E i ). Such sections determine a Deligne cocycle (ζ, Λ) ∈Ž 2 (S 1 X ) as follows. The first component is determined by E itself, that is ζ-twisted by construction. Moreover, the fixed sections determine the corresponding transition functions g ij : U ij → U(H), therefore, from the universal connection θ on U(H), we set Λ ij := g * ij θ. It follows from (127) and (128) that (ζ, Λ) is a cocycle.
Let us show that any cocycle (ζ, Λ) can be reached in this way. In fact, we already know that for any ζ there exists a ζ-twisted bundle E. Inducing any (ζ, Λ 0 ) through sections {s n i }, for any fixed Λ there exists a cochain (1, λ) such that (ζ, Λ) = (ζ, Λ 0 ) ·Ď 1 (1, λ) = ({ζ ijk }, {Λ 0 ij − λ i + λ j }). We can replace the sections {s n i }, inducing transition functions g ij , through any change of basis h i : U i → U(H), so that we get the transition functions g ′ ij := h i g ij h −1 j . We choose h i such that h * i θ = −λ i , that is always possible, as we have shown in the remarks after formula (129). It follows from (127) that (g ′ ij ) * θ = Λ 0 ij − λ i + λ j = Λ ij , as desired.
Differential twisted K-theory. The spaces Gr p and Fred(H) are Banach manifolds, smoothly homotopically equivalent among each other. In particular, they are all homotopic to the Hilbert manifold Gr 2 , therefore their de-Rham cohomology is well-defined using the complex of smooth differential forms, as in the finite-dimensional setting, and there is a canonical isomorphism from the de-Rham cohomology with complex coefficients to singular cohomology with complex coefficients. On Gr p , using Quillen's superconnections, we can fix smooth even-degree differential forms Φ 2n ∈ Ω 2n (Gr p ) such that Φ ev := ∞ n=0 Φ 2n represents the Chern character of the canonical K-theory class, the latter being the class represented by the identity [12, Section 2.2].
Let us fix a ζ-twisted Hilbert bundle E = ({E i }, {ϕ ij }) and a section ψ ∈ Γ(F ′ P(E) ). Given a Deligne cocycle (ζ, Λ, B), we fix a system of local sections {s n i : U i → E i } n∈N , inducing the cocycle (ζ, Λ) as we have seen above. Such sections identify ψ with a family of local functions ψ i : U i → Gr p , therefore we get the local pull-backs ψ * i Φ ev ∈ Ω ev (U i ). This implies that the local sections determine at the same time the local forms ψ * i Φ ev and the local potentials B i up to a global formB, therefore it is not surprising that these two data glue to the global form e B i ∧ ψ * i Φ ev , that is d H -closed. Let us show more in detail why. If a function h i : U i → U(H) acts on ψ i by conjugation (i.e. if we apply the representation ρ ′ defined above), then the behaviour of ψ * i Φ ev is the following one: It follows that, conjugating ψ i by the transition function g ij (i.e. acting with ρ ′ (g ij )), we have that This justifies the following definition. • ψ is a smooth section of F ′ P(E) ; • η ∈ Ω odd (X)/Im(d H ); • a homotopy between (ψ, η) and (ψ ′ , η ′ ) is a homotopy of sections Ψ : ψ ∼ ψ ′ such that where '∼ d H ' denotes that they are equal up to a d H -exact form. 16 We get the following functors: and θ ζ is the isomorphism (126); for any section ψ.
The extension to any negative degree is realized through definition 3.5 in this context as well, and, because of the Bott periodicity, we setK n (ζ,Λ,B) (X) :=K −n (ζ,Λ,B) (X) for any n > 0. The reader can verify that axioms R1-R3 are satisfied. Product and S 1 -integration are defined similarly to the finite-order framework.

Dependence on the cocycle
Let us analyse the dependence ofK (ζ,Λ,B) (X) on (ζ, Λ, B) in steps. We will get the same picture of section 3.2.3.
Dependence on the cocycle -Part I. Definition 3.10 seems to depend on the triple (E, s, B), but we can show that it canonically depends only on (ζ, Λ, B) as follows. We call K (E,s,B) (X) the group defined in 3.10, assuming that it depends on E and s. Moreover, we call Γ(F ′ P(E) ) the set of smooth sections of F ′ P(E) (we recall that F ′ P(E) := P P(E) × ρ ′ Gr p ) and we set Γ ′ (F ′ P(E) ) := Γ(F ′ P(E) ) × (Ω odd (X)/Im d H ). Using the notation of definition 3.10, it follows that (ψ, η) ∈ Γ ′ (F ′ P(E) ), henceK (E,s,B) (X) is the quotient of Γ ′ (F ′ P(E) ) up to homotopy. We observe that, fixing (ζ, Λ, B), the space Γ ′ (F ′ P(E) ) only depends on E, while the notion of homotopy is determined by s, since s induces the local forms Ψ i in equation (131). Therefore, we write:K where '∼ s ' is the equivalence relation induced by the existence of a homotopy. Now we have to analyse the dependence on E and on s.
In this case such isomorphism is unique, since the condition s ′ = f * s completely determines f . We get the isomorphism: The reader can verify that it is well-defined, since, if (ψ 1 , η 1 ) ∼ s (ψ 2 , η 2 ) through the homotopy Ψ, then (f • ψ 1 , η 1 ) ∼ s ′ (f • ψ 2 , η 2 ) through the homotopyf • Ψ. The isomorphism f # , induced by the unique isomorphism f , would be enough for our purposes if the condition g * ij θ = Λ ij completely determined the transition functions g ij , but this is not the case in general. Nevertheless, at least it shows that, fixing (ζ, Λ, B), we are free to choose any ζ-twisted bundle E. In fact, given any triple (E ′ , s ′ , B) and fixing E, we can choose an isomorphism f : E → E ′ and set s := f * s ′ , so thatK (E ′ ,s ′ ,B) (X) is canonically isomorphic toK (E,s,B) (X) through f # .
Dependence on s. Given two triples of the form (E, s, B) and (E, s ′ , B), both inducing (ζ, Λ, B), we fix a base change {h i : U i → U(H)} from s to s ′ and we set α i := h * i θ. Calling g and g ′ the transition functions induced by s and s ′ respectively, since g * ij θ = g ′ * ij θ = Λ ij and g ′ ij = h i g ij h −1 j , it follows from (127) that h * i θ = h * j θ, hence the local forms α i glue to a global 1-form α on X. We get the following isomorphism: The reader can verify that, if (ψ 1 , η 1 ) ∼ s (ψ 2 , η 2 ) through the homotopy Ψ, then (ψ 1 , η 1 ∧ e dα ) ∼ s ′ (ψ 2 , η 2 ∧ e −dα ) through the same homotopy Ψ, hence Ξ s,s ′ is well-defined. 17 Canonicity. We did not show the canonical dependence on (ζ, Λ, B) yet, since the isomorphisms (133) and (134) do not agree when they are both defined. In fact, given (E, s, B) and (E, s ′ , B), such that g = g ′ , the isomorphism f # , induced by the unique automorphism f of E such that f * s = s ′ , is different from Ξ s,s ′ in general. For this reason, we argue as follows. We have seen in the previous paragraph that, from s and s ′ , we get the global 1-form α, that here we denote by α s,s ′ . If dα s,s ′ = 0, then it follows from formula (130) that the equivalence relations '∼ s ' and '∼ s ′ ' in formula (132) coincide (since the induced forms Ψ i , appearing in the definition of homotopy, are the same), thusK (E,s,B) (X) =K (E,s ′ ,B) (X). Coherently, it easily follows from definition (134) that Ξ s,s ′ is the identity in this case. For this reason, we introduce the following equivalence relation in the set of local trivializations of E inducing Λ: we set s ∼ s ′ if and only α s,s ′ = 0 (up to now it would be enough to require dα s,s ′ = 0, but we will need α itself vanishing). It follows thatK (E,[s],B) (X) is well-defined.
Given an automorphism f : E → E, we set α f := α s,f * s for any local trivialization s. We introduce the following equivalence relation in Aut(E): we set f ∼ f ′ if and only if α f = α f ′ (equivalently, f ∼ id if and only if α f = 0). Coherently, given two ζ-twisted bundles E and We have that the last equality being due to formula (131). The functions ψ i are defined from where, in the equality (⋆), we called H i : U i → U(H) the change of basis from s i toF * s i . It follows that (F where, in the equality (⋆⋆), we applied formula (130), the local forms H * i θ glueing to the global 1-form A. Hence, from formula (135) we get: We have that where we set . Hence, the integral in formula (136) becomes: From Stoke's formula we have: Formula (137) Existence is a consequence of the following property: ( * ) For any 1-form α on X and any ζ-twisted bundle E, there exits an automorphism f : E → E such that α f = α.
In fact, given (E, s, B) and (E ′ , s ′ , B), we choose any isomorphism f ′ : E → E ′ and, applying ( * ), we choose an automorphism g ′ : In order to prove ( * ), we start from the following property of the universal connection θ. Let us fix a ζ-twisted bundle E, a ζ ′ -twisted bundle E ′ , an automorphism f : E → E and an automorphism f ′ : Let us call H the trivial (non-twisted) bundle. For any ζ-twisted bundle E, we have that E ≃ E ⊗ H. An automorphism of H is a global function f : X → U(H) and, since θ is a universal connection, for any 1-form α on X there exists f such that f * θ = α. It follows that α id⊗f = α as required.
This completes the proof thatK Dependence on the cocycle -Part II. Now we can show thatK (ζ,Λ,B) (X) only depends on the cohomology class [(ζ, Λ, B)] up to non-canonical isomorphism. We fix a cochain (ξ, λ) and we set (ζ ′ , Λ ′ , B ′ ) := (ζ, Λ, B) +Ď 1 (ξ, λ). We get the isomorphism defined as follows. We consider separately cochains of the form (ξ, 0) and (1, λ). Action of (ξ, 0). We fix a representativeK (E,s,B) (X) ofK (ζ,Λ,B) (X). We set 0) as the identity between these two representatives. We remark that the transition functions determined by s in the two cases are not the same. In particular, we have that ij , as desired. Action of (1, λ). We fix a representativeK (E,[s],B) (X) ofK (ζ,Λ,B) (X). We fix any change of basis h i : U i → U(H) such that h * i θ = λ i and we set s ′ i := h i (s i ). We have thatK (E,s,B) (X) =K (E,s ′ ,B ′ ) (X), therefore we define Φ (1,λ) as the identity between these two representatives. In fact, considering formula (132), we have that Γ ′ (F ′ P(E) ) is the same in the two cases, since it only depends on E and H. It remains to show that ∼ s and ∼ s ′ coincide. In fact, given a section ψ = [s i , ψ i ] with respect to (E, s, B), we have that hence ∼ s and ∼ s ′ coincide. This also shows that (138) does not change the curvature. Coboundaries and canonicity. Let us show that, if (ξ, λ) =Ď 0 (υ), then (138) is the identity. In fact, we have that (ξ, λ) = (δ 0 υ,dυ). Starting fromK (E,s,B) (X) and applying Φ (δ 0 υ,0) we get the identity fromK (E,s,B) (X) toK (Φδ 0 υ (E),s,B) (X). Then, we choose h i = υ i , so that h * i θ =dυ, hence, applying Φ (1,dυ) , we get the identity toK (Φδ 0 υ (E),υ·s,B) (X). Let us show that id :K (E,s,B) (X) →K (Φδ 0 υ (E),υ·s,B) (X) is exactly the canonical isomorphism considered in the definition ofK (ζ,Λ,B) (X). In fact, we fix f : ) . Moreover, in the domain we quotient up to s, while in the codomain we quotient up to f * s = υs, coherently with Φ (δ 0 υ,dυ) . It follows that the set of isomorphisms of the form (138)  Compact support. All of the constructions shown in this section can be easily extended to the compactly-supported framework. In particular, we define a compactly-supported representative as a pair (ψ, η) such that there exists a compact subset K ⊂ X such that ψ i = ǫ for every U i ⊂ X \K and any local sections s i . Since ǫ is invariant by conjugation, this condition does not depend on s i . A homotopy between two such pairs must be compactlysupported too. In this way we defineK (ζ,Λ,B),cpt (X), that only depends on [(ζ, Λ, B)] up to isomorphism, the latter being non-canonical unless H 1 (X; Z) = 0.

Equivalence for torsion twisting
We sketch how to show that, when the twisting cocycle represents a finite-order class, the models through Fredholm operators or Schatten Grasmannians agree with the model through twisted vector bundles. We defer the details to a future paper.
Topological framework. We assume that X is compact and U is finite. Moreover, we assume that each element of U has compact closure. We fix a cocycle ζ ∈Ž 2 (U, U(1)), that represents a finite-order cohomology class, and a ζ-twisted rank-N vector bundleĒ := ({U i × C N }, {g ij }), for any suitable N ∈ N. 18 We consider the ζ-twisted Hilbert bundle E :=Ē⊗H, where H is the trivial bundle. This means that E : In this section we set K E (X) :=Γ(F P(E) ), following definition 3.8, while we denote by K ζ (X) the model through finite-dimensional twisted bundles, as defined in section 3.1. Let us consider a section ψ ∈ Γ(F P(E) ). It follows that [ψ] ∈Γ(F P(E) ) = K E (X). Since the local bundles U i × (C N ⊗ H) are already trivialized, the section ψ corresponds to a family of sections ψ i : U i → Fred(C N ⊗ H) such that ψ i = g ij · ψ j · g −1 ij . We have that C N ⊗ H ≃ H ⊕N and we call π 1 , . . . , π N : H ⊕N → H the canonical projections. For each x ∈Ū i , we consider the following space V x,i ⊂ H: Such a space is closed and finite-codimensional. In fact, Ker ψ i (x) is finite-dimensional, since ψ i (x) is Fredholm, hence each projection π k (Ker ψ i (x)) is finite-dimensional too. It follows that the orthogonal complement is closed and finite-codimensional, hence the same holds about the finite intersection V x,i . Thus, V ⊕N x,i is closed and finite-codimensional in x,i and w := (w 1 , . . . , w N ) ∈ Ker ψ i (x), then w k ∈ π k (Ker ψ i (x)) and v k ∈ π k (Ker ψ i (x)) ⊥ , hence v k , w k = 0, therefore v, w = 0.
Following the proof of [1, Prop. A5], for each x ∈Ū i there exists a neighbourhood It follows V ⊕N is closed and finite-codimensional and V ⊕N ∩ Ker(ψ i (x)) = {0} for every x ∈ U i and for every i. Moreover, g ij (V ⊕N ) = V ⊕N for every i and j, since the transition functions act as N × N complex matrices on H ⊕N . Projecting to the quotient, we get the pointwise isomorphismḡ ij : H ⊕N /V → H ⊕N /V , therefore we get the following well-defined ζ-twisted finite-dimensional vector bundle on X: We set H ⊕N /ψ i (V ) : it is a trivial vector bundle. Moreover, we get a well-defined isomorphismḡ ij : , therefore we get the following ζ-twisted finite-dimensional vector bundle on X: With these data, we get the following isomorphism: The proof that (141) is actually an isomorphism follows the same line of the appendix of [1], adapted to the twisted framework.
Differential extension. Now we extend the isomorphism (141) to the differential framework. For this aim, we choose a cocyle of the form (ζ, 0, B), with ζ constant. Since e B is a global form, the local form ψ * i Φ ev in definition 3.10 glue to a global form ψ * Φ ev . Let us show that, in this case, we can give a definition similar to 3.10, considering the space Fred(H) (as in the topological setting) instead of Gr p .
We fix two homotopy equivalences ξ : Fred(H ⊕N ) → Gr p (H ⊕N ) andξ : Gr p (H ⊕N ) → Fred(H ⊕N ) inverse to each other, so that we have two homotopies Θ : ξ •ξ ≃ id Gr p and Θ :ξ•ξ ≃ id Fred(H ⊕N ) . We construct these maps in such a way that they are U(N)-equivariant. Moreover, we setΦ ev := ξ * Φ ev on Fred(H ⊕N ) and φ odd := I Θ * Φ ev on Gr p . We definê K (E,s,B) (X) as in definition 3.10, but replacing F ′ P(E) by F P(E) , which means replacing the fibre Gr p by Fred(H ⊕N ), and e B i ∧ Ψ * i Φ ev by e B ∧ Ψ * Φ ev . The following functions are isomorphisms inverse to each other: the compositions ξ • ψ andξ • ψ and the pull-back ψ * φ odd being defined through the local functions ψ i induced by s. The twisted bundles (139) and (140) can be endowed with a ζ-twisted connection, induced by orthogonal projection by any fixed ζ-twisted connection on E, extended toĒ ⊗ H by tensor product with the trivial connection. Therefore, we extend (141) to the following isomorphism: Similar isomorphisms hold for compactly-supported classes.

Topological K-theory twisted by a Deligne class
We have shown the definition of ζ-twisted topological K-theory and of (ζ, Λ, B)-twisted differential K-theory. Actually, we can define topological (ζ, Λ, B)-twisted K-theory, that we denote by K • (ζ,Λ,B) (X), both in the model through twisted bundles and through Schatten Grassmannians. Considering the transformation a : Ω odd (X)/Im(d H ) →K (ζ,Λ,B) (X), introduced after definition 3.10, we set: We get the natural isomorphism K • (ζ,Λ,B) (X) , where θ ζ is the isomorphism (126). Analogously, in the twisted-bundle model, the isomorphism is given by The advantage of definition (143)

Twisted K-characters and K-homology
We now introduce the notion of twisted K-character, that is the suitable tool in order to define rigorously the world-volume of a D-brane and the Wess-Zumino action. We start recalling the notions of orientation and integration in the framework of twisted differential K-theory, following [9, sec. 4.8-4.10] and [18, chap. 6].

Orientation and integration
We show how to orient a vector bundle, a smooth map and a smooth manifold, then we discuss the integration map. In this section every manifold is simply-connected by hypothesis. 19 I. Orientation of vector bundles. We fix a real vector bundle π : E → X of rank n on a compact manifold X. We have seen in section 3.1.4 that a topological K • -orientation of E is a w 2 (E)-twisted Thom class u of E. We get the Thom isomorphism that is the canonical version of (91) through definition (143). We call integration map its inverse E/X : The n-degree component of the Chern character ch u (see formulas (101) and (102) with H = 0) defines an orientation of E in the usual sense, hence it is possible to integrate a compactly-supported form fibre-wise. We define the Todd class Td(u) := E/X ch u ∈ H ev dR (X). The following formula holds: In order to orient a vector bundle with respect to differential K-theory, one just has to refine a Thom class u to a differential Thom class.
Definition 4.1 A differential Thom class of E is a classû ∈K n w 2 (E),cpt (E) such that I(û) is a topological Thom class.

Lemma 4.3 (2x3 principle)
Given two real vector bundles E, F → X, with projections p E : E ⊕ F → E and p F : E ⊕ F → F , we consider a triple (û,v,ŵ) of differential Thom classes on E, F and E ⊕F respectively, such thatŵ is homotopic to p * Eû ·p * Fv . Two elements of such a triple uniquely determine the third one up to homotopy.
II. Orientation of smooth maps. As in the non-twisted framework, we start defining a representative of an orientation as follows.
Definition 4.4 A representative of aK • -orientation of a smooth neat map f : Y → X between compact manifolds is the datum of: • a neat embedding ι : Y ֒→ X × R N , for any N ∈ N, such that π X • ι = f ; • a differential Thom classû of the normal bundle N : We have the following natural map on differential forms, called curvature map: When f is a submersion, we can choose a proper representative [18], i.e. we can choose the diffeomorphism ϕ in such a way that the image of the fibre of N y is contained in {f (y)}×R N . In this case: We can introduce a suitable equivalence relation on representatives, induced by homotopy and stabilization [18]. In particular, the curvature map (147) [18]. The following lemma is a consequence of lemma 4.3 and of the uniqueness up to homotopy and stabilization of the embedding ι.  III. Orientation of smooth manifolds. In this subsection we discuss separately the cases of manifolds without boundary and with boundary. Definition 4.7 AK • -orientation of a manifold without boundary X is aK • -orientation of the map p X : X → {pt}.
By definition, an orientation of p X is essentially a Thom classû on the (stable) normal bundle of X; whenû has been fixed, we set Td(X) := Td(û).
Given a manifold with boundary X, we recall that a defining function for the boundary is a smooth neat map Φ : X → I such that ∂X = Φ −1 {0} (by neatness, it follows that Φ −1 {1} = ∅). It is easy to verify that any two defining functions are neatly homotopic. Nevertheless, a defining function is not a submersion in general, hence we have to modify slightly the definition of orientation. Following definition (147), the curvature map should be ω → I×R N /I i * ϕ * (R(û) ∧ π * ω), but we also integrate on I the result: Definition 4.8 AK • -orientation on a smooth manifold with boundary X is a homotopy class ofK • -oriented defining functions for the boundary, considering the curvature map (149) in the definition of homotopy.
Formula (148) keeps on holding, in the following form: With this definition of the curvature map, an orientation of a manifold without boundary can be thought of as a particular case of an orientation of a manifold with boundary.
Remark 4.9 It follows from remark 4.5 that an orientation on a manifold with boundary canonically induces an orientation on the boundary. In particular, let us fix a defining function Φ : X → I and an orientation [ι,û], with ι : X ֒→ I × R N . We call i ∂X : ∂X ֒→ X the natural embedding and we set ι ′ := ι • i ∂X : ∂X ֒→ {0} × R N andû ′ :=û| ∂X . We get the orientation [ι ′ ,û ′ ] of ∂X.

IV.
Integration. Let f : Y → X be aK • -oriented neat submersion. We set w 2 (f ) := w 2 (Y ) + f * w 2 (X) and . Moreover, we fix twisting classes [(ξ, Θ, C)] in Y and [(ζ, Λ, B)] in X and we denote by w ′ 2 (f ) the image of w 2 (f ) trough the morphism in cohomology induced by Z 2 ֒→ U(1). The integration map (or Gysin map) induced by f is defined under the following hypothesis: Computing the first Chern class of each term, we get the topological condition We remark that, for a fixed [(ζ, Λ, B)], there exists a unique class [(ξ, Θ, C)] such that (151) holds. In this case, we define: where n = dim(Y ) − dim(X). The integration map with respect to R N is defined as follows.
The open embedding j : R N ֒→ (S 1 ) N , defined through the embedding R ֒→ R + ≃ S 1 in each coordinate, induces the push-forward (id × j) * in compactly-supported K-theory, thus we set R Nβ := S 1 · · · S 1 (id × j) * β . Condition (151) is due to the fact that the class w 2 (N ), where N is the normal bundle considered in definition 4.4, coincides with w 2 (f ), therefore the classû · π * α is twisted by Such a formula is due to the fact that all the structures involved in the definition of (∂f ) ! are the restrictions to the boundary of the corresponding structures for f ! . It is easy to prove from the axioms that: thus the following diagram commutes: Theorem 4.10 Let f : Y → X be a neatK • -oriented submersion between compact manifolds.
• The Gysin map f ! only depends on the homotopy class of f as anK • -oriented map.
• Given another neatK • -oriented map g : Z → Y and endowing f • g of the naturally induced orientation, we have (f • g) ! = f ! • g ! .
• We have that: For the proof see [ Up to now we supposed that f is oriented. If f : Y → X is a neat submersion between K • -oriented manifolds without boundary, then, since p Y = p X • f , it follows from lemma 4.6 that f inherits a unique orientation from the ones of X and Y . Hence, the integration map f ! is well-defined. If X and Y have boundary, the same result follows from the analogous 2x3 principle.
V. Topological Integration. In the differential framework it is essential to consider proper submersions, since only in this setting some natural tools of algebraic topology (in particular, the 2x3 rule and homotopy invariance) keep on holding. The main reason is that the curvature is relevant as a single differential form, not only as a representative of a cohomology class. Of course this does not happen in the topological framework, therefore all of the previous definitions and constructions can be applied to any K • -oriented smooth map (and to any bundle or manifold as well). In particular, if condition (151) holds, we get the Gysin map for any K • -oriented f : Y → X, applying definition (143), i.e. quotienting out by the image of the natural transformation a. If X and Y are K • -oriented, then f inherits canonically an orientation, so that (153) is well-defined. The integration map, stated with the present language, is coherent with the the topological one constructed in [11] and with the the differential one constructed (for embeddings) in [12]. We defer the details of the comparison to a future paper.

Twisted K-Characters
In [25] the author provides a geometric model of the homology theory dual to a given cohomology theory. One item of that definition uses the "vector bundle modification", that has been replaced in [16] by the (more natural) integration map. We briefly recall the definition of this model, in the specific case of (non-twisted) K-homology. We only consider smooth manifolds, since we are interested in the differential extension, but the construction holds for any space with the homotopy type of a finite CW-complex.

Definition 4.11
On a compact manifold X we define: • the group of n-pre-cycles as the free abelian group generated by quadruples of the form (M, u, α, f ), with: -M a smooth compact spin c manifold (without boundary) with Thom class u, whose connected components {M i } have dimension n + q i , with q i arbitrary; -f : M → X a smooth map; • the group of n-cycles, denoted by Z n (X), as the quotient of the group of n-pre-cycles by the free subgroup generated by elements of the form: • the group of n-boundaries, denoted by B n (X), as the subgroup of Z n (X) generated by the cycles which are representable by a pre-cycle (M, u, α, f ), such that there exits a quadruple (W, U, A, F ), where W is a manifold and M = ∂W , U is a Thom class of W and U| M = u, A ∈ K • (W ) and A| M = α, F : W → X is a smooth map satisfying We set K n (X) := Z n (X)/B n (X).
We cannot generalize 4.11 to the twisted framework directly, since we have seen that the integration map is canonically defined only in the category of manifolds with vanishing firstdegree homology. A stronger but more natural condition consists of simply-connectedness, and it turns out to be the correct choice. Let us show how. We define the groups Z sc n (X) and B sc n (X) as in definition 4.11, but imposing that M is simply-connected in every pre-cycle (hence in every cycle). In the definition of boundaries, W must be simply-connected too. We set K sc n (X) := Z sc n (X)/B sc n (X). The inclusion of simply-connected pre-cycles in the set of all pre-cycles induces a natural morphism υ : K sc n (X) → K n (X). The following lemma justifies the definition of K sc n (X).

Lemma 4.12
If X is simply-connected, then υ : K sc n (X) Sketch of the proof: We begin with surjectivity, i.e. we show that any class [(M, u, α, f )] ∈ K n (X) can be represented by (M ′ , u ′ , α ′ , f ′ ) such that M ′ is simply-connected. We prove it in steps.
Step I. We can suppose that M is connected, dim(M) ≥ 4 and α ∈ K 0 (M). In fact, if M is not connected, we just analyse separately each connected component. About the dimension, if it is not high enough, we replace M byM := M × T n , for any n, and α by a classᾱ ∈ K • (M ) such that T nᾱ = α. Moreover, we choose the canonically induced Thom classū andf := f • π, where π :M → M is the natural projection. It follows that [(M ,ū,ᾱ,f )] = [(M, u, π !ᾱ ,f )] = [(M, u, α, f )], hence M has been replaced byM , of arbitrarily high dimension. Moreover, ifᾱ ∈ K 1 (M ), then we just replace T n by T n+1 , so thatᾱ ∈ K 0 (M).
Step II. M is cobordant to a simply-connected manifold M ′ . The following argument 20 holds since we can suppose dim(M) ≥ 4 by step I. Let us fix a loop γ in M that represents a generator of π 1 (M). We fix a tubular neighbourhood of γ, that we identify with S 1 × D n−1 . We consider the trivial cobordism M × I and, in M × {1}, we glue D 2 × D n−1 to S 1 × D n−1 .
Cheeger-Simons character for any cohomology theory. In particular, in definition 4.11, one just has to replace each orientation u by a differential orientationû and each K-theory class α by a differential K-theory classα. Of course the topological Gysin map is replaced by its differential version. For this reason, in the third item of the definition of cycle, the map ϕ : N → M must be a submersion, as we have seen above. We callẐ n (X) andB n (X) the corresponding groups of differential cycles and boundaries and we set K ′ n (X) :=Ẑ n (X)/B n (X). The natural morphism K ′ n (X) → K n (X), [(M,û,α, f )] → [(M, I(û), I(α), f )], turns out to be an isomorphism, hence we can identify K ′ n (X) with K n (X). Moreover, we defineẐ sc n (X) andB sc n (X) in the same way, but only considering simply-connected manifolds, and lemma 4.12 keeps on holding.
Now we adapt this construction to the case of twisted K-theory. Considering definition (143), it is natural to start from the differential framework. Since twisted K-theory canonically depends on the cohomology class only on simply-connected manifolds, lemma 4.12 naturally suggests the following definition.
Definition 4.14 On a compact simply-connected manifold X with a fixed Deligne class [(ζ, Λ, B)] ∈Ĥ 2 (X), we define: • the group of n-pre-cycles as the free abelian group generated by quadruples of the form (M,û,α, f ), with: -M a smooth compact simply-connected manifold (without boundary) with (w 2 (M)twisted) Thom classû, whose connected components {M i } have dimension n + q i , with q i arbitrary; Condition (158) will be clarified soon. Now we are ready to define Cheeger-Simons characters in the framework of twisted differential K-theory.
The following theorem can be considered as the definition of holonomy of a differential K-theory class on a cycle.

Twisted K-homology
We can provide a definition analogous to 4.14, but replacing every differential class by the underlying topological one, generalizing 4.11 to twisted K-homology through definition (143). We get the group K [(ζ,Λ,B)],n (X), that is canonically isomorphic to the group K ′ [(ζ,Λ,B)],n (X) defined in 4.14. Such a group will be the natural one to classify topological D-brane charges. Fixing an orientationū of X, we get the twisted Poincaré duality, defined as follows (we set x := dim X):

D-Branes charges and Wess-Zumino action
Now we have all the tools to correctly define the world-volume, the Wess-Zumino action and the topological charge of a D-brane with the language of twisted K-theory. We start with a brief review of the cohomological classification, in order to compare it with the K-theoretical framework. We assume for simplicity that the space-time X is compact, but this hypothesis can be removed quite easily.

Cohomological classification
Using the language of ordinary (i.e. singular) differential cohomology, we suppose that the B-field vanishes on the whole space-time, since the construction of the twisted version is more problematic than the K-theoretical counter-part. In this setting, the local Ramond-Ramond potentials C p are (a part of) a connection on an abelian (p − 1)-gerbe, the latter being described by a Deligne cohomology classα ∈Ĥ p+1 (X). Concretely,α is described by the local data (2), the corresponding curvature being the field strength G p+1 . Moreover, a D(p − 1)-brane world-volume is thought of as a p-dimensional submanifold W of the spacetime X, that, via a suitable triangulation, defines a singular p-cycle. When the numerical charge is q ∈ Z, we think of a stack of q D-branes (anti-branes if q < 0), whose underlying cycle is qW . The topological charge of the D-brane is the Poincaré dual of the underlying homology class [qW ] ∈ H p (X; Z), therefore it belongs to H n−p (X; Z), where n := dim X. The Wess-Zumino action, usually written as W C p , is the holonomy ofα on the cycle W . In the particular case I(α| W ) = 0, we have thatα| W = a(C p ), where C p is a global form on W , and the holonomy ofα coincides with W C p mod Z. Nevertheless, for a generic clasŝ α, such an integral is meaningless. Moreover, assuming for simplicity that W is the only world-volume in X, the violated Bianchi identity is: This implies that G n−p−1 is a closed form in the complement of W and, if L is a linking manifold of W , with linking number l, we get 1 l L G n−p−1 = q ∈ Z. That's why the Ramond-Ramond field strength is quantized, so that it is correct to think of it as the curvature of a connection.

K-Theoretical classification
In the K-theoretical framework, we impose no constraints on the B-field, but we start supposing that the space-time and each D-brane world-volume are simply-connected, since, in this case, all of the twisted groups are canonically defined, in the sense that they only depend on the cohomology classes involved.
Ramond-Ramond fields. In order to describe the Ramond-Ramond fields, the two meaningful groups areK The previous definition perfectly fits to formalize the notion of D-brane world-volume. In fact, M is the underlying submanifold, that we usually denote by W when referring to a D-brane, u is a spin structure, that must be fixed as a part of the background data, ρ is the embedding in the space time (the latter being endowed with the space-time metric), (E, ∇) is the Chan-Patton bundle with the corresponding gauge theory and, when ρ * H does not vanish, the formB is necessary to fix the bundle up to large gauge transformations, as we have seen in remark 2.17. Actually, we consider one bundle pre-cycle made by all the evendimensional world-volumes or one made by all the odd-dimensional ones, depending whether we are considering the type IIA or type IIB theory. Moreover, we have seen in remark 5.3 that a bundle pre-cycle canonically induces a differential cycle, on which we can compute the holonomy of a differential K-theory class through formula (162). When the differential K-theory class represents the Ramond-Ramond fields of the corresponding degree, such a holonomy is by definition the Wess-Zumino action. Summarizing: Usually the Wess-Zumino action is written supposing thatα is topologically trivial, so that it can be described by a global form C up to twisted K-integral ones. It has the following form [30], using the notation of definition 5.5: This easily follows from the right formula of (156), normalizing C with Td(X) − 1 2 . In fact: We stated definition 5.5 under the hypothesis that the space-time X and the world-volume submanifold W are simply-connected. Actually, definitions 4.14 and 4.15 and theorem 4.16 hold with no hypotheses on X and only requiring that H 1 (W ; Z) = 0, the disadvantage being that they do not correctly generalize the analogous notions in the non-twisted setting under these weaker assumptions. Nevertheless, this does not affect definition 5.5 in any way, therefore the mathematical formalization of the notions of world-volume and Wess-Zumino action, that we provided in this section, hold for any space-time and for world-volumes with vanishing first-degree homology. When H 1 (W ; Z) does not vanish, definition 5.5 is canonical up to a flat line bundle on W , this ambiguity being physically motivated, since it perfectly corresponds to the ambiguity arising from the classification of the possible gauge theories on W , realized in section 2.7. Of course, when the B-field vanishes in the whole X and W is spin, then we have no ambiguity in any case, since ordinary K-homology and K-characters are always well-defined.
Remark 5.6 This picture completes (and partially corrects) some statements in [15]. In fact, first of all it is not necessary to choose the differential refinement of the orientation u on W as a part of the background datum, since, as we have seen above, it is induced by the space-time metric. Moreover, in [15] we stated that, if the B-field vanishes, then we can use ordinary K-theory and K-homology. This is "canonically true" only if the world-volume is spin, while the Freed-Witten anomaly in this case only imposes that it is spin c . In [15] we got ordinary K-theory fixing a spin c structure on W , instead of a w 2 (W )-twisted spin structure. As we have seen in the paragraph before section 3.2, this corresponds to the choice of an isomorphism between w 2 (W )-twisted K-theory and ordinary K-theory. Nevertheless, the half-integral line bundle, involved in the choice of a spin c -structure, is better included in the gauge theory, hence it is more natural to fix a twisted spin structure. It follows that, only if w 2 (W ) = 0, we can canonically avoid any twisting.
Quantization. Using classical cohomology, the integral of the field-strength along a linking manifold is the numerical charge of the D-brane. A linking manifold L of W is the boundary of a manifold S that intersects W transversely in a finite number of points of the interior.
The number of such points is the linking number. In the twisted K-theoretical framework, we consider a linking K-cycle (L, u, F, ι). Here L is a "generalized" linking manifold, i.e., L is the boundary of a manifold S such that S and W intersect transversely in a submanifold (without boundary) contained in the interior of S. If S ∩W is 0-dimensional, we get a linking manifold in the usual sense. Moreover, the twisted bundle F represents a K-theory class and ι : L ֒→ X is the embedding. We consider the even-dimensional field-strengths G ev , the discussion about G odd being analogous. The violated Bianchi identity is [30]: Equation (167) implies that G ev ∧ Td(X) − 1 2 is K-quantized and the pairing with a linking K-cycle gives the corresponding charge. In fact, choosing any connection ∇ ′ on F , since G ev is H-closed and ch(∇ ′ ) is −H-closed, we have:

Comparing the two frameworks
Now we have all the elements in order to draw a complete parallel between the two classification schemes of D-branes. Table 1 shows such a parallel.

Singular cohomology
Twisted K-theory